Code coverage tests

This page documents the degree to which the PARI/GP source code is tested by our public test suite, distributed with the source distribution in directory src/test/. This is measured by the gcov utility; we then process gcov output using the lcov frond-end.

We test a few variants depending on Configure flags on the pari.math.u-bordeaux.fr machine (x86_64 architecture), and agregate them in the final report:

The target is to exceed 90% coverage for all mathematical modules (given that branches depending on DEBUGLEVEL or DEBUGMEM are not covered). This script is run to produce the results below.

LCOV - code coverage report
Current view: top level - modules - algebras.c (source / functions) Hit Total Coverage
Test: PARI/GP v2.16.1 lcov report (development 28904-c3aa21e911) Lines: 3501 3527 99.3 %
Date: 2023-12-04 07:51:13 Functions: 305 306 99.7 %
Legend: Lines: hit not hit

          Line data    Source code
       1             : /* Copyright (C) 2000  The PARI group.
       2             : 
       3             : This file is part of the PARI/GP package.
       4             : 
       5             : PARI/GP is free software; you can redistribute it and/or modify it under the
       6             : terms of the GNU General Public License as published by the Free Software
       7             : Foundation; either version 2 of the License, or (at your option) any later
       8             : version. It is distributed in the hope that it will be useful, but WITHOUT
       9             : ANY WARRANTY WHATSOEVER.
      10             : 
      11             : Check the License for details. You should have received a copy of it, along
      12             : with the package; see the file 'COPYING'. If not, write to the Free Software
      13             : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
      14             : #include "pari.h"
      15             : #include "paripriv.h"
      16             : 
      17             : #define DEBUGLEVEL DEBUGLEVEL_alg
      18             : 
      19             : #define dbg_printf(lvl) if (DEBUGLEVEL >= (lvl) + 3) err_printf
      20             : 
      21             : /********************************************************************/
      22             : /**                                                                **/
      23             : /**           ASSOCIATIVE ALGEBRAS, CENTRAL SIMPLE ALGEBRAS        **/
      24             : /**                 contributed by Aurel Page (2014)               **/
      25             : /**                                                                **/
      26             : /********************************************************************/
      27             : static GEN alg_subalg(GEN al, GEN basis);
      28             : static GEN alg_maximal_primes(GEN al, GEN P);
      29             : static GEN algnatmultable(GEN al, long D);
      30             : static GEN _tablemul_ej(GEN mt, GEN x, long j);
      31             : static GEN _tablemul_ej_Fp(GEN mt, GEN x, long j, GEN p);
      32             : static GEN _tablemul_ej_Fl(GEN mt, GEN x, long j, ulong p);
      33             : static ulong algtracei(GEN mt, ulong p, ulong expo, ulong modu);
      34             : static GEN alg_pmaximal(GEN al, GEN p);
      35             : static GEN alg_maximal(GEN al);
      36             : static GEN algtracematrix(GEN al);
      37             : static GEN algtableinit_i(GEN mt0, GEN p);
      38             : static GEN algbasisrightmultable(GEN al, GEN x);
      39             : static GEN algabstrace(GEN al, GEN x);
      40             : static GEN algbasismul(GEN al, GEN x, GEN y);
      41             : static GEN algbasismultable(GEN al, GEN x);
      42             : static GEN algbasismultable_Flm(GEN mt, GEN x, ulong m);
      43             : 
      44             : static void H_compo(GEN x, GEN* a, GEN* b, GEN* c, GEN* d);
      45             : static GEN H_add(GEN x, GEN y);
      46             : static GEN H_charpoly(GEN x, long v, long abs);
      47             : static GEN H_divl_i(GEN x, GEN y);
      48             : static GEN H_inv(GEN x);
      49             : static GEN H_mul(GEN x, GEN y);
      50             : static GEN H_neg(GEN x);
      51             : static GEN H_norm(GEN x, long abs);
      52             : static GEN H_random(GEN b);
      53             : static GEN H_sqr(GEN x);
      54             : static GEN H_tomatrix(GEN x, long abs);
      55             : static GEN H_trace(GEN x, long abs);
      56             : static GEN mk_C();
      57             : 
      58             : static int
      59      903061 : checkalg_i(GEN al)
      60             : {
      61             :   GEN mt, rnf;
      62             :   long t;
      63      903061 :   if (typ(al) != t_VEC || lg(al) != 12) return 0;
      64      902844 :   mt = alg_get_multable(al);
      65      902844 :   if (typ(mt) != t_VEC || lg(mt) == 1 || typ(gel(mt,1)) != t_MAT) return 0;
      66      902823 :   rnf = alg_get_splittingfield(al);
      67      902823 :   if (isintzero(rnf) || !gequal0(alg_get_char(al)))
      68      437735 :     return 1;
      69      465088 :   if (typ(gel(al,2)) != t_VEC || lg(gel(al,2)) == 1) return 0;
      70             :   /* not checkrnf_i: beware placeholder from alg_csa_table */
      71      465081 :   t = typ(rnf);
      72      465081 :   return t==t_COMPLEX || t==t_REAL || (t==t_VEC && lg(rnf)==13);
      73             : }
      74             : void
      75     1032414 : checkalg(GEN al)
      76             : {
      77     1032414 :   if (al && !checkalg_i(al))
      78         112 :     pari_err_TYPE("checkalg [please apply alginit()]",al);
      79     1032302 : }
      80             : 
      81             : static int
      82      180992 : checklat_i(GEN al, GEN lat)
      83             : {
      84             :   long N,i,j;
      85             :   GEN m,t,c;
      86      180992 :   if (typ(lat)!=t_VEC || lg(lat) != 3) return 0;
      87      180992 :   t = gel(lat,2);
      88      180992 :   if (typ(t) != t_INT && typ(t) != t_FRAC) return 0;
      89      180992 :   if (gsigne(t)<=0) return 0;
      90      180992 :   m = gel(lat,1);
      91      180992 :   if (typ(m) != t_MAT) return 0;
      92      180992 :   N = alg_get_absdim(al);
      93      180992 :   if (lg(m)-1 != N || lg(gel(m,1))-1 != N) return 0;
      94     1628886 :   for (i=1; i<=N; i++)
      95    13031067 :     for (j=1; j<=N; j++) {
      96    11583173 :       c = gcoeff(m,i,j);
      97    11583173 :       if (typ(c) != t_INT) return 0;
      98    11583173 :       if (j<i && signe(gcoeff(m,i,j))) return 0;
      99    11583173 :       if (i==j && !signe(gcoeff(m,i,j))) return 0;
     100             :     }
     101      180985 :   return 1;
     102             : }
     103      180992 : void checklat(GEN al, GEN lat)
     104      180992 : { if (!checklat_i(al,lat)) pari_err_TYPE("checklat [please apply alglathnf()]", lat); }
     105             : 
     106             : /**  ACCESSORS  **/
     107             : long
     108     5906032 : alg_type(GEN al)
     109             : {
     110             :   long t;
     111     5906032 :   if (!al) return al_REAL;
     112     5775881 :   t = typ(alg_get_splittingfield(al));
     113     5775881 :   if (t==t_REAL || t==t_COMPLEX) return al_REAL;
     114     5772465 :   if (isintzero(alg_get_splittingfield(al)) || !gequal0(alg_get_char(al))) return al_TABLE;
     115     3952388 :   switch(typ(gmael(al,2,1))) {
     116      933436 :     case t_MAT: return al_CSA;
     117     3018917 :     case t_INT:
     118             :     case t_FRAC:
     119             :     case t_POL:
     120     3018917 :     case t_POLMOD: return al_CYCLIC;
     121          35 :     default: return al_NULL;
     122             :   }
     123             :   return -1; /*LCOV_EXCL_LINE*/
     124             : }
     125             : long
     126         224 : algtype(GEN al)
     127         224 : { return checkalg_i(al)? alg_type(al): al_NULL; }
     128             : 
     129             : /* absdim == dim for al_TABLE. */
     130             : static long
     131         238 : algreal_dim(GEN al)
     132             : {
     133         238 :   switch(lg(alg_get_multable(al))) {
     134         154 :     case 2: case 3: return 1;
     135          77 :     case 5: return 4;
     136           7 :     default: pari_err_TYPE("algreal_dim", al);
     137             :   }
     138             :   return -1; /*LCOV_EXCL_LINE*/
     139             : }
     140             : long
     141      225335 : alg_get_dim(GEN al)
     142             : {
     143             :   long d;
     144      225335 :   if (!al) return 4;
     145      225335 :   switch(alg_type(al)) {
     146       10876 :     case al_TABLE: return lg(alg_get_multable(al))-1;
     147      214347 :     case al_CSA: return lg(alg_get_relmultable(al))-1;
     148          77 :     case al_CYCLIC: d = alg_get_degree(al); return d*d;
     149          28 :     case al_REAL: return algreal_dim(al);
     150           7 :     default: pari_err_TYPE("alg_get_dim", al);
     151             :   }
     152             :   return -1; /*LCOV_EXCL_LINE*/
     153             : }
     154             : 
     155             : long
     156     1686565 : alg_get_absdim(GEN al)
     157             : {
     158     1686565 :   if (!al) return 4;
     159     1639847 :   switch(alg_type(al)) {
     160      758769 :     case al_TABLE: case al_REAL: return lg(alg_get_multable(al))-1;
     161      113351 :     case al_CSA: return alg_get_dim(al)*nf_get_degree(alg_get_center(al));
     162      767720 :     case al_CYCLIC:
     163      767720 :       return rnf_get_absdegree(alg_get_splittingfield(al))*alg_get_degree(al);
     164           7 :     default: pari_err_TYPE("alg_get_absdim", al);
     165             :   }
     166             :   return -1;/*LCOV_EXCL_LINE*/
     167             : }
     168             : 
     169             : long
     170        1778 : algdim(GEN al, long abs)
     171             : {
     172        1778 :   checkalg(al);
     173        1757 :   if (abs) return alg_get_absdim(al);
     174        1526 :   return alg_get_dim(al);
     175             : }
     176             : 
     177             : /* only cyclic */
     178             : GEN
     179       14308 : alg_get_auts(GEN al)
     180             : {
     181       14308 :   long ta = alg_type(al);
     182       14308 :   if (ta != al_CYCLIC && ta != al_REAL)
     183           0 :     pari_err_TYPE("alg_get_auts [noncyclic algebra]", al);
     184       14308 :   return gel(al,2);
     185             : }
     186             : GEN
     187         112 : alg_get_aut(GEN al)
     188             : {
     189         112 :   long ta = alg_type(al);
     190         112 :   if (ta != al_CYCLIC && ta != al_REAL)
     191           7 :     pari_err_TYPE("alg_get_aut [noncyclic algebra]", al);
     192         105 :   return gel(alg_get_auts(al),1);
     193             : }
     194             : GEN
     195          42 : algaut(GEN al) { checkalg(al); return alg_get_aut(al); }
     196             : GEN
     197       14329 : alg_get_b(GEN al)
     198             : {
     199       14329 :   long ta = alg_type(al);
     200       14329 :   if (ta != al_CYCLIC && ta != al_REAL)
     201           7 :     pari_err_TYPE("alg_get_b [noncyclic algebra]", al);
     202       14322 :   return gel(al,3);
     203             : }
     204             : GEN
     205          56 : algb(GEN al) { checkalg(al); return alg_get_b(al); }
     206             : 
     207             : /* only CSA */
     208             : GEN
     209      216475 : alg_get_relmultable(GEN al)
     210             : {
     211      216475 :   if (alg_type(al) != al_CSA)
     212          14 :     pari_err_TYPE("alg_get_relmultable [algebra not given via mult. table]", al);
     213      216461 :   return gel(al,2);
     214             : }
     215             : GEN
     216          49 : algrelmultable(GEN al) { checkalg(al); return alg_get_relmultable(al); }
     217             : GEN
     218          56 : alg_get_splittingdata(GEN al)
     219             : {
     220          56 :   if (alg_type(al) != al_CSA)
     221          14 :     pari_err_TYPE("alg_get_splittingdata [algebra not given via mult. table]",al);
     222          42 :   return gel(al,3);
     223             : }
     224             : GEN
     225          56 : algsplittingdata(GEN al) { checkalg(al); return alg_get_splittingdata(al); }
     226             : GEN
     227        4102 : alg_get_splittingbasis(GEN al)
     228             : {
     229        4102 :   if (alg_type(al) != al_CSA)
     230           0 :     pari_err_TYPE("alg_get_splittingbasis [algebra not given via mult. table]",al);
     231        4102 :   return gmael(al,3,2);
     232             : }
     233             : GEN
     234        4102 : alg_get_splittingbasisinv(GEN al)
     235             : {
     236        4102 :   if (alg_type(al) != al_CSA)
     237           0 :     pari_err_TYPE("alg_get_splittingbasisinv [algebra not given via mult. table]",al);
     238        4102 :   return gmael(al,3,3);
     239             : }
     240             : 
     241             : /* only cyclic and CSA */
     242             : GEN
     243    14902376 : alg_get_splittingfield(GEN al) { return gel(al,1); }
     244             : GEN
     245         119 : algsplittingfield(GEN al)
     246             : {
     247             :   long ta;
     248         119 :   checkalg(al);
     249         119 :   ta = alg_type(al);
     250         119 :   if (ta != al_CYCLIC && ta != al_CSA && ta != al_REAL)
     251           7 :     pari_err_TYPE("alg_get_splittingfield [use alginit]",al);
     252         112 :   return alg_get_splittingfield(al);
     253             : }
     254             : long
     255     1217074 : alg_get_degree(GEN al)
     256             : {
     257             :   long ta;
     258     1217074 :   ta = alg_type(al);
     259     1217074 :   if (ta == al_REAL) return algreal_dim(al)==1? 1 : 2;
     260     1216990 :   if (ta != al_CYCLIC && ta != al_CSA)
     261          21 :     pari_err_TYPE("alg_get_degree [use alginit]",al);
     262     1216969 :   return rnf_get_degree(alg_get_splittingfield(al));
     263             : }
     264             : long
     265         322 : algdegree(GEN al)
     266             : {
     267         322 :   checkalg(al);
     268         315 :   return alg_get_degree(al);
     269             : }
     270             : 
     271             : GEN
     272      302961 : alg_get_center(GEN al)
     273             : {
     274             :   long ta;
     275      302961 :   ta = alg_type(al);
     276      302961 :   if (ta == al_REAL)
     277             :   {
     278          21 :     if (algreal_dim(al) != 4) return alg_get_splittingfield(al);
     279           7 :     return stor(1,3);
     280             :   }
     281      302940 :   if (ta != al_CSA && ta != al_CYCLIC)
     282           7 :     pari_err_TYPE("alg_get_center [use alginit]",al);
     283      302933 :   return rnf_get_nf(alg_get_splittingfield(al));
     284             : }
     285             : GEN
     286          70 : alg_get_splitpol(GEN al)
     287             : {
     288          70 :   long ta = alg_type(al);
     289          70 :   if (ta != al_CYCLIC && ta != al_CSA)
     290           0 :     pari_err_TYPE("alg_get_splitpol [use alginit]",al);
     291          70 :   return rnf_get_pol(alg_get_splittingfield(al));
     292             : }
     293             : GEN
     294       72038 : alg_get_abssplitting(GEN al)
     295             : {
     296       72038 :   long ta = alg_type(al), prec;
     297       72038 :   if (ta != al_CYCLIC && ta != al_CSA)
     298           0 :     pari_err_TYPE("alg_get_abssplitting [use alginit]",al);
     299       72038 :   prec = nf_get_prec(alg_get_center(al));
     300       72038 :   return rnf_build_nfabs(alg_get_splittingfield(al), prec);
     301             : }
     302             : GEN
     303        1204 : alg_get_hasse_i(GEN al)
     304             : {
     305        1204 :   long ta = alg_type(al);
     306        1204 :   if (ta != al_CYCLIC && ta != al_CSA && ta != al_REAL)
     307           7 :     pari_err_TYPE("alg_get_hasse_i [use alginit]",al);
     308        1197 :   if (ta == al_CSA) pari_err_IMPL("computation of Hasse invariants over table CSA");
     309        1190 :   return gel(al,4);
     310             : }
     311             : GEN
     312         231 : alghassei(GEN al) { checkalg(al); return alg_get_hasse_i(al); }
     313             : GEN
     314        1988 : alg_get_hasse_f(GEN al)
     315             : {
     316        1988 :   long ta = alg_type(al);
     317             :   GEN hf;
     318        1988 :   if (ta != al_CYCLIC && ta != al_CSA)
     319           7 :     pari_err_TYPE("alg_get_hasse_f [use alginit]",al);
     320        1981 :   if (ta == al_CSA) pari_err_IMPL("computation of Hasse invariants over table CSA");
     321        1974 :   hf = gel(al,5);
     322        1974 :   if (typ(hf) == t_INT) /* could be computed on the fly */
     323          28 :     pari_err(e_MISC, "Hasse invariants were not computed for this algebra");
     324        1946 :   return hf;
     325             : }
     326             : GEN
     327         336 : alghassef(GEN al) { checkalg(al); return alg_get_hasse_f(al); }
     328             : 
     329             : /* all types */
     330             : GEN
     331        2744 : alg_get_basis(GEN al) { return gel(al,7); }
     332             : GEN
     333          91 : algbasis(GEN al) { checkalg(al); return alg_get_basis(al); }
     334             : GEN
     335       61881 : alg_get_invbasis(GEN al) { return gel(al,8); }
     336             : GEN
     337          63 : alginvbasis(GEN al) { checkalg(al); return alg_get_invbasis(al); }
     338             : GEN
     339     2500461 : alg_get_multable(GEN al) { return gel(al,9); }
     340             : GEN
     341         245 : algmultable(GEN al) { checkalg(al); return alg_get_multable(al); }
     342             : GEN
     343     6119767 : alg_get_char(GEN al) { if (!al) return gen_0; return gel(al,10); }
     344             : GEN
     345         112 : algchar(GEN al) { checkalg(al); return alg_get_char(al); }
     346             : GEN
     347      246427 : alg_get_tracebasis(GEN al) { return gel(al,11); }
     348             : 
     349             : /* lattices */
     350             : GEN
     351      244314 : alglat_get_primbasis(GEN lat) { return gel(lat,1); }
     352             : GEN
     353      289905 : alglat_get_scalar(GEN lat) { return gel(lat,2); }
     354             : 
     355             : /** ADDITIONAL **/
     356             : 
     357             : /* is N=smooth*prime? */
     358       14682 : static int Z_easyfactor(GEN N, ulong lim)
     359             : {
     360             :   GEN fa;
     361       14682 :   if (lgefint(N) <= 3) return 1;
     362       13629 :   fa = absZ_factor_limit(N, lim);
     363       13629 :   return BPSW_psp(veclast(gel(fa,1)));
     364             : }
     365             : 
     366             : /* no garbage collection */
     367             : static GEN
     368        1106 : backtrackfacto(GEN y0, long n, GEN red, GEN pl, GEN nf, GEN data, int (*test)(GEN,GEN), GEN* fa, GEN N, GEN I)
     369             : {
     370             :   long b, i;
     371        1106 :   ulong lim = 1UL << 17;
     372        1106 :   long *v = new_chunk(n+1);
     373        1106 :   pari_sp av = avma;
     374        1106 :   for (b = 0;; b += (2*b)/(3*n) + 1)
     375         316 :   {
     376             :     GEN ny, y1, y2;
     377        1422 :     set_avma(av);
     378        4242 :     for (i = 1; i <= n; i++) v[i] = -b;
     379        1422 :     v[n]--;
     380             :     for(;;)
     381             :     {
     382       15035 :       i = n;
     383       15629 :       while (i > 0)
     384       15313 :       { if (v[i] == b) v[i--] = -b; else { v[i]++; break; } }
     385       15035 :       if (i==0) break;
     386             : 
     387       14719 :       y1 = y0;
     388       31835 :       for (i = 1; i <= n; i++) y1 = nfadd(nf, y1, ZC_z_mul(gel(red,i), v[i]));
     389       14719 :       if (!nfchecksigns(nf, y1, pl)) continue;
     390             : 
     391       14682 :       ny = absi_shallow(nfnorm(nf, y1));
     392       14682 :       if (!signe(ny)) continue;
     393       14682 :       ny = diviiexact(ny, gcdii(ny, N));
     394       14682 :       if (!Z_easyfactor(ny, lim)) continue;
     395             : 
     396        1661 :       y2 = idealdivexact(nf, y1, idealadd(nf,y1,I));
     397        1661 :       *fa = idealfactor(nf, y2);
     398        1661 :       if (!data || test(data,*fa)) return y1;
     399             :     }
     400             :   }
     401             : }
     402             : 
     403             : /* if data == NULL, the test is skipped */
     404             : /* in the test, the factorization does not contain the known factors */
     405             : static GEN
     406        1106 : factoredextchinesetest(GEN nf, GEN x, GEN y, GEN pl, GEN* fa, GEN data, int (*test)(GEN,GEN))
     407             : {
     408        1106 :   pari_sp av = avma;
     409             :   long n,i;
     410        1106 :   GEN x1, y0, y1, red, N, I, P = gel(x,1), E = gel(x,2);
     411        1106 :   n = nf_get_degree(nf);
     412        1106 :   x = idealchineseinit(nf, mkvec2(x,pl));
     413        1106 :   x1 = gel(x,1);
     414        1106 :   red = lg(x1) == 1? matid(n): gmael(x1,1,1);
     415        1106 :   y0 = idealchinese(nf, x, y);
     416             : 
     417        1106 :   E = shallowcopy(E);
     418        1106 :   if (!gequal0(y0))
     419        5839 :     for (i=1; i<lg(E); i++)
     420             :     {
     421        4733 :       long v = nfval(nf,y0,gel(P,i));
     422        4733 :       if (cmpsi(v, gel(E,i)) < 0) gel(E,i) = stoi(v);
     423             :     }
     424             :   /* N and I : known factors */
     425        1106 :   I = factorbackprime(nf, P, E);
     426        1106 :   N = idealnorm(nf,I);
     427             : 
     428        1106 :   y1 = backtrackfacto(y0, n, red, pl, nf, data, test, fa, N, I);
     429             : 
     430             :   /* restore known factors */
     431        5839 :   for (i=1; i<lg(E); i++) gel(E,i) = stoi(nfval(nf,y1,gel(P,i)));
     432        1106 :   *fa = famat_reduce(famat_mul_shallow(*fa, mkmat2(P, E)));
     433        1106 :   return gc_all(av, 2, &y1, fa);
     434             : }
     435             : 
     436             : static GEN
     437         833 : factoredextchinese(GEN nf, GEN x, GEN y, GEN pl, GEN* fa)
     438         833 : { return factoredextchinesetest(nf,x,y,pl,fa,NULL,NULL); }
     439             : 
     440             : /** OPERATIONS ON ASSOCIATIVE ALGEBRAS algebras.c **/
     441             : 
     442             : /*
     443             : Convention:
     444             : (K/F,sigma,b) = sum_{i=0..n-1} u^i*K
     445             : t*u = u*sigma(t)
     446             : 
     447             : Natural basis:
     448             : 1<=i<=d*n^2
     449             : b_i = u^((i-1)/(dn))*ZKabs.((i-1)%(dn)+1)
     450             : 
     451             : Integral basis:
     452             : Basis of some order.
     453             : 
     454             : al:
     455             : 1- rnf of the cyclic splitting field of degree n over the center nf of degree d
     456             : 2- VEC of aut^i 1<=i<=n if n>1, or i=0 if n=1
     457             : 3- b in nf
     458             : 4- infinite hasse invariants (mod n) : VECSMALL of size r1, values only 0 or n/2 (if integral)
     459             : 5- finite hasse invariants (mod n) : VEC[VEC of primes, VECSMALL of hasse inv mod n]
     460             : 6- nf of the splitting field (absolute)
     461             : 7* dn^2*dn^2 matrix expressing the integral basis in terms of the natural basis
     462             : 8* dn^2*dn^2 matrix expressing the natural basis in terms of the integral basis
     463             : 9* VEC of dn^2 matrices giving the dn^2*dn^2 left multiplication tables of the integral basis
     464             : 10* characteristic of the base field (used only for algebras given by a multiplication table)
     465             : 11* trace of basis elements
     466             : 
     467             : If al is given by a multiplication table (al_TABLE), only the * fields are present.
     468             : */
     469             : 
     470             : /* assumes same center and same variable */
     471             : /* currently only works for coprime degrees */
     472             : GEN
     473          84 : algtensor(GEN al1, GEN al2, long flag) {
     474          84 :   pari_sp av = avma;
     475             :   long v, k, d1, d2;
     476             :   GEN nf, P1, P2, aut1, aut2, b1, b2, C, rnf, aut, b, x1, x2, al, rnfpol;
     477             : 
     478          84 :   checkalg(al1);
     479          70 :   checkalg(al2);
     480          63 :   if (alg_type(al1) != al_CYCLIC  || alg_type(al2) != al_CYCLIC)
     481          21 :     pari_err_IMPL("tensor of noncyclic algebras"); /* TODO: do it. */
     482             : 
     483          42 :   nf = alg_get_center(al1);
     484          42 :   if (!gequal(alg_get_center(al2),nf))
     485           7 :     pari_err_OP("tensor product [not the same center]", al1, al2);
     486             : 
     487          35 :   P1=alg_get_splitpol(al1); aut1=alg_get_aut(al1); b1=alg_get_b(al1);
     488          35 :   P2=alg_get_splitpol(al2); aut2=alg_get_aut(al2); b2=alg_get_b(al2);
     489          35 :   v=varn(P1);
     490             : 
     491          35 :   d1=alg_get_degree(al1);
     492          35 :   d2=alg_get_degree(al2);
     493          35 :   if (ugcd(d1,d2) != 1)
     494           7 :     pari_err_IMPL("tensor of cyclic algebras of noncoprime degrees"); /* TODO */
     495             : 
     496          28 :   if (d1==1) return gcopy(al2);
     497          21 :   if (d2==1) return gcopy(al1);
     498             : 
     499          14 :   C = nfcompositum(nf, P1, P2, 3);
     500          14 :   rnfpol = gel(C,1);
     501          14 :   if (!(flag & al_FACTOR)) rnfpol = mkvec2(rnfpol, stoi(1<<20));
     502          14 :   rnf = rnfinit(nf, rnfpol);
     503             :   /* TODO use integral basis of P1 and P2 to get that of C */
     504          14 :   x1 = gel(C,2);
     505          14 :   x2 = gel(C,3);
     506          14 :   k = itos(gel(C,4));
     507          14 :   aut = gadd(gsubst(aut2,v,x2),gmulsg(k,gsubst(aut1,v,x1)));
     508          14 :   b = nfmul(nf,nfpow_u(nf,b1,d2),nfpow_u(nf,b2,d1));
     509          14 :   al = alg_cyclic(rnf, aut, b, flag);
     510          14 :   return gerepilecopy(av,al);
     511             : }
     512             : 
     513             : /* M an n x d Flm of rank d, n >= d. Initialize Mx = y solver */
     514             : static GEN
     515        4454 : Flm_invimage_init(GEN M, ulong p)
     516             : {
     517        4454 :   GEN v = Flm_indexrank(M, p), perm = gel(v,1);
     518        4454 :   GEN MM = rowpermute(M, perm); /* square invertible */
     519        4454 :   return mkvec2(Flm_inv(MM,p), perm);
     520             : }
     521             : /* assume Mx = y has a solution, v = Flm_invimage_init(M,p); return x */
     522             : static GEN
     523      245283 : Flm_invimage_pre(GEN v, GEN y, ulong p)
     524             : {
     525      245283 :   GEN inv = gel(v,1), perm = gel(v,2);
     526      245283 :   return Flm_Flc_mul(inv, vecsmallpermute(y, perm), p);
     527             : }
     528             : 
     529             : GEN
     530        6342 : algradical(GEN al)
     531             : {
     532        6342 :   pari_sp av = avma;
     533             :   GEN I, x, traces, K, MT, P, mt;
     534             :   long l,i,ni, n;
     535             :   ulong modu, expo, p;
     536        6342 :   checkalg(al);
     537        6342 :   if (alg_type(al) != al_TABLE) return gen_0;
     538        6251 :   P = alg_get_char(al);
     539        6251 :   mt = alg_get_multable(al);
     540        6251 :   n = alg_get_absdim(al);
     541        6251 :   dbg_printf(1)("algradical: char=%Ps, dim=%d\n", P, n);
     542        6251 :   traces = algtracematrix(al);
     543        6251 :   if (!signe(P))
     544             :   {
     545         518 :     dbg_printf(2)(" char 0, computing kernel...\n");
     546         518 :     K = ker(traces);
     547         518 :     dbg_printf(2)(" ...done.\n");
     548         518 :     ni = lg(K)-1; if (!ni) return gc_const(av, gen_0);
     549          70 :     return gerepileupto(av, K);
     550             :   }
     551        5733 :   dbg_printf(2)(" char>0, computing kernel...\n");
     552        5733 :   K = FpM_ker(traces, P);
     553        5733 :   dbg_printf(2)(" ...done.\n");
     554        5733 :   ni = lg(K)-1; if (!ni) return gc_const(av, gen_0);
     555        3768 :   if (abscmpiu(P,n)>0) return gerepileupto(av, K);
     556             : 
     557             :   /* tough case, p <= n. Ronyai's algorithm */
     558        2418 :   p = P[2]; l = 1;
     559        2418 :   expo = p; modu = p*p;
     560        2418 :   dbg_printf(2)(" char>0, hard case.\n");
     561        4895 :   while (modu<=(ulong)n) { l++; modu *= p; }
     562        2418 :   MT = ZMV_to_FlmV(mt, modu);
     563        2418 :   I = ZM_to_Flm(K,p); /* I_0 */
     564        6536 :   for (i=1; i<=l; i++) {/*compute I_i, expo = p^i, modu = p^(l+1) > n*/
     565             :     long j, lig,col;
     566        4454 :     GEN v = cgetg(ni+1, t_VECSMALL);
     567        4454 :     GEN invI = Flm_invimage_init(I, p);
     568        4454 :     dbg_printf(2)(" computing I_%d:\n", i);
     569        4454 :     traces = cgetg(ni+1,t_MAT);
     570       29356 :     for (j = 1; j <= ni; j++)
     571             :     {
     572       24902 :       GEN M = algbasismultable_Flm(MT, gel(I,j), modu);
     573       24902 :       uel(v,j) = algtracei(M, p,expo,modu);
     574             :     }
     575       29356 :     for (col=1; col<=ni; col++)
     576             :     {
     577       24902 :       GEN t = cgetg(n+1,t_VECSMALL); gel(traces,col) = t;
     578       24902 :       x = gel(I, col); /*col-th basis vector of I_{i-1}*/
     579      270185 :       for (lig=1; lig<=n; lig++)
     580             :       {
     581      245283 :         GEN y = _tablemul_ej_Fl(MT,x,lig,p);
     582      245283 :         GEN z = Flm_invimage_pre(invI, y, p);
     583      245283 :         uel(t,lig) = Flv_dotproduct(v, z, p);
     584             :       }
     585             :     }
     586        4454 :     dbg_printf(2)(" computing kernel...\n");
     587        4454 :     K = Flm_ker(traces, p);
     588        4454 :     dbg_printf(2)(" ...done.\n");
     589        4454 :     ni = lg(K)-1; if (!ni) return gc_const(av, gen_0);
     590        4118 :     I = Flm_mul(I,K,p);
     591        4118 :     expo *= p;
     592             :   }
     593        2082 :   return Flm_to_ZM(I);
     594             : }
     595             : 
     596             : /* compute the multiplication table of the element x, where mt is a
     597             :  * multiplication table in an arbitrary ring */
     598             : static GEN
     599         427 : Rgmultable(GEN mt, GEN x)
     600             : {
     601         427 :   long i, l = lg(x);
     602         427 :   GEN z = NULL;
     603        5796 :   for (i = 1; i < l; i++)
     604             :   {
     605        5369 :     GEN c = gel(x,i);
     606        5369 :     if (!gequal0(c))
     607             :     {
     608         644 :       GEN M = RgM_Rg_mul(gel(mt,i),c);
     609         644 :       z = z? RgM_add(z, M): M;
     610             :     }
     611             :   }
     612         427 :   return z;
     613             : }
     614             : 
     615             : static GEN
     616          49 : change_Rgmultable(GEN mt, GEN P, GEN Pi)
     617             : {
     618             :   GEN mt2;
     619          49 :   long lmt = lg(mt), i;
     620          49 :   mt2 = cgetg(lmt,t_VEC);
     621         476 :   for (i=1;i<lmt;i++) {
     622         427 :     GEN mti = Rgmultable(mt,gel(P,i));
     623         427 :     gel(mt2,i) = RgM_mul(Pi, RgM_mul(mti,P));
     624             :   }
     625          49 :   return mt2;
     626             : }
     627             : 
     628             : static GEN
     629       21463 : alg_quotient0(GEN al, GEN S, GEN Si, long nq, GEN p, long maps)
     630             : {
     631       21463 :   GEN mt = cgetg(nq+1,t_VEC), P, Pi, d;
     632             :   long i;
     633       21463 :   dbg_printf(3)("  alg_quotient0: char=%Ps, dim=%d, dim I=%d\n", p, alg_get_absdim(al), lg(S)-1);
     634       84903 :   for (i=1; i<=nq; i++) {
     635       63440 :     GEN mti = algbasismultable(al,gel(S,i));
     636       63440 :     if (signe(p)) gel(mt,i) = FpM_mul(Si, FpM_mul(mti,S,p), p);
     637        5390 :     else          gel(mt,i) = RgM_mul(Si, RgM_mul(mti,S));
     638             :   }
     639       21463 :   if (!signe(p) && !isint1(Q_denom(mt))) {
     640          35 :     dbg_printf(3)("  bad case: denominator=%Ps\n", Q_denom(mt));
     641          35 :     P = Q_remove_denom(Si,&d);
     642          35 :     P = ZM_hnf(P);
     643          35 :     P = RgM_Rg_div(P,d);
     644          35 :     Pi = RgM_inv(P);
     645          35 :     mt = change_Rgmultable(mt,P,Pi);
     646          35 :     Si = RgM_mul(P,Si);
     647          35 :     S = RgM_mul(S,Pi);
     648             :   }
     649       21463 :   al = algtableinit_i(mt,p);
     650       21463 :   if (maps) al = mkvec3(al,Si,S); /* algebra, proj, lift */
     651       21463 :   return al;
     652             : }
     653             : 
     654             : /* quotient of an algebra by a nontrivial two-sided ideal */
     655             : GEN
     656        3523 : alg_quotient(GEN al, GEN I, long maps)
     657             : {
     658        3523 :   pari_sp av = avma;
     659             :   GEN p, IS, ISi, S, Si;
     660             :   long n, ni;
     661             : 
     662        3523 :   checkalg(al);
     663        3523 :   if (alg_type(al) != al_TABLE) pari_err_TYPE("alg_quotient [not a table algebra]", al);
     664        3516 :   p = alg_get_char(al);
     665        3516 :   n = alg_get_absdim(al);
     666        3516 :   ni = lg(I)-1;
     667             : 
     668             :   /* force first vector of complement to be the identity */
     669        3516 :   IS = shallowconcat(I, gcoeff(alg_get_multable(al),1,1));
     670        3516 :   if (signe(p)) {
     671        3488 :     IS = FpM_suppl(IS,p);
     672        3488 :     ISi = FpM_inv(IS,p);
     673             :   }
     674             :   else {
     675          28 :     IS = suppl(IS);
     676          28 :     ISi = RgM_inv(IS);
     677             :   }
     678        3516 :   S = vecslice(IS, ni+1, n);
     679        3516 :   Si = rowslice(ISi, ni+1, n);
     680        3516 :   return gerepilecopy(av, alg_quotient0(al, S, Si, n-ni, p, maps));
     681             : }
     682             : 
     683             : static GEN
     684       28187 : image_keep_first(GEN m, GEN p) /* assume first column is nonzero or m==0, no GC */
     685             : {
     686             :   GEN ir, icol, irow, M, c, x;
     687             :   long i;
     688       28187 :   if (gequal0(gel(m,1))) return zeromat(nbrows(m),0);
     689             : 
     690       28173 :   if (signe(p)) ir = FpM_indexrank(m,p);
     691        1498 :   else          ir = indexrank(m);
     692             : 
     693       28173 :   icol = gel(ir,2);
     694       28173 :   if (icol[1]==1) return extract0(m,icol,NULL);
     695             : 
     696           7 :   irow = gel(ir,1);
     697           7 :   M = extract0(m, irow, icol);
     698           7 :   c = extract0(gel(m,1), irow, NULL);
     699           7 :   if (signe(p)) x = FpM_FpC_invimage(M,c,p);
     700           0 :   else          x = inverseimage(M,c); /* TODO modulo a small prime */
     701             : 
     702           7 :   for (i=1; i<lg(x); i++)
     703             :   {
     704           7 :     if (!gequal0(gel(x,i)))
     705             :     {
     706           7 :       icol[i] = 1;
     707           7 :       vecsmall_sort(icol);
     708           7 :       return extract0(m,icol,NULL);
     709             :     }
     710             :   }
     711             : 
     712             :   return NULL; /* LCOV_EXCL_LINE */
     713             : }
     714             : 
     715             : /* z[1],...z[nz] central elements such that z[1]A + z[2]A + ... + z[nz]A = A
     716             :  * is a direct sum. idempotents ==> first basis element is identity */
     717             : GEN
     718        8633 : alg_centralproj(GEN al, GEN z, long maps)
     719             : {
     720        8633 :   pari_sp av = avma;
     721             :   GEN S, U, Ui, alq, p;
     722        8633 :   long i, iu, lz = lg(z), ta;
     723             : 
     724        8633 :   checkalg(al);
     725        8633 :   ta = alg_type(al);
     726        8633 :   if (ta != al_TABLE) pari_err_TYPE("algcentralproj [not a table algebra]", al);
     727        8626 :   if (typ(z) != t_VEC) pari_err_TYPE("alcentralproj",z);
     728        8619 :   p = alg_get_char(al);
     729        8619 :   dbg_printf(3)("  alg_centralproj: char=%Ps, dim=%d, #z=%d\n", p, alg_get_absdim(al), lz-1);
     730        8619 :   S = cgetg(lz,t_VEC); /* S[i] = Im(z_i) */
     731       26580 :   for (i=1; i<lz; i++)
     732             :   {
     733       17961 :     GEN mti = algbasismultable(al, gel(z,i));
     734       17961 :     gel(S,i) = image_keep_first(mti,p);
     735             :   }
     736        8619 :   U = shallowconcat1(S); /* U = [Im(z_1)|Im(z_2)|...|Im(z_nz)], n x n */
     737        8619 :   if (lg(U)-1 < alg_get_absdim(al)) pari_err_TYPE("alcentralproj [z[i]'s not surjective]",z);
     738        8612 :   if (signe(p)) Ui = FpM_inv(U,p);
     739         749 :   else          Ui = RgM_inv(U);
     740             :   if (!Ui) pari_err_BUG("alcentralproj"); /*LCOV_EXCL_LINE*/
     741             : 
     742        8612 :   alq = cgetg(lz,t_VEC);
     743       26559 :   for (iu=0,i=1; i<lz; i++)
     744             :   {
     745       17947 :     long nq = lg(gel(S,i))-1, ju = iu + nq;
     746       17947 :     GEN Si = rowslice(Ui, iu+1, ju);
     747       17947 :     gel(alq, i) = alg_quotient0(al,gel(S,i),Si,nq,p,maps);
     748       17947 :     iu = ju;
     749             :   }
     750        8612 :   return gerepilecopy(av, alq);
     751             : }
     752             : 
     753             : /* al is an al_TABLE */
     754             : static GEN
     755       19658 : algtablecenter(GEN al)
     756             : {
     757       19658 :   pari_sp av = avma;
     758             :   long n, i, j, k, ic;
     759             :   GEN C, cij, mt, p;
     760             : 
     761       19658 :   n = alg_get_absdim(al);
     762       19658 :   mt = alg_get_multable(al);
     763       19658 :   p = alg_get_char(al);
     764       19658 :   C = cgetg(n+1,t_MAT);
     765       94151 :   for (j=1; j<=n; j++)
     766             :   {
     767       74493 :     gel(C,j) = cgetg(n*n-n+1,t_COL);
     768       74493 :     ic = 1;
     769      600051 :     for (i=2; i<=n; i++) {
     770      525558 :       if (signe(p)) cij = FpC_sub(gmael(mt,i,j),gmael(mt,j,i),p);
     771       52318 :       else          cij = RgC_sub(gmael(mt,i,j),gmael(mt,j,i));
     772     7351662 :       for (k=1; k<=n; k++, ic++) gcoeff(C,ic,j) = gel(cij, k);
     773             :     }
     774             :   }
     775       19658 :   if (signe(p)) return gerepileupto(av, FpM_ker(C,p));
     776        1645 :   else          return gerepileupto(av, ker(C));
     777             : }
     778             : 
     779             : GEN
     780        4886 : algcenter(GEN al)
     781             : {
     782        4886 :   checkalg(al);
     783        4886 :   if (alg_type(al)==al_TABLE) return algtablecenter(al);
     784          49 :   return alg_get_center(al);
     785             : }
     786             : 
     787             : /* Only in positive characteristic. Assumes that al is semisimple. */
     788             : GEN
     789        4995 : algprimesubalg(GEN al)
     790             : {
     791        4995 :   pari_sp av = avma;
     792             :   GEN p, Z, F, K;
     793             :   long nz, i;
     794        4995 :   checkalg(al);
     795        4995 :   p = alg_get_char(al);
     796        4995 :   if (!signe(p)) pari_err_DOMAIN("algprimesubalg","characteristic","=",gen_0,p);
     797             : 
     798        4981 :   Z = algtablecenter(al);
     799        4981 :   nz = lg(Z)-1;
     800        4981 :   if (nz==1) return Z;
     801             : 
     802        3602 :   F = cgetg(nz+1, t_MAT);
     803       17105 :   for (i=1; i<=nz; i++) {
     804       13503 :     GEN zi = gel(Z,i);
     805       13503 :     gel(F,i) = FpC_sub(algpow(al,zi,p),zi,p);
     806             :   }
     807        3602 :   K = FpM_ker(F,p);
     808        3602 :   return gerepileupto(av, FpM_mul(Z,K,p));
     809             : }
     810             : 
     811             : static GEN
     812       14873 : out_decompose(GEN t, GEN Z, GEN P, GEN p)
     813             : {
     814       14873 :   GEN ali = gel(t,1), projm = gel(t,2), liftm = gel(t,3), pZ;
     815       14873 :   if (signe(p)) pZ = FpM_image(FpM_mul(projm,Z,p),p);
     816        1407 :   else          pZ = image(RgM_mul(projm,Z));
     817       14873 :   return mkvec5(ali, projm, liftm, pZ, P);
     818             : }
     819             : /* fa factorization of charpol(x) */
     820             : static GEN
     821        7475 : alg_decompose_from_facto(GEN al, GEN x, GEN fa, GEN Z, long mini)
     822             : {
     823        7475 :   long k = lgcols(fa)-1, k2 = mini? 1: k/2;
     824        7475 :   GEN v1 = rowslice(fa,1,k2);
     825        7475 :   GEN v2 = rowslice(fa,k2+1,k);
     826        7475 :   GEN alq, P, Q, p = alg_get_char(al);
     827        7475 :   dbg_printf(3)("  alg_decompose_from_facto\n");
     828        7475 :   if (signe(p)) {
     829        6754 :     P = FpXV_factorback(gel(v1,1), gel(v1,2), p, 0);
     830        6754 :     Q = FpXV_factorback(gel(v2,1), gel(v2,2), p, 0);
     831        6754 :     P = FpX_mul(P, FpXQ_inv(P,Q,p), p);
     832             :   }
     833             :   else {
     834         721 :     P = factorback(v1);
     835         721 :     Q = factorback(v2);
     836         721 :     P = RgX_mul(P, RgXQ_inv(P,Q));
     837             :   }
     838        7475 :   P = algpoleval(al, P, x);
     839        7475 :   if (signe(p)) Q = FpC_sub(col_ei(lg(P)-1,1), P, p);
     840         721 :   else          Q = gsub(gen_1, P);
     841        7475 :   if (gequal0(P) || gequal0(Q)) return NULL;
     842        7475 :   alq = alg_centralproj(al, mkvec2(P,Q), 1);
     843             : 
     844        7475 :   P = out_decompose(gel(alq,1), Z, P, p); if (mini) return P;
     845        7398 :   Q = out_decompose(gel(alq,2), Z, Q, p);
     846        7398 :   return mkvec2(P,Q);
     847             : }
     848             : 
     849             : static GEN
     850       11934 : random_pm1(long n)
     851             : {
     852       11934 :   GEN z = cgetg(n+1,t_VECSMALL);
     853             :   long i;
     854       52286 :   for (i = 1; i <= n; i++) z[i] = random_bits(5)%3 - 1;
     855       11934 :   return z;
     856             : }
     857             : 
     858             : static GEN alg_decompose(GEN al, GEN Z, long mini, GEN* pt_primelt);
     859             : /* Try to split al using x's charpoly. Return gen_0 if simple, NULL if failure.
     860             :  * And a splitting otherwise
     861             :  * If pt_primelt!=NULL, compute a primitive element of the center when simple */
     862             : static GEN
     863       13958 : try_fact(GEN al, GEN x, GEN zx, GEN Z, GEN Zal, long mini, GEN* pt_primelt)
     864             : {
     865       13958 :   GEN z, dec0, dec1, cp = algcharpoly(Zal,zx,0,1), fa, p = alg_get_char(al);
     866             :   long nfa, e;
     867       13958 :   dbg_printf(3)("  try_fact: zx=%Ps\n", zx);
     868       13958 :   if (signe(p)) fa = FpX_factor(cp,p);
     869        1281 :   else          fa = factor(cp);
     870       13958 :   dbg_printf(3)("  charpoly=%Ps\n", fa);
     871       13958 :   nfa = nbrows(fa);
     872       13958 :   if (nfa == 1) {
     873        6483 :     if (signe(p)) e = gel(fa,2)[1];
     874         560 :     else          e = itos(gcoeff(fa,1,2));
     875        6483 :     if (e == 1) {
     876        3689 :       if (pt_primelt != NULL) *pt_primelt = mkvec2(x, cp);
     877        3689 :       return gen_0;
     878             :     }
     879        2794 :     else return NULL;
     880             :   }
     881        7475 :   dec0 = alg_decompose_from_facto(al, x, fa, Z, mini);
     882        7475 :   if (!dec0) return NULL;
     883        7475 :   if (!mini) return dec0;
     884          77 :   dec1 = alg_decompose(gel(dec0,1), gel(dec0,4), 1, pt_primelt);
     885          77 :   z = gel(dec0,5);
     886          77 :   if (!isintzero(dec1)) {
     887           7 :     if (signe(p)) z = FpM_FpC_mul(gel(dec0,3),dec1,p);
     888           7 :     else          z = RgM_RgC_mul(gel(dec0,3),dec1);
     889             :   }
     890          77 :   return z;
     891             : }
     892             : static GEN
     893           7 : randcol(long n, GEN b)
     894             : {
     895           7 :   GEN N = addiu(shifti(b,1), 1);
     896             :   long i;
     897           7 :   GEN res =  cgetg(n+1,t_COL);
     898          63 :   for (i=1; i<=n; i++)
     899             :   {
     900          56 :     pari_sp av = avma;
     901          56 :     gel(res,i) = gerepileuptoint(av, subii(randomi(N),b));
     902             :   }
     903           7 :   return res;
     904             : }
     905             : /* Return gen_0 if already simple. mini: only returns a central idempotent
     906             :  * corresponding to one simple factor
     907             :  * if pt_primelt!=NULL, sets it to a primitive element of the center when simple */
     908             : static GEN
     909       20500 : alg_decompose(GEN al, GEN Z, long mini, GEN* pt_primelt)
     910             : {
     911             :   pari_sp av;
     912             :   GEN Zal, x, zx, rand, dec0, B, p;
     913       20500 :   long i, nz = lg(Z)-1;
     914             : 
     915       20500 :   if (nz == 1) {
     916        9336 :     if (pt_primelt != 0) *pt_primelt = mkvec2(zerocol(alg_get_dim(al)), pol_x(0));
     917        9336 :     return gen_0;
     918             :   }
     919       11164 :   p = alg_get_char(al);
     920       11164 :   dbg_printf(2)(" alg_decompose: char=%Ps, dim=%d, dim Z=%d\n", p, alg_get_absdim(al), nz);
     921       11164 :   Zal = alg_subalg(al,Z);
     922       11164 :   Z = gel(Zal,2);
     923       11164 :   Zal = gel(Zal,1);
     924       11164 :   av = avma;
     925             : 
     926       11164 :   rand = random_pm1(nz);
     927       11164 :   zx = zc_to_ZC(rand);
     928       11164 :   if (signe(p)) {
     929       10191 :     zx = FpC_red(zx,p);
     930       10191 :     x = ZM_zc_mul(Z,rand);
     931       10191 :     x = FpC_red(x,p);
     932             :   }
     933         973 :   else x = RgM_zc_mul(Z,rand);
     934       11164 :   dec0 = try_fact(al,x,zx,Z,Zal,mini,pt_primelt);
     935       11164 :   if (dec0) return dec0;
     936        2738 :   set_avma(av);
     937             : 
     938        2794 :   for (i=2; i<=nz; i++)
     939             :   {
     940        2787 :     dec0 = try_fact(al,gel(Z,i),col_ei(nz,i),Z,Zal,mini,pt_primelt);
     941        2787 :     if (dec0) return dec0;
     942          56 :     set_avma(av);
     943             :   }
     944           7 :   B = int2n(10);
     945             :   for (;;)
     946           0 :   {
     947           7 :     GEN x = randcol(nz,B), zx = ZM_ZC_mul(Z,x);
     948           7 :     dec0 = try_fact(al,x,zx,Z,Zal,mini,pt_primelt);
     949           7 :     if (dec0) return dec0;
     950           0 :     set_avma(av);
     951             :   }
     952             : }
     953             : 
     954             : static GEN
     955       16902 : alg_decompose_total(GEN al, GEN Z, long maps)
     956             : {
     957             :   GEN dec, sc, p;
     958             :   long i;
     959             : 
     960       16902 :   dec = alg_decompose(al, Z, 0, NULL);
     961       16902 :   if (isintzero(dec))
     962             :   {
     963        9504 :     if (maps) {
     964        6872 :       long n = alg_get_absdim(al);
     965        6872 :       al = mkvec3(al, matid(n), matid(n));
     966             :     }
     967        9504 :     return mkvec(al);
     968             :   }
     969        7398 :   p = alg_get_char(al); if (!signe(p)) p = NULL;
     970        7398 :   sc = cgetg(lg(dec), t_VEC);
     971       22194 :   for (i=1; i<lg(sc); i++) {
     972       14796 :     GEN D = gel(dec,i), a = gel(D,1), Za = gel(D,4);
     973       14796 :     GEN S = alg_decompose_total(a, Za, maps);
     974       14796 :     gel(sc,i) = S;
     975       14796 :     if (maps)
     976             :     {
     977       10540 :       GEN projm = gel(D,2), liftm = gel(D,3);
     978       10540 :       long j, lS = lg(S);
     979       28534 :       for (j=1; j<lS; j++)
     980             :       {
     981       17994 :         GEN Sj = gel(S,j), p2 = gel(Sj,2), l2 = gel(Sj,3);
     982       17994 :         if (p) p2 = FpM_mul(p2, projm, p);
     983          49 :         else   p2 = RgM_mul(p2, projm);
     984       17994 :         if (p) l2 = FpM_mul(liftm, l2, p);
     985          49 :         else   l2 = RgM_mul(liftm, l2);
     986       17994 :         gel(Sj,2) = p2;
     987       17994 :         gel(Sj,3) = l2;
     988             :       }
     989             :     }
     990             :   }
     991        7398 :   return shallowconcat1(sc);
     992             : }
     993             : 
     994             : static GEN
     995       11220 : alg_subalg(GEN al, GEN basis)
     996             : {
     997       11220 :   GEN invbasis, mt, p = alg_get_char(al);
     998       11220 :   long i, j, n = lg(basis)-1;
     999             : 
    1000       11220 :   if (!signe(p)) p = NULL;
    1001       11220 :   basis = shallowmatconcat(mkvec2(col_ei(n,1), basis));
    1002       11220 :   if (p)
    1003             :   {
    1004       10226 :     basis = image_keep_first(basis,p);
    1005       10226 :     invbasis = FpM_inv(basis,p);
    1006             :   }
    1007             :   else
    1008             :   { /* FIXME use an integral variant of image_keep_first */
    1009         994 :     basis = QM_ImQ_hnf(basis);
    1010         994 :     invbasis = RgM_inv(basis);
    1011             :   }
    1012       11220 :   mt = cgetg(n+1,t_VEC);
    1013       11220 :   gel(mt,1) = matid(n);
    1014       37671 :   for (i = 2; i <= n; i++)
    1015             :   {
    1016       26451 :     GEN mtx = cgetg(n+1,t_MAT), x = gel(basis,i);
    1017       26451 :     gel(mtx,1) = col_ei(n,i);
    1018      167776 :     for (j = 2; j <= n; j++)
    1019             :     {
    1020      141325 :       GEN xy = algmul(al, x, gel(basis,j));
    1021      141325 :       if (p) gel(mtx,j) = FpM_FpC_mul(invbasis, xy, p);
    1022       29701 :       else   gel(mtx,j) = RgM_RgC_mul(invbasis, xy);
    1023             :     }
    1024       26451 :     gel(mt,i) = mtx;
    1025             :   }
    1026       11220 :   return mkvec2(algtableinit_i(mt,p), basis);
    1027             : }
    1028             : 
    1029             : GEN
    1030          70 : algsubalg(GEN al, GEN basis)
    1031             : {
    1032          70 :   pari_sp av = avma;
    1033             :   GEN p;
    1034          70 :   checkalg(al);
    1035          70 :   if (alg_type(al) == al_REAL) pari_err_TYPE("algsubalg [real algebra]", al);
    1036          63 :   if (typ(basis) != t_MAT) pari_err_TYPE("algsubalg",basis);
    1037          56 :   p = alg_get_char(al);
    1038          56 :   if (signe(p)) basis = RgM_to_FpM(basis,p);
    1039          56 :   return gerepilecopy(av, alg_subalg(al,basis));
    1040             : }
    1041             : 
    1042             : static int
    1043       11962 : cmp_algebra(GEN x, GEN y)
    1044             : {
    1045             :   long d;
    1046       11962 :   d = gel(x,1)[1] - gel(y,1)[1]; if (d) return d < 0? -1: 1;
    1047       10751 :   d = gel(x,1)[2] - gel(y,1)[2]; if (d) return d < 0? -1: 1;
    1048       10751 :   return cmp_universal(gel(x,2), gel(y,2));
    1049             : }
    1050             : 
    1051             : GEN
    1052        5093 : algsimpledec_ss(GEN al, long maps)
    1053             : {
    1054        5093 :   pari_sp av = avma;
    1055             :   GEN Z, p, r, res, perm;
    1056             :   long i, l, n;
    1057        5093 :   checkalg(al);
    1058        5093 :   p = alg_get_char(al);
    1059        5093 :   dbg_printf(1)("algsimpledec_ss: char=%Ps, dim=%d\n", p, alg_get_absdim(al));
    1060        5093 :   if (signe(p))                     Z = algprimesubalg(al);
    1061         266 :   else if (alg_type(al)!=al_TABLE)  Z = gen_0;
    1062         245 :   else                              Z = algtablecenter(al);
    1063             : 
    1064        5093 :   if (lg(Z) == 2) {/* dim Z = 1 */
    1065        2987 :     n = alg_get_absdim(al);
    1066        2987 :     set_avma(av);
    1067        2987 :     if (!maps) return mkveccopy(al);
    1068        2840 :     retmkvec(mkvec3(gcopy(al), matid(n), matid(n)));
    1069             :   }
    1070        2106 :   res = alg_decompose_total(al, Z, maps);
    1071        2106 :   l = lg(res); r = cgetg(l, t_VEC);
    1072       11610 :   for (i = 1; i < l; i++)
    1073             :   {
    1074        9504 :     GEN A = maps? gmael(res,i,1): gel(res,i);
    1075        9504 :     gel(r,i) = mkvec2(mkvecsmall2(alg_get_dim(A), lg(algtablecenter(A))),
    1076             :                       alg_get_multable(A));
    1077             :   }
    1078        2106 :   perm = gen_indexsort(r, (void*)cmp_algebra, &cmp_nodata);
    1079        2106 :   return gerepilecopy(av, vecpermute(res, perm));
    1080             : }
    1081             : 
    1082             : GEN
    1083         777 : algsimpledec(GEN al, long maps)
    1084             : {
    1085         777 :   pari_sp av = avma;
    1086             :   int ss;
    1087         777 :   GEN rad, dec, res, proj=NULL, lift=NULL;
    1088         777 :   rad = algradical(al);
    1089         777 :   ss = gequal0(rad);
    1090         777 :   if (!ss)
    1091             :   {
    1092          42 :     al = alg_quotient(al, rad, maps);
    1093          42 :     if (maps) {
    1094          14 :       proj = gel(al,2);
    1095          14 :       lift = gel(al,3);
    1096          14 :       al = gel(al,1);
    1097             :     }
    1098             :   }
    1099         777 :   dec = algsimpledec_ss(al, maps);
    1100         777 :   if (!ss && maps) /* update maps */
    1101             :   {
    1102          14 :     GEN p = alg_get_char(al);
    1103             :     long i;
    1104          42 :     for (i=1; i<lg(dec); i++)
    1105             :     {
    1106          28 :       if (signe(p))
    1107             :       {
    1108          14 :         gmael(dec,i,2) = FpM_mul(gmael(dec,i,2), proj, p);
    1109          14 :         gmael(dec,i,3) = FpM_mul(lift, gmael(dec,i,3), p);
    1110             :       }
    1111             :       else
    1112             :       {
    1113          14 :         gmael(dec,i,2) = RgM_mul(gmael(dec,i,2), proj);
    1114          14 :         gmael(dec,i,3) = RgM_mul(lift, gmael(dec,i,3));
    1115             :       }
    1116             :     }
    1117             :   }
    1118         777 :   res = mkvec2(rad, dec);
    1119         777 :   return gerepilecopy(av,res);
    1120             : }
    1121             : 
    1122             : static GEN alg_idempotent(GEN al, long n, long d);
    1123             : static GEN
    1124        6482 : try_split(GEN al, GEN x, long n, long d)
    1125             : {
    1126        6482 :   GEN cp, p = alg_get_char(al), fa, e, pol, exp, P, Q, U, u, mx, mte, ire;
    1127        6482 :   long nfa, i, smalldim = alg_get_absdim(al)+1, dim, smalli = 0;
    1128        6482 :   cp = algcharpoly(al,x,0,1);
    1129        6482 :   fa = FpX_factor(cp,p);
    1130        6482 :   nfa = nbrows(fa);
    1131        6482 :   if (nfa == 1) return NULL;
    1132        3052 :   pol = gel(fa,1);
    1133        3052 :   exp = gel(fa,2);
    1134             : 
    1135             :   /* charpoly is always a d-th power */
    1136        9254 :   for (i=1; i<lg(exp); i++) {
    1137        6209 :     if (exp[i]%d) pari_err(e_MISC, "the algebra must be simple (try_split 1)");
    1138        6202 :     exp[i] /= d;
    1139             :   }
    1140        3045 :   cp = FpXV_factorback(gel(fa,1), gel(fa,2), p, 0);
    1141             : 
    1142             :   /* find smallest Fp-dimension of a characteristic space */
    1143        9247 :   for (i=1; i<lg(pol); i++) {
    1144        6202 :     dim = degree(gel(pol,i))*exp[i];
    1145        6202 :     if (dim < smalldim) {
    1146        3115 :       smalldim = dim;
    1147        3115 :       smalli = i;
    1148             :     }
    1149             :   }
    1150        3045 :   i = smalli;
    1151        3045 :   if (smalldim != n) return NULL;
    1152             :   /* We could also compute e*al*e and try again with this smaller algebra */
    1153             :   /* Fq-rank 1 = Fp-rank n idempotent: success */
    1154             : 
    1155             :   /* construct idempotent */
    1156        3031 :   mx = algbasismultable(al,x);
    1157        3031 :   P = gel(pol,i);
    1158        3031 :   P = FpX_powu(P, exp[i], p);
    1159        3031 :   Q = FpX_div(cp, P, p);
    1160        3031 :   e = algpoleval(al, Q, mkvec2(x,mx));
    1161        3031 :   U = FpXQ_inv(Q, P, p);
    1162        3031 :   u = algpoleval(al, U, mkvec2(x,mx));
    1163        3031 :   e = algbasismul(al, e, u);
    1164        3031 :   mte = algbasisrightmultable(al,e);
    1165        3031 :   ire = FpM_indexrank(mte,p);
    1166        3031 :   if (lg(gel(ire,1))-1 != smalldim*d) pari_err(e_MISC, "the algebra must be simple (try_split 2)");
    1167             : 
    1168        3024 :   return mkvec3(e,mte,ire);
    1169             : }
    1170             : 
    1171             : /*
    1172             :  * Given a simple algebra al of dimension d^2 over its center of degree n,
    1173             :  * find an idempotent e in al with rank n (which is minimal).
    1174             : */
    1175             : static GEN
    1176        3038 : alg_idempotent(GEN al, long n, long d)
    1177             : {
    1178        3038 :   pari_sp av = avma;
    1179        3038 :   long i, N = alg_get_absdim(al);
    1180        3038 :   GEN e, p = alg_get_char(al), x;
    1181        6377 :   for(i=2; i<=N; i++) {
    1182        6321 :     x = col_ei(N,i);
    1183        6321 :     e = try_split(al, x, n, d);
    1184        6307 :     if (e) return e;
    1185        3339 :     set_avma(av);
    1186             :   }
    1187             :   for(;;) {
    1188         161 :     x = random_FpC(N,p);
    1189         161 :     e = try_split(al, x, n, d);
    1190         161 :     if (e) return e;
    1191         105 :     set_avma(av);
    1192             :   }
    1193             : }
    1194             : 
    1195             : static GEN
    1196        3857 : try_descend(GEN M, GEN B, GEN p, long m, long n, long d)
    1197             : {
    1198        3857 :   GEN B2 = cgetg(m+1,t_MAT), b;
    1199        3857 :   long i, j, k=0;
    1200       11011 :   for (i=1; i<=d; i++)
    1201             :   {
    1202        7154 :     k++;
    1203        7154 :     b = gel(B,i);
    1204        7154 :     gel(B2,k) = b;
    1205       17248 :     for (j=1; j<n; j++)
    1206             :     {
    1207       10094 :       k++;
    1208       10094 :       b = FpM_FpC_mul(M,b,p);
    1209       10094 :       gel(B2,k) = b;
    1210             :     }
    1211             :   }
    1212        3857 :   if (!signe(FpM_det(B2,p))) return NULL;
    1213        3437 :   return FpM_inv(B2,p);
    1214             : }
    1215             : 
    1216             : /* Given an m*m matrix M with irreducible charpoly over F of degree n,
    1217             :  * let K = F(M), which is a field, and write m=d*n.
    1218             :  * Compute the d-dimensional K-vector space structure on V=F^m induced by M.
    1219             :  * Return [B,C] where:
    1220             :  *  - B is m*d matrix over F giving a K-basis b_1,...,b_d of V
    1221             :  *  - C is d*m matrix over F[x] expressing the canonical F-basis of V on the b_i
    1222             :  * Currently F = Fp TODO extend this. */
    1223             : static GEN
    1224        3437 : descend_i(GEN M, long n, GEN p)
    1225             : {
    1226             :   GEN B, C;
    1227             :   long m,d,i;
    1228             :   pari_sp av;
    1229        3437 :   m = lg(M)-1;
    1230        3437 :   d = m/n;
    1231        3437 :   B = cgetg(d+1,t_MAT);
    1232        3437 :   av = avma;
    1233             : 
    1234             :   /* try a subset of the canonical basis */
    1235        9751 :   for (i=1; i<=d; i++)
    1236        6314 :     gel(B,i) = col_ei(m,n*(i-1)+1);
    1237        3437 :   C = try_descend(M,B,p,m,n,d);
    1238        3437 :   if (C) return mkvec2(B,C);
    1239         385 :   set_avma(av);
    1240             : 
    1241             :   /* try smallish elements */
    1242        1155 :   for (i=1; i<=d; i++)
    1243         770 :     gel(B,i) = FpC_red(zc_to_ZC(random_pm1(m)),p);
    1244         385 :   C = try_descend(M,B,p,m,n,d);
    1245         385 :   if (C) return mkvec2(B,C);
    1246          35 :   set_avma(av);
    1247             : 
    1248             :   /* try random elements */
    1249             :   for (;;)
    1250             :   {
    1251         105 :     for (i=1; i<=d; i++)
    1252          70 :       gel(B,i) = random_FpC(m,p);
    1253          35 :     C = try_descend(M,B,p,m,n,d);
    1254          35 :     if (C) return mkvec2(B,C);
    1255           0 :     set_avma(av);
    1256             :   }
    1257             : }
    1258             : static GEN
    1259       15568 : RgC_contract(GEN C, long n, long v) /* n>1 */
    1260             : {
    1261             :   GEN C2, P;
    1262             :   long m, d, i, j;
    1263       15568 :   m = lg(C)-1;
    1264       15568 :   d = m/n;
    1265       15568 :   C2 = cgetg(d+1,t_COL);
    1266       43344 :   for (i=1; i<=d; i++)
    1267             :   {
    1268       27776 :     P = pol_xn(n-1,v);
    1269      105728 :     for (j=1; j<=n; j++)
    1270       77952 :       gel(P,j+1) = gel(C,n*(i-1)+j);
    1271       27776 :     P = normalizepol(P);
    1272       27776 :     gel(C2,i) = P;
    1273             :   }
    1274       15568 :   return C2;
    1275             : }
    1276             : static GEN
    1277        3437 : RgM_contract(GEN A, long n, long v) /* n>1 */
    1278             : {
    1279        3437 :   GEN A2 = cgetg(lg(A),t_MAT);
    1280             :   long i;
    1281       19005 :   for (i=1; i<lg(A2); i++)
    1282       15568 :     gel(A2,i) = RgC_contract(gel(A,i),n,v);
    1283        3437 :   return A2;
    1284             : }
    1285             : static GEN
    1286        3437 : descend(GEN M, long n, GEN p, long v)
    1287             : {
    1288        3437 :   GEN res = descend_i(M,n,p);
    1289        3437 :   gel(res,2) = RgM_contract(gel(res,2),n,v);
    1290        3437 :   return res;
    1291             : }
    1292             : 
    1293             : /* isomorphism of Fp-vector spaces M_d(F_p^n) -> (F_p)^(d^2*n) */
    1294             : static GEN
    1295       29939 : Fq_mat2col(GEN M, long d, long n)
    1296             : {
    1297       29939 :   long N = d*d*n, i, j, k;
    1298       29939 :   GEN C = cgetg(N+1, t_COL);
    1299       90160 :   for (i=1; i<=d; i++)
    1300      191632 :     for (j=1; j<=d; j++)
    1301      400526 :       for (k=0; k<n; k++)
    1302      269115 :         gel(C,n*(d*(i-1)+j-1)+k+1) = polcoef_i(gcoeff(M,i,j),k,-1);
    1303       29939 :   return C;
    1304             : }
    1305             : 
    1306             : static GEN
    1307        3752 : alg_finite_csa_split(GEN al, long v)
    1308             : {
    1309             :   GEN Z, e, mte, ire, primelt, b, T, M, proje, lifte, extre, p, B, C, mt, mx, map, mapi, T2, ro;
    1310        3752 :   long n, d, N = alg_get_absdim(al), i;
    1311        3752 :   p = alg_get_char(al);
    1312             :   /* compute the center */
    1313        3752 :   Z = algcenter(al);
    1314             :   /* TODO option to give the center as input instead of computing it */
    1315        3752 :   n = lg(Z)-1;
    1316             : 
    1317             :   /* compute a minimal rank idempotent e */
    1318        3752 :   if (n==N) {
    1319         707 :     d = 1;
    1320         707 :     e = col_ei(N,1);
    1321         707 :     mte = matid(N);
    1322         707 :     ire = mkvec2(identity_perm(n),identity_perm(n));
    1323             :   }
    1324             :   else {
    1325        3045 :     d = usqrt(N/n);
    1326        3045 :     if (d*d*n != N) pari_err(e_MISC, "the algebra must be simple (alg_finite_csa_split 1)");
    1327        3038 :     e = alg_idempotent(al,n,d);
    1328        3024 :     mte = gel(e,2);
    1329        3024 :     ire = gel(e,3);
    1330        3024 :     e = gel(e,1);
    1331             :   }
    1332             : 
    1333             :   /* identify the center */
    1334        3731 :   if (n==1)
    1335             :   {
    1336         287 :     T = pol_x(v);
    1337         287 :     primelt = gen_0;
    1338             :   }
    1339             :   else
    1340             :   {
    1341        3444 :     b = alg_decompose(al, Z, 1, &primelt);
    1342        3444 :     if (!gequal0(b)) pari_err(e_MISC, "the algebra must be simple (alg_finite_csa_split 2)");
    1343        3437 :     T = gel(primelt,2);
    1344        3437 :     primelt = gel(primelt,1);
    1345        3437 :     setvarn(T,v);
    1346             :   }
    1347             : 
    1348             :   /* use the ffinit polynomial */
    1349        3724 :   if (n>1)
    1350             :   {
    1351        3437 :     T2 = init_Fq(p,n,v);
    1352        3437 :     setvarn(T,fetch_var_higher());
    1353        3437 :     ro = FpXQX_roots(T2,T,p);
    1354        3437 :     ro = gel(ro,1);
    1355        3437 :     primelt = algpoleval(al,ro,primelt);
    1356        3437 :     T = T2;
    1357             :   }
    1358             : 
    1359             :   /* descend al*e to a vector space over the center */
    1360             :   /* lifte: al*e -> al ; proje: al*e -> al */
    1361        3724 :   lifte = shallowextract(mte,gel(ire,2));
    1362        3724 :   extre = shallowmatextract(mte,gel(ire,1),gel(ire,2));
    1363        3724 :   extre = FpM_inv(extre,p);
    1364        3724 :   proje = rowpermute(mte,gel(ire,1));
    1365        3724 :   proje = FpM_mul(extre,proje,p);
    1366        3724 :   if (n==1)
    1367             :   {
    1368         287 :     B = lifte;
    1369         287 :     C = proje;
    1370             :   }
    1371             :   else
    1372             :   {
    1373        3437 :     M = algbasismultable(al,primelt);
    1374        3437 :     M = FpM_mul(M,lifte,p);
    1375        3437 :     M = FpM_mul(proje,M,p);
    1376        3437 :     B = descend(M,n,p,v);
    1377        3437 :     C = gel(B,2);
    1378        3437 :     B = gel(B,1);
    1379        3437 :     B = FpM_mul(lifte,B,p);
    1380        3437 :     C = FqM_mul(C,proje,T,p);
    1381             :   }
    1382             : 
    1383             :   /* compute the isomorphism */
    1384        3724 :   mt = alg_get_multable(al);
    1385        3724 :   map = cgetg(N+1,t_VEC);
    1386        3724 :   M = cgetg(N+1,t_MAT);
    1387       33663 :   for (i=1; i<=N; i++)
    1388             :   {
    1389       29939 :     mx = gel(mt,i);
    1390       29939 :     mx = FpM_mul(mx,B,p);
    1391       29939 :     mx = FqM_mul(C,mx,T,p);
    1392       29939 :     gel(map,i) = mx;
    1393       29939 :     gel(M,i) = Fq_mat2col(mx,d,n);
    1394             :   }
    1395        3724 :   mapi = FpM_inv(M,p);
    1396        3724 :   if (!mapi) pari_err(e_MISC, "the algebra must be simple (alg_finite_csa_split 3)");
    1397        3717 :   return mkvec3(T,map,mapi);
    1398             : }
    1399             : 
    1400             : GEN
    1401        3766 : algsplit(GEN al, long v)
    1402             : {
    1403        3766 :   pari_sp av = avma;
    1404             :   GEN res, T, map, mapi, ff, p;
    1405             :   long i,j,k,li,lj;
    1406        3766 :   checkalg(al);
    1407        3759 :   p = alg_get_char(al);
    1408        3759 :   if (gequal0(p))
    1409           7 :     pari_err_IMPL("splitting a characteristic 0 algebra over its center");
    1410        3752 :   res = alg_finite_csa_split(al, v);
    1411        3717 :   T = gel(res,1);
    1412        3717 :   map = gel(res,2);
    1413        3717 :   mapi = gel(res,3);
    1414        3717 :   ff = Tp_to_FF(T,p);
    1415       33593 :   for (i=1; i<lg(map); i++)
    1416             :   {
    1417       29876 :     li = lg(gel(map,i));
    1418       89908 :     for (j=1; j<li; j++)
    1419             :     {
    1420       60032 :       lj = lg(gmael(map,i,j));
    1421      190876 :       for (k=1; k<lj; k++)
    1422      130844 :         gmael3(map,i,j,k) = Fq_to_FF(gmael3(map,i,j,k),ff);
    1423             :     }
    1424             :   }
    1425             : 
    1426        3717 :   return gerepilecopy(av, mkvec2(map,mapi));
    1427             : }
    1428             : 
    1429             : /* multiplication table sanity checks */
    1430             : static GEN
    1431       38549 : check_mt_noid(GEN mt, GEN p)
    1432             : {
    1433             :   long i, l;
    1434       38549 :   GEN MT = cgetg_copy(mt, &l);
    1435       38549 :   if (typ(MT) != t_VEC || l == 1) return NULL;
    1436      185517 :   for (i = 1; i < l; i++)
    1437             :   {
    1438      147017 :     GEN M = gel(mt,i);
    1439      147017 :     if (typ(M) != t_MAT || lg(M) != l || lgcols(M) != l) return NULL;
    1440      146989 :     if (p) M = RgM_to_FpM(M,p);
    1441      146989 :     gel(MT,i) = M;
    1442             :   }
    1443       38500 :   return MT;
    1444             : }
    1445             : static GEN
    1446       38045 : check_mt(GEN mt, GEN p)
    1447             : {
    1448             :   long i;
    1449             :   GEN MT;
    1450       38045 :   MT = check_mt_noid(mt, p);
    1451       38045 :   if (!MT || !ZM_isidentity(gel(MT,1))) return NULL;
    1452      143846 :   for (i=2; i<lg(MT); i++)
    1453      105829 :     if (ZC_is_ei(gmael(MT,i,1)) != i) return NULL;
    1454       38017 :   return MT;
    1455             : }
    1456             : 
    1457             : static GEN
    1458         175 : check_relmt(GEN nf, GEN mt)
    1459             : {
    1460         175 :   long i, l = lg(mt), j, k;
    1461         175 :   GEN MT = gcopy(mt), a, b, d;
    1462         175 :   if (typ(MT) != t_VEC || l == 1) return NULL;
    1463         693 :   for (i = 1; i < l; i++)
    1464             :   {
    1465         539 :     GEN M = gel(MT,i);
    1466         539 :     if (typ(M) != t_MAT || lg(M) != l || lgcols(M) != l) return NULL;
    1467        2758 :     for (k = 1; k < l; k++)
    1468       13643 :       for (j = 1; j < l; j++)
    1469             :       {
    1470       11424 :         a = gcoeff(M,j,k);
    1471       11424 :         if (typ(a)==t_INT) continue;
    1472        1771 :         b = algtobasis(nf,a);
    1473        1771 :         d = Q_denom(b);
    1474        1771 :         if (!isint1(d))
    1475          14 :           pari_err_DOMAIN("alg_csa_table", "denominator(mt)", "!=", gen_1, mt);
    1476        1757 :         gcoeff(M,j,k) = lift(basistoalg(nf,b));
    1477             :       }
    1478         525 :     if (i > 1 && RgC_is_ei(gel(M,1)) != i) return NULL; /* i = 1 checked at end */
    1479         518 :     gel(MT,i) = M;
    1480             :   }
    1481         154 :   if (!RgM_isidentity(gel(MT,1))) return NULL;
    1482         154 :   return MT;
    1483             : }
    1484             : 
    1485             : int
    1486         511 : algisassociative(GEN mt0, GEN p)
    1487             : {
    1488         511 :   pari_sp av = avma;
    1489             :   long i, j, k, n;
    1490             :   GEN M, mt;
    1491             : 
    1492         511 :   if (checkalg_i(mt0)) { p = alg_get_char(mt0); mt0 = alg_get_multable(mt0); }
    1493         511 :   if (typ(p) != t_INT) pari_err_TYPE("algisassociative",p);
    1494         504 :   mt = check_mt_noid(mt0, isintzero(p)? NULL: p);
    1495         504 :   if (!mt) pari_err_TYPE("algisassociative (mult. table)", mt0);
    1496         469 :   if (!ZM_isidentity(gel(mt,1))) return gc_bool(av,0);
    1497         455 :   n = lg(mt)-1;
    1498         455 :   M = cgetg(n+1,t_MAT);
    1499        3542 :   for (j=1; j<=n; j++) gel(M,j) = cgetg(n+1,t_COL);
    1500        3542 :   for (i=1; i<=n; i++)
    1501             :   {
    1502        3087 :     GEN mi = gel(mt,i);
    1503       35182 :     for (j=1; j<=n; j++) gcoeff(M,i,j) = gel(mi,j); /* ei.ej */
    1504             :   }
    1505        3073 :   for (i=2; i<=n; i++) {
    1506        2625 :     GEN mi = gel(mt,i);
    1507       28973 :     for (j=2; j<=n; j++) {
    1508      368291 :       for (k=2; k<=n; k++) {
    1509             :         GEN x, y;
    1510      341943 :         if (signe(p)) {
    1511      242039 :           x = _tablemul_ej_Fp(mt,gcoeff(M,i,j),k,p);
    1512      242039 :           y = FpM_FpC_mul(mi,gcoeff(M,j,k),p);
    1513             :         }
    1514             :         else {
    1515       99904 :           x = _tablemul_ej(mt,gcoeff(M,i,j),k);
    1516       99904 :           y = RgM_RgC_mul(mi,gcoeff(M,j,k));
    1517             :         }
    1518             :         /* not cmp_universal: must not fail on 0 == Mod(0,2) for instance */
    1519      341943 :         if (!gequal(x,y)) return gc_bool(av,0);
    1520             :       }
    1521             :     }
    1522             :   }
    1523         448 :   return gc_bool(av,1);
    1524             : }
    1525             : 
    1526             : int
    1527         371 : algiscommutative(GEN al) /* assumes e_1 = 1 */
    1528             : {
    1529             :   long i,j,k,N,sp;
    1530             :   GEN mt,a,b,p;
    1531         371 :   checkalg(al);
    1532         371 :   if (alg_type(al) != al_TABLE) return alg_get_degree(al)==1;
    1533         308 :   N = alg_get_absdim(al);
    1534         308 :   mt = alg_get_multable(al);
    1535         308 :   p = alg_get_char(al);
    1536         308 :   sp = signe(p);
    1537        1449 :   for (i=2; i<=N; i++)
    1538        9464 :     for (j=2; j<=N; j++)
    1539       85820 :       for (k=1; k<=N; k++) {
    1540       77553 :         a = gcoeff(gel(mt,i),k,j);
    1541       77553 :         b = gcoeff(gel(mt,j),k,i);
    1542       77553 :         if (sp) {
    1543       73423 :           if (cmpii(Fp_red(a,p), Fp_red(b,p))) return 0;
    1544             :         }
    1545        4130 :         else if (gcmp(a,b)) return 0;
    1546             :       }
    1547         252 :   return 1;
    1548             : }
    1549             : 
    1550             : int
    1551         371 : algissemisimple(GEN al)
    1552             : {
    1553         371 :   pari_sp av = avma;
    1554             :   GEN rad;
    1555         371 :   checkalg(al);
    1556         371 :   if (alg_type(al) != al_TABLE) return 1;
    1557         308 :   rad = algradical(al);
    1558         308 :   set_avma(av);
    1559         308 :   return gequal0(rad);
    1560             : }
    1561             : 
    1562             : /* ss : known to be semisimple */
    1563             : int
    1564         280 : algissimple(GEN al, long ss)
    1565             : {
    1566         280 :   pari_sp av = avma;
    1567             :   GEN Z, dec, p;
    1568         280 :   checkalg(al);
    1569         280 :   if (alg_type(al) != al_TABLE) return 1;
    1570         224 :   if (!ss && !algissemisimple(al)) return 0;
    1571             : 
    1572         182 :   p = alg_get_char(al);
    1573         182 :   if (signe(p)) Z = algprimesubalg(al);
    1574          91 :   else          Z = algtablecenter(al);
    1575             : 
    1576         182 :   if (lg(Z) == 2) {/* dim Z = 1 */
    1577         105 :     set_avma(av);
    1578         105 :     return 1;
    1579             :   }
    1580          77 :   dec = alg_decompose(al, Z, 1, NULL);
    1581          77 :   set_avma(av);
    1582          77 :   return gequal0(dec);
    1583             : }
    1584             : 
    1585             : static long
    1586         329 : is_place_emb(GEN nf, GEN pl)
    1587             : {
    1588             :   long r, r1, r2;
    1589         329 :   if (typ(pl) != t_INT) pari_err_TYPE("is_place_emb", pl);
    1590         315 :   if (signe(pl)<=0) pari_err_DOMAIN("is_place_emb", "pl", "<=", gen_0, pl);
    1591         308 :   nf_get_sign(nf,&r1,&r2); r = r1+r2;
    1592         308 :   if (cmpiu(pl,r)>0) pari_err_DOMAIN("is_place_emb", "pl", ">", utoi(r), pl);
    1593         294 :   return itou(pl);
    1594             : }
    1595             : 
    1596             : static long
    1597         294 : alghasse_emb(GEN al, long emb)
    1598             : {
    1599         294 :   GEN nf = alg_get_center(al);
    1600         294 :   long r1 = nf_get_r1(nf);
    1601         294 :   return (emb <= r1)? alg_get_hasse_i(al)[emb]: 0;
    1602             : }
    1603             : 
    1604             : static long
    1605         413 : alghasse_pr(GEN al, GEN pr)
    1606             : {
    1607         413 :   GEN hf = alg_get_hasse_f(al);
    1608         406 :   long i = tablesearch(gel(hf,1), pr, &cmp_prime_ideal);
    1609         406 :   return i? gel(hf,2)[i]: 0;
    1610             : }
    1611             : 
    1612             : static long
    1613         777 : alghasse_0(GEN al, GEN pl)
    1614             : {
    1615             :   long ta;
    1616             :   GEN pr, nf;
    1617         777 :   ta = alg_type(al);
    1618         777 :   if (ta == al_REAL) return algreal_dim(al)!=1;
    1619         756 :   if (!pl)
    1620           7 :     pari_err(e_MISC, "must provide a place pl");
    1621         749 :   if (ta == al_CSA)
    1622           7 :     pari_err_IMPL("computation of Hasse invariants over table CSA");
    1623         742 :   if ((pr = get_prid(pl))) return alghasse_pr(al, pr);
    1624         329 :   nf = alg_get_center(al);
    1625         329 :   return alghasse_emb(al, is_place_emb(nf, pl));
    1626             : }
    1627             : GEN
    1628         252 : alghasse(GEN al, GEN pl)
    1629             : {
    1630             :   long h;
    1631         252 :   checkalg(al);
    1632         252 :   if (alg_type(al) == al_TABLE) pari_err_TYPE("alghasse [use alginit]",al);
    1633         245 :   h = alghasse_0(al,pl);
    1634         189 :   return sstoQ(h, alg_get_degree(al));
    1635             : }
    1636             : 
    1637             : /* h >= 0, d >= 0 */
    1638             : static long
    1639         819 : indexfromhasse(long h, long d) { return d/ugcd(h,d); }
    1640             : 
    1641             : long
    1642         819 : algindex(GEN al, GEN pl)
    1643             : {
    1644             :   long d, res, i, l, ta;
    1645             :   GEN hi, hf;
    1646             : 
    1647         819 :   checkalg(al);
    1648         812 :   ta = alg_type(al);
    1649         812 :   if (ta == al_TABLE) pari_err_TYPE("algindex [use alginit]",al);
    1650         805 :   if (ta == al_REAL) return algreal_dim(al)==1 ? 1 : 2;
    1651         721 :   d = alg_get_degree(al);
    1652         721 :   if (pl) return indexfromhasse(alghasse_0(al,pl), d);
    1653             : 
    1654             :   /* else : global index */
    1655         189 :   res = 1;
    1656         189 :   hi = alg_get_hasse_i(al); l = lg(hi);
    1657         322 :   for (i=1; i<l && res!=d; i++) res = ulcm(res, indexfromhasse(hi[i],d));
    1658         189 :   hf = gel(alg_get_hasse_f(al), 2); l = lg(hf);
    1659         336 :   for (i=1; i<l && res!=d; i++) res = ulcm(res, indexfromhasse(hf[i],d));
    1660         182 :   return res;
    1661             : }
    1662             : 
    1663             : int
    1664         224 : algisdivision(GEN al, GEN pl)
    1665             : {
    1666         224 :   checkalg(al);
    1667         224 :   if (alg_type(al) == al_TABLE) {
    1668          21 :     if (!algissimple(al,0)) return 0;
    1669          14 :     if (algiscommutative(al)) return 1;
    1670           7 :     pari_err_IMPL("algisdivision for table algebras");
    1671             :   }
    1672         203 :   return algindex(al,pl) == alg_get_degree(al);
    1673             : }
    1674             : 
    1675             : int
    1676         406 : algissplit(GEN al, GEN pl)
    1677             : {
    1678         406 :   checkalg(al);
    1679         406 :   if (alg_type(al) == al_TABLE) pari_err_TYPE("algissplit [use alginit]", al);
    1680         392 :   return algindex(al,pl) == 1;
    1681             : }
    1682             : 
    1683             : int
    1684         203 : algisramified(GEN al, GEN pl) { return !algissplit(al,pl); }
    1685             : 
    1686             : GEN
    1687         105 : algramifiedplaces(GEN al)
    1688             : {
    1689         105 :   pari_sp av = avma;
    1690             :   GEN ram, hf, hi, Lpr;
    1691             :   long r1, count, i, ta;
    1692         105 :   checkalg(al);
    1693         105 :   ta = alg_type(al);
    1694         105 :   if (ta != al_CSA && ta != al_CYCLIC)
    1695          14 :     pari_err_TYPE("algramifiedplaces [not a central simple algebra"
    1696             :         " over a number field]", al);
    1697          91 :   r1 = nf_get_r1(alg_get_center(al));
    1698          91 :   hi = alg_get_hasse_i(al);
    1699          91 :   hf = alg_get_hasse_f(al);
    1700          84 :   Lpr = gel(hf,1);
    1701          84 :   hf = gel(hf,2);
    1702          84 :   ram = cgetg(r1+lg(Lpr), t_VEC);
    1703          84 :   count = 0;
    1704         280 :   for (i=1; i<=r1; i++)
    1705         196 :     if (hi[i]) {
    1706          91 :       count++;
    1707          91 :       gel(ram,count) = stoi(i);
    1708             :     }
    1709         272 :   for (i=1; i<lg(Lpr); i++)
    1710         188 :     if (hf[i]) {
    1711          77 :       count++;
    1712          77 :       gel(ram,count) = gel(Lpr,i);
    1713             :     }
    1714          84 :   setlg(ram, count+1);
    1715          84 :   return gerepilecopy(av, ram);
    1716             : }
    1717             : 
    1718             : /** OPERATIONS ON ELEMENTS operations.c **/
    1719             : 
    1720             : static long
    1721     1125365 : alg_model0(GEN al, GEN x)
    1722             : {
    1723     1125365 :   long t, N = alg_get_absdim(al), lx = lg(x), d, n, D, i;
    1724     1125365 :   if (typ(x) == t_MAT) return al_MATRIX;
    1725     1079242 :   if (typ(x) != t_COL) return al_INVALID;
    1726     1079179 :   if (N == 1) {
    1727        2842 :     if (lx != 2) return al_INVALID;
    1728        2821 :     switch(typ(gel(x,1)))
    1729             :     {
    1730        1799 :       case t_INT: case t_FRAC: return al_TRIVIAL; /* cannot distinguish basis and alg from size */
    1731        1015 :       case t_POL: case t_POLMOD: return al_ALGEBRAIC;
    1732           7 :       default: return al_INVALID;
    1733             :     }
    1734             :   }
    1735             : 
    1736     1076337 :   switch(alg_type(al)) {
    1737      643512 :     case al_TABLE:
    1738      643512 :       if (lx != N+1) return al_INVALID;
    1739      643491 :       return al_BASIS;
    1740      346746 :     case al_CYCLIC:
    1741      346746 :       d = alg_get_degree(al);
    1742      346746 :       if (lx == N+1) return al_BASIS;
    1743       94872 :       if (lx == d+1) return al_ALGEBRAIC;
    1744          14 :       return al_INVALID;
    1745       86079 :     case al_CSA:
    1746       86079 :       D = alg_get_dim(al);
    1747       86079 :       n = nf_get_degree(alg_get_center(al));
    1748       86079 :       if (n == 1) {
    1749        1323 :         if (lx != D+1) return al_INVALID;
    1750        4144 :         for (i=1; i<=D; i++) {
    1751        3437 :           t = typ(gel(x,i));
    1752        3437 :           if (t == t_POL || t == t_POLMOD)  return al_ALGEBRAIC;
    1753             :             /* TODO t_COL for coefficients in basis form ? */
    1754             :         }
    1755         707 :         return al_BASIS;
    1756             :       }
    1757             :       else {
    1758       84756 :         if (lx == N+1) return al_BASIS;
    1759       22582 :         if (lx == D+1) return al_ALGEBRAIC;
    1760           7 :         return al_INVALID;
    1761             :       }
    1762             :   }
    1763             :   return al_INVALID; /* LCOV_EXCL_LINE */
    1764             : }
    1765             : 
    1766             : static void
    1767     1125225 : checkalgx(GEN x, long model)
    1768             : {
    1769             :   long t, i;
    1770     1125225 :   switch(model) {
    1771      958246 :     case al_BASIS:
    1772     9570662 :       for (i=1; i<lg(x); i++) {
    1773     8612423 :         t = typ(gel(x,i));
    1774     8612423 :         if (t != t_INT && t != t_FRAC)
    1775           7 :           pari_err_TYPE("checkalgx", gel(x,i));
    1776             :       }
    1777      958239 :       return;
    1778      120856 :     case al_TRIVIAL:
    1779             :     case al_ALGEBRAIC:
    1780      409230 :       for (i=1; i<lg(x); i++) {
    1781      288381 :         t = typ(gel(x,i));
    1782      288381 :         if (t != t_INT && t != t_FRAC && t != t_POL && t != t_POLMOD)
    1783             :           /* TODO t_COL ? */
    1784           7 :           pari_err_TYPE("checkalgx", gel(x,i));
    1785             :       }
    1786      120849 :       return;
    1787             :   }
    1788             : }
    1789             : 
    1790             : long
    1791     1125365 : alg_model(GEN al, GEN x)
    1792             : {
    1793     1125365 :   long res = alg_model0(al, x);
    1794     1125365 :   if (res == al_INVALID) pari_err_TYPE("alg_model", x);
    1795     1125225 :   checkalgx(x, res); return res;
    1796             : }
    1797             : 
    1798             : static long
    1799      462630 : H_model0(GEN x)
    1800             : {
    1801             :   long i;
    1802      462630 :   switch(typ(x))
    1803             :   {
    1804       15218 :     case t_INT:
    1805             :     case t_FRAC:
    1806             :     case t_REAL:
    1807             :     case t_COMPLEX:
    1808       15218 :       return H_SCALAR;
    1809       10157 :     case t_MAT:
    1810       10157 :       return H_MATRIX;
    1811      437143 :     case t_COL:
    1812      437143 :       if (lg(x)!=5) return H_INVALID;
    1813     2185603 :       for (i=1; i<=4; i++) if (!is_real_t(typ(gel(x,i)))) return H_INVALID;
    1814      437115 :       return H_QUATERNION;
    1815         112 :     default:
    1816         112 :       return al_INVALID;
    1817             :   }
    1818             : }
    1819             : 
    1820             : static long
    1821      462630 : H_model(GEN x)
    1822             : {
    1823      462630 :   long res = H_model0(x);
    1824      462630 :   if (res == H_INVALID) pari_err_TYPE("H_model", x);
    1825      462490 :   return res;
    1826             : }
    1827             : 
    1828             : static GEN
    1829         756 : alC_add_i(GEN al, GEN x, GEN y, long lx)
    1830             : {
    1831         756 :   GEN A = cgetg(lx, t_COL);
    1832             :   long i;
    1833        2296 :   for (i=1; i<lx; i++) gel(A,i) = algadd(al, gel(x,i), gel(y,i));
    1834         749 :   return A;
    1835             : }
    1836             : static GEN
    1837         406 : alM_add(GEN al, GEN x, GEN y)
    1838             : {
    1839         406 :   long lx = lg(x), l, j;
    1840             :   GEN z;
    1841         406 :   if (lg(y) != lx) pari_err_DIM("alM_add (rows)");
    1842         392 :   if (lx == 1) return cgetg(1, t_MAT);
    1843         385 :   z = cgetg(lx, t_MAT); l = lgcols(x);
    1844         385 :   if (lgcols(y) != l) pari_err_DIM("alM_add (columns)");
    1845        1127 :   for (j = 1; j < lx; j++) gel(z,j) = alC_add_i(al, gel(x,j), gel(y,j), l);
    1846         371 :   return z;
    1847             : }
    1848             : static GEN
    1849       17745 : H_add(GEN x, GEN y)
    1850             : {
    1851       17745 :   long tx = H_model(x), ty = H_model(y);
    1852       17724 :   if ((tx==H_MATRIX) ^ (ty==H_MATRIX)) pari_err_TYPE2("H_add", x, y);
    1853       17710 :   if (tx>ty) { swap(x,y); lswap(tx,ty); }
    1854       17710 :   switch (tx)
    1855             :   {
    1856         105 :     case H_MATRIX: /* both H_MATRIX */ return alM_add(NULL, x, y);
    1857       16681 :     case H_QUATERNION: /* both H_QUATERNION */ return gadd(x,y);
    1858         924 :     case H_SCALAR:
    1859         924 :       if (ty == H_SCALAR) return gadd(x,y);
    1860             :       else /* ty == H_QUATERNION */
    1861             :       {
    1862         217 :         pari_sp av = avma;
    1863         217 :         GEN res = gcopy(y), im;
    1864         217 :         gel(res,1) = gadd(gel(res,1), real_i(x));
    1865         217 :         im = imag_i(x);
    1866         217 :         if (im != gen_0) gel(res,2) = gadd(gel(res,2), im);
    1867         217 :         return gerepileupto(av, res);
    1868             :       }
    1869             :   }
    1870             :   return NULL; /*LCOV_EXCL_LINE*/
    1871             : }
    1872             : GEN
    1873       54845 : algadd(GEN al, GEN x, GEN y)
    1874             : {
    1875       54845 :   pari_sp av = avma;
    1876             :   long tx, ty;
    1877             :   GEN p;
    1878       54845 :   checkalg(al);
    1879       54845 :   if (alg_type(al)==al_REAL) return H_add(x,y);
    1880       37100 :   tx = alg_model(al,x);
    1881       37093 :   ty = alg_model(al,y);
    1882       37093 :   p = alg_get_char(al);
    1883       37093 :   if (signe(p)) return FpC_add(x,y,p);
    1884       36960 :   if (tx==ty) {
    1885       36078 :     if (tx!=al_MATRIX) return gadd(x,y);
    1886         301 :     return gerepilecopy(av, alM_add(al,x,y));
    1887             :   }
    1888         882 :   if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
    1889         882 :   if (ty==al_ALGEBRAIC) y = algalgtobasis(al,y);
    1890         882 :   return gerepileupto(av, gadd(x,y));
    1891             : }
    1892             : 
    1893             : static GEN
    1894          98 : H_neg(GEN x)
    1895             : {
    1896          98 :   (void)H_model(x);
    1897          70 :   return gneg(x);
    1898             : }
    1899             : 
    1900             : GEN
    1901         245 : algneg(GEN al, GEN x)
    1902             : {
    1903         245 :   checkalg(al);
    1904         245 :   if (alg_type(al)==al_REAL) return H_neg(x);
    1905         147 :   (void)alg_model(al,x);
    1906         140 :   return gneg(x);
    1907             : }
    1908             : 
    1909             : static GEN
    1910         210 : alC_sub_i(GEN al, GEN x, GEN y, long lx)
    1911             : {
    1912             :   long i;
    1913         210 :   GEN A = cgetg(lx, t_COL);
    1914         630 :   for (i=1; i<lx; i++) gel(A,i) = algsub(al, gel(x,i), gel(y,i));
    1915         210 :   return A;
    1916             : }
    1917             : static GEN
    1918         126 : alM_sub(GEN al, GEN x, GEN y)
    1919             : {
    1920         126 :   long lx = lg(x), l, j;
    1921             :   GEN z;
    1922         126 :   if (lg(y) != lx) pari_err_DIM("alM_sub (rows)");
    1923         119 :   if (lx == 1) return cgetg(1, t_MAT);
    1924         112 :   z = cgetg(lx, t_MAT); l = lgcols(x);
    1925         112 :   if (lgcols(y) != l) pari_err_DIM("alM_sub (columns)");
    1926         315 :   for (j = 1; j < lx; j++) gel(z,j) = alC_sub_i(al, gel(x,j), gel(y,j), l);
    1927         105 :   return z;
    1928             : }
    1929             : GEN
    1930        1120 : algsub(GEN al, GEN x, GEN y)
    1931             : {
    1932             :   long tx, ty;
    1933        1120 :   pari_sp av = avma;
    1934             :   GEN p;
    1935        1120 :   checkalg(al);
    1936        1120 :   if (alg_type(al)==al_REAL) return gerepileupto(av, algadd(NULL,x,gneg(y)));
    1937         966 :   tx = alg_model(al,x);
    1938         959 :   ty = alg_model(al,y);
    1939         959 :   p = alg_get_char(al);
    1940         959 :   if (signe(p)) return FpC_sub(x,y,p);
    1941         868 :   if (tx==ty) {
    1942         546 :     if (tx != al_MATRIX) return gsub(x,y);
    1943         126 :     return gerepilecopy(av, alM_sub(al,x,y));
    1944             :   }
    1945         322 :   if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
    1946         322 :   if (ty==al_ALGEBRAIC) y = algalgtobasis(al,y);
    1947         322 :   return gerepileupto(av, gsub(x,y));
    1948             : }
    1949             : 
    1950             : static GEN
    1951        1659 : algalgmul_cyc(GEN al, GEN x, GEN y)
    1952             : {
    1953        1659 :   pari_sp av = avma;
    1954        1659 :   long n = alg_get_degree(al), i, k;
    1955             :   GEN xalg, yalg, res, rnf, auts, sum, b, prod, autx;
    1956        1659 :   rnf = alg_get_splittingfield(al);
    1957        1659 :   auts = alg_get_auts(al);
    1958        1659 :   b = alg_get_b(al);
    1959             : 
    1960        1659 :   xalg = cgetg(n+1, t_COL);
    1961        4935 :   for (i=0; i<n; i++)
    1962        3276 :     gel(xalg,i+1) = lift_shallow(rnfbasistoalg(rnf,gel(x,i+1)));
    1963             : 
    1964        1659 :   yalg = cgetg(n+1, t_COL);
    1965        4935 :   for (i=0; i<n; i++) gel(yalg,i+1) = rnfbasistoalg(rnf,gel(y,i+1));
    1966             : 
    1967        1659 :   res = cgetg(n+1,t_COL);
    1968        4935 :   for (k=0; k<n; k++) {
    1969        3276 :     gel(res,k+1) = gmul(gel(xalg,k+1),gel(yalg,1));
    1970        5166 :     for (i=1; i<=k; i++) {
    1971        1890 :       autx = poleval(gel(xalg,k-i+1),gel(auts,i));
    1972        1890 :       prod = gmul(autx,gel(yalg,i+1));
    1973        1890 :       gel(res,k+1) = gadd(gel(res,k+1), prod);
    1974             :     }
    1975             : 
    1976        3276 :     sum = gen_0;
    1977        5166 :     for (; i<n; i++) {
    1978        1890 :       autx = poleval(gel(xalg,k+n-i+1),gel(auts,i));
    1979        1890 :       prod = gmul(autx,gel(yalg,i+1));
    1980        1890 :       sum = gadd(sum,prod);
    1981             :     }
    1982        3276 :     sum = gmul(b,sum);
    1983             : 
    1984        3276 :     gel(res,k+1) = gadd(gel(res,k+1),sum);
    1985             :   }
    1986             : 
    1987        1659 :   return gerepilecopy(av, res);
    1988             : }
    1989             : 
    1990             : static GEN
    1991      205394 : _tablemul(GEN mt, GEN x, GEN y)
    1992             : {
    1993      205394 :   pari_sp av = avma;
    1994      205394 :   long D = lg(mt)-1, i;
    1995      205394 :   GEN res = NULL;
    1996     1930922 :   for (i=1; i<=D; i++) {
    1997     1725528 :     GEN c = gel(x,i);
    1998     1725528 :     if (!gequal0(c)) {
    1999      990409 :       GEN My = RgM_RgC_mul(gel(mt,i),y);
    2000      990409 :       GEN t = RgC_Rg_mul(My,c);
    2001      990409 :       res = res? RgC_add(res,t): t;
    2002             :     }
    2003             :   }
    2004      205394 :   if (!res) { set_avma(av); return zerocol(D); }
    2005      204491 :   return gerepileupto(av, res);
    2006             : }
    2007             : 
    2008             : static GEN
    2009      263319 : _tablemul_Fp(GEN mt, GEN x, GEN y, GEN p)
    2010             : {
    2011      263319 :   pari_sp av = avma;
    2012      263319 :   long D = lg(mt)-1, i;
    2013      263319 :   GEN res = NULL;
    2014     2654754 :   for (i=1; i<=D; i++) {
    2015     2391435 :     GEN c = gel(x,i);
    2016     2391435 :     if (signe(c)) {
    2017      456187 :       GEN My = FpM_FpC_mul(gel(mt,i),y,p);
    2018      456187 :       GEN t = FpC_Fp_mul(My,c,p);
    2019      456187 :       res = res? FpC_add(res,t,p): t;
    2020             :     }
    2021             :   }
    2022      263319 :   if (!res) { set_avma(av); return zerocol(D); }
    2023      262780 :   return gerepileupto(av, res);
    2024             : }
    2025             : 
    2026             : /* x*ej */
    2027             : static GEN
    2028       99904 : _tablemul_ej(GEN mt, GEN x, long j)
    2029             : {
    2030       99904 :   pari_sp av = avma;
    2031       99904 :   long D = lg(mt)-1, i;
    2032       99904 :   GEN res = NULL;
    2033     1563793 :   for (i=1; i<=D; i++) {
    2034     1463889 :     GEN c = gel(x,i);
    2035     1463889 :     if (!gequal0(c)) {
    2036      116718 :       GEN My = gel(gel(mt,i),j);
    2037      116718 :       GEN t = RgC_Rg_mul(My,c);
    2038      116718 :       res = res? RgC_add(res,t): t;
    2039             :     }
    2040             :   }
    2041       99904 :   if (!res) { set_avma(av); return zerocol(D); }
    2042       99764 :   return gerepileupto(av, res);
    2043             : }
    2044             : static GEN
    2045      242039 : _tablemul_ej_Fp(GEN mt, GEN x, long j, GEN p)
    2046             : {
    2047      242039 :   pari_sp av = avma;
    2048      242039 :   long D = lg(mt)-1, i;
    2049      242039 :   GEN res = NULL;
    2050     4364787 :   for (i=1; i<=D; i++) {
    2051     4122748 :     GEN c = gel(x,i);
    2052     4122748 :     if (!gequal0(c)) {
    2053      289954 :       GEN My = gel(gel(mt,i),j);
    2054      289954 :       GEN t = FpC_Fp_mul(My,c,p);
    2055      289954 :       res = res? FpC_add(res,t,p): t;
    2056             :     }
    2057             :   }
    2058      242039 :   if (!res) { set_avma(av); return zerocol(D); }
    2059      241927 :   return gerepileupto(av, res);
    2060             : }
    2061             : 
    2062             : static GEN
    2063      245283 : _tablemul_ej_Fl(GEN mt, GEN x, long j, ulong p)
    2064             : {
    2065      245283 :   pari_sp av = avma;
    2066      245283 :   long D = lg(mt)-1, i;
    2067      245283 :   GEN res = NULL;
    2068     3945192 :   for (i=1; i<=D; i++) {
    2069     3699909 :     ulong c = x[i];
    2070     3699909 :     if (c) {
    2071      394085 :       GEN My = gel(gel(mt,i),j);
    2072      394085 :       GEN t = Flv_Fl_mul(My,c, p);
    2073      394085 :       res = res? Flv_add(res,t, p): t;
    2074             :     }
    2075             :   }
    2076      245283 :   if (!res) { set_avma(av); return zero_Flv(D); }
    2077      245283 :   return gerepileupto(av, res);
    2078             : }
    2079             : 
    2080             : static GEN
    2081         686 : algalgmul_csa(GEN al, GEN x, GEN y)
    2082             : {
    2083         686 :   GEN z, nf = alg_get_center(al);
    2084             :   long i;
    2085         686 :   z = _tablemul(alg_get_relmultable(al), x, y);
    2086        2485 :   for (i=1; i<lg(z); i++)
    2087        1799 :     gel(z,i) = basistoalg(nf,gel(z,i));
    2088         686 :   return z;
    2089             : }
    2090             : 
    2091             : /* assumes x and y in algebraic form */
    2092             : static GEN
    2093        2345 : algalgmul(GEN al, GEN x, GEN y)
    2094             : {
    2095        2345 :   switch(alg_type(al))
    2096             :   {
    2097        1659 :     case al_CYCLIC: return algalgmul_cyc(al, x, y);
    2098         686 :     case al_CSA: return algalgmul_csa(al, x, y);
    2099             :   }
    2100             :   return NULL; /*LCOV_EXCL_LINE*/
    2101             : }
    2102             : 
    2103             : static GEN
    2104      468027 : algbasismul(GEN al, GEN x, GEN y)
    2105             : {
    2106      468027 :   GEN mt = alg_get_multable(al), p = alg_get_char(al);
    2107      468027 :   if (signe(p)) return _tablemul_Fp(mt, x, y, p);
    2108      204708 :   return _tablemul(mt, x, y);
    2109             : }
    2110             : 
    2111             : /* x[i,]*y. Assume lg(x) > 1 and 0 < i < lgcols(x) */
    2112             : static GEN
    2113      119651 : alMrow_alC_mul_i(GEN al, GEN x, GEN y, long i, long lx)
    2114             : {
    2115      119651 :   pari_sp av = avma;
    2116      119651 :   GEN c = algmul(al,gcoeff(x,i,1),gel(y,1)), ZERO;
    2117             :   long k;
    2118      119651 :   ZERO = zerocol(alg_get_absdim(al));
    2119      273308 :   for (k = 2; k < lx; k++)
    2120             :   {
    2121      153657 :     GEN t = algmul(al, gcoeff(x,i,k), gel(y,k));
    2122      153657 :     if (!gequal(t,ZERO)) c = algadd(al, c, t);
    2123             :   }
    2124      119651 :   return gerepilecopy(av, c);
    2125             : }
    2126             : /* return x * y, 1 < lx = lg(x), l = lgcols(x) */
    2127             : static GEN
    2128       54502 : alM_alC_mul_i(GEN al, GEN x, GEN y, long lx, long l)
    2129             : {
    2130       54502 :   GEN z = cgetg(l,t_COL);
    2131             :   long i;
    2132      174153 :   for (i=1; i<l; i++) gel(z,i) = alMrow_alC_mul_i(al,x,y,i,lx);
    2133       54502 :   return z;
    2134             : }
    2135             : static GEN
    2136       25627 : alM_mul(GEN al, GEN x, GEN y)
    2137             : {
    2138       25627 :   long j, l, lx=lg(x), ly=lg(y);
    2139             :   GEN z;
    2140       25627 :   if (ly==1) return cgetg(1,t_MAT);
    2141       25529 :   if (lx != lgcols(y)) pari_err_DIM("alM_mul");
    2142       25508 :   if (lx==1) return zeromat(0, ly-1);
    2143       25501 :   l = lgcols(x); z = cgetg(ly,t_MAT);
    2144       80003 :   for (j=1; j<ly; j++) gel(z,j) = alM_alC_mul_i(al,x,gel(y,j),lx,l);
    2145       25501 :   return z;
    2146             : }
    2147             : 
    2148             : static void
    2149      205583 : H_compo(GEN x, GEN* a, GEN* b, GEN* c, GEN* d)
    2150             : {
    2151      205583 :   switch(H_model(x))
    2152             :   {
    2153        5173 :     case H_SCALAR:
    2154        5173 :       *a = real_i(x);
    2155        5173 :       *b = imag_i(x);
    2156        5173 :       *c = gen_0;
    2157        5173 :       *d = gen_0;
    2158        5173 :       return;
    2159      200410 :     case H_QUATERNION:
    2160      200410 :       *a = gel(x,1);
    2161      200410 :       *b = gel(x,2);
    2162      200410 :       *c = gel(x,3);
    2163      200410 :       *d = gel(x,4);
    2164      200410 :       return;
    2165             :     default: *a = *b = *c = *d = NULL; return; /*LCOV_EXCL_LINE*/
    2166             :   }
    2167             : }
    2168             : static GEN
    2169      108101 : H_mul(GEN x, GEN y)
    2170             : {
    2171      108101 :   pari_sp av = avma;
    2172             :   GEN a,b,c,d,u,v,w,z;
    2173      108101 :   long tx = H_model(x), ty = H_model(y);
    2174      108087 :   if ((tx==H_MATRIX) ^ (ty==H_MATRIX)) pari_err_TYPE2("H_mul", x, y);
    2175      108080 :   if (tx == H_MATRIX) /* both H_MATRIX */ return alM_mul(NULL, x, y);
    2176      103789 :   if (tx == H_SCALAR && ty == H_SCALAR) return gmul(x,y);
    2177      102592 :   H_compo(x,&a,&b,&c,&d);
    2178      102592 :   H_compo(y,&u,&v,&w,&z);
    2179      102592 :   return gerepilecopy(av,mkcol4(
    2180             :         gsub(gmul(a,u), gadd(gadd(gmul(b,v),gmul(c,w)),gmul(d,z))),
    2181             :         gsub(gadd(gmul(a,v),gadd(gmul(b,u),gmul(c,z))), gmul(d,w)),
    2182             :         gsub(gadd(gmul(a,w),gadd(gmul(c,u),gmul(d,v))), gmul(b,z)),
    2183             :         gsub(gadd(gmul(a,z),gadd(gmul(b,w),gmul(d,u))), gmul(c,v))
    2184             :         ));
    2185             : }
    2186             : 
    2187             : GEN
    2188      491011 : algmul(GEN al, GEN x, GEN y)
    2189             : {
    2190      491011 :   pari_sp av = avma;
    2191             :   long tx, ty;
    2192      491011 :   checkalg(al);
    2193      491011 :   if (alg_type(al)==al_REAL) return H_mul(x,y);
    2194      383162 :   tx = alg_model(al,x);
    2195      383148 :   ty = alg_model(al,y);
    2196      383148 :   if (tx==al_MATRIX) {
    2197       20832 :     if (ty==al_MATRIX) return alM_mul(al,x,y);
    2198           7 :     pari_err_TYPE("algmul", y);
    2199             :   }
    2200      362316 :   if (signe(alg_get_char(al))) return algbasismul(al,x,y);
    2201      205135 :   if (tx==al_TRIVIAL) retmkcol(gmul(gel(x,1),gel(y,1)));
    2202      205030 :   if (tx==al_ALGEBRAIC && ty==al_ALGEBRAIC) return algalgmul(al,x,y);
    2203      203504 :   if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
    2204      203504 :   if (ty==al_ALGEBRAIC) y = algalgtobasis(al,y);
    2205      203504 :   return gerepileupto(av,algbasismul(al,x,y));
    2206             : }
    2207             : 
    2208             : static GEN
    2209         329 : H_sqr(GEN x)
    2210             : {
    2211         329 :   pari_sp av = avma;
    2212         329 :   long tx = H_model(x);
    2213             :   GEN a,b,c,d;
    2214         308 :   if (tx == H_SCALAR) return gsqr(x);
    2215         224 :   if (tx == H_MATRIX) return H_mul(x,x);
    2216         119 :   H_compo(x,&a,&b,&c,&d);
    2217         119 :   return gerepilecopy(av, mkcol4(
    2218             :         gsub(gsqr(a), gadd(gsqr(b),gadd(gsqr(c),gsqr(d)))),
    2219             :         gshift(gmul(a,b),1),
    2220             :         gshift(gmul(a,c),1),
    2221             :         gshift(gmul(a,d),1)
    2222             :         ));
    2223             : }
    2224             : 
    2225             : GEN
    2226      106257 : algsqr(GEN al, GEN x)
    2227             : {
    2228      106257 :   pari_sp av = avma;
    2229             :   long tx;
    2230      106257 :   checkalg(al);
    2231      106222 :   if (alg_type(al)==al_REAL) return H_sqr(x);
    2232      105893 :   tx = alg_model(al,x);
    2233      105823 :   if (tx==al_MATRIX) return gerepilecopy(av,alM_mul(al,x,x));
    2234      105312 :   if (signe(alg_get_char(al))) return algbasismul(al,x,x);
    2235        2205 :   if (tx==al_TRIVIAL) retmkcol(gsqr(gel(x,1)));
    2236        2023 :   if (tx==al_ALGEBRAIC) return algalgmul(al,x,x);
    2237        1204 :   return gerepileupto(av,algbasismul(al,x,x));
    2238             : }
    2239             : 
    2240             : static GEN
    2241        9380 : algmtK2Z_cyc(GEN al, GEN m)
    2242             : {
    2243        9380 :   pari_sp av = avma;
    2244        9380 :   GEN nf = alg_get_abssplitting(al), res, mt, rnf = alg_get_splittingfield(al), c, dc;
    2245        9380 :   long n = alg_get_degree(al), N = nf_get_degree(nf), Nn, i, j, i1, j1;
    2246        9380 :   Nn = N*n;
    2247        9380 :   res = zeromatcopy(Nn,Nn);
    2248       42364 :   for (i=0; i<n; i++)
    2249      196168 :   for (j=0; j<n; j++) {
    2250      163184 :     c = gcoeff(m,i+1,j+1);
    2251      163184 :     if (!gequal0(c)) {
    2252       32984 :       c = rnfeltreltoabs(rnf,c);
    2253       32984 :       c = algtobasis(nf,c);
    2254       32984 :       c = Q_remove_denom(c,&dc);
    2255       32984 :       mt = zk_multable(nf,c);
    2256       32984 :       if (dc) mt = ZM_Z_div(mt,dc);
    2257      302162 :       for (i1=1; i1<=N; i1++)
    2258     2945936 :       for (j1=1; j1<=N; j1++)
    2259     2676758 :         gcoeff(res,i*N+i1,j*N+j1) = gcoeff(mt,i1,j1);
    2260             :     }
    2261             :   }
    2262        9380 :   return gerepilecopy(av,res);
    2263             : }
    2264             : 
    2265             : static GEN
    2266         945 : algmtK2Z_csa(GEN al, GEN m)
    2267             : {
    2268         945 :   pari_sp av = avma;
    2269         945 :   GEN nf = alg_get_center(al), res, mt, c, dc;
    2270         945 :   long d2 = alg_get_dim(al), n = nf_get_degree(nf), D, i, j, i1, j1;
    2271         945 :   D = d2*n;
    2272         945 :   res = zeromatcopy(D,D);
    2273        5502 :   for (i=0; i<d2; i++)
    2274       31122 :   for (j=0; j<d2; j++) {
    2275       26565 :     c = gcoeff(m,i+1,j+1);
    2276       26565 :     if (!gequal0(c)) {
    2277        3906 :       c = algtobasis(nf,c);
    2278        3906 :       c = Q_remove_denom(c,&dc);
    2279        3906 :       mt = zk_multable(nf,c);
    2280        3906 :       if (dc) mt = ZM_Z_div(mt,dc);
    2281       13076 :       for (i1=1; i1<=n; i1++)
    2282       32564 :       for (j1=1; j1<=n; j1++)
    2283       23394 :         gcoeff(res,i*n+i1,j*n+j1) = gcoeff(mt,i1,j1);
    2284             :     }
    2285             :   }
    2286         945 :   return gerepilecopy(av,res);
    2287             : }
    2288             : 
    2289             : /* assumes al is a CSA or CYCLIC */
    2290             : static GEN
    2291       10325 : algmtK2Z(GEN al, GEN m)
    2292             : {
    2293       10325 :   switch(alg_type(al))
    2294             :   {
    2295        9380 :     case al_CYCLIC: return algmtK2Z_cyc(al, m);
    2296         945 :     case al_CSA: return algmtK2Z_csa(al, m);
    2297             :   }
    2298             :   return NULL; /*LCOV_EXCL_LINE*/
    2299             : }
    2300             : 
    2301             : /* left multiplication table, as a vector space of dimension n over the splitting field (by right multiplication) */
    2302             : static GEN
    2303       11998 : algalgmultable_cyc(GEN al, GEN x)
    2304             : {
    2305       11998 :   pari_sp av = avma;
    2306       11998 :   long n = alg_get_degree(al), i, j;
    2307             :   GEN res, rnf, auts, b, pol;
    2308       11998 :   rnf = alg_get_splittingfield(al);
    2309       11998 :   auts = alg_get_auts(al);
    2310       11998 :   b = alg_get_b(al);
    2311       11998 :   pol = rnf_get_pol(rnf);
    2312             : 
    2313       11998 :   res = zeromatcopy(n,n);
    2314       50288 :   for (i=0; i<n; i++)
    2315       38290 :     gcoeff(res,i+1,1) = lift_shallow(rnfbasistoalg(rnf,gel(x,i+1)));
    2316             : 
    2317       50288 :   for (i=0; i<n; i++) {
    2318      106386 :     for (j=1; j<=i; j++)
    2319       68096 :       gcoeff(res,i+1,j+1) = gmodulo(poleval(gcoeff(res,i-j+1,1),gel(auts,j)),pol);
    2320      106386 :     for (; j<n; j++)
    2321       68096 :       gcoeff(res,i+1,j+1) = gmodulo(gmul(b,poleval(gcoeff(res,n+i-j+1,1),gel(auts,j))), pol);
    2322             :   }
    2323             : 
    2324       50288 :   for (i=0; i<n; i++)
    2325       38290 :     gcoeff(res,i+1,1) = gmodulo(gcoeff(res,i+1,1),pol);
    2326             : 
    2327       11998 :   return gerepilecopy(av, res);
    2328             : }
    2329             : 
    2330             : static GEN
    2331        1393 : elementmultable(GEN mt, GEN x)
    2332             : {
    2333        1393 :   pari_sp av = avma;
    2334        1393 :   long D = lg(mt)-1, i;
    2335        1393 :   GEN z = NULL;
    2336        7448 :   for (i=1; i<=D; i++)
    2337             :   {
    2338        6055 :     GEN c = gel(x,i);
    2339        6055 :     if (!gequal0(c))
    2340             :     {
    2341        2163 :       GEN M = RgM_Rg_mul(gel(mt,i),c);
    2342        2163 :       z = z? RgM_add(z, M): M;
    2343             :     }
    2344             :   }
    2345        1393 :   if (!z) { set_avma(av); return zeromatcopy(D,D); }
    2346        1393 :   return gerepileupto(av, z);
    2347             : }
    2348             : /* mt a t_VEC of Flm modulo m */
    2349             : static GEN
    2350       24902 : algbasismultable_Flm(GEN mt, GEN x, ulong m)
    2351             : {
    2352       24902 :   pari_sp av = avma;
    2353       24902 :   long D = lg(gel(mt,1))-1, i;
    2354       24902 :   GEN z = NULL;
    2355      270185 :   for (i=1; i<=D; i++)
    2356             :   {
    2357      245283 :     ulong c = x[i];
    2358      245283 :     if (c)
    2359             :     {
    2360       34126 :       GEN M = Flm_Fl_mul(gel(mt,i),c, m);
    2361       34126 :       z = z? Flm_add(z, M, m): M;
    2362             :     }
    2363             :   }
    2364       24902 :   if (!z) { set_avma(av); return zero_Flm(D,D); }
    2365       24902 :   return gerepileupto(av, z);
    2366             : }
    2367             : static GEN
    2368      225834 : elementabsmultable_Z(GEN mt, GEN x)
    2369             : {
    2370      225834 :   long i, l = lg(x);
    2371      225834 :   GEN z = NULL;
    2372     2325357 :   for (i = 1; i < l; i++)
    2373             :   {
    2374     2099523 :     GEN c = gel(x,i);
    2375     2099523 :     if (signe(c))
    2376             :     {
    2377      809817 :       GEN M = ZM_Z_mul(gel(mt,i),c);
    2378      809817 :       z = z? ZM_add(z, M): M;
    2379             :     }
    2380             :   }
    2381      225834 :   return z;
    2382             : }
    2383             : static GEN
    2384      114964 : elementabsmultable(GEN mt, GEN x)
    2385             : {
    2386      114964 :   GEN d, z = elementabsmultable_Z(mt, Q_remove_denom(x,&d));
    2387      114964 :   return (z && d)? ZM_Z_div(z, d): z;
    2388             : }
    2389             : static GEN
    2390      110870 : elementabsmultable_Fp(GEN mt, GEN x, GEN p)
    2391             : {
    2392      110870 :   GEN z = elementabsmultable_Z(mt, x);
    2393      110870 :   return z? FpM_red(z, p): z;
    2394             : }
    2395             : static GEN
    2396      225834 : algbasismultable(GEN al, GEN x)
    2397             : {
    2398      225834 :   pari_sp av = avma;
    2399      225834 :   GEN z, p = alg_get_char(al), mt = alg_get_multable(al);
    2400      225834 :   z = signe(p)? elementabsmultable_Fp(mt, x, p): elementabsmultable(mt, x);
    2401      225834 :   if (!z)
    2402             :   {
    2403         699 :     long D = lg(mt)-1;
    2404         699 :     set_avma(av); return zeromat(D,D);
    2405             :   }
    2406      225135 :   return gerepileupto(av, z);
    2407             : }
    2408             : 
    2409             : static GEN
    2410        1393 : algalgmultable_csa(GEN al, GEN x)
    2411             : {
    2412        1393 :   GEN nf = alg_get_center(al), m;
    2413             :   long i,j;
    2414        1393 :   m = elementmultable(alg_get_relmultable(al), x);
    2415        7448 :   for (i=1; i<lg(m); i++)
    2416       38318 :     for(j=1; j<lg(m); j++)
    2417       32263 :       gcoeff(m,i,j) = basistoalg(nf,gcoeff(m,i,j));
    2418        1393 :   return m;
    2419             : }
    2420             : 
    2421             : /* assumes x in algebraic form */
    2422             : static GEN
    2423       13097 : algalgmultable(GEN al, GEN x)
    2424             : {
    2425       13097 :   switch(alg_type(al))
    2426             :   {
    2427       11998 :     case al_CYCLIC: return algalgmultable_cyc(al, x);
    2428        1099 :     case al_CSA: return algalgmultable_csa(al, x);
    2429             :   }
    2430             :   return NULL; /*LCOV_EXCL_LINE*/
    2431             : }
    2432             : 
    2433             : /* on the natural basis */
    2434             : /* assumes x in algebraic form */
    2435             : static GEN
    2436       10325 : algZmultable(GEN al, GEN x) {
    2437       10325 :   pari_sp av = avma;
    2438       10325 :   return gerepileupto(av, algmtK2Z(al,algalgmultable(al,x)));
    2439             : }
    2440             : 
    2441             : /* x integral */
    2442             : static GEN
    2443       36575 : algbasisrightmultable(GEN al, GEN x)
    2444             : {
    2445       36575 :   long N = alg_get_absdim(al), i,j,k;
    2446       36575 :   GEN res = zeromatcopy(N,N), c, mt = alg_get_multable(al), p = alg_get_char(al);
    2447       36575 :   if (gequal0(p)) p = NULL;
    2448      330960 :   for (i=1; i<=N; i++) {
    2449      294385 :     c = gel(x,i);
    2450      294385 :     if (!gequal0(c)) {
    2451      892773 :       for (j=1; j<=N; j++)
    2452     7582134 :       for(k=1; k<=N; k++) {
    2453     6785842 :         if (p) gcoeff(res,k,j) = Fp_add(gcoeff(res,k,j), Fp_mul(c, gcoeff(gel(mt,j),k,i), p), p);
    2454     5160974 :         else gcoeff(res,k,j) = addii(gcoeff(res,k,j), mulii(c, gcoeff(gel(mt,j),k,i)));
    2455             :       }
    2456             :     }
    2457             :   }
    2458       36575 :   return res;
    2459             : }
    2460             : 
    2461             : /* basis for matrices : 1, E_{i,j} for (i,j)!=(1,1) */
    2462             : /* index : ijk = ((i-1)*N+j-1)*n + k */
    2463             : /* square matrices only, coefficients in basis form, shallow function */
    2464             : static GEN
    2465       23961 : algmat2basis(GEN al, GEN M)
    2466             : {
    2467       23961 :   long n = alg_get_absdim(al), N = lg(M)-1, i, j, k, ij, ijk;
    2468             :   GEN res, x;
    2469       23961 :   res = zerocol(N*N*n);
    2470       75131 :   for (i=1; i<=N; i++) {
    2471      163310 :     for (j=1, ij=(i-1)*N+1; j<=N; j++, ij++) {
    2472      112140 :       x = gcoeff(M,i,j);
    2473      819532 :       for (k=1, ijk=(ij-1)*n+1; k<=n; k++, ijk++) {
    2474      707392 :         gel(res, ijk) = gel(x, k);
    2475      707392 :         if (i>1 && i==j) gel(res, ijk) = gsub(gel(res,ijk), gel(res,k));
    2476             :       }
    2477             :     }
    2478             :   }
    2479             : 
    2480       23961 :   return res;
    2481             : }
    2482             : 
    2483             : static GEN
    2484         294 : algbasis2mat(GEN al, GEN M, long N)
    2485             : {
    2486         294 :   long n = alg_get_absdim(al), i, j, k, ij, ijk;
    2487             :   GEN res, x;
    2488         294 :   res = zeromatcopy(N,N);
    2489         882 :   for (i=1; i<=N; i++)
    2490        1764 :   for (j=1; j<=N; j++)
    2491        1176 :     gcoeff(res,i,j) = zerocol(n);
    2492             : 
    2493         882 :   for (i=1; i<=N; i++) {
    2494        1764 :     for (j=1, ij=(i-1)*N+1; j<=N; j++, ij++) {
    2495        1176 :       x = gcoeff(res,i,j);
    2496        9240 :       for (k=1, ijk=(ij-1)*n+1; k<=n; k++, ijk++) {
    2497        8064 :         gel(x,k) = gel(M,ijk);
    2498        8064 :         if (i>1 && i==j) gel(x,k) = gadd(gel(x,k), gel(M,k));
    2499             :       }
    2500             :     }
    2501             :   }
    2502             : 
    2503         294 :   return res;
    2504             : }
    2505             : 
    2506             : static GEN
    2507       23884 : algmatbasis_ei(GEN al, long ijk, long N)
    2508             : {
    2509       23884 :   long n = alg_get_absdim(al), i, j, k, ij;
    2510             :   GEN res;
    2511             : 
    2512       23884 :   res = zeromatcopy(N,N);
    2513       74900 :   for (i=1; i<=N; i++)
    2514      162848 :   for (j=1; j<=N; j++)
    2515      111832 :     gcoeff(res,i,j) = zerocol(n);
    2516             : 
    2517       23884 :   k = ijk%n;
    2518       23884 :   if (k==0) k=n;
    2519       23884 :   ij = (ijk-k)/n+1;
    2520             : 
    2521       23884 :   if (ij==1) {
    2522       16947 :     for (i=1; i<=N; i++)
    2523       11410 :       gcoeff(res,i,i) = col_ei(n,k);
    2524        5537 :     return res;
    2525             :   }
    2526             : 
    2527       18347 :   j = ij%N;
    2528       18347 :   if (j==0) j=N;
    2529       18347 :   i = (ij-j)/N+1;
    2530             : 
    2531       18347 :   gcoeff(res,i,j) = col_ei(n,k);
    2532       18347 :   return res;
    2533             : }
    2534             : 
    2535             : /* FIXME lazy implementation! */
    2536             : static GEN
    2537         910 : algleftmultable_mat(GEN al, GEN M)
    2538             : {
    2539         910 :   long N = lg(M)-1, n = alg_get_absdim(al), D = N*N*n, j;
    2540             :   GEN res, x, Mx;
    2541         910 :   if (N == 0) return cgetg(1, t_MAT);
    2542         903 :   if (N != nbrows(M)) pari_err_DIM("algleftmultable_mat (nonsquare)");
    2543         882 :   res = cgetg(D+1, t_MAT);
    2544       24766 :   for (j=1; j<=D; j++) {
    2545       23884 :     x = algmatbasis_ei(al, j, N);
    2546       23884 :     Mx = algmul(al, M, x);
    2547       23884 :     gel(res, j) = algmat2basis(al, Mx);
    2548             :   }
    2549         882 :   return res;
    2550             : }
    2551             : 
    2552             : /* left multiplication table on integral basis */
    2553             : static GEN
    2554        6951 : algleftmultable(GEN al, GEN x)
    2555             : {
    2556        6951 :   pari_sp av = avma;
    2557             :   long tx;
    2558             :   GEN res;
    2559             : 
    2560        6951 :   checkalg(al);
    2561        6951 :   tx = alg_model(al,x);
    2562        6944 :   switch(tx) {
    2563          98 :     case al_TRIVIAL : res = mkmatcopy(mkcol(gel(x,1))); break;
    2564         196 :     case al_ALGEBRAIC : x = algalgtobasis(al,x);
    2565        6328 :     case al_BASIS : res = algbasismultable(al,x); break;
    2566         518 :     case al_MATRIX : res = algleftmultable_mat(al,x); break;
    2567             :     default : return NULL; /* LCOV_EXCL_LINE */
    2568             :   }
    2569        6937 :   return gerepileupto(av,res);
    2570             : }
    2571             : 
    2572             : static GEN
    2573        4102 : algbasissplittingmatrix_csa(GEN al, GEN x)
    2574             : {
    2575        4102 :   long d = alg_get_degree(al), i, j;
    2576        4102 :   GEN rnf = alg_get_splittingfield(al), splba = alg_get_splittingbasis(al), splbainv = alg_get_splittingbasisinv(al), M;
    2577        4102 :   M = algbasismultable(al,x);
    2578        4102 :   M = RgM_mul(M, splba); /* TODO best order ? big matrix /Q vs small matrix /nf */
    2579        4102 :   M = RgM_mul(splbainv, M);
    2580       12131 :   for (i=1; i<=d; i++)
    2581       23912 :   for (j=1; j<=d; j++)
    2582       15883 :     gcoeff(M,i,j) = rnfeltabstorel(rnf, gcoeff(M,i,j));
    2583        4102 :   return M;
    2584             : }
    2585             : 
    2586             : static GEN
    2587         728 : algmat_tomatrix(GEN al, GEN x) /* abs = 0 */
    2588             : {
    2589             :   GEN res;
    2590             :   long i,j;
    2591         728 :   if (lg(x) == 1) return cgetg(1, t_MAT);
    2592         700 :   res = zeromatcopy(nbrows(x),lg(x)-1);
    2593        2212 :   for (j=1; j<lg(x); j++)
    2594        4879 :   for (i=1; i<lgcols(x); i++)
    2595        3367 :     gcoeff(res,i,j) = algtomatrix(al,gcoeff(x,i,j),0);
    2596         700 :   return shallowmatconcat(res);
    2597             : }
    2598             : 
    2599             : static GEN
    2600          42 : R_tomatrix(GEN x)
    2601             : {
    2602          42 :   long t = H_model(x);
    2603          42 :   if (t == H_QUATERNION) pari_err_TYPE("R_tomatrix", x);
    2604          35 :   if (t == H_MATRIX) return x;
    2605          21 :   return mkmat(mkcol(x));
    2606             : }
    2607             : static GEN
    2608          84 : C_tomatrix(GEN z, long abs)
    2609             : {
    2610             :   GEN x,y;
    2611          84 :   long t = H_model(z), nrows, ncols;
    2612          84 :   if (t == H_QUATERNION) pari_err_TYPE("C_tomatrix", z);
    2613          77 :   if (!abs)
    2614             :   {
    2615          14 :     if (t == H_MATRIX) return z;
    2616           7 :     return mkmat(mkcol(z));
    2617             :   }
    2618          63 :   if (t == H_MATRIX)
    2619             :   {
    2620             :     /* Warning: this is not the same choice of basis as for other algebras */
    2621             :     GEN res, a, b;
    2622             :     long i,j;
    2623          56 :     RgM_dimensions(z,&nrows,&ncols);
    2624          56 :     res = zeromatcopy(2*nrows,2*ncols);
    2625         168 :     for (i=1; i<=nrows; i++)
    2626         336 :       for (j=1; j<=ncols; j++)
    2627             :       {
    2628         224 :         a = real_i(gcoeff(z,i,j));
    2629         224 :         b = imag_i(gcoeff(z,i,j));
    2630         224 :         gcoeff(res,2*i-1,2*j-1) = a;
    2631         224 :         gcoeff(res,2*i,2*j) = a;
    2632         224 :         gcoeff(res,2*i-1,2*j) = gneg(b);
    2633         224 :         gcoeff(res,2*i,2*j-1) = b;
    2634             :       }
    2635          56 :     return res;
    2636             :   }
    2637           7 :   x = real_i(z);
    2638           7 :   y = imag_i(z);
    2639           7 :   return mkmat22(x,gneg(y),y,x);
    2640             : }
    2641             : static GEN
    2642        2331 : H_tomatrix(GEN x, long abs)
    2643             : {
    2644        2331 :   long tx = H_model(x);
    2645        2324 :   GEN a = NULL, b =NULL, c = NULL, d = NULL, md = NULL, M = NULL;
    2646        2324 :   if (abs) {
    2647         287 :     if (tx == H_MATRIX) return algleftmultable_mat(NULL,x);
    2648         154 :     switch(tx)
    2649             :     {
    2650          35 :       case H_SCALAR:
    2651          35 :         a = real_i(x);
    2652          35 :         b = imag_i(x);
    2653          35 :         c = gen_0;
    2654          35 :         d = gen_0;
    2655          35 :         break;
    2656         119 :       case H_QUATERNION:
    2657         119 :         a = gel(x,1);
    2658         119 :         b = gel(x,2);
    2659         119 :         c = gel(x,3);
    2660         119 :         d = gel(x,4);
    2661         119 :         break;
    2662             :     }
    2663         154 :     M = scalarmat(a,4);
    2664         154 :     gcoeff(M,2,1) = gcoeff(M,4,3) = b;
    2665         154 :     gcoeff(M,1,2) = gcoeff(M,3,4) = gneg(b);
    2666         154 :     gcoeff(M,3,1) = gcoeff(M,2,4) = c;
    2667         154 :     gcoeff(M,4,2) = gcoeff(M,1,3) = gneg(c);
    2668         154 :     gcoeff(M,4,1) = gcoeff(M,3,2) = d;
    2669         154 :     gcoeff(M,2,3) = gcoeff(M,1,4) = gneg(d);
    2670             :   }
    2671             :   else /* abs == 0 */
    2672             :   {
    2673        2037 :     if (tx == H_MATRIX) return algmat_tomatrix(NULL,x);
    2674        1778 :     switch(tx)
    2675             :     {
    2676         273 :       case H_SCALAR:
    2677         273 :         M = mkmat22(
    2678             :             x,      gen_0,
    2679             :             gen_0,  conj_i(x)
    2680             :             );
    2681         273 :         break;
    2682        1505 :       case H_QUATERNION:
    2683        1505 :         a = gel(x,1);
    2684        1505 :         b = gel(x,2);
    2685        1505 :         c = gel(x,3);
    2686        1505 :         md = gneg(gel(x,4));
    2687        1505 :         M = mkmat22(
    2688             :             mkcomplex(a,b),     mkcomplex(gneg(c),md),
    2689             :             mkcomplex(c,md),    mkcomplex(a,gneg(b))
    2690             :             );
    2691             :     }
    2692        1932 :   }
    2693        1932 :   return M;
    2694             : }
    2695             : 
    2696             : GEN
    2697        9667 : algtomatrix(GEN al, GEN x, long abs)
    2698             : {
    2699        9667 :   pari_sp av = avma;
    2700        9667 :   GEN res = NULL;
    2701             :   long ta, tx;
    2702        9667 :   checkalg(al);
    2703        9667 :   ta = alg_type(al);
    2704        9667 :   if (ta==al_REAL)
    2705             :   {
    2706        2268 :     switch(alg_get_absdim(al)) {
    2707          42 :       case 1: res = R_tomatrix(x); break;
    2708          84 :       case 2: res = C_tomatrix(x,abs); break;
    2709        2135 :       case 4: res = H_tomatrix(x,abs); break;
    2710           7 :       default: pari_err_TYPE("algtomatrix [apply alginit]", al);
    2711             :     }
    2712        2240 :     return gerepilecopy(av, res);
    2713             :   }
    2714        7399 :   if (abs || ta==al_TABLE) return algleftmultable(al,x);
    2715        6622 :   tx = alg_model(al,x);
    2716        6622 :   if (tx == al_MATRIX) res = algmat_tomatrix(al,x);
    2717        6153 :   else switch (alg_type(al))
    2718             :   {
    2719        2051 :     case al_CYCLIC:
    2720        2051 :       if (tx==al_BASIS) x = algbasistoalg(al,x);
    2721        2051 :       res = algalgmultable(al,x);
    2722        2051 :       break;
    2723        4102 :     case al_CSA:
    2724        4102 :       if (tx==al_ALGEBRAIC) x = algalgtobasis(al,x);
    2725        4102 :       res = algbasissplittingmatrix_csa(al,x);
    2726        4102 :       break;
    2727             :     default: return NULL; /*LCOV_EXCL_LINE*/
    2728             :   }
    2729        6622 :   return gerepilecopy(av,res);
    2730             : }
    2731             : 
    2732             : /*  x^(-1)*y, NULL if no solution */
    2733             : static GEN
    2734         112 : C_divl_i(GEN x, GEN y)
    2735             : {
    2736         112 :   long tx = H_model(x), ty = H_model(y);
    2737         112 :   if (tx != ty) pari_err_TYPE2("C_divl", x, y);
    2738         105 :   switch (tx) {
    2739          42 :     case H_SCALAR:
    2740          42 :       if (gequal0(x)) return gequal0(y) ? gen_0 : NULL;
    2741          14 :       else return gdiv(y,x);
    2742          56 :     case H_MATRIX:
    2743          56 :       if ((lg(x)>1 && lg(x) != lgcols(x)) || (lg(y)>1 && lg(y) != lgcols(y)))
    2744           7 :         pari_err_DIM("C_divl (nonsquare)");
    2745          49 :       if (lg(x) != lg(y)) pari_err_DIM("C_divl");
    2746          42 :       if (lg(y) == 1) return cgetg(1, t_MAT);
    2747          42 :       return RgM_invimage(x, y);
    2748           7 :     default: pari_err_TYPE("C_divl", x); return NULL;
    2749             :   }
    2750             : }
    2751             : /* H^k -> C^2k */
    2752             : static GEN
    2753         140 : HC_to_CC(GEN v)
    2754             : {
    2755         140 :   long l = lg(v), i;
    2756         140 :   GEN w = cgetg(2*l-1, t_COL), a, b, c, d;
    2757         420 :   for (i=1; i<l; i++)
    2758             :   {
    2759         280 :     H_compo(gel(v,i),&a,&b,&c,&d);
    2760         280 :     gel(w,2*i-1) = mkcomplex(a,b);
    2761         280 :     gel(w,2*i) = mkcomplex(c,gneg(d));
    2762             :   }
    2763         140 :   return w;
    2764             : }
    2765             : /* C^2k -> H^k */
    2766             : static GEN
    2767          98 : CC_to_HC(GEN w)
    2768             : {
    2769          98 :   long l = lg(w), i, lv = (l+1)/2;
    2770          98 :   GEN v = cgetg(lv, t_COL), ab, cd;
    2771         294 :   for (i=1; i<lv; i++)
    2772             :   {
    2773         196 :     ab = gel(w,2*i-1);
    2774         196 :     cd = gel(w,2*i);
    2775         196 :     gel(v,i) = mkcol4(real_i(ab),imag_i(ab),real_i(cd),gneg(imag_i(cd)));
    2776             :   }
    2777          98 :   return v;
    2778             : }
    2779             : /* M_{k,n}(H) -> M_{2k,n}(C) */
    2780             : static GEN
    2781         210 : HM_to_CM(GEN x) pari_APPLY_same(HC_to_CC(gel(x,i)));
    2782             : /* M_{2k,n}(C) -> M_{k,n}(H) */
    2783             : static GEN
    2784         147 : CM_to_HM(GEN x) pari_APPLY_same(CC_to_HC(gel(x,i)));
    2785             : /*  x^(-1)*y, NULL if no solution */
    2786             : static GEN
    2787         203 : H_divl_i(GEN x, GEN y)
    2788             : {
    2789         203 :   pari_sp av = avma;
    2790         203 :   long tx = H_model(x), ty = H_model(y);
    2791         189 :   if ((tx==H_MATRIX) ^ (ty==H_MATRIX)) pari_err_TYPE2("H_divl", x, y);
    2792         168 :   if (tx==H_MATRIX)
    2793             :   {
    2794             :     GEN mx, my, mxdivy;
    2795          98 :     if ((lg(x)>1 && lg(x) != lgcols(x)) || (lg(y)>1 && lg(y) != lgcols(y)))
    2796          14 :       pari_err_DIM("H_divl (nonsquare)");
    2797          84 :     if (lg(x) != lg(y)) pari_err_DIM("H_divl");
    2798          77 :     if (lg(y) == 1) return cgetg(1, t_MAT);
    2799          70 :     mx = H_tomatrix(x,0);
    2800          70 :     my = HM_to_CM(y);
    2801          70 :     mxdivy = RgM_invimage(mx, my);
    2802          70 :     if (!mxdivy) return gc_NULL(av);
    2803          49 :     return gerepilecopy(av,CM_to_HM(mxdivy));
    2804             :   }
    2805          70 :   if (gequal0(y)) return gen_0;
    2806          56 :   if (gequal0(x)) return NULL;
    2807          42 :   return gerepilecopy(av,H_mul(H_inv(x),y));
    2808             : }
    2809             : /*  x^(-1)*y, NULL if no solution */
    2810             : static GEN
    2811        1715 : algdivl_i(GEN al, GEN x, GEN y, long tx, long ty) {
    2812        1715 :   pari_sp av = avma;
    2813        1715 :   GEN res, p = alg_get_char(al), mtx;
    2814        1715 :   if (tx != ty) {
    2815         343 :     if (tx==al_ALGEBRAIC) { x = algalgtobasis(al,x); tx=al_BASIS; }
    2816         343 :     if (ty==al_ALGEBRAIC) { y = algalgtobasis(al,y); ty=al_BASIS; }
    2817             :   }
    2818        1715 :   if (ty == al_MATRIX)
    2819             :   {
    2820          77 :     if (alg_type(al) != al_TABLE) y = algalgtobasis(al,y);
    2821          77 :     y = algmat2basis(al,y);
    2822             :   }
    2823        1715 :   if (signe(p)) res = FpM_FpC_invimage(algbasismultable(al,x),y,p);
    2824             :   else
    2825             :   {
    2826        1526 :     if (ty==al_ALGEBRAIC)   mtx = algalgmultable(al,x);
    2827         819 :     else                    mtx = algleftmultable(al,x);
    2828        1526 :     res = inverseimage(mtx,y);
    2829             :   }
    2830        1715 :   if (!res || lg(res)==1) return gc_NULL(av);
    2831        1687 :   if (tx == al_MATRIX) {
    2832         294 :     res = algbasis2mat(al, res, lg(x)-1);
    2833         294 :     return gerepilecopy(av,res);
    2834             :   }
    2835        1393 :   return gerepileupto(av,res);
    2836             : }
    2837             : static GEN
    2838        1001 : algdivl_i2(GEN al, GEN x, GEN y)
    2839             : {
    2840             :   long tx, ty;
    2841        1001 :   checkalg(al);
    2842        1001 :   if (alg_type(al)==al_REAL) switch(alg_get_absdim(al)) {
    2843         112 :     case 1: case 2: return C_divl_i(x,y);
    2844         147 :     case 4: return H_divl_i(x,y);
    2845             :   }
    2846         742 :   tx = alg_model(al,x);
    2847         735 :   ty = alg_model(al,y);
    2848         735 :   if (tx == al_MATRIX) {
    2849         140 :     if (ty != al_MATRIX) pari_err_TYPE2("\\", x, y);
    2850         133 :     if ((lg(x)>1 && lg(x) != lgcols(x)) || (lg(y)>1 && lg(y) != lgcols(y)))
    2851          28 :       pari_err_DIM("algdivl (nonsquare)");
    2852         105 :     if (lg(x) != lg(y)) pari_err_DIM("algdivl");
    2853          84 :     if (lg(y) == 1) return cgetg(1, t_MAT);
    2854             :   }
    2855         672 :   return algdivl_i(al,x,y,tx,ty);
    2856             : }
    2857             : 
    2858         875 : GEN algdivl(GEN al, GEN x, GEN y)
    2859             : {
    2860             :   GEN z;
    2861         875 :   z = algdivl_i2(al,x,y);
    2862         728 :   if (!z) pari_err_INV("algdivl", x);
    2863         714 :   return z;
    2864             : }
    2865             : 
    2866             : int
    2867         126 : algisdivl(GEN al, GEN x, GEN y, GEN* ptz)
    2868             : {
    2869         126 :   pari_sp av = avma;
    2870         126 :   GEN z = algdivl_i2(al,x,y);
    2871         126 :   if (!z) return gc_bool(av,0);
    2872          84 :   if (ptz != NULL) *ptz = z;
    2873          84 :   return 1;
    2874             : }
    2875             : 
    2876             : static GEN
    2877         140 : C_inv(GEN x)
    2878             : {
    2879         140 :   switch (H_model(x))
    2880             :   {
    2881          63 :     case H_SCALAR: return gequal0(x) ? NULL : ginv(x);
    2882          70 :     case H_MATRIX: return RgM_inv(x);
    2883           7 :     default: pari_err_TYPE("alginv_i", x);
    2884             :   }
    2885             :   return NULL; /*LCOV_EXCL_LINE*/
    2886             : }
    2887             : static GEN
    2888         259 : H_inv(GEN x)
    2889             : {
    2890         259 :   pari_sp av = avma;
    2891             :   GEN nm, xi;
    2892             :   long i;
    2893         259 :   switch (H_model(x))
    2894             :   {
    2895          28 :     case H_SCALAR:
    2896          28 :       if (gequal0(x)) return NULL;
    2897          14 :       return ginv(x);
    2898         161 :     case H_QUATERNION:
    2899         161 :       if (gequal0(x)) return NULL;
    2900         154 :       nm = H_norm(x, 0);
    2901         154 :       xi = gdiv(x,nm);
    2902         616 :       for(i=2; i<=4; i++) gel(xi,i) = gneg(gel(xi,i));
    2903         154 :       return gerepilecopy(av,xi);
    2904          63 :     case H_MATRIX:
    2905          63 :       if (lg(x)==1) return cgetg(1,t_MAT);
    2906          56 :       return H_divl_i(x, matid(lg(x)-1));
    2907             :   }
    2908             :   return NULL; /*LCOV_EXCL_LINE*/
    2909             : }
    2910             : static GEN
    2911        1512 : alginv_i(GEN al, GEN x)
    2912             : {
    2913        1512 :   pari_sp av = avma;
    2914        1512 :   GEN res = NULL, p = alg_get_char(al);
    2915             :   long tx, n, ta;
    2916        1512 :   ta = alg_type(al);
    2917        1512 :   if (ta==al_REAL) switch(alg_get_absdim(al)) {
    2918         140 :     case 1: case 2: return C_inv(x);
    2919         217 :     case 4: return H_inv(x);
    2920           7 :     default: pari_err_TYPE("alginv_i [apply alginit]", al);
    2921             :   }
    2922        1148 :   tx = alg_model(al,x);
    2923        1127 :   switch(tx) {
    2924          63 :     case al_TRIVIAL :
    2925          63 :       if (signe(p)) { res = mkcol(Fp_inv(gel(x,1),p)); break; }
    2926          49 :       else          { res = mkcol(ginv(gel(x,1))); break; }
    2927         455 :     case al_ALGEBRAIC :
    2928             :       switch(ta) {
    2929         350 :         case al_CYCLIC: n = alg_get_degree(al); break;
    2930         105 :         case al_CSA: n = alg_get_dim(al); break;
    2931             :         default: return NULL; /* LCOV_EXCL_LINE */
    2932             :       }
    2933         455 :       res = algdivl_i(al, x, col_ei(n,1), tx, al_ALGEBRAIC); break;
    2934         371 :     case al_BASIS : res = algdivl_i(al, x, col_ei(alg_get_absdim(al),1), tx,
    2935         371 :                                                             al_BASIS); break;
    2936         238 :     case al_MATRIX :
    2937         238 :       n = lg(x)-1;
    2938         238 :       if (n==0) return cgetg(1, t_MAT);
    2939         224 :       if (n != nbrows(x)) pari_err_DIM("alginv_i (nonsquare)");
    2940         217 :       res = algdivl_i(al, x, col_ei(n*n*alg_get_absdim(al),1), tx, al_BASIS);
    2941             :         /* cheat on type because wrong dimension */
    2942             :   }
    2943        1106 :   if (!res) return gc_NULL(av);
    2944        1092 :   return gerepilecopy(av,res);
    2945             : }
    2946             : GEN
    2947        1323 : alginv(GEN al, GEN x)
    2948             : {
    2949             :   GEN z;
    2950        1323 :   checkalg(al);
    2951        1323 :   z = alginv_i(al,x);
    2952        1274 :   if (!z) pari_err_INV("alginv", x);
    2953        1239 :   return z;
    2954             : }
    2955             : 
    2956             : int
    2957         189 : algisinv(GEN al, GEN x, GEN* ptix)
    2958             : {
    2959         189 :   pari_sp av = avma;
    2960             :   GEN ix;
    2961         189 :   if (al) checkalg(al);
    2962         189 :   ix = alginv_i(al,x);
    2963         189 :   if (!ix) return gc_bool(av,0);
    2964         133 :   if (ptix != NULL) *ptix = ix;
    2965         133 :   return 1;
    2966             : }
    2967             : 
    2968             : /*  x*y^(-1)  */
    2969             : GEN
    2970         469 : algdivr(GEN al, GEN x, GEN y) { return algmul(al, x, alginv(al, y)); }
    2971             : 
    2972       41315 : static GEN _mul(void* data, GEN x, GEN y) { return algmul((GEN)data,x,y); }
    2973      105025 : static GEN _sqr(void* data, GEN x) { return algsqr((GEN)data,x); }
    2974             : 
    2975             : static GEN
    2976          21 : algmatid(GEN al, long N)
    2977             : {
    2978          21 :   long n = alg_get_absdim(al), i, j;
    2979             :   GEN res, one, zero;
    2980             : 
    2981          21 :   res = zeromatcopy(N,N);
    2982          21 :   one = col_ei(n,1);
    2983          21 :   zero = zerocol(n);
    2984          49 :   for (i=1; i<=N; i++)
    2985          84 :   for (j=1; j<=N; j++)
    2986          56 :     gcoeff(res,i,j) = i==j ? one : zero;
    2987          21 :   return res;
    2988             : }
    2989             : 
    2990             : GEN
    2991       14490 : algpow(GEN al, GEN x, GEN n)
    2992             : {
    2993       14490 :   pari_sp av = avma;
    2994             :   GEN res;
    2995       14490 :   long s = signe(n);
    2996       14490 :   checkalg(al);
    2997       14490 :   if (!s && alg_type(al)==al_REAL)
    2998             :   {
    2999          63 :     if (H_model(x) == H_MATRIX) return matid(lg(x)-1);
    3000          35 :     else                        return gen_1;
    3001             :   }
    3002       14427 :   switch (s) {
    3003          28 :     case 0:
    3004          28 :       if (alg_model(al,x) == al_MATRIX)
    3005          21 :         res = algmatid(al,lg(x)-1);
    3006             :       else
    3007           7 :         res = col_ei(alg_get_absdim(al),1);
    3008          28 :       return res;
    3009       14252 :     case 1:
    3010       14252 :       res = gen_pow_i(x, n, (void*)al, _sqr, _mul); break;
    3011         147 :     default: /* -1 */
    3012         147 :       res = gen_pow_i(alginv(al,x), gneg(n), (void*)al, _sqr, _mul);
    3013             :   }
    3014       14385 :   return gerepilecopy(av,res);
    3015             : }
    3016             : 
    3017             : static GEN
    3018         378 : algredcharpoly_i(GEN al, GEN x, long v)
    3019             : {
    3020         378 :   GEN rnf = alg_get_splittingfield(al);
    3021         378 :   GEN cp = charpoly(algtomatrix(al,x,0),v);
    3022         371 :   long i, m = lg(cp);
    3023        1540 :   for (i=2; i<m; i++) gel(cp,i) = rnfeltdown(rnf, gel(cp,i));
    3024         371 :   return cp;
    3025             : }
    3026             : 
    3027             : /* assumes al is CSA or CYCLIC */
    3028             : static GEN
    3029         385 : algredcharpoly(GEN al, GEN x, long v)
    3030             : {
    3031         385 :   pari_sp av = avma;
    3032         385 :   long w = gvar(rnf_get_pol(alg_get_center(al)));
    3033         385 :   if (varncmp(v,w)>=0) pari_err_PRIORITY("algredcharpoly",pol_x(v),">=",w);
    3034         378 :   switch(alg_type(al))
    3035             :   {
    3036         378 :     case al_CYCLIC:
    3037             :     case al_CSA:
    3038         378 :       return gerepileupto(av, algredcharpoly_i(al, x, v));
    3039             :   }
    3040             :   return NULL; /*LCOV_EXCL_LINE*/
    3041             : }
    3042             : 
    3043             : static GEN
    3044       21189 : algbasischarpoly(GEN al, GEN x, long v)
    3045             : {
    3046       21189 :   pari_sp av = avma;
    3047       21189 :   GEN p = alg_get_char(al), mx;
    3048       21189 :   if (alg_model(al,x) == al_MATRIX) mx = algleftmultable_mat(al,x);
    3049       21098 :   else                              mx = algbasismultable(al,x);
    3050       21182 :   if (signe(p)) {
    3051       19278 :     GEN res = FpM_charpoly(mx,p);
    3052       19278 :     setvarn(res,v);
    3053       19278 :     return gerepileupto(av, res);
    3054             :   }
    3055        1904 :   return gerepileupto(av, charpoly(mx,v));
    3056             : }
    3057             : 
    3058             : static GEN
    3059          35 : R_charpoly(GEN x, long v, long abs)
    3060             : {
    3061          35 :   pari_sp av = avma;
    3062          35 :   GEN res = NULL;
    3063          35 :   switch (H_model(x))
    3064             :   {
    3065          14 :     case H_SCALAR: res = mkpoln(2, gen_1, gneg(x)); break;
    3066          14 :     case H_MATRIX:
    3067          14 :       res = charpoly(x,v);
    3068          14 :       if (abs) res = gpowgs(res,nbrows(x));
    3069          14 :       break;
    3070           7 :     default: pari_err_TYPE("R_charpoly", x);
    3071             :   }
    3072          28 :   if (v) setvarn(res, v);
    3073          28 :   return gerepilecopy(av, res);
    3074             : }
    3075             : static GEN
    3076          35 : C_charpoly(GEN x, long v, long abs)
    3077             : {
    3078          35 :   pari_sp av = avma;
    3079          35 :   GEN res = NULL;
    3080          35 :   switch (H_model(x))
    3081             :   {
    3082          14 :     case H_SCALAR:
    3083          14 :       if (abs)  res = mkpoln(3, gen_1, gneg(gshift(real_i(x),1)), cxnorm(x));
    3084           7 :       else      res = mkpoln(2, gen_1, gneg(x));
    3085          14 :       break;
    3086          14 :     case H_MATRIX:
    3087          14 :       res = charpoly(x,v);
    3088          14 :       if (abs) res = gpowgs(real_i(gmul(res,gconj(res))),nbrows(x));
    3089          14 :       break;
    3090           7 :     default: pari_err_TYPE("C_charpoly", x);
    3091             :   }
    3092          28 :   if (v) setvarn(res, v);
    3093          28 :   return gerepilecopy(av, res);
    3094             : }
    3095             : static GEN
    3096          98 : H_charpoly(GEN x, long v, long abs)
    3097             : {
    3098          98 :   pari_sp av = avma;
    3099             :   GEN res;
    3100          98 :   if (H_model(x) == H_MATRIX) return greal(charpoly(H_tomatrix(x,abs),v));
    3101          70 :   res = mkpoln(3, gen_1, gneg(H_trace(x,0)), H_norm(x,0));
    3102          70 :   if (v) setvarn(res, v);
    3103          70 :   if (abs) res = gsqr(res);
    3104          70 :   return gerepilecopy(av, res);
    3105             : }
    3106             : 
    3107             : GEN
    3108       21385 : algcharpoly(GEN al, GEN x, long v, long abs)
    3109             : {
    3110             :   long ta;
    3111       21385 :   if (v<0) v=0;
    3112       21385 :   checkalg(al);
    3113       21385 :   ta = alg_type(al);
    3114       21385 :   if (ta == al_REAL) switch (alg_get_absdim(al)) {
    3115          35 :     case 1: return R_charpoly(x, v, abs);
    3116          35 :     case 2: return C_charpoly(x, v, abs);
    3117          98 :     case 4: return H_charpoly(x, v, abs);
    3118           7 :     default: pari_err_TYPE("algcharpoly [apply alginit]", al);
    3119             :   }
    3120             : 
    3121             :   /* gneg(x[1]) left on stack */
    3122       21210 :   if (alg_model(al,x) == al_TRIVIAL) {
    3123          56 :     GEN p = alg_get_char(al);
    3124          56 :     if (signe(p)) return deg1pol(gen_1,Fp_neg(gel(x,1),p),v);
    3125          42 :     return deg1pol(gen_1,gneg(gel(x,1)),v);
    3126             :   }
    3127             : 
    3128       21147 :   switch(ta) {
    3129         490 :     case al_CYCLIC: case al_CSA:
    3130         490 :       if (abs)
    3131             :       {
    3132         105 :         if (alg_model(al,x)==al_ALGEBRAIC) x = algalgtobasis(al,x);
    3133             :       }
    3134         385 :       else return algredcharpoly(al,x,v);
    3135       20762 :     case al_TABLE: return algbasischarpoly(al,x,v);
    3136             :     default : return NULL; /* LCOV_EXCL_LINE */
    3137             :   }
    3138             : }
    3139             : 
    3140             : /* assumes x in basis form */
    3141             : static GEN
    3142      246581 : algabstrace(GEN al, GEN x)
    3143             : {
    3144      246581 :   pari_sp av = avma;
    3145      246581 :   GEN res = NULL, p = alg_get_char(al);
    3146      246581 :   if (signe(p)) return FpV_dotproduct(x, alg_get_tracebasis(al), p);
    3147       36645 :   switch(alg_model(al,x)) {
    3148         154 :     case al_TRIVIAL: return gcopy(gel(x,1)); break;
    3149       36491 :     case al_BASIS: res = RgV_dotproduct(x, alg_get_tracebasis(al)); break;
    3150             :   }
    3151       36491 :   return gerepileupto(av,res);
    3152             : }
    3153             : 
    3154             : static GEN
    3155        1372 : algredtrace(GEN al, GEN x)
    3156             : {
    3157        1372 :   pari_sp av = avma;
    3158        1372 :   GEN res = NULL;
    3159        1372 :   switch(alg_model(al,x)) {
    3160          35 :     case al_TRIVIAL: return gcopy(gel(x,1)); break;
    3161         490 :     case al_BASIS: return algredtrace(al, algbasistoalg(al,x));
    3162             :                    /* TODO precompute too? */
    3163         847 :     case al_ALGEBRAIC:
    3164         847 :       switch(alg_type(al))
    3165             :       {
    3166         553 :         case al_CYCLIC:
    3167         553 :           res = rnfelttrace(alg_get_splittingfield(al),gel(x,1));
    3168         553 :           break;
    3169         294 :         case al_CSA:
    3170         294 :           res = gtrace(algalgmultable_csa(al,x));
    3171         294 :           res = gdiv(res, stoi(alg_get_degree(al)));
    3172         294 :           break;
    3173             :         default: return NULL; /* LCOV_EXCL_LINE */
    3174             :       }
    3175         847 :   }
    3176         847 :   return gerepileupto(av,res);
    3177             : }
    3178             : 
    3179             : static GEN
    3180         469 : algtrace_mat(GEN al, GEN M, long abs) {
    3181         469 :   pari_sp av = avma;
    3182         469 :   long N = lg(M)-1, i;
    3183         469 :   GEN res, p = alg_get_char(al);
    3184         469 :   if (N == 0) return gen_0;
    3185         448 :   if (N != nbrows(M)) pari_err_DIM("algtrace_mat (nonsquare)");
    3186             : 
    3187         434 :   if (!signe(p)) p = NULL;
    3188         434 :   if (alg_type(al) == al_TABLE) abs = 1;
    3189         434 :   res = algtrace(al, gcoeff(M,1,1), abs);
    3190         896 :   for (i=2; i<=N; i++) {
    3191         462 :     if (p)  res = Fp_add(res, algtrace(al,gcoeff(M,i,i),abs), p);
    3192         455 :     else    res = gadd(res, algtrace(al,gcoeff(M,i,i),abs));
    3193             :   }
    3194         434 :   if (abs) res = gmulgu(res, N); /* absolute trace */
    3195         434 :   return gerepileupto(av, res);
    3196             : }
    3197             : 
    3198             : static GEN
    3199          35 : R_trace(GEN x, long abs)
    3200             : {
    3201          35 :   pari_sp av = avma;
    3202          35 :   GEN res = NULL;
    3203          35 :   switch (H_model(x))
    3204             :   {
    3205          14 :     case H_SCALAR: res = gcopy(x); break;
    3206          14 :     case H_MATRIX: res = abs? mulrs(gtrace(x),nbrows(x)) : gtrace(x); break;
    3207           7 :     default: pari_err_TYPE("R_trace", x);
    3208             :   }
    3209          28 :   return gerepilecopy(av, res);
    3210             : }
    3211             : static GEN
    3212          35 : C_trace(GEN x, long abs)
    3213             : {
    3214          35 :   pari_sp av = avma;
    3215          35 :   GEN res = NULL;
    3216          35 :   switch (H_model(x))
    3217             :   {
    3218          14 :     case H_SCALAR: res = abs ? gshift(real_i(x),1) : x; break;
    3219          14 :     case H_MATRIX:
    3220          14 :       res = abs ? mulrs(real_i(gtrace(x)),2*nbrows(x)) : gtrace(x); break;
    3221           7 :     default: pari_err_TYPE("C_trace", x);
    3222             :   }
    3223          28 :   return gerepilecopy(av, res);
    3224             : }
    3225             : static GEN
    3226         567 : H_trace(GEN x, long abs)
    3227             : {
    3228         567 :   long s = abs? 2 : 1;
    3229         567 :   switch (H_model(x))
    3230             :   {
    3231         154 :     case H_SCALAR: return gshift(real_i(x),s);
    3232         329 :     case H_QUATERNION: return gshift(gel(x,1),s);
    3233          77 :     case H_MATRIX:
    3234          77 :       return algtrace_mat(NULL, x, abs);
    3235             :   }
    3236             :   return NULL; /*LCOV_EXCL_LINE*/
    3237             : }
    3238             : 
    3239             : GEN
    3240        2632 : algtrace(GEN al, GEN x, long abs)
    3241             : {
    3242             :   long ta;
    3243        2632 :   checkalg(al);
    3244        2632 :   ta = alg_type(al);
    3245        2632 :   if (ta==al_REAL) switch (alg_get_absdim(al)) {
    3246          35 :     case 1: return R_trace(x,abs);
    3247          35 :     case 2: return C_trace(x,abs);
    3248         497 :     case 4: return H_trace(x,abs);
    3249           7 :     default: pari_err_TYPE("algtrace [apply alginit]", al);
    3250             :   }
    3251        2058 :   if (alg_model(al,x) == al_MATRIX) return algtrace_mat(al,x,abs);
    3252        1666 :   switch(ta) {
    3253        1526 :     case al_CYCLIC: case al_CSA:
    3254        1526 :       if (!abs) return algredtrace(al,x);
    3255         644 :       if (alg_model(al,x)==al_ALGEBRAIC) x = algalgtobasis(al,x);
    3256         784 :     case al_TABLE: return algabstrace(al,x);
    3257             :     default : return NULL; /* LCOV_EXCL_LINE */
    3258             :   }
    3259             : }
    3260             : 
    3261             : static GEN
    3262       42764 : ZM_trace(GEN x)
    3263             : {
    3264       42764 :   long i, lx = lg(x);
    3265             :   GEN t;
    3266       42764 :   if (lx < 3) return lx == 1? gen_0: gcopy(gcoeff(x,1,1));
    3267       41959 :   t = gcoeff(x,1,1);
    3268      700515 :   for (i = 2; i < lx; i++) t = addii(t, gcoeff(x,i,i));
    3269       41959 :   return t;
    3270             : }
    3271             : static GEN
    3272      131162 : FpM_trace(GEN x, GEN p)
    3273             : {
    3274      131162 :   long i, lx = lg(x);
    3275             :   GEN t;
    3276      131162 :   if (lx < 3) return lx == 1? gen_0: gcopy(gcoeff(x,1,1));
    3277      123218 :   t = gcoeff(x,1,1);
    3278      895546 :   for (i = 2; i < lx; i++) t = Fp_add(t, gcoeff(x,i,i), p);
    3279      123218 :   return t;
    3280             : }
    3281             : 
    3282             : static GEN
    3283       40770 : algtracebasis(GEN al)
    3284             : {
    3285       40770 :   pari_sp av = avma;
    3286       40770 :   GEN mt = alg_get_multable(al), p = alg_get_char(al);
    3287       40770 :   long i, l = lg(mt);
    3288       40770 :   GEN v = cgetg(l, t_VEC);
    3289      171932 :   if (signe(p)) for (i=1; i < l; i++) gel(v,i) = FpM_trace(gel(mt,i), p);
    3290       48436 :   else          for (i=1; i < l; i++) gel(v,i) = ZM_trace(gel(mt,i));
    3291       40770 :   return gerepileupto(av,v);
    3292             : }
    3293             : 
    3294             : /* Assume: i > 0, expo := p^i <= absdim, x contained in I_{i-1} given by mult
    3295             :  * table modulo modu=p^(i+1). Return Tr(x^(p^i)) mod modu */
    3296             : static ulong
    3297       24902 : algtracei(GEN mt, ulong p, ulong expo, ulong modu)
    3298             : {
    3299       24902 :   pari_sp av = avma;
    3300       24902 :   long j, l = lg(mt);
    3301       24902 :   ulong tr = 0;
    3302       24902 :   mt = Flm_powu(mt,expo,modu);
    3303      270185 :   for (j=1; j<l; j++) tr += ucoeff(mt,j,j);
    3304       24902 :   return gc_ulong(av, (tr/expo) % p);
    3305             : }
    3306             : 
    3307             : static GEN
    3308          42 : R_norm(GEN x, long abs)
    3309             : {
    3310          42 :   pari_sp av = avma;
    3311          42 :   GEN res = NULL;
    3312          42 :   switch (H_model(x))
    3313             :   {
    3314          14 :     case H_SCALAR: res = gcopy(x); break;
    3315          21 :     case H_MATRIX: res = abs ? powrs(det(x),nbrows(x)) : det(x); break;
    3316           7 :     default: pari_err_TYPE("R_norm", x);
    3317             :   }
    3318          35 :   return gerepilecopy(av,res);
    3319             : }
    3320             : static GEN
    3321          42 : C_norm(GEN x, long abs)
    3322             : {
    3323          42 :   pari_sp av = avma;
    3324          42 :   GEN res = NULL;
    3325          42 :   switch (H_model(x))
    3326             :   {
    3327          14 :     case H_SCALAR: res = abs ? cxnorm(x) : x; break;
    3328          21 :     case H_MATRIX: res = abs ? powrs(cxnorm(det(x)),nbrows(x)) : det(x); break;
    3329           7 :     default: pari_err_TYPE("C_norm", x);
    3330             :   }
    3331          35 :   return gerepilecopy(av,res);
    3332             : }
    3333             : static GEN
    3334         434 : H_norm(GEN x, long abs)
    3335             : {
    3336         434 :   pari_sp av = avma;
    3337         434 :   switch (H_model(x))
    3338             :   {
    3339          42 :     case H_SCALAR:
    3340          42 :       if (abs)  return gerepilecopy(av,gsqr(gnorm(x)));
    3341          35 :       else      return gnorm(x);
    3342         322 :     case H_QUATERNION:
    3343         322 :       if (abs)  return gerepilecopy(av,gsqr(gnorml2(x)));
    3344         294 :       else      return gnorml2(x);
    3345          63 :     case H_MATRIX:
    3346          63 :       return gerepilecopy(av,real_i(det(H_tomatrix(x,abs))));
    3347             :   }
    3348             :   return NULL; /*LCOV_EXCL_LINE*/
    3349             : }
    3350             : 
    3351             : GEN
    3352        1253 : algnorm(GEN al, GEN x, long abs)
    3353             : {
    3354        1253 :   pari_sp av = avma;
    3355             :   long tx, ta;
    3356             :   GEN p, rnf, res, mx;
    3357        1253 :   checkalg(al);
    3358        1253 :   ta = alg_type(al);
    3359        1253 :   if (ta==al_REAL) switch (alg_get_absdim(al)) {
    3360          42 :     case 1: return R_norm(x,abs);
    3361          42 :     case 2: return C_norm(x,abs);
    3362         210 :     case 4: return H_norm(x,abs);
    3363           7 :     default: pari_err_TYPE("algnorm [apply alginit]", al);
    3364             :   }
    3365         952 :   p = alg_get_char(al);
    3366         952 :   tx = alg_model(al,x);
    3367         952 :   if (signe(p)) {
    3368          21 :     if (tx == al_MATRIX)    mx = algleftmultable_mat(al,x);
    3369          14 :     else                    mx = algbasismultable(al,x);
    3370          21 :     return gerepileupto(av, FpM_det(mx,p));
    3371             :   }
    3372         931 :   if (tx == al_TRIVIAL) return gcopy(gel(x,1));
    3373             : 
    3374         889 :   switch(ta) {
    3375         819 :     case al_CYCLIC: case al_CSA:
    3376         819 :       if (abs)
    3377             :       {
    3378         196 :         if (alg_model(al,x)==al_ALGEBRAIC) x = algalgtobasis(al,x);
    3379             :       }
    3380             :       else
    3381             :       {
    3382         623 :         rnf = alg_get_splittingfield(al);
    3383         623 :         res = rnfeltdown(rnf, det(algtomatrix(al,x,0)));
    3384         616 :         break;
    3385             :       }
    3386             :     case al_TABLE:
    3387         266 :       if (tx == al_MATRIX)  mx = algleftmultable_mat(al,x);
    3388         105 :       else                  mx = algbasismultable(al,x);
    3389         259 :       res = det(mx);
    3390         259 :       break;
    3391             :     default: return NULL; /* LCOV_EXCL_LINE */
    3392             :   }
    3393         875 :   return gerepileupto(av, res);
    3394             : }
    3395             : 
    3396             : static GEN
    3397       50443 : algalgtonat_cyc(GEN al, GEN x)
    3398             : {
    3399       50443 :   pari_sp av = avma;
    3400       50443 :   GEN nf = alg_get_abssplitting(al), rnf = alg_get_splittingfield(al), res, c;
    3401       50443 :   long n = alg_get_degree(al), N = nf_get_degree(nf), i, i1;
    3402       50443 :   res = zerocol(N*n);
    3403      154570 :   for (i=0; i<n; i++) {
    3404      104127 :     c = gel(x,i+1);
    3405      104127 :     c = rnfeltreltoabs(rnf,c);
    3406      104127 :     if (!gequal0(c)) {
    3407       78156 :       c = algtobasis(nf,c);
    3408      412954 :       for (i1=1; i1<=N; i1++) gel(res,i*N+i1) = gel(c,i1);
    3409             :     }
    3410             :   }
    3411       50443 :   return gerepilecopy(av, res);
    3412             : }
    3413             : 
    3414             : static GEN
    3415       11375 : algalgtonat_csa(GEN al, GEN x)
    3416             : {
    3417       11375 :   pari_sp av = avma;
    3418       11375 :   GEN nf = alg_get_center(al), res, c;
    3419       11375 :   long d2 = alg_get_dim(al), n = nf_get_degree(nf), i, i1;
    3420       11375 :   res = zerocol(d2*n);
    3421       56644 :   for (i=0; i<d2; i++) {
    3422       45269 :     c = gel(x,i+1);
    3423       45269 :     if (!gequal0(c)) {
    3424       31563 :       c = algtobasis(nf,c);
    3425       95095 :       for (i1=1; i1<=n; i1++) gel(res,i*n+i1) = gel(c,i1);
    3426             :     }
    3427             :   }
    3428       11375 :   return gerepilecopy(av, res);
    3429             : }
    3430             : 
    3431             : /* assumes al CSA or CYCLIC */
    3432             : static GEN
    3433       61818 : algalgtonat(GEN al, GEN x)
    3434             : {
    3435       61818 :   switch(alg_type(al))
    3436             :   {
    3437       50443 :     case al_CYCLIC: return algalgtonat_cyc(al, x);
    3438       11375 :     case al_CSA: return algalgtonat_csa(al, x);
    3439             :   }
    3440             :   return NULL; /*LCOV_EXCL_LINE*/
    3441             : }
    3442             : 
    3443             : static GEN
    3444       11669 : algnattoalg_cyc(GEN al, GEN x)
    3445             : {
    3446       11669 :   pari_sp av = avma;
    3447       11669 :   GEN nf = alg_get_abssplitting(al), rnf = alg_get_splittingfield(al), res, c;
    3448       11669 :   long n = alg_get_degree(al), N = nf_get_degree(nf), i, i1;
    3449       11669 :   res = zerocol(n);
    3450       11669 :   c = zerocol(N);
    3451       49154 :   for (i=0; i<n; i++) {
    3452      324527 :     for (i1=1; i1<=N; i1++) gel(c,i1) = gel(x,i*N+i1);
    3453       37485 :     gel(res,i+1) = rnfeltabstorel(rnf,basistoalg(nf,c));
    3454             :   }
    3455       11669 :   return gerepilecopy(av, res);
    3456             : }
    3457             : 
    3458             : static GEN
    3459        1309 : algnattoalg_csa(GEN al, GEN x)
    3460             : {
    3461        1309 :   pari_sp av = avma;
    3462        1309 :   GEN nf = alg_get_center(al), res, c;
    3463        1309 :   long d2 = alg_get_dim(al), n = nf_get_degree(nf), i, i1;
    3464        1309 :   res = zerocol(d2);
    3465        1309 :   c = zerocol(n);
    3466        7028 :   for (i=0; i<d2; i++) {
    3467       19390 :     for (i1=1; i1<=n; i1++) gel(c,i1) = gel(x,i*n+i1);
    3468        5719 :     gel(res,i+1) = basistoalg(nf,c);
    3469             :   }
    3470        1309 :   return gerepilecopy(av, res);
    3471             : }
    3472             : 
    3473             : /* assumes al CSA or CYCLIC */
    3474             : static GEN
    3475       12978 : algnattoalg(GEN al, GEN x)
    3476             : {
    3477       12978 :   switch(alg_type(al))
    3478             :   {
    3479       11669 :     case al_CYCLIC: return algnattoalg_cyc(al, x);
    3480        1309 :     case al_CSA: return algnattoalg_csa(al, x);
    3481             :   }
    3482             :   return NULL; /*LCOV_EXCL_LINE*/
    3483             : }
    3484             : 
    3485             : static GEN
    3486         182 : algalgtobasis_mat(GEN al, GEN x) /* componentwise */
    3487             : {
    3488         182 :   pari_sp av = avma;
    3489             :   long lx, lxj, i, j;
    3490             :   GEN res;
    3491         182 :   lx = lg(x);
    3492         182 :   res = cgetg(lx, t_MAT);
    3493         546 :   for (j=1; j<lx; j++) {
    3494         364 :     lxj = lg(gel(x,j));
    3495         364 :     gel(res,j) = cgetg(lxj, t_COL);
    3496        1092 :     for (i=1; i<lxj; i++)
    3497         728 :       gcoeff(res,i,j) = algalgtobasis(al,gcoeff(x,i,j));
    3498             :   }
    3499         182 :   return gerepilecopy(av,res);
    3500             : }
    3501             : GEN
    3502       62280 : algalgtobasis(GEN al, GEN x)
    3503             : {
    3504             :   pari_sp av;
    3505             :   long tx, ta;
    3506       62280 :   checkalg(al);
    3507       62280 :   ta = alg_type(al);
    3508       62280 :   if (ta != al_CYCLIC && ta != al_CSA) pari_err_TYPE("algalgtobasis [use alginit]", al);
    3509       62259 :   tx = alg_model(al,x);
    3510       62259 :   if (tx==al_BASIS) return gcopy(x);
    3511       62000 :   if (tx==al_MATRIX) return algalgtobasis_mat(al,x);
    3512       61818 :   av = avma;
    3513       61818 :   x = algalgtonat(al,x);
    3514       61818 :   x = RgM_RgC_mul(alg_get_invbasis(al),x);
    3515       61818 :   return gerepileupto(av, x);
    3516             : }
    3517             : 
    3518             : static GEN
    3519         119 : algbasistoalg_mat(GEN al, GEN x) /* componentwise */
    3520             : {
    3521         119 :   long j, lx = lg(x);
    3522         119 :   GEN res = cgetg(lx, t_MAT);
    3523         357 :   for (j=1; j<lx; j++) {
    3524         238 :     long i, lxj = lg(gel(x,j));
    3525         238 :     gel(res,j) = cgetg(lxj, t_COL);
    3526         714 :     for (i=1; i<lxj; i++) gcoeff(res,i,j) = algbasistoalg(al,gcoeff(x,i,j));
    3527             :   }
    3528         119 :   return res;
    3529             : }
    3530             : GEN
    3531        2926 : algbasistoalg(GEN al, GEN x)
    3532             : {
    3533             :   pari_sp av;
    3534             :   long tx, ta;
    3535        2926 :   checkalg(al);
    3536        2926 :   ta = alg_type(al);
    3537        2926 :   if (ta != al_CYCLIC && ta != al_CSA) pari_err_TYPE("algbasistoalg [use alginit]", al);
    3538        2905 :   tx = alg_model(al,x);
    3539        2905 :   if (tx==al_ALGEBRAIC) return gcopy(x);
    3540        2772 :   if (tx==al_MATRIX) return algbasistoalg_mat(al,x);
    3541        2653 :   av = avma;
    3542        2653 :   x = RgM_RgC_mul(alg_get_basis(al),x);
    3543        2653 :   x = algnattoalg(al,x);
    3544        2653 :   return gerepileupto(av, x);
    3545             : }
    3546             : 
    3547             : static GEN
    3548        4466 : R_random(GEN b)
    3549             : {
    3550        4466 :   pari_sp av = avma;
    3551        4466 :   long prec = realprec(b);
    3552        4466 :   GEN z = randomr(prec); shiftr_inplace(z, 1);
    3553        4466 :   return gerepileuptoleaf(av, mulrr(b,addsr(-1, z)));
    3554             : }
    3555             : static GEN
    3556         182 : C_random(GEN b)
    3557             : {
    3558         182 :   retmkcomplex(R_random(b), R_random(b));
    3559             : }
    3560             : static GEN
    3561         980 : H_random(GEN b)
    3562             : {
    3563         980 :   GEN res = cgetg(5, t_COL);
    3564             :   long i;
    3565        4900 :   for (i=1; i<=4; i++) gel(res,i) = R_random(b);
    3566         980 :   return res;
    3567             : }
    3568             : GEN
    3569       19677 : algrandom(GEN al, GEN b)
    3570             : {
    3571       19677 :   GEN res = NULL, p, N;
    3572             :   long i, n;
    3573       19677 :   checkalg(al);
    3574       19663 :   if (alg_type(al)==al_REAL)
    3575             :   {
    3576        1365 :     if (typ(b) != t_REAL) pari_err_TYPE("algrandom",b);
    3577        1358 :     if (signe(b) < 0) pari_err_DOMAIN("algrandom", "b", "<", gen_0, b);
    3578        1351 :     switch(alg_get_absdim(al))
    3579             :     {
    3580         182 :       case 1: res = R_random(b); break;
    3581         182 :       case 2: res = C_random(b); break;
    3582         980 :       case 4: res = H_random(b); break;
    3583           7 :       default: pari_err_TYPE("algrandom [apply alginit]", al);
    3584             :     }
    3585        1344 :     return res;
    3586             :   }
    3587       18298 :   if (typ(b) != t_INT) pari_err_TYPE("algrandom",b);
    3588       18291 :   if (signe(b) < 0) pari_err_DOMAIN("algrandom", "b", "<", gen_0, b);
    3589       18284 :   n = alg_get_absdim(al);
    3590       18284 :   N = addiu(shifti(b,1), 1); /* left on stack */
    3591       18284 :   p = alg_get_char(al); if (!signe(p)) p = NULL;
    3592       18284 :   res = cgetg(n+1,t_COL);
    3593      163828 :   for (i = 1; i <= n; i++)
    3594             :   {
    3595      145544 :     pari_sp av = avma;
    3596      145544 :     GEN t = subii(randomi(N),b);
    3597      145544 :     if (p) t = modii(t, p);
    3598      145544 :     gel(res,i) = gerepileuptoint(av, t);
    3599             :   }
    3600       18284 :   return res;
    3601             : }
    3602             : 
    3603             : static GEN
    3604          77 : H_poleval(GEN pol, GEN x)
    3605             : {
    3606          77 :   pari_sp av = avma;
    3607             :   GEN res;
    3608             :   long i;
    3609          77 :   switch (H_model(x))
    3610             :   {
    3611          21 :     case H_SCALAR: return RgX_cxeval(pol, x, NULL);
    3612          42 :     case H_QUATERNION: break;
    3613           7 :     default: pari_err_TYPE("H_poleval", x);
    3614             :   }
    3615             : 
    3616          42 :   res = zerocol(4);
    3617         189 :   for (i=lg(pol)-1; i>1; i--)
    3618             :   {
    3619         147 :     gel(res,1) = gadd(gel(res,1), gel(pol,i));
    3620         147 :     if (i>2) res = H_mul(x, res);
    3621             :   }
    3622             : 
    3623          42 :   return gerepilecopy(av,res);
    3624             : }
    3625             : 
    3626             : /* Assumes pol has coefficients in the same ring as the COL x; x either
    3627             :  * in basis or algebraic form or [x,mx] where mx is the mult. table of x.
    3628             :  TODO more general version: pol with coeffs in center and x in basis form */
    3629             : GEN
    3630       17324 : algpoleval(GEN al, GEN pol, GEN x)
    3631             : {
    3632       17324 :   pari_sp av = avma;
    3633       17324 :   GEN p, mx = NULL, res;
    3634             :   long i;
    3635       17324 :   if (typ(pol) != t_POL) pari_err_TYPE("algpoleval", pol);
    3636       17310 :   checkalg(al);
    3637       17310 :   if (alg_type(al)==al_REAL) return H_poleval(pol,x);
    3638       17233 :   p = alg_get_char(al);
    3639       17233 :   if (typ(x) == t_VEC)
    3640             :   {
    3641        6097 :     if (lg(x) != 3) pari_err_TYPE("algpoleval [vector must be of length 2]", x);
    3642        6090 :     mx = gel(x,2);
    3643        6090 :     x = gel(x,1);
    3644        6090 :     if (typ(mx)!=t_MAT || !gequal(x,gel(mx,1)))
    3645          21 :       pari_err_TYPE("algpoleval [mx must be the multiplication table of x]", mx);
    3646             :   }
    3647             :   else
    3648             :   {
    3649       11136 :     switch(alg_model(al,x))
    3650             :     {
    3651          14 :       case al_ALGEBRAIC: mx = algalgmultable(al,x); break;
    3652       11094 :       case al_BASIS: if (!RgX_is_QX(pol))
    3653           7 :         pari_err_IMPL("algpoleval with x in basis form and pol not in Q[x]");
    3654       11101 :       case al_TRIVIAL: mx = algbasismultable(al,x); break;
    3655           7 :       default: pari_err_TYPE("algpoleval", x);
    3656             :     }
    3657             :   }
    3658       17184 :   res = zerocol(lg(mx)-1);
    3659       17184 :   if (signe(p)) {
    3660       64486 :     for (i=lg(pol)-1; i>1; i--)
    3661             :     {
    3662       48100 :       gel(res,1) = Fp_add(gel(res,1), gel(pol,i), p);
    3663       48100 :       if (i>2) res = FpM_FpC_mul(mx, res, p);
    3664             :     }
    3665             :   }
    3666             :   else {
    3667        4879 :     for (i=lg(pol)-1; i>1; i--)
    3668             :     {
    3669        4081 :       gel(res,1) = gadd(gel(res,1), gel(pol,i));
    3670        4081 :       if (i>2) res = RgM_RgC_mul(mx, res);
    3671             :     }
    3672             :   }
    3673       17184 :   return gerepileupto(av, res);
    3674             : }
    3675             : 
    3676             : /** GRUNWALD-WANG **/
    3677             : /*
    3678             : Song Wang's PhD thesis (pdf pages)
    3679             : p.25 definition of chi_b. K^Ker(chi_b) = K(b^(1/m))
    3680             : p.26 bound on the conductor (also Cohen adv. GTM 193 p.166)
    3681             : p.21 & p.34 description special case, also on wikipedia:
    3682             : http://en.wikipedia.org/wiki/Grunwald%E2%80%93Wang_theorem#Special_fields
    3683             : p.77 Kummer case
    3684             : */
    3685             : 
    3686             : /* n > 0. Is n = 2^k ? */
    3687             : static int
    3688         329 : uispow2(ulong n) { return !(n &(n-1)); }
    3689             : 
    3690             : static GEN
    3691         378 : get_phi0(GEN bnr, GEN Lpr, GEN Ld, GEN pl, long *pr, long *pn)
    3692             : {
    3693         378 :   const long NTRY = 10; /* FIXME: magic constant */
    3694         378 :   const long n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
    3695         378 :   GEN S = bnr_get_cyc(bnr);
    3696             :   GEN Sst, G, globGmod, loc, X, Rglob, Rloc, H, U, Lconj;
    3697             :   long i, j, r, nbfrob, nbloc, nz, t;
    3698             : 
    3699         378 :   *pn = n;
    3700         378 :   *pr = r = lg(S)-1;
    3701         378 :   if (!r) return NULL;
    3702         329 :   Sst = cgetg(r+1, t_VECSMALL); /* Z/n-dual */
    3703        1589 :   for (i=1; i<=r; i++) Sst[i] = ugcdiu(gel(S,i), n);
    3704         329 :   if (Sst[1] != n) return NULL;
    3705         329 :   Lconj = NULL;
    3706         329 :   nbloc = nbfrob = lg(Lpr)-1;
    3707         329 :   if (uispow2(n))
    3708             :   {
    3709         259 :     long l = lg(pl), k = 0;
    3710         259 :     GEN real = cgetg(l, t_VECSMALL);
    3711         973 :     for (i = 1; i < l; i++)
    3712         714 :       if (pl[i] == -1) real[++k] = i;
    3713         259 :     if (k)
    3714             :     {
    3715         259 :       GEN nf = bnr_get_nf(bnr), I = bid_get_fact(bnr_get_bid(bnr));
    3716         259 :       GEN v, y, C = idealchineseinit(bnr, I);
    3717         259 :       long r1 = nf_get_r1(nf), n = nbrows(I);
    3718         259 :       nbloc += k;
    3719         259 :       Lconj = cgetg(k+1, t_VEC);
    3720         259 :       v = const_vecsmall(r1, 1);
    3721         259 :       y = const_vec(n, gen_1);
    3722         707 :       for (i = 1; i <= k; i++)
    3723             :       {
    3724         448 :         v[real[i]] = -1; gel(Lconj,i) = idealchinese(nf, mkvec2(C,v), y);
    3725         448 :         v[real[i]] = 1;
    3726             :       }
    3727             :     }
    3728             :   }
    3729         329 :   globGmod = cgetg(r+1,t_MAT);
    3730         329 :   G = cgetg(r+1,t_VECSMALL);
    3731        1589 :   for (i = 1; i <= r; i++)
    3732             :   {
    3733        1260 :     G[i] = n / Sst[i]; /* pairing between S and Sst */
    3734        1260 :     gel(globGmod,i) = cgetg(nbloc+1,t_VECSMALL);
    3735             :   }
    3736             : 
    3737             :   /* compute images of Frobenius elements (and complex conjugation) */
    3738         329 :   loc = cgetg(nbloc+1,t_VECSMALL);
    3739         700 :   for (i = 1; i <= nbloc; i++)
    3740             :   {
    3741             :     long L;
    3742         539 :     if (i <= nbfrob)
    3743             :     {
    3744         224 :       X = gel(Lpr, i);
    3745         224 :       L = Ld[i];
    3746             :     }
    3747             :     else
    3748             :     { /* X = 1 (mod f), sigma_i(x) < 0, positive at all other real places */
    3749         315 :       X = gel(Lconj, i-nbfrob);
    3750         315 :       L = 2;
    3751             :     }
    3752         539 :     X = ZV_to_Flv(isprincipalray(bnr,X), n);
    3753        2275 :     for (nz=0,j=1; j<=r; j++)
    3754             :     {
    3755        1736 :       ulong c = (X[j] * G[j]) % L;
    3756        1736 :       ucoeff(globGmod,i,j) = c;
    3757        1736 :       if (c) nz = 1;
    3758             :     }
    3759         539 :     if (!nz) return NULL;
    3760         371 :     loc[i] = L;
    3761             :   }
    3762             : 
    3763             :   /* try some random elements in the dual */
    3764         161 :   Rglob = cgetg(r+1,t_VECSMALL);
    3765         443 :   for (t=0; t<NTRY; t++) {
    3766        1656 :     for (j = 1; j <= r; j++) Rglob[j] = random_Fl(Sst[j]);
    3767         436 :     Rloc = zm_zc_mul(globGmod,Rglob);
    3768         968 :     for (i = 1; i <= nbloc; i++)
    3769         814 :       if (Rloc[i] % loc[i] == 0) break;
    3770         436 :     if (i > nbloc) return zv_to_ZV(Rglob);
    3771             :   }
    3772             : 
    3773             :   /* try to realize some random elements of the product of the local duals */
    3774           7 :   H = ZM_hnfall_i(shallowconcat(zm_to_ZM(globGmod),
    3775             :                                 diagonal_shallow(zv_to_ZV(loc))), &U, 2);
    3776             :   /* H,U nbloc x nbloc */
    3777           7 :   Rloc = cgetg(nbloc+1,t_COL);
    3778          77 :   for (t = 0; t < NTRY; t++)
    3779             :   { /* nonzero random coordinate */ /* TODO add special case ? */
    3780         560 :     for (i = 1; i <= nbloc; i++) gel(Rloc,i) = stoi(1 + random_Fl(loc[i]-1));
    3781          70 :     Rglob = hnf_invimage(H, Rloc);
    3782          70 :     if (Rglob)
    3783             :     {
    3784           0 :       Rglob = ZM_ZC_mul(U,Rglob);
    3785           0 :       return vecslice(Rglob,1,r);
    3786             :     }
    3787             :   }
    3788           7 :   return NULL;
    3789             : }
    3790             : 
    3791             : static GEN
    3792         378 : bnrgwsearch(GEN bnr, GEN Lpr, GEN Ld, GEN pl)
    3793             : {
    3794         378 :   pari_sp av = avma;
    3795             :   long n, r;
    3796         378 :   GEN phi0 = get_phi0(bnr,Lpr,Ld,pl, &r,&n), gn, v, H,U;
    3797         378 :   if (!phi0) return gc_const(av, gen_0);
    3798         154 :   gn = stoi(n);
    3799             :   /* compute kernel of phi0 */
    3800         154 :   v = ZV_extgcd(vec_append(phi0, gn));
    3801         154 :   U = vecslice(gel(v,2), 1,r);
    3802         154 :   H = ZM_hnfmodid(rowslice(U, 1,r), gn);
    3803         154 :   return gerepileupto(av, H);
    3804             : }
    3805             : 
    3806             : GEN
    3807         154 : bnfgwgeneric(GEN bnf, GEN Lpr, GEN Ld, GEN pl, long var)
    3808             : {
    3809         154 :   pari_sp av = avma;
    3810         154 :   const long n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
    3811             :   forprime_t S;
    3812         154 :   GEN bnr = NULL, ideal = gen_1, nf, dec, H = gen_0, finf, pol;
    3813             :   ulong ell, p;
    3814             :   long deg, i, degell;
    3815         154 :   (void)uisprimepower(n, &ell);
    3816         154 :   nf = bnf_get_nf(bnf);
    3817         154 :   deg = nf_get_degree(nf);
    3818         154 :   degell = ugcd(deg,ell-1);
    3819         154 :   finf = cgetg(lg(pl),t_VEC);
    3820         427 :   for (i=1; i<lg(pl); i++) gel(finf,i) = pl[i]==-1 ? gen_1 : gen_0;
    3821             : 
    3822         154 :   u_forprime_init(&S, 2, ULONG_MAX);
    3823         679 :   while ((p = u_forprime_next(&S))) {
    3824         679 :     if (Fl_powu(p % ell, degell, ell) != 1) continue; /* ell | p^deg-1 ? */
    3825         364 :     dec = idealprimedec(nf, utoipos(p));
    3826         700 :     for (i=1; i<lg(dec); i++) {
    3827         490 :       GEN pp = gel(dec,i);
    3828         490 :       if (RgV_isin(Lpr,pp)) continue;
    3829             :         /* TODO also accept the prime ideals at which there is a condition
    3830             :          * (use local Artin)? */
    3831         434 :       if (smodis(idealnorm(nf,pp),ell) != 1) continue; /* ell | N(pp)-1 ? */
    3832         378 :       ideal = idealmul(bnf,ideal,pp);
    3833             :       /* TODO: give factorization ? */
    3834         378 :       bnr = Buchray(bnf, mkvec2(ideal,finf), nf_INIT);
    3835         378 :       H = bnrgwsearch(bnr,Lpr,Ld,pl);
    3836         378 :       if (H != gen_0)
    3837             :       {
    3838         154 :         pol = rnfkummer(bnr,H,nf_get_prec(nf));
    3839         154 :         setvarn(pol, var);
    3840         154 :         return gerepileupto(av,pol);
    3841             :       }
    3842             :     }
    3843             :   }
    3844             :   pari_err_BUG("bnfgwgeneric (no suitable p)"); /*LCOV_EXCL_LINE*/
    3845             :   return NULL;/*LCOV_EXCL_LINE*/
    3846             : }
    3847             : 
    3848             : /* pr.p != ell */
    3849             : static GEN
    3850        1554 : localextdeg(GEN nf, GEN pr, long d, ulong ell, long n)
    3851             : {
    3852             :   GEN modpr, T, p, gen, k;
    3853        1554 :   if (d == 1) return gen_1;
    3854        1540 :   k = powuu(ell, Z_lval(subiu(pr_norm(pr),1), ell));
    3855        1540 :   k = divis(k, n / d);
    3856        1540 :   modpr = nf_to_Fq_init(nf, &pr, &T, &p);
    3857        1540 :   (void)Fq_sqrtn(gen_1, k, T, p, &gen);
    3858        1540 :   return Fq_to_nf(gen, modpr);
    3859             : }
    3860             : /* pr.p = ell */
    3861             : static GEN
    3862         133 : localextdegell(GEN nf, GEN pr, GEN E, long d, long n)
    3863             : {
    3864             :   GEN x;
    3865         133 :   if (d == 1) return gen_1;
    3866         126 :   x = nfadd(nf, gen_1, pr_get_gen(pr));
    3867         126 :   return nfpowmodideal(nf, x, stoi(n / d), idealpow(nf, pr, E));
    3868             : }
    3869             : 
    3870             : /* Ld[i] must be nontrivial powers of the same prime ell */
    3871             : /* pl : -1 at real places at which the extension must ramify, 0 elsewhere */
    3872             : GEN
    3873         210 : nfgwkummer(GEN nf, GEN Lpr, GEN Ld, GEN pl, long var)
    3874             : {
    3875         210 :   const long n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
    3876             :   ulong ell;
    3877         210 :   long i, l = lg(Lpr), v = uisprimepower(n, &ell);
    3878         210 :   GEN E = cgetg(l, t_COL), y = cgetg(l, t_VEC), fa;
    3879             : 
    3880        1897 :   for (i = 1; i < l; i++)
    3881             :   {
    3882        1687 :     GEN pr = gel(Lpr,i), p = pr_get_p(pr);
    3883        1687 :     if (!absequalui(ell, p))
    3884             :     {
    3885        1554 :       gel(E, i) = gen_1;
    3886        1554 :       gel(y, i) = localextdeg(nf, pr, Ld[i], ell, n);
    3887             :     }
    3888             :     else
    3889             :     {
    3890         133 :       long e = pr_get_e(pr);
    3891         133 :       gel(E, i) = addui(1 + v*e, divsi(e, subiu(p,1)));
    3892         133 :       gel(y, i) = localextdegell(nf, pr, gel(E,i), Ld[i], n);
    3893             :     }
    3894             :   }
    3895         210 :   y = factoredextchinese(nf, mkmat2(shallowtrans(Lpr),E), y, pl, &fa);
    3896         210 :   return gsub(gpowgs(pol_x(var),n), basistoalg(nf, y));
    3897             : }
    3898             : 
    3899             : static GEN
    3900         833 : get_vecsmall(GEN v)
    3901             : {
    3902         833 :   switch(typ(v))
    3903             :   {
    3904         707 :     case t_VECSMALL: return v;
    3905         119 :     case t_VEC: if (RgV_is_ZV(v)) return ZV_to_zv(v);
    3906             :   }
    3907           7 :   pari_err_TYPE("nfgrunwaldwang",v);
    3908             :   return NULL;/*LCOV_EXCL_LINE*/
    3909             : }
    3910             : GEN
    3911         462 : nfgrunwaldwang(GEN nf0, GEN Lpr, GEN Ld, GEN pl, long var)
    3912             : {
    3913             :   ulong n, ell, ell2;
    3914         462 :   pari_sp av = avma;
    3915             :   GEN nf, bnf;
    3916             :   long t, w, i, vnf;
    3917             : 
    3918         462 :   if (var < 0) var = 0;
    3919         462 :   nf = get_nf(nf0,&t);
    3920         462 :   if (!nf) pari_err_TYPE("nfgrunwaldwang",nf0);
    3921         462 :   vnf = nf_get_varn(nf);
    3922         462 :   if (varncmp(var, vnf) >= 0)
    3923           7 :     pari_err_PRIORITY("nfgrunwaldwang", pol_x(var), ">=", vnf);
    3924         455 :   if (typ(Lpr) != t_VEC) pari_err_TYPE("nfgrunwaldwang",Lpr);
    3925         441 :   if (lg(Lpr) != lg(Ld)) pari_err_DIM("nfgrunwaldwang [#Lpr != #Ld]");
    3926         434 :   if (nf_get_degree(nf)==1) Lpr = shallowcopy(Lpr);
    3927        2359 :   for (i=1; i<lg(Lpr); i++) {
    3928        1932 :     GEN pr = gel(Lpr,i);
    3929        1932 :     if (nf_get_degree(nf)==1 && typ(pr)==t_INT)
    3930          77 :       gel(Lpr,i) = gel(idealprimedec(nf,pr), 1);
    3931        1855 :     else checkprid(pr);
    3932             :   }
    3933         427 :   if (lg(pl)-1 != nf_get_r1(nf))
    3934           7 :     pari_err_DOMAIN("nfgrunwaldwang [pl should have r1 components]", "#pl",
    3935           7 :         "!=", stoi(nf_get_r1(nf)), stoi(lg(pl)-1));
    3936             : 
    3937         420 :   Ld = get_vecsmall(Ld);
    3938         413 :   pl = get_vecsmall(pl);
    3939         413 :   bnf = get_bnf(nf0,&t);
    3940         413 :   n = (lg(Ld)==1)? 2: vecsmall_max(Ld);
    3941             : 
    3942         413 :   if (!uisprimepower(n, &ell))
    3943           7 :     pari_err_IMPL("nfgrunwaldwang for non prime-power local degrees (a)");
    3944        2296 :   for (i=1; i<lg(Ld); i++)
    3945        1897 :     if (Ld[i]!=1 && (!uisprimepower(Ld[i],&ell2) || ell2!=ell))
    3946           7 :       pari_err_IMPL("nfgrunwaldwang for non prime-power local degrees (b)");
    3947        1043 :   for (i=1; i<lg(pl); i++)
    3948         651 :     if (pl[i]==-1 && ell%2)
    3949           7 :       pari_err_IMPL("nfgrunwaldwang for non prime-power local degrees (c)");
    3950             : 
    3951         392 :   w = bnf? bnf_get_tuN(bnf): itos(gel(nfrootsof1(nf),1));
    3952             : 
    3953             :   /* TODO choice between kummer and generic ? Let user choose between speed
    3954             :    * and size */
    3955         392 :   if (w%n==0 && lg(Ld)>1)
    3956         210 :     return gerepileupto(av, nfgwkummer(nf,Lpr,Ld,pl,var));
    3957         182 :   if (ell==n)
    3958             :   {
    3959         154 :     if (!bnf) bnf = Buchall(nf, nf_FORCE, 0);
    3960         154 :     return gerepileupto(av, bnfgwgeneric(bnf,Lpr,Ld,pl,var));
    3961             :   }
    3962          28 :   pari_err_IMPL("nfgrunwaldwang for nonprime degree");
    3963             :   return NULL; /*LCOV_EXCL_LINE*/
    3964             : }
    3965             : 
    3966             : /** HASSE INVARIANTS **/
    3967             : 
    3968             : /* TODO long -> ulong + uel */
    3969             : static GEN
    3970        1064 : hasseconvert(GEN H, long n)
    3971             : {
    3972             :   GEN h, c;
    3973             :   long i, l;
    3974        1064 :   switch(typ(H)) {
    3975         994 :     case t_VEC:
    3976         994 :       l = lg(H); h = cgetg(l,t_VECSMALL);
    3977         994 :       if (l == 1) return h;
    3978         882 :       c = gel(H,1);
    3979         882 :       if (typ(c) == t_VEC && l == 3)
    3980         336 :         return mkvec2(gel(H,1),hasseconvert(gel(H,2),n));
    3981        2891 :       for (i=1; i<l; i++)
    3982             :       {
    3983        2373 :         c = gel(H,i);
    3984        2373 :         switch(typ(c)) {
    3985         728 :           case t_INT:  break;
    3986           7 :           case t_INTMOD:
    3987           7 :             c = gel(c,2); break;
    3988        1617 :           case t_FRAC :
    3989        1617 :             c = gmulgs(c,n);
    3990        1617 :             if (typ(c) == t_INT) break;
    3991           7 :             pari_err_DOMAIN("hasseconvert [degree should be a denominator of the invariant]", "denom(h)", "ndiv", stoi(n), Q_denom(gel(H,i)));
    3992          21 :           default : pari_err_TYPE("Hasse invariant", c);
    3993             :         }
    3994        2345 :         h[i] = smodis(c,n);
    3995             :       }
    3996         518 :       return h;
    3997          63 :     case t_VECSMALL: return H;
    3998             :   }
    3999           7 :   pari_err_TYPE("Hasse invariant", H);
    4000             :   return NULL;/*LCOV_EXCL_LINE*/
    4001             : }
    4002             : 
    4003             : /* assume f >= 2 */
    4004             : static long
    4005         385 : cyclicrelfrob0(GEN nf, GEN aut, GEN pr, GEN q, long f, long g)
    4006             : {
    4007         385 :   GEN T, p, a, b, modpr = nf_to_Fq_init(nf,&pr,&T,&p);
    4008             :   long s;
    4009             : 
    4010         385 :   a = pol_x(nf_get_varn(nf));
    4011         385 :   b = galoisapply(nf, aut, modpr_genFq(modpr));
    4012         385 :   b = nf_to_Fq(nf, b, modpr);
    4013        1169 :   for (s = 0; !ZX_equal(a, b); s++) a = Fq_pow(a, q, T, p);
    4014         385 :   return g * Fl_inv(s, f); /* < n */
    4015             : }
    4016             : 
    4017             : static long
    4018        2471 : cyclicrelfrob(GEN rnf, GEN auts, GEN pr)
    4019             : {
    4020        2471 :   pari_sp av = avma;
    4021        2471 :   long f,g,frob, n = rnf_get_degree(rnf);
    4022        2471 :   GEN P = rnfidealprimedec(rnf, pr);
    4023             : 
    4024        2471 :   if (pr_get_e(gel(P,1)) > pr_get_e(pr))
    4025           0 :     pari_err_DOMAIN("cyclicrelfrob","e(PR/pr)",">",gen_1,pr);
    4026        2471 :   g = lg(P) - 1;
    4027        2471 :   f = n / g;
    4028             : 
    4029        2471 :   if (f <= 2) frob = g % n;
    4030             :   else {
    4031         385 :     GEN nf2, PR = gel(P,1);
    4032         385 :     GEN autabs = rnfeltreltoabs(rnf,gel(auts,g));
    4033         385 :     nf2 = obj_check(rnf,rnf_NFABS);
    4034         385 :     autabs = nfadd(nf2, autabs, gmul(rnf_get_k(rnf), rnf_get_alpha(rnf)));
    4035         385 :     frob = cyclicrelfrob0(nf2, autabs, PR, pr_norm(pr), f, g);
    4036             :   }
    4037        2471 :   return gc_long(av, frob);
    4038             : }
    4039             : 
    4040             : static long
    4041         623 : localhasse(GEN rnf, GEN cnd, GEN pl, GEN auts, GEN b, long k)
    4042             : {
    4043         623 :   pari_sp av = avma;
    4044             :   long v, m, h, lfa, frob, n, i;
    4045             :   GEN previous, y, pr, nf, q, fa;
    4046         623 :   nf = rnf_get_nf(rnf);
    4047         623 :   n = rnf_get_degree(rnf);
    4048         623 :   pr = gcoeff(cnd,k,1);
    4049         623 :   v = nfval(nf, b, pr);
    4050         623 :   m = lg(cnd)>1 ? nbrows(cnd) : 0;
    4051             : 
    4052             :   /* add the valuation of b to the conductor... */
    4053         623 :   previous = gcoeff(cnd,k,2);
    4054         623 :   gcoeff(cnd,k,2) = addis(previous, v);
    4055             : 
    4056         623 :   y = const_vec(m, gen_1);
    4057         623 :   gel(y,k) = b;
    4058             :   /* find a factored element y congruent to b mod pr^(vpr(b)+vpr(cnd)) and to 1 mod the conductor. */
    4059         623 :   y = factoredextchinese(nf, cnd, y, pl, &fa);
    4060         623 :   h = 0;
    4061         623 :   lfa = nbrows(fa);
    4062             :   /* sum of all Hasse invariants of (rnf/nf,aut,y) is 0, Hasse invariants at q!=pr are easy, Hasse invariant at pr is the same as for al=(rnf/nf,aut,b). */
    4063        1225 :   for (i=1; i<=lfa; i++) {
    4064         602 :     q = gcoeff(fa,i,1);
    4065         602 :     if (cmp_prime_ideal(pr,q)) {
    4066         567 :       frob = cyclicrelfrob(rnf, auts, q);
    4067         567 :       frob = Fl_mul(frob,umodiu(gcoeff(fa,i,2),n),n);
    4068         567 :       h = Fl_add(h,frob,n);
    4069             :     }
    4070             :   }
    4071             :   /* ...then restore it. */
    4072         623 :   gcoeff(cnd,k,2) = previous;
    4073         623 :   return gc_long(av, Fl_neg(h,n));
    4074             : }
    4075             : 
    4076             : static GEN
    4077         819 : allauts(GEN rnf, GEN aut)
    4078             : {
    4079         819 :   long n = rnf_get_degree(rnf), i;
    4080         819 :   GEN pol = rnf_get_pol(rnf), vaut;
    4081         819 :   if (n==1) n=2;
    4082         819 :   vaut = cgetg(n,t_VEC);
    4083         819 :   aut = lift_shallow(rnfbasistoalg(rnf,aut));
    4084         819 :   if (typ(aut) != t_POL || varn(pol) != varn(aut))
    4085           0 :     pari_err_TYPE("alg_cyclic", aut);
    4086         819 :   gel(vaut,1) = aut;
    4087        1141 :   for (i=1; i<n-1; i++)
    4088         322 :     gel(vaut,i+1) = RgX_rem(poleval(gel(vaut,i), aut), pol);
    4089         819 :   return vaut;
    4090             : }
    4091             : 
    4092             : static GEN
    4093         273 : clean_factor(GEN fa)
    4094             : {
    4095         273 :   GEN P2,E2, P = gel(fa,1), E = gel(fa,2);
    4096         273 :   long l = lg(P), i, j = 1;
    4097         273 :   P2 = cgetg(l, t_COL);
    4098         273 :   E2 = cgetg(l, t_COL);
    4099        2339 :   for (i = 1;i < l; i++)
    4100        2066 :     if (signe(gel(E,i))) {
    4101         526 :       gel(P2,j) = gel(P,i);
    4102         526 :       gel(E2,j) = gel(E,i); j++;
    4103             :     }
    4104         273 :   setlg(P2,j);
    4105         273 :   setlg(E2,j); return mkmat2(P2,E2);
    4106             : }
    4107             : 
    4108             : /* shallow concat x[1],...x[nx],y[1], ... y[ny], returning a t_COL. To be
    4109             :  * used when we do not know whether x,y are t_VEC or t_COL */
    4110             : static GEN
    4111         546 : colconcat(GEN x, GEN y)
    4112             : {
    4113         546 :   long i, lx = lg(x), ly = lg(y);
    4114         546 :   GEN z=cgetg(lx+ly-1, t_COL);
    4115        3696 :   for (i=1; i<lx; i++) z[i]     = x[i];
    4116        1528 :   for (i=1; i<ly; i++) z[lx+i-1]= y[i];
    4117         546 :   return z;
    4118             : }
    4119             : 
    4120             : /* return v(x) at all primes in listpr, replace x by cofactor */
    4121             : static GEN
    4122        1092 : nfmakecoprime(GEN nf, GEN *px, GEN listpr)
    4123             : {
    4124        1092 :   long j, l = lg(listpr);
    4125        1092 :   GEN x1, x = *px, L = cgetg(l, t_COL);
    4126             : 
    4127        1092 :   if (typ(x) != t_MAT)
    4128             :   { /* scalar, divide at the end (fast valuation) */
    4129         945 :     x1 = NULL;
    4130        4943 :     for (j=1; j<l; j++)
    4131             :     {
    4132        3998 :       GEN pr = gel(listpr,j), e;
    4133        3998 :       long v = nfval(nf, x, pr);
    4134        3998 :       e = stoi(v); gel(L,j) = e;
    4135        5587 :       if (v) x1 = x1? idealmulpowprime(nf, x1, pr, e)
    4136        1589 :                     : idealpow(nf, pr, e);
    4137             :     }
    4138         945 :     if (x1) x = idealdivexact(nf, idealhnf(nf,x), x1);
    4139             :   }
    4140             :   else
    4141             :   { /* HNF, divide as we proceed (reduce size) */
    4142         273 :     for (j=1; j<l; j++)
    4143             :     {
    4144         126 :       GEN pr = gel(listpr,j);
    4145         126 :       long v = idealval(nf, x, pr);
    4146         126 :       gel(L,j) = stoi(v);
    4147         126 :       if (v) x = idealmulpowprime(nf, x, pr, stoi(-v));
    4148             :     }
    4149             :   }
    4150        1092 :   *px = x; return L;
    4151             : }
    4152             : 
    4153             : /* Caveat: factorizations are not sorted wrt cmp_prime_ideal: Lpr comes first */
    4154             : static GEN
    4155         273 : computecnd(GEN rnf, GEN Lpr)
    4156             : {
    4157             :   GEN id, nf, fa, Le, P,E;
    4158         273 :   long n = rnf_get_degree(rnf);
    4159             : 
    4160         273 :   nf = rnf_get_nf(rnf);
    4161         273 :   id = rnf_get_idealdisc(rnf);
    4162         273 :   Le = nfmakecoprime(nf, &id, Lpr);
    4163         273 :   fa = idealfactor(nf, id); /* part of D_{L/K} coprime with Lpr */
    4164         273 :   P =  colconcat(Lpr,gel(fa,1));
    4165         273 :   E =  colconcat(Le, gel(fa,2));
    4166         273 :   fa = mkmat2(P, gdiventgs(E, eulerphiu(n)));
    4167         273 :   return mkvec2(fa, clean_factor(fa));
    4168             : }
    4169             : 
    4170             : /* h >= 0 */
    4171             : static void
    4172          35 : nextgen(GEN gene, long h, GEN* gens, GEN* hgens, long* ngens, long* curgcd) {
    4173          35 :   long nextgcd = ugcd(h,*curgcd);
    4174          35 :   if (nextgcd == *curgcd) return;
    4175          35 :   (*ngens)++;
    4176          35 :   gel(*gens,*ngens) = gene;
    4177          35 :   gel(*hgens,*ngens) = utoi(h);
    4178          35 :   *curgcd = nextgcd;
    4179          35 :   return;
    4180             : }
    4181             : 
    4182             : static int
    4183          49 : dividesmod(long d, long h, long n) { return !(h%cgcd(d,n)); }
    4184             : 
    4185             : /* ramified prime with nontrivial Hasse invariant */
    4186             : static GEN
    4187          35 : localcomplete(GEN rnf, GEN pl, GEN cnd, GEN auts, long j, long n, long h, long* v)
    4188             : {
    4189             :   GEN nf, gens, hgens, pr, modpr, T, p, sol, U, b, gene, randg, pu;
    4190             :   long ngens, i, d, np, d1, d2, hg, dnf, vcnd, curgcd;
    4191          35 :   nf = rnf_get_nf(rnf);
    4192          35 :   pr = gcoeff(cnd,j,1);
    4193          35 :   np = umodiu(pr_norm(pr), n);
    4194          35 :   dnf = nf_get_degree(nf);
    4195          35 :   vcnd = itos(gcoeff(cnd,j,2));
    4196          35 :   ngens = 13+dnf;
    4197          35 :   gens = zerovec(ngens);
    4198          35 :   hgens = zerovec(ngens);
    4199          35 :   *v = 0;
    4200          35 :   curgcd = 0;
    4201          35 :   ngens = 0;
    4202             : 
    4203          35 :   if (!uisprime(n)) {
    4204           0 :     gene =  pr_get_gen(pr);
    4205           0 :     hg = localhasse(rnf, cnd, pl, auts, gene, j);
    4206           0 :     nextgen(gene, hg, &gens, &hgens, &ngens, &curgcd);
    4207             :   }
    4208             : 
    4209          35 :   if (ugcd(np,n) != 1) { /* GCD(Np,n) != 1 */
    4210          35 :     pu = idealprincipalunits(nf,pr,vcnd);
    4211          35 :     pu = abgrp_get_gen(pu);
    4212          70 :     for (i=1; i<lg(pu) && !dividesmod(curgcd,h,n); i++) {
    4213          35 :       gene = gel(pu,i);
    4214          35 :       hg = localhasse(rnf, cnd, pl, auts, gene, j);
    4215          35 :       nextgen(gene, hg, &gens, &hgens, &ngens, &curgcd);
    4216             :     }
    4217             :   }
    4218             : 
    4219          35 :   d = ugcd(np-1,n);
    4220          35 :   if (d != 1) { /* GCD(Np-1,n) != 1 */
    4221           7 :     modpr = nf_to_Fq_init(nf, &pr, &T, &p);
    4222           7 :     while (!dividesmod(curgcd,h,n)) { /* TODO gener_FpXQ_local */
    4223           0 :       if (T==NULL) randg = randomi(p);
    4224           0 :       else randg = random_FpX(degpol(T), varn(T),p);
    4225             : 
    4226           0 :       if (!gequal0(randg) && !gequal1(randg)) {
    4227           0 :         gene = Fq_to_nf(randg, modpr);
    4228           0 :         hg = localhasse(rnf, cnd, pl, auts, gene, j);
    4229           0 :         nextgen(gene, hg, &gens, &hgens, &ngens, &curgcd);
    4230             :       }
    4231             :     }
    4232             :   }
    4233             : 
    4234          35 :   setlg(gens,ngens+1);
    4235          35 :   setlg(hgens,ngens+1);
    4236             : 
    4237          35 :   sol = ZV_extgcd(hgens);
    4238          35 :   U = ZV_to_Flv(gmael(sol,2,ngens), n);
    4239          35 :   d = itou(gel(sol,1));
    4240          35 :   d1 = ugcd(d, n);
    4241          35 :   d2 = d / d1;
    4242          35 :   d = Fl_mul(h / d1, Fl_inv(d2,n), n);
    4243          35 :   if (d != 1) U = Flv_Fl_mul(U, d, n);
    4244          70 :   for (i = 1, b = gen_1; i <= ngens; i++)
    4245          35 :     if (U[i]) b = nfmul(nf, b, nfpow_u(nf, gel(gens,i), U[i]));
    4246          35 :   *v = U[1]; return b;
    4247             : }
    4248             : 
    4249             : static int
    4250         828 : testsplits(GEN data, GEN fa)
    4251             : {
    4252         828 :   GEN rnf = gel(data,1), forbid = gel(data,2), P = gel(fa,1), E = gel(fa,2);
    4253         828 :   long i, n, l = lg(P);
    4254             : 
    4255        1167 :   for (i = 1; i < l; i++)
    4256             :   {
    4257         823 :     GEN pr = gel(P,i);
    4258         823 :     if (tablesearch(forbid, pr, &cmp_prime_ideal)) return 0;
    4259             :   }
    4260         344 :   n = rnf_get_degree(rnf);
    4261         510 :   for (i = 1; i < l; i++)
    4262             :   {
    4263         237 :     long e = itos(gel(E,i)) % n;
    4264         237 :     if (e)
    4265             :     {
    4266         223 :       GEN L = rnfidealprimedec(rnf, gel(P,i));
    4267         223 :       long g = lg(L) - 1;
    4268         223 :       if ((e * g) % n) return 0;
    4269             :     }
    4270             :   }
    4271         273 :   return 1;
    4272             : }
    4273             : 
    4274             : /* remove entries with Hasse invariant 0 */
    4275             : static GEN
    4276         574 : hassereduce(GEN hf)
    4277             : {
    4278         574 :   GEN pr,h, PR = gel(hf,1), H = gel(hf,2);
    4279         574 :   long i, j, l = lg(PR);
    4280             : 
    4281         574 :   pr= cgetg(l, t_VEC);
    4282         574 :   h = cgetg(l, t_VECSMALL);
    4283        4081 :   for (i = j = 1; i < l; i++)
    4284        3507 :     if (H[i]) {
    4285        3178 :       gel(pr,j) = gel(PR,i);
    4286        3178 :       h[j] = H[i]; j++;
    4287             :     }
    4288         574 :   setlg(pr,j);
    4289         574 :   setlg(h,j); return mkvec2(pr,h);
    4290             : }
    4291             : 
    4292             : /* rnf complete */
    4293             : static GEN
    4294         273 : alg_complete0(GEN rnf, GEN aut, GEN hf, GEN hi, long flag)
    4295             : {
    4296         273 :   pari_sp av = avma;
    4297             :   GEN nf, pl, pl2, cnd, prcnd, cnds, y, Lpr, auts, b, fa, data, hfe;
    4298             :   GEN forbid, al, ind;
    4299             :   long D, n, d, i, j, l;
    4300         273 :   nf = rnf_get_nf(rnf);
    4301         273 :   n = rnf_get_degree(rnf);
    4302         273 :   d = nf_get_degree(nf);
    4303         273 :   D = d*n*n;
    4304         273 :   checkhasse(nf,hf,hi,n);
    4305         273 :   hf = hassereduce(hf);
    4306         273 :   Lpr = gel(hf,1);
    4307         273 :   hfe = gel(hf,2);
    4308             : 
    4309         273 :   auts = allauts(rnf,aut);
    4310             : 
    4311         273 :   pl = leafcopy(hi); /* conditions on the final b */
    4312         273 :   pl2 = leafcopy(hi); /* conditions for computing local Hasse invariants */
    4313         273 :   l = lg(pl); ind = cgetg(l, t_VECSMALL);
    4314         686 :   for (i = j = 1; i < l; i++)
    4315         413 :     if (hi[i]) { pl[i] = -1; pl2[i] = 1; } else ind[j++] = i;
    4316         273 :   setlg(ind, j);
    4317         273 :   y = nfpolsturm(nf, rnf_get_pol(rnf), ind);
    4318         511 :   for (i = 1; i < j; i++)
    4319         238 :     if (!signe(gel(y,i))) { pl[ind[i]] = 1; pl2[ind[i]] = 1; }
    4320             : 
    4321         273 :   cnds = computecnd(rnf,Lpr);
    4322         273 :   prcnd = gel(cnds,1);
    4323         273 :   cnd = gel(cnds,2);
    4324         273 :   y = cgetg(lgcols(prcnd),t_VEC);
    4325         273 :   forbid = vectrunc_init(lg(Lpr));
    4326        1848 :   for (i=j=1; i<lg(Lpr); i++)
    4327             :   {
    4328        1575 :     GEN pr = gcoeff(prcnd,i,1), yi;
    4329        1575 :     long v, e = itou( gcoeff(prcnd,i,2) );
    4330        1575 :     if (!e) {
    4331        1540 :       long frob = cyclicrelfrob(rnf,auts,pr), f1 = ugcd(frob,n);
    4332        1540 :       vectrunc_append(forbid, pr);
    4333        1540 :       yi = gen_0;
    4334        1540 :       v = ((hfe[i]/f1) * Fl_inv(frob/f1,n)) % n;
    4335             :     }
    4336             :     else
    4337          35 :       yi = localcomplete(rnf, pl2, cnd, auts, j++, n, hfe[i], &v);
    4338        1575 :     gel(y,i) = yi;
    4339        1575 :     gcoeff(prcnd,i,2) = stoi(e + v);
    4340             :   }
    4341         764 :   for (; i<lgcols(prcnd); i++) gel(y,i) = gen_1;
    4342         273 :   gen_sort_inplace(forbid, (void*)&cmp_prime_ideal, &cmp_nodata, NULL);
    4343         273 :   data = mkvec2(rnf,forbid);
    4344         273 :   b = factoredextchinesetest(nf,prcnd,y,pl,&fa,data,testsplits);
    4345             : 
    4346         273 :   al = cgetg(12, t_VEC);
    4347         273 :   gel(al,10)= gen_0; /* must be set first */
    4348         273 :   gel(al,1) = rnf;
    4349         273 :   gel(al,2) = auts;
    4350         273 :   gel(al,3) = basistoalg(nf,b);
    4351         273 :   gel(al,4) = hi;
    4352             :   /* add primes | disc or b with trivial Hasse invariant to hf */
    4353         273 :   Lpr = gel(prcnd,1); y = b;
    4354         273 :   (void)nfmakecoprime(nf, &y, Lpr);
    4355         273 :   Lpr = shallowconcat(Lpr, gel(idealfactor(nf,y), 1));
    4356         273 :   settyp(Lpr,t_VEC);
    4357         273 :   hf = mkvec2(Lpr, shallowconcat(hfe, const_vecsmall(lg(Lpr)-lg(hfe), 0)));
    4358         273 :   gel(al,5) = hf;
    4359         273 :   gel(al,6) = gen_0;
    4360         273 :   gel(al,7) = matid(D);
    4361         273 :   gel(al,8) = matid(D); /* TODO modify 7, 8 et 9 once LLL added */
    4362         273 :   gel(al,9) = algnatmultable(al,D);
    4363         273 :   gel(al,11)= algtracebasis(al);
    4364         273 :   if (flag & al_MAXORD) al = alg_maximal_primes(al, prV_primes(Lpr));
    4365         273 :   return gerepilecopy(av, al);
    4366             : }
    4367             : 
    4368             : GEN
    4369           0 : alg_complete(GEN rnf, GEN aut, GEN hf, GEN hi, long flag)
    4370             : {
    4371           0 :   long n = rnf_get_degree(rnf);
    4372           0 :   rnfcomplete(rnf);
    4373           0 :   return alg_complete0(rnf, aut, hasseconvert(hf,n), hasseconvert(hi,n), flag);
    4374             : }
    4375             : 
    4376             : void
    4377        1421 : checkhasse(GEN nf, GEN hf, GEN hi, long n)
    4378             : {
    4379             :   GEN Lpr, Lh;
    4380             :   long i, sum;
    4381        1421 :   if (typ(hf) != t_VEC || lg(hf) != 3) pari_err_TYPE("checkhasse [hf]", hf);
    4382        1414 :   Lpr = gel(hf,1);
    4383        1414 :   Lh = gel(hf,2);
    4384        1414 :   if (typ(Lpr) != t_VEC) pari_err_TYPE("checkhasse [Lpr]", Lpr);
    4385        1414 :   if (typ(Lh) != t_VECSMALL) pari_err_TYPE("checkhasse [Lh]", Lh);
    4386        1414 :   if (typ(hi) != t_VECSMALL) pari_err_TYPE("checkhasse [hi]", hi);
    4387        1414 :   if ((nf && lg(hi) != nf_get_r1(nf)+1))
    4388           7 :     pari_err_DOMAIN("checkhasse [hi should have r1 components]","#hi","!=",stoi(nf_get_r1(nf)),stoi(lg(hi)-1));
    4389        1407 :   if (lg(Lpr) != lg(Lh))
    4390           7 :     pari_err_DIM("checkhasse [Lpr and Lh should have same length]");
    4391        7455 :   for (i=1; i<lg(Lpr); i++) checkprid(gel(Lpr,i));
    4392        1400 :   if (lg(gen_sort_uniq(Lpr, (void*)cmp_prime_ideal, cmp_nodata)) < lg(Lpr))
    4393           7 :     pari_err(e_MISC, "error in checkhasse [duplicate prime ideal]");
    4394        1393 :   sum = 0;
    4395        7434 :   for (i=1; i<lg(Lh); i++) sum = (sum+Lh[i])%n;
    4396        3234 :   for (i=1; i<lg(hi); i++) {
    4397        1855 :       if (hi[i] && 2*hi[i] != n) pari_err_DOMAIN("checkhasse", "Hasse invariant at real place [must be 0 or 1/2]", "!=", n%2? gen_0 : stoi(n/2), stoi(hi[i]));
    4398        1841 :       sum = (sum+hi[i])%n;
    4399             :   }
    4400        1379 :   if (sum<0) sum = n+sum;
    4401        1379 :   if (sum != 0)
    4402           7 :     pari_err_DOMAIN("checkhasse","sum(Hasse invariants)","!=",gen_0,Lh);
    4403        1372 : }
    4404             : 
    4405             : static GEN
    4406         371 : hassecoprime(GEN hf, GEN hi, long n)
    4407             : {
    4408         371 :   pari_sp av = avma;
    4409             :   long l, i, j, lk, inv;
    4410             :   GEN fa, P,E, res, hil, hfl;
    4411         371 :   hi = hasseconvert(hi, n);
    4412         357 :   hf = hasseconvert(hf, n);
    4413         336 :   checkhasse(NULL,hf,hi,n);
    4414         294 :   fa = factoru(n);
    4415         294 :   P = gel(fa,1); l = lg(P);
    4416         294 :   E = gel(fa,2);
    4417         294 :   res = cgetg(l,t_VEC);
    4418         595 :   for (i=1; i<l; i++) {
    4419         301 :     lk = upowuu(P[i],E[i]);
    4420         301 :     inv = Fl_invsafe((n/lk)%lk, lk);
    4421         301 :     hil = gcopy(hi);
    4422         301 :     hfl = gcopy(hf);
    4423             : 
    4424         301 :     if (P[i] == 2)
    4425         651 :       for (j=1; j<lg(hil); j++) hil[j] = hi[j]==0 ? 0 : lk/2;
    4426             :     else
    4427          98 :       for (j=1; j<lg(hil); j++) hil[j] = 0;
    4428        2233 :     for (j=1; j<lgcols(hfl); j++) gel(hfl,2)[j] = (gel(hf,2)[j]*inv)%lk;
    4429         301 :     hfl = hassereduce(hfl);
    4430         301 :     gel(res,i) = mkvec3(hfl,hil,utoi(lk));
    4431             :   }
    4432             : 
    4433         294 :   return gerepilecopy(av, res);
    4434             : }
    4435             : 
    4436             : /* no garbage collection */
    4437             : static GEN
    4438          77 : genefrob(GEN nf, GEN gal, GEN r)
    4439             : {
    4440             :   long i;
    4441          77 :   GEN g = identity_perm(nf_get_degree(nf)), fa = Z_factor(r), p, pr, frob;
    4442         126 :   for (i=1; i<lgcols(fa); i++) {
    4443          49 :     p = gcoeff(fa,i,1);
    4444          49 :     pr = idealprimedec(nf, p);
    4445          49 :     pr = gel(pr,1);
    4446          49 :     frob = idealfrobenius(nf, gal, pr);
    4447          49 :     g = perm_mul(g, perm_pow(frob, gcoeff(fa,i,2)));
    4448             :   }
    4449          77 :   return g;
    4450             : }
    4451             : 
    4452             : static GEN
    4453         273 : rnfcycaut(GEN rnf)
    4454             : {
    4455         273 :   GEN nf2 = obj_check(rnf, rnf_NFABS);
    4456             :   GEN L, alpha, pol, salpha, s, sj, polabs, k, X, pol0, nf;
    4457             :   long i, d, j;
    4458         273 :   d = rnf_get_degree(rnf);
    4459         273 :   L = galoisconj(nf2,NULL);
    4460         273 :   alpha = lift_shallow(rnf_get_alpha(rnf));
    4461         273 :   pol = rnf_get_pol(rnf);
    4462         273 :   k = rnf_get_k(rnf);
    4463         273 :   polabs = rnf_get_polabs(rnf);
    4464         273 :   nf = rnf_get_nf(rnf);
    4465         273 :   pol0 = nf_get_pol(nf);
    4466         273 :   X = RgX_rem(pol_x(varn(pol0)), pol0);
    4467             : 
    4468             :   /* TODO check mod prime of degree 1 */
    4469         386 :   for (i=1; i<lg(L); i++) {
    4470         386 :     s = gel(L,i);
    4471         386 :     salpha = RgX_RgXQ_eval(alpha,s,polabs);
    4472         386 :     if (!gequal(alpha,salpha)) continue;
    4473             : 
    4474         336 :     s = lift_shallow(rnfeltabstorel(rnf,s));
    4475         336 :     sj = s = gsub(s, gmul(k,X));
    4476         651 :     for (j=1; !gequal0(gsub(sj,pol_x(varn(s)))); j++)
    4477         315 :       sj = RgX_RgXQ_eval(sj,s,pol);
    4478         336 :     if (j<d) continue;
    4479         273 :     return s;
    4480             :   }
    4481             :   return NULL; /*LCOV_EXCL_LINE*/
    4482             : }
    4483             : 
    4484             : /* returns the smallest prime not in P */
    4485             : static GEN
    4486          84 : extraprime(GEN P)
    4487             : {
    4488             :   forprime_t T;
    4489             :   GEN p;
    4490          84 :   forprime_init(&T, gen_2, NULL);
    4491          98 :   while ((p = forprime_next(&T))) if (!ZV_search(P, p)) break;
    4492          84 :   return p;
    4493             : }
    4494             : 
    4495             : /* true nf */
    4496             : GEN
    4497         385 : alg_hasse(GEN nf, long n, GEN hf, GEN hi, long var, long flag)
    4498             : {
    4499         385 :   pari_sp av = avma;
    4500         385 :   GEN primary, al = gen_0, al2, rnf, hil, hfl, Ld, pl, pol, Lpr, aut, Lpr2, Ld2;
    4501             :   long i, lk, j, maxdeg;
    4502         385 :   dbg_printf(1)("alg_hasse\n");
    4503         385 :   if (n<=1) pari_err_DOMAIN("alg_hasse", "degree", "<=", gen_1, stoi(n));
    4504         371 :   primary = hassecoprime(hf, hi, n);
    4505         574 :   for (i=1; i<lg(primary); i++) {
    4506         301 :     lk = itos(gmael(primary,i,3));
    4507         301 :     hfl = gmael(primary,i,1);
    4508         301 :     hil = gmael(primary,i,2);
    4509         301 :     checkhasse(nf, hfl, hil, lk);
    4510         294 :     dbg_printf(1)("alg_hasse: i=%d hf=%Ps hi=%Ps lk=%d\n", i, hfl, hil, lk);
    4511             : 
    4512         294 :     if (lg(gel(hfl,1))>1 || lk%2==0) {
    4513         287 :       maxdeg = 1;
    4514         287 :       Lpr = gel(hfl,1);
    4515         287 :       Ld = gcopy(gel(hfl,2));
    4516        1876 :       for (j=1; j<lg(Ld); j++)
    4517             :       {
    4518        1589 :         Ld[j] = lk/ugcd(lk,Ld[j]);
    4519        1589 :         maxdeg = maxss(Ld[j],maxdeg);
    4520             :       }
    4521         287 :       pl = leafcopy(hil);
    4522         714 :       for (j=1; j<lg(pl); j++) if(pl[j])
    4523             :       {
    4524         175 :         pl[j] = -1;
    4525         175 :         maxdeg = maxss(maxdeg,2);
    4526             :       }
    4527             : 
    4528         287 :       Lpr2 = Lpr;
    4529         287 :       Ld2 = Ld;
    4530         287 :       if (maxdeg<lk)
    4531             :       {
    4532         154 :         if (maxdeg==1 && lk==2 && lg(pl)>1) pl[1] = -1;
    4533             :         else
    4534             :         {
    4535          84 :           GEN p = extraprime(prV_primes(Lpr));
    4536          84 :           Lpr2 = vec_append(Lpr2, idealprimedec_galois(nf, p));
    4537          84 :           Ld2 = vecsmall_append(Ld2, lk);
    4538             :         }
    4539             :       }
    4540             : 
    4541         287 :       dbg_printf(2)("alg_hasse: calling nfgrunwaldwang Lpr=%Ps Pd=%Ps pl=%Ps\n",
    4542             :           Lpr, Ld, pl);
    4543         287 :       pol = nfgrunwaldwang(nf, Lpr2, Ld2, pl, var);
    4544         273 :       dbg_printf(2)("alg_hasse: calling rnfinit(%Ps)\n", pol);
    4545         273 :       rnf = rnfinit0(nf,pol,1);
    4546         273 :       dbg_printf(2)("alg_hasse: computing automorphism\n");
    4547         273 :       aut = rnfcycaut(rnf);
    4548         273 :       dbg_printf(2)("alg_hasse: calling alg_complete\n");
    4549         273 :       al2 = alg_complete0(rnf, aut, hfl, hil, flag);
    4550             :     }
    4551           7 :     else al2 = alg_matrix(nf, lk, var, flag);
    4552             : 
    4553         280 :     if (i==1) al = al2;
    4554           7 :     else      al = algtensor(al,al2,flag);
    4555             :   }
    4556         273 :   return gerepilecopy(av,al);
    4557             : }
    4558             : 
    4559             : /** CYCLIC ALGEBRA WITH GIVEN HASSE INVARIANTS **/
    4560             : 
    4561             : /* no garbage collection */
    4562             : static GEN
    4563          77 : subcycloindep(GEN nf, long n, long v, GEN *pr)
    4564             : {
    4565             :   pari_sp av;
    4566             :   forprime_t S;
    4567             :   ulong p;
    4568          77 :   u_forprime_arith_init(&S, 1, ULONG_MAX, 1, n);
    4569          77 :   av = avma;
    4570          84 :   while ((p = u_forprime_next(&S)))
    4571             :   {
    4572          84 :     ulong r = pgener_Fl(p);
    4573          84 :     GEN pol = galoissubcyclo(utoipos(p), utoipos(Fl_powu(r,n,p)), 0, v);
    4574          84 :     GEN fa = nffactor(nf, pol);
    4575          84 :     if (lgcols(fa) == 2) { *pr = utoipos(r); return pol; }
    4576           7 :     set_avma(av);
    4577             :   }
    4578             :   pari_err_BUG("subcycloindep (no suitable prime = 1(mod n))"); /*LCOV_EXCL_LINE*/
    4579             :   *pr = NULL; return NULL; /*LCOV_EXCL_LINE*/
    4580             : }
    4581             : 
    4582             : GEN
    4583          84 : alg_matrix(GEN nf, long n, long v, long flag)
    4584             : {
    4585          84 :   pari_sp av = avma;
    4586             :   GEN pol, gal, rnf, cyclo, g, r, aut;
    4587          84 :   dbg_printf(1)("alg_matrix\n");
    4588          84 :   if (n<=0) pari_err_DOMAIN("alg_matrix", "n", "<=", gen_0, stoi(n));
    4589          77 :   pol = subcycloindep(nf, n, v, &r);
    4590          77 :   rnf = rnfinit(nf, pol);
    4591          77 :   cyclo = nfinit(pol, nf_get_prec(nf));
    4592          77 :   gal = galoisinit(cyclo, NULL);
    4593          77 :   g = genefrob(cyclo,gal,r);
    4594          77 :   aut = galoispermtopol(gal,g);
    4595          77 :   return gerepileupto(av, alg_cyclic(rnf, aut, gen_1, flag));
    4596             : }
    4597             : 
    4598             : GEN
    4599         329 : alg_hilbert(GEN nf, GEN a, GEN b, long v, long flag)
    4600             : {
    4601         329 :   pari_sp av = avma;
    4602             :   GEN rnf, aut, rnfpol;
    4603         329 :   dbg_printf(1)("alg_hilbert\n");
    4604         329 :   if (!isint1(Q_denom(a)))
    4605           7 :     pari_err_DOMAIN("alg_hilbert", "denominator(a)", "!=", gen_1,a);
    4606         322 :   if (!isint1(Q_denom(b)))
    4607           7 :     pari_err_DOMAIN("alg_hilbert", "denominator(b)", "!=", gen_1,b);
    4608             : 
    4609         315 :   if (v < 0) v = 0;
    4610         315 :   rnfpol = deg2pol_shallow(gen_1, gen_0, gneg(a), v);
    4611         315 :   if (!(flag & al_FACTOR)) rnfpol = mkvec2(rnfpol, stoi(1<<20));
    4612         315 :   rnf = rnfinit(nf, rnfpol);
    4613         308 :   aut = gneg(pol_x(v));
    4614         308 :   return gerepileupto(av, alg_cyclic(rnf, aut, b, flag));
    4615             : }
    4616             : 
    4617             : /* return a structure representing the algebra of real numbers */
    4618             : static GEN
    4619          14 : mk_R()
    4620             : {
    4621          14 :   pari_sp av = avma;
    4622             :   GEN al;
    4623          14 :   al = zerovec(11);
    4624          14 :   gel(al,1) = stor(1,3);
    4625          14 :   gel(al,2) = mkvec(gel(al,1));
    4626          14 :   gel(al,3) = gen_1;
    4627          14 :   gel(al,4) = mkvecsmall(0);
    4628          14 :   gel(al,8) = gel(al,7) = matid(1);
    4629          14 :   gel(al,9) = mkvec(matid(1));
    4630          14 :   return gerepilecopy(av,al);
    4631             : }
    4632             : /* return a structure representing the algebra of complex numbers */
    4633             : static GEN
    4634          14 : mk_C()
    4635             : {
    4636          14 :   pari_sp av = avma;
    4637             :   GEN al, I;
    4638          14 :   al = zerovec(11);
    4639          14 :   I = gen_I();
    4640          14 :   gel(al,1) = I;
    4641          14 :   gel(al,2) = mkvec(I);
    4642          14 :   gel(al,3) = gen_1;
    4643          14 :   gel(al,4) = cgetg(1,t_VECSMALL);
    4644          14 :   gel(al,8) = gel(al,7) = matid(2);
    4645          14 :   gel(al,9) = mkvec2(
    4646             :     matid(2),
    4647             :     mkmat22(gen_0,gen_m1,gen_1,gen_0)
    4648             :   );
    4649          14 :   return gerepilecopy(av,al);
    4650             : }
    4651             : /* return a structure representing the Hamilton quaternion algebra */
    4652             : static GEN
    4653          14 : mk_H()
    4654             : {
    4655          14 :   pari_sp av = avma;
    4656             :   GEN al, I;
    4657          14 :   al = zerovec(11);
    4658          14 :   I = gen_I();
    4659          14 :   gel(al,1) = I;
    4660          14 :   gel(al,2) = mkvec(gconj(I));
    4661          14 :   gel(al,3) = gen_m1;
    4662          14 :   gel(al,4) = mkvecsmall(1);
    4663          14 :   gel(al,8) = gel(al,7) = matid(4);
    4664          14 :   gel(al,9) = mkvec4(
    4665             :     matid(4),
    4666             :     H_tomatrix(I,1),
    4667             :     H_tomatrix(mkcol4(gen_0,gen_0,gen_1,gen_0),1),
    4668             :     H_tomatrix(mkcol4(gen_0,gen_0,gen_0,gen_1),1)
    4669             :   );
    4670          14 :   return gerepilecopy(av,al);
    4671             : }
    4672             : 
    4673             : GEN
    4674        1239 : alginit(GEN A, GEN B, long v, long flag)
    4675             : {
    4676             :   long w;
    4677        1239 :   if (typ(A) == t_COMPLEX) return mk_C();
    4678        1225 :   if (typ(A) == t_REAL)
    4679             :   {
    4680          35 :     if (is_scalar_t(typ(B)) && gequal0(B)) return mk_R();
    4681          21 :     if (typ(B) == t_FRAC && gequal(B, mkfrac(gen_1,gen_2))) return mk_H();
    4682           7 :     pari_err_DOMAIN("alginit", "real Hasse invariant [must be 0 or 1/2]", "", NULL, B);
    4683             :   }
    4684        1190 :   switch(nftyp(A))
    4685             :   {
    4686        1001 :     case typ_NF:
    4687        1001 :       if (v<0) v=0;
    4688        1001 :       w = gvar(nf_get_pol(A));
    4689        1001 :       if (varncmp(v,w)>=0) pari_err_PRIORITY("alginit", pol_x(v), ">=", w);
    4690         987 :       switch(typ(B))
    4691             :       {
    4692             :         long nB;
    4693          77 :         case t_INT: return alg_matrix(A, itos(B), v, flag);
    4694         903 :         case t_VEC:
    4695         903 :           nB = lg(B)-1;
    4696         903 :           if (nB && typ(gel(B,1)) == t_MAT) return alg_csa_table(A,B,v,flag);
    4697             :           switch(nB)
    4698             :           {
    4699         329 :             case 2: return alg_hilbert(A, gel(B,1), gel(B,2), v, flag);
    4700         392 :             case 3:
    4701         392 :               if (typ(gel(B,1))!=t_INT)
    4702           7 :                   pari_err_TYPE("alginit [degree should be an integer]", gel(B,1));
    4703         385 :               return alg_hasse(A, itos(gel(B,1)), gel(B,2), gel(B,3), v,
    4704             :                                                                       flag);
    4705             :           }
    4706             :       }
    4707          14 :       pari_err_TYPE("alginit", B); break;
    4708             : 
    4709         175 :     case typ_RNF:
    4710         175 :       if (typ(B) != t_VEC || lg(B) != 3) pari_err_TYPE("alginit", B);
    4711         161 :       return alg_cyclic(A, gel(B,1), gel(B,2), flag);
    4712             :   }
    4713          14 :   pari_err_TYPE("alginit", A);
    4714             :   return NULL;/*LCOV_EXCL_LINE*/
    4715             : }
    4716             : 
    4717             : /* assumes al CSA or CYCLIC */
    4718             : static GEN
    4719         966 : algnatmultable(GEN al, long D)
    4720             : {
    4721             :   GEN res, x;
    4722             :   long i;
    4723         966 :   res = cgetg(D+1,t_VEC);
    4724       11291 :   for (i=1; i<=D; i++) {
    4725       10325 :     x = algnattoalg(al,col_ei(D,i));
    4726       10325 :     gel(res,i) = algZmultable(al,x);
    4727             :   }
    4728         966 :   return res;
    4729             : }
    4730             : 
    4731         140 : static int normfact_is_partial(GEN nf, GEN x, GEN fax)
    4732             : {
    4733             :   long i;
    4734             :   GEN nfx;
    4735         140 :   nfx = RgM_shallowcopy(fax);
    4736         385 :   for (i=1; i<lg(gel(nfx,1)); i++)
    4737         245 :     gcoeff(nfx,i,1) = idealnorm(nf, gcoeff(nfx,i,1));
    4738         140 :   nfx = factorback(nfx);
    4739         140 :   return !gequal(idealnorm(nf, x), nfx);
    4740             : }
    4741             : /* no garbage collection */
    4742             : static void
    4743         546 : algcomputehasse(GEN al, long flag)
    4744             : {
    4745             :   int partialfact;
    4746             :   long r1, k, n, m, m1, m2, m3, i, m23, m123;
    4747             :   GEN rnf, nf, b, fab, disc2, cnd, fad, auts, pr, pl, perm, y, hi, PH, H, L;
    4748             : 
    4749         546 :   rnf = alg_get_splittingfield(al);
    4750         546 :   n = rnf_get_degree(rnf);
    4751         546 :   nf = rnf_get_nf(rnf);
    4752         546 :   b = alg_get_b(al);
    4753         546 :   r1 = nf_get_r1(nf);
    4754         546 :   auts = alg_get_auts(al);
    4755         546 :   (void)alg_get_abssplitting(al);
    4756             : 
    4757         546 :   y = nfpolsturm(nf, rnf_get_pol(rnf), NULL);
    4758         546 :   pl = cgetg(r1+1, t_VECSMALL);
    4759             :   /* real places where rnf/nf ramifies */
    4760        1134 :   for (k = 1; k <= r1; k++) pl[k] = !signe(gel(y,k));
    4761             : 
    4762             :   /* infinite Hasse invariants */
    4763         546 :   if (odd(n)) hi = const_vecsmall(r1, 0);
    4764             :   else
    4765             :   {
    4766         462 :     GEN s = nfsign(nf, b);
    4767         462 :     hi = cgetg(r1+1, t_VECSMALL);
    4768         994 :     for (k = 1; k<=r1; k++) hi[k] = (s[k] && pl[k]) ? (n/2) : 0;
    4769             :   }
    4770         546 :   gel(al,4) = hi;
    4771             : 
    4772         546 :   partialfact = 0;
    4773         546 :   if (flag & al_FACTOR)
    4774         462 :     fab = idealfactor(nf, b);
    4775             :   else {
    4776          84 :     fab = idealfactor_limit(nf, b, 1<<20);
    4777             :     /* does not report whether factorisation was partial; check it */
    4778          84 :     partialfact = normfact_is_partial(nf, b, fab);
    4779             :   }
    4780             : 
    4781         546 :   disc2 = rnf_get_idealdisc(rnf);
    4782         546 :   L = nfmakecoprime(nf, &disc2, gel(fab,1));
    4783         546 :   m = lg(L)-1;
    4784             :   /* m1 = #{pr|b: pr \nmid disc}, m3 = #{pr|b: pr | disc} */
    4785         546 :   perm = cgetg(m+1, t_VECSMALL);
    4786        1029 :   for (i=1, m1=m, k=1; k<=m; k++)
    4787         483 :     if (signe(gel(L,k))) perm[m1--] = k; else perm[i++] = k;
    4788         546 :   m3 = m - m1;
    4789             : 
    4790             :   /* disc2 : factor of disc coprime to b */
    4791         546 :   if (flag & al_FACTOR)
    4792         462 :     fad = idealfactor(nf, disc2);
    4793             :   else {
    4794          84 :     fad = idealfactor_limit(nf, disc2, 1<<20);
    4795          84 :     partialfact = partialfact || normfact_is_partial(nf, disc2, fad);
    4796             :   }
    4797             : 
    4798             :   /* if factorisation is partial, do not compute Hasse invariants */
    4799             :   /* we could compute their sum at composite factors */
    4800         546 :   if (partialfact)
    4801             :   {
    4802          35 :     if (!(flag & al_MAXORD))
    4803             :     {
    4804          28 :       gel(al,5) = gen_0;
    4805          35 :       return;
    4806             :     }
    4807             :     /* but transmit list of factors found for computation of maximal order */
    4808           7 :     PH = prV_primes(shallowconcat(gel(fab,1), gel(fad,1)));
    4809           7 :     gel(al,5) = mkvec2(PH, gen_0);;
    4810           7 :     return;
    4811             :   }
    4812             : 
    4813             :   /* m2 : number of prime factors of disc not dividing b */
    4814         511 :   m2 = nbrows(fad);
    4815         511 :   m23 = m2+m3;
    4816         511 :   m123 = m1+m2+m3;
    4817             : 
    4818             :   /* initialize the possibly ramified primes (hasse) and the factored conductor of rnf/nf (cnd) */
    4819         511 :   cnd = zeromatcopy(m23,2);
    4820         511 :   PH = cgetg(m123+1, t_VEC); /* ramified primes */
    4821         511 :   H = cgetg(m123+1, t_VECSMALL); /* Hasse invariant */
    4822             :   /* compute Hasse invariant at primes that are unramified in rnf/nf */
    4823         875 :   for (k=1; k<=m1; k++) {/* pr | b, pr \nmid disc */
    4824         364 :     long frob, e, j = perm[k];
    4825         364 :     pr = gcoeff(fab,j,1);
    4826         364 :     e = itos(gcoeff(fab,j,2));
    4827         364 :     frob = cyclicrelfrob(rnf, auts, pr);
    4828         364 :     gel(PH,k) = pr;
    4829         364 :     H[k] = Fl_mul(frob, e, n);
    4830             :   }
    4831             :   /* compute Hasse invariant at primes that are ramified in rnf/nf */
    4832        1064 :   for (k=1; k<=m2; k++) {/* pr \nmid b, pr | disc */
    4833         553 :     pr = gcoeff(fad,k,1);
    4834         553 :     gel(PH,k+m1) = pr;
    4835         553 :     gcoeff(cnd,k,1) = pr;
    4836         553 :     gcoeff(cnd,k,2) = gcoeff(fad,k,2);
    4837             :   }
    4838         546 :   for (k=1; k<=m3; k++) { /* pr | (b, disc) */
    4839          35 :     long j = perm[k+m1];
    4840          35 :     pr = gcoeff(fab,j,1);
    4841          35 :     gel(PH,k+m1+m2) = pr;
    4842          35 :     gcoeff(cnd,k+m2,1) = pr;
    4843          35 :     gcoeff(cnd,k+m2,2) = gel(L,j);
    4844             :   }
    4845         511 :   gel(cnd,2) = gdiventgs(gel(cnd,2), eulerphiu(n));
    4846        1099 :   for (k=1; k<=m23; k++) H[k+m1] = localhasse(rnf, cnd, pl, auts, b, k);
    4847         511 :   perm = gen_indexsort(PH, (void*)&cmp_prime_ideal, &cmp_nodata);
    4848         511 :   gel(al,5) = mkvec2(vecpermute(PH,perm),vecsmallpermute(H,perm));
    4849         511 :   checkhasse(nf, alg_get_hasse_f(al), alg_get_hasse_i(al), n);
    4850             : }
    4851             : 
    4852             : static GEN
    4853         805 : alg_maximal_primes(GEN al, GEN P)
    4854             : {
    4855         805 :   pari_sp av = avma;
    4856         805 :   long l = lg(P), i;
    4857        2855 :   for (i=1; i<l; i++)
    4858             :   {
    4859        2050 :     if (i != 1) al = gerepilecopy(av, al);
    4860        2050 :     al = alg_pmaximal(al,gel(P,i));
    4861             :   }
    4862         805 :   return al;
    4863             : }
    4864             : 
    4865             : GEN
    4866         560 : alg_cyclic(GEN rnf, GEN aut, GEN b, long flag)
    4867             : {
    4868         560 :   pari_sp av = avma;
    4869             :   GEN al, nf;
    4870             :   long D, n, d;
    4871         560 :   dbg_printf(1)("alg_cyclic\n");
    4872         560 :   checkrnf(rnf); nf = rnf_get_nf(rnf);
    4873         560 :   b = nf_to_scalar_or_basis(nf, b);
    4874         553 :   if (typ(b) == t_FRAC || (typ(b) == t_COL && !RgV_is_ZV(b)))
    4875           7 :     pari_err_DOMAIN("alg_cyclic", "denominator(b)", "!=", gen_1,b);
    4876             : 
    4877         546 :   n = rnf_get_degree(rnf);
    4878         546 :   d = nf_get_degree(nf);
    4879         546 :   D = d*n*n;
    4880             : 
    4881         546 :   al = cgetg(12,t_VEC);
    4882         546 :   gel(al,10)= gen_0; /* must be set first */
    4883         546 :   gel(al,1) = rnf;
    4884         546 :   gel(al,2) = allauts(rnf, aut);
    4885         546 :   gel(al,3) = basistoalg(nf,b);
    4886         546 :   rnf_build_nfabs(rnf, nf_get_prec(nf));
    4887         546 :   gel(al,6) = gen_0;
    4888         546 :   gel(al,7) = matid(D);
    4889         546 :   gel(al,8) = matid(D); /* TODO modify 7, 8 et 9 once LLL added */
    4890         546 :   gel(al,9) = algnatmultable(al,D);
    4891         546 :   gel(al,11)= algtracebasis(al);
    4892             : 
    4893         546 :   algcomputehasse(al, flag);
    4894             : 
    4895         546 :   if (flag & al_MAXORD) {
    4896         448 :     GEN hf = alg_get_hasse_f(al), pr = gel(hf,1);
    4897         448 :     if (typ(gel(hf,2)) == t_INT) /* factorisation was partial */
    4898           7 :       gel(al,5) = gen_0;
    4899         441 :     else pr = prV_primes(pr);
    4900         448 :     al = alg_maximal_primes(al, pr);
    4901             :   }
    4902         546 :   return gerepilecopy(av, al);
    4903             : }
    4904             : 
    4905             : static int
    4906         427 : ismaximalsubfield(GEN al, GEN x, GEN d, long v, GEN *pt_minpol)
    4907             : {
    4908         427 :   GEN cp = algbasischarpoly(al, x, v), lead;
    4909         427 :   if (!ispower(cp, d, pt_minpol)) return 0;
    4910         427 :   lead = leading_coeff(*pt_minpol);
    4911         427 :   if (isintm1(lead)) *pt_minpol = gneg(*pt_minpol);
    4912         427 :   return ZX_is_irred(*pt_minpol);
    4913             : }
    4914             : 
    4915             : static GEN
    4916         147 : findmaximalsubfield(GEN al, GEN d, long v)
    4917             : {
    4918         147 :   long count, nb=2, i, N = alg_get_absdim(al), n = nf_get_degree(alg_get_center(al));
    4919         147 :   GEN x, minpol, maxc = gen_1;
    4920             : 
    4921         238 :   for (i=n+1; i<=N; i+=n) {
    4922         399 :     for (count=0; count<2 && i+count<=N; count++) {
    4923         308 :       x = col_ei(N,i+count);
    4924         308 :       if (ismaximalsubfield(al, x, d, v, &minpol)) return mkvec2(x,minpol);
    4925             :     }
    4926             :   }
    4927             : 
    4928             :   while(1) {
    4929         119 :     x = zerocol(N);
    4930         504 :     for (count=0; count<nb; count++)
    4931             :     {
    4932         385 :       i = random_Fl(N)+1;
    4933         385 :       gel(x,i) = addiu(randomi(maxc),1);
    4934         385 :       if (random_bits(1)) gel(x,i) = negi(gel(x,i));
    4935             :     }
    4936         119 :     if (ismaximalsubfield(al, x, d, v, &minpol)) return mkvec2(x,minpol);
    4937          56 :     if (!random_bits(3)) maxc = addiu(maxc,1);
    4938          56 :     if (nb<N) nb++;
    4939             :   }
    4940             : 
    4941             :   return NULL; /* LCOV_EXCL_LINE */
    4942             : }
    4943             : 
    4944             : static GEN
    4945         147 : frobeniusform(GEN al, GEN x)
    4946             : {
    4947             :   GEN M, FP, P, Pi;
    4948             : 
    4949             :   /* /!\ has to be the *right* multiplication table */
    4950         147 :   M = algbasisrightmultable(al, x);
    4951             : 
    4952         147 :   FP = matfrobenius(M,2,0); /* M = P^(-1)*F*P */
    4953         147 :   P = gel(FP,2);
    4954         147 :   Pi = RgM_inv(P);
    4955         147 :   return mkvec2(P, Pi);
    4956             : }
    4957             : 
    4958             : static void
    4959         147 : computesplitting(GEN al, long d, long v, long flag)
    4960             : {
    4961         147 :   GEN subf, x, pol, polabs, basis, P, Pi, nf = alg_get_center(al), rnf, Lbasis, Lbasisinv, Q, pows;
    4962         147 :   long i, n = nf_get_degree(nf), nd = n*d, N = alg_get_absdim(al), j, j2;
    4963             : 
    4964         147 :   subf = findmaximalsubfield(al, utoipos(d), v);
    4965         147 :   x = gel(subf, 1);
    4966         147 :   polabs = gel(subf, 2);
    4967             : 
    4968             :   /* Frobenius form to obtain L-vector space structure */
    4969         147 :   basis = frobeniusform(al, x);
    4970         147 :   P = gel(basis, 1);
    4971         147 :   Pi = gel(basis, 2);
    4972             : 
    4973             :   /* construct rnf of splitting field */
    4974         147 :   pol = nffactor(nf,polabs);
    4975         147 :   pol = gcoeff(pol,1,1);
    4976         147 :   if (!(flag & al_FACTOR)) pol = mkvec2(pol, stoi(1<<20));
    4977         147 :   gel(al,1) = rnf = rnfinit(nf, pol);
    4978             :   /* since pol is irreducible over Q, we have k=0 in rnf. */
    4979         147 :   if (!gequal0(rnf_get_k(rnf)))
    4980             :     pari_err_BUG("computesplitting (k!=0)"); /*LCOV_EXCL_LINE*/
    4981         147 :   gel(al,6) = gen_0;
    4982         147 :   rnf_build_nfabs(rnf, nf_get_prec(nf));
    4983             : 
    4984             :   /* construct splitting data */
    4985         147 :   Lbasis = cgetg(d+1, t_MAT);
    4986         399 :   for (j=j2=1; j<=d; j++, j2+=nd)
    4987         252 :     gel(Lbasis,j) = gel(Pi,j2);
    4988             : 
    4989         147 :   Q = zeromatcopy(d,N);
    4990         147 :   pows = pol_x_powers(nd,v);
    4991         399 :   for (i=j=1; j<=N; j+=nd, i++)
    4992        1197 :   for (j2=0; j2<nd; j2++)
    4993         945 :     gcoeff(Q,i,j+j2) = mkpolmod(gel(pows,j2+1),polabs);
    4994         147 :   Lbasisinv = RgM_mul(Q,P);
    4995             : 
    4996         147 :   gel(al,3) = mkvec3(x,Lbasis,Lbasisinv);
    4997         147 : }
    4998             : 
    4999             : /* assumes that mt defines a central simple algebra over nf */
    5000             : GEN
    5001         175 : alg_csa_table(GEN nf, GEN mt0, long v, long flag)
    5002             : {
    5003         175 :   pari_sp av = avma;
    5004             :   GEN al, mt;
    5005         175 :   long n, D, d2 = lg(mt0)-1, d = usqrt(d2);
    5006         175 :   dbg_printf(1)("alg_csa_table\n");
    5007             : 
    5008         175 :   mt = check_relmt(nf,mt0);
    5009         161 :   if (!mt) pari_err_TYPE("alg_csa_table", mt0);
    5010         154 :   n = nf_get_degree(nf);
    5011         154 :   D = n*d2;
    5012         154 :   if (d*d != d2)
    5013           7 :     pari_err_DOMAIN("alg_csa_table","(nonsquare) dimension","!=",stoi(d*d),mt);
    5014             : 
    5015         147 :   al = cgetg(12, t_VEC);
    5016         147 :   gel(al,10) = gen_0; /* must be set first */
    5017         147 :   gel(al,1) = zerovec(12); gmael(al,1,10) = nf;
    5018         147 :   gmael(al,1,1) = gpowgs(pol_x(0), d); /* placeholder before splitting field */
    5019         147 :   gel(al,2) = mt;
    5020         147 :   gel(al,3) = gen_0; /* placeholder */
    5021         147 :   gel(al,4) = gel(al,5) = gen_0; /* TODO Hasse invariants if flag&al_FACTOR */
    5022         147 :   gel(al,5) = gel(al,6) = gen_0; /* placeholder */
    5023         147 :   gel(al,7) = matid(D);
    5024         147 :   gel(al,8) = matid(D);
    5025         147 :   gel(al,9) = algnatmultable(al,D);
    5026         147 :   gel(al,11)= algtracebasis(al);
    5027         147 :   if (flag & al_MAXORD) al = alg_maximal(al);
    5028         147 :   computesplitting(al, d, v, flag);
    5029         147 :   return gerepilecopy(av, al);
    5030             : }
    5031             : 
    5032             : static GEN
    5033       38003 : algtableinit_i(GEN mt0, GEN p)
    5034             : {
    5035             :   GEN al, mt;
    5036             :   long i, n;
    5037             : 
    5038       38003 :   if (p && !signe(p)) p = NULL;
    5039       38003 :   mt = check_mt(mt0,p);
    5040       38003 :   if (!mt) pari_err_TYPE("algtableinit", mt0);
    5041       37996 :   if (!p && !isint1(Q_denom(mt0)))
    5042           7 :     pari_err_DOMAIN("algtableinit", "denominator(mt)", "!=", gen_1, mt0);
    5043       37989 :   n = lg(mt)-1;
    5044       37989 :   al = cgetg(12, t_VEC);
    5045      265923 :   for (i=1; i<=6; i++) gel(al,i) = gen_0;
    5046       37989 :   gel(al,7) = matid(n);
    5047       37989 :   gel(al,8) = matid(n);
    5048       37989 :   gel(al,9) = mt;
    5049       37989 :   gel(al,10) = p? p: gen_0;
    5050       37989 :   gel(al,11)= algtracebasis(al);
    5051       37989 :   return al;
    5052             : }
    5053             : GEN
    5054        4200 : algtableinit(GEN mt0, GEN p)
    5055             : {
    5056        4200 :   pari_sp av = avma;
    5057        4200 :   if (p)
    5058             :   {
    5059        4074 :     if (typ(p) != t_INT) pari_err_TYPE("algtableinit",p);
    5060        4067 :     if (signe(p) && !BPSW_psp(p)) pari_err_PRIME("algtableinit",p);
    5061             :   }
    5062        4179 :   return gerepilecopy(av, algtableinit_i(mt0, p));
    5063             : }
    5064             : 
    5065             : /** REPRESENTATIONS OF GROUPS **/
    5066             : 
    5067             : static GEN
    5068         294 : list_to_regular_rep(GEN elts, long n)
    5069             : {
    5070             :   GEN reg, elts2, g;
    5071             :   long i,j;
    5072         294 :   elts = shallowcopy(elts);
    5073         294 :   gen_sort_inplace(elts, (void*)&vecsmall_lexcmp, &cmp_nodata, NULL);
    5074         294 :   reg = cgetg(n+1, t_VEC);
    5075         294 :   gel(reg,1) = identity_perm(n);
    5076        3857 :   for (i=2; i<=n; i++) {
    5077        3563 :     g = perm_inv(gel(elts,i));
    5078        3563 :     elts2 = cgetg(n+1, t_VEC);
    5079       74543 :     for (j=1; j<=n; j++) gel(elts2,j) = perm_mul(g,gel(elts,j));
    5080        3563 :     gen_sort_inplace(elts2, (void*)&vecsmall_lexcmp, &cmp_nodata, &gel(reg,i));
    5081             :   }
    5082         294 :   return reg;
    5083             : }
    5084             : 
    5085             : static GEN
    5086        3857 : matrix_perm(GEN perm, long n)
    5087             : {
    5088             :   GEN m;
    5089             :   long j;
    5090        3857 :   m = cgetg(n+1, t_MAT);
    5091       78694 :   for (j=1; j<=n; j++) {
    5092       74837 :     gel(m,j) = col_ei(n,perm[j]);
    5093             :   }
    5094        3857 :   return m;
    5095             : }
    5096             : 
    5097             : GEN
    5098         847 : conjclasses_algcenter(GEN cc, GEN p)
    5099             : {
    5100         847 :   GEN mt, elts = gel(cc,1), conjclass = gel(cc,2), rep = gel(cc,3), card;
    5101         847 :   long i, nbcl = lg(rep)-1, n = lg(elts)-1;
    5102             :   pari_sp av;
    5103             : 
    5104         847 :   card = zero_Flv(nbcl);
    5105       14819 :   for (i=1; i<=n; i++) card[conjclass[i]]++;
    5106             : 
    5107             :   /* multiplication table of the center of Z[G] (class functions) */
    5108         847 :   mt = cgetg(nbcl+1,t_VEC);
    5109        7217 :   for (i=1;i<=nbcl;i++) gel(mt,i) = zero_Flm_copy(nbcl,nbcl);
    5110         847 :   av = avma;
    5111        7217 :   for (i=1;i<=nbcl;i++)
    5112             :   {
    5113        6370 :     GEN xi = gel(elts,rep[i]), mi = gel(mt,i);
    5114             :     long j,k;
    5115      132244 :     for (j=1;j<=n;j++)
    5116             :     {
    5117      125874 :       GEN xj = gel(elts,j);
    5118      125874 :       k = vecsearch(elts, perm_mul(xi,xj), NULL);
    5119      125874 :       ucoeff(mi, conjclass[k], conjclass[j])++;
    5120             :     }
    5121       70238 :     for (k=1; k<=nbcl; k++)
    5122      852362 :       for (j=1; j<=nbcl; j++)
    5123             :       {
    5124      788494 :         ucoeff(mi,k,j) *= card[i];
    5125      788494 :         ucoeff(mi,k,j) /= card[k];
    5126             :       }
    5127        6370 :     set_avma(av);
    5128             :   }
    5129        7217 :   for (i=1;i<=nbcl;i++) gel(mt,i) = Flm_to_ZM(gel(mt,i));
    5130         847 :   return algtableinit_i(mt,p);
    5131             : }
    5132             : 
    5133             : GEN
    5134         329 : alggroupcenter(GEN G, GEN p, GEN *pcc)
    5135             : {
    5136         329 :   pari_sp av = avma;
    5137         329 :   GEN cc = group_to_cc(G), al = conjclasses_algcenter(cc, p);
    5138         315 :   if (!pcc) return gerepilecopy(av,al);
    5139           7 :   *pcc = cc; return gc_all(av, 2, &al, pcc);
    5140             : }
    5141             : 
    5142             : static GEN
    5143         294 : groupelts_algebra(GEN elts, GEN p)
    5144             : {
    5145         294 :   pari_sp av = avma;
    5146             :   GEN mt;
    5147         294 :   long i, n = lg(elts)-1;
    5148         294 :   elts = list_to_regular_rep(elts,n);
    5149         294 :   mt = cgetg(n+1, t_VEC);
    5150        4151 :   for (i=1; i<=n; i++) gel(mt,i) = matrix_perm(gel(elts,i),n);
    5151         294 :   return gerepilecopy(av, algtableinit_i(mt,p));
    5152             : }
    5153             : 
    5154             : GEN
    5155         329 : alggroup(GEN gal, GEN p)
    5156             : {
    5157         329 :   GEN elts = checkgroupelts(gal);
    5158         294 :   return groupelts_algebra(elts, p);
    5159             : }
    5160             : 
    5161             : /** MAXIMAL ORDER **/
    5162             : 
    5163             : static GEN
    5164       51461 : mattocol(GEN M, long n)
    5165             : {
    5166       51461 :   GEN C = cgetg(n*n+1, t_COL);
    5167             :   long i,j,ic;
    5168       51461 :   ic = 1;
    5169      899534 :   for (i=1; i<=n; i++)
    5170    19807952 :   for (j=1; j<=n; j++, ic++) gel(C,ic) = gcoeff(M,i,j);
    5171       51461 :   return C;
    5172             : }
    5173             : 
    5174             : /* Ip is a lift of a left O/pO-ideal where O is the integral basis of al */
    5175             : static GEN
    5176        4775 : algleftordermodp(GEN al, GEN Ip, GEN p)
    5177             : {
    5178        4775 :   pari_sp av = avma;
    5179             :   GEN I, Ii, M, mt, K, imi, p2;
    5180             :   long n, i;
    5181        4775 :   n = alg_get_absdim(al);
    5182        4775 :   mt = alg_get_multable(al);
    5183        4775 :   p2 = sqri(p);
    5184             : 
    5185        4775 :   I = ZM_hnfmodid(Ip, p);
    5186        4775 :   Ii = ZM_inv(I,NULL);
    5187             : 
    5188        4775 :   M = cgetg(n+1, t_MAT);
    5189       56236 :   for (i=1; i<=n; i++) {
    5190       51461 :     imi = FpM_mul(Ii, FpM_mul(gel(mt,i), I, p2), p2);
    5191       51461 :     imi = ZM_Z_divexact(imi, p);
    5192       51461 :     gel(M,i) = mattocol(imi, n);
    5193             :   }
    5194        4775 :   K = FpM_ker(M, p);
    5195        4775 :   if (lg(K)==1) { set_avma(av); return matid(n); }
    5196        1815 :   K = ZM_hnfmodid(K,p);
    5197             : 
    5198        1815 :   return gerepileupto(av, ZM_Z_div(K,p));
    5199             : }
    5200             : 
    5201             : static GEN
    5202        6796 : alg_ordermodp(GEN al, GEN p)
    5203             : {
    5204             :   GEN alp;
    5205        6796 :   long i, N = alg_get_absdim(al);
    5206        6796 :   alp = cgetg(12, t_VEC);
    5207       61164 :   for (i=1; i<=8; i++) gel(alp,i) = gen_0;
    5208        6796 :   gel(alp,9) = cgetg(N+1, t_VEC);
    5209       69764 :   for (i=1; i<=N; i++) gmael(alp,9,i) = FpM_red(gmael(al,9,i), p);
    5210        6796 :   gel(alp,10) = p;
    5211        6796 :   gel(alp,11) = cgetg(N+1, t_VEC);
    5212       69764 :   for (i=1; i<=N; i++) gmael(alp,11,i) = Fp_red(gmael(al,11,i), p);
    5213             : 
    5214        6796 :   return alp;
    5215             : }
    5216             : 
    5217             : static GEN
    5218        3865 : algpradical_i(GEN al, GEN p, GEN zprad, GEN projs)
    5219             : {
    5220        3865 :   pari_sp av = avma;
    5221        3865 :   GEN alp = alg_ordermodp(al, p), liftrad, projrad, alq, alrad, res, Lalp, radq;
    5222             :   long i;
    5223        3865 :   if (lg(zprad)==1) {
    5224        2854 :     liftrad = NULL;
    5225        2854 :     projrad = NULL;
    5226             :   }
    5227             :   else {
    5228        1011 :     alq = alg_quotient(alp, zprad, 1);
    5229        1011 :     alp = gel(alq,1);
    5230        1011 :     projrad = gel(alq,2);
    5231        1011 :     liftrad = gel(alq,3);
    5232             :   }
    5233             : 
    5234        3865 :   if (projs) {
    5235         572 :     if (projrad) {
    5236          28 :       projs = gcopy(projs);
    5237          84 :       for (i=1; i<lg(projs); i++)
    5238          56 :         gel(projs,i) = FpM_FpC_mul(projrad, gel(projs,i), p);
    5239             :     }
    5240         572 :     Lalp = alg_centralproj(alp, projs, 1);
    5241             : 
    5242         572 :     alrad = cgetg(lg(Lalp),t_VEC);
    5243        2088 :     for (i=1; i<lg(Lalp); i++) {
    5244        1516 :       alq = gel(Lalp,i);
    5245        1516 :       radq = algradical(gel(alq,1));
    5246        1516 :       if (gequal0(radq))
    5247         880 :         gel(alrad,i) = cgetg(1,t_MAT);
    5248             :       else {
    5249         636 :         radq = FpM_mul(gel(alq,3),radq,p);
    5250         636 :         gel(alrad,i) = radq;
    5251             :       }
    5252             :     }
    5253         572 :     alrad = shallowmatconcat(alrad);
    5254         572 :     alrad = FpM_image(alrad,p);
    5255             :   }
    5256        3293 :   else alrad = algradical(alp);
    5257             : 
    5258        3865 :   if (!gequal0(alrad)) {
    5259        3061 :     if (liftrad) alrad = FpM_mul(liftrad, alrad, p);
    5260        3061 :     res = shallowmatconcat(mkvec2(alrad, zprad));
    5261        3061 :     res = FpM_image(res,p);
    5262             :   }
    5263         804 :   else res = lg(zprad)==1 ? gen_0 : zprad;
    5264        3865 :   return gerepilecopy(av, res);
    5265             : }
    5266             : 
    5267             : static GEN
    5268        2931 : algpdecompose0(GEN al, GEN prad, GEN p, GEN projs)
    5269             : {
    5270        2931 :   pari_sp av = avma;
    5271        2931 :   GEN alp, quo, ss, liftm = NULL, projm = NULL, dec, res, I, Lss, deci;
    5272             :   long i, j;
    5273             : 
    5274        2931 :   alp = alg_ordermodp(al, p);
    5275        2931 :   if (!gequal0(prad)) {
    5276        2400 :     quo = alg_quotient(alp, prad, 1);
    5277        2400 :     ss = gel(quo,1);
    5278        2400 :     projm = gel(quo,2);
    5279        2400 :     liftm = gel(quo,3);
    5280             :   }
    5281         531 :   else ss = alp;
    5282             : 
    5283        2931 :   if (projs) {
    5284         502 :     if (projm) {
    5285        1263 :       for (i=1; i<lg(projs); i++)
    5286         914 :         gel(projs,i) = FpM_FpC_mul(projm, gel(projs,i), p);
    5287             :     }
    5288         502 :     Lss = alg_centralproj(ss, projs, 1);
    5289             : 
    5290         502 :     dec = cgetg(lg(Lss),t_VEC);
    5291        1857 :     for (i=1; i<lg(Lss); i++) {
    5292        1355 :       gel(dec,i) = algsimpledec_ss(gmael(Lss,i,1), 1);
    5293        1355 :       deci = gel(dec,i);
    5294        3066 :       for (j=1; j<lg(deci); j++)
    5295        1711 :        gmael(deci,j,3) = FpM_mul(gmael(Lss,i,3), gmael(deci,j,3), p);
    5296             :     }
    5297         502 :     dec = shallowconcat1(dec);
    5298             :   }
    5299        2429 :   else dec = algsimpledec_ss(ss,1);
    5300             : 
    5301        2931 :   res = cgetg(lg(dec),t_VEC);
    5302        7834 :   for (i=1; i<lg(dec); i++) {
    5303        4903 :     I = gmael(dec,i,3);
    5304        4903 :     if (liftm) I = FpM_mul(liftm,I,p);
    5305        4903 :     I = shallowmatconcat(mkvec2(I,prad));
    5306        4903 :     gel(res,i) = I;
    5307             :   }
    5308             : 
    5309        2931 :   return gerepilecopy(av, res);
    5310             : }
    5311             : 
    5312             : /* finds a nontrivial ideal of O/prad or gen_0 if there is none. */
    5313             : static GEN
    5314         881 : algpdecompose_i(GEN al, GEN p, GEN zprad, GEN projs)
    5315             : {
    5316         881 :   pari_sp av = avma;
    5317         881 :   GEN prad = algpradical_i(al,p,zprad,projs);
    5318         881 :   return gerepileupto(av, algpdecompose0(al, prad, p, projs));
    5319             : }
    5320             : 
    5321             : /* ord is assumed to be in hnf wrt the integral basis of al. */
    5322             : /* assumes that alg_get_invbasis(al) is integral. */
    5323             : static GEN
    5324        1815 : alg_change_overorder_shallow(GEN al, GEN ord)
    5325             : {
    5326             :   GEN al2, mt, iord, mtx, den, den2, div;
    5327             :   long i, n;
    5328        1815 :   n = alg_get_absdim(al);
    5329             : 
    5330        1815 :   iord = QM_inv(ord);
    5331        1815 :   al2 = shallowcopy(al);
    5332        1815 :   ord = Q_remove_denom(ord,&den);
    5333             : 
    5334        1815 :   gel(al2,7) = Q_remove_denom(gel(al,7), &den2);
    5335        1815 :   if (den2) div = mulii(den,den2);
    5336         693 :   else      div = den;
    5337        1815 :   gel(al2,7) = ZM_Z_div(ZM_mul(gel(al2,7), ord), div);
    5338             : 
    5339        1815 :   gel(al2,8) = ZM_mul(iord, gel(al,8));
    5340             : 
    5341        1815 :   mt = cgetg(n+1,t_VEC);
    5342        1815 :   gel(mt,1) = matid(n);
    5343        1815 :   div = sqri(den);
    5344       19846 :   for (i=2; i<=n; i++) {
    5345       18031 :     mtx = algbasismultable(al,gel(ord,i));
    5346       18031 :     gel(mt,i) = ZM_mul(iord, ZM_mul(mtx, ord));
    5347       18031 :     gel(mt,i) = ZM_Z_divexact(gel(mt,i), div);
    5348             :   }
    5349        1815 :   gel(al2,9) = mt;
    5350             : 
    5351        1815 :   gel(al2,11) = algtracebasis(al2);
    5352             : 
    5353        1815 :   return al2;
    5354             : }
    5355             : 
    5356             : static GEN
    5357       12209 : algfromcenter(GEN al, GEN x)
    5358             : {
    5359       12209 :   GEN nf = alg_get_center(al);
    5360             :   long n;
    5361       12209 :   switch(alg_type(al)) {
    5362       11075 :     case al_CYCLIC:
    5363       11075 :       n = alg_get_degree(al);
    5364       11075 :       break;
    5365        1134 :     case al_CSA:
    5366        1134 :       n = alg_get_dim(al);
    5367        1134 :       break;
    5368             :     default: return NULL; /*LCOV_EXCL_LINE*/
    5369             :   }
    5370       12209 :   return algalgtobasis(al, scalarcol(basistoalg(nf, x), n));
    5371             : }
    5372             : 
    5373             : /* x is an ideal of the center in hnf form */
    5374             : static GEN
    5375        3865 : algfromcenterhnf(GEN al, GEN x)
    5376             : {
    5377             :   GEN res;
    5378             :   long i;
    5379        3865 :   res = cgetg(lg(x), t_MAT);
    5380       11265 :   for (i=1; i<lg(x); i++) gel(res,i) = algfromcenter(al, gel(x,i));
    5381        3865 :   return res;
    5382             : }
    5383             : 
    5384             : /* assumes al is CSA or CYCLIC */
    5385             : static GEN
    5386        2050 : algcenter_precompute(GEN al, GEN p)
    5387             : {
    5388        2050 :   GEN fa, pdec, nfprad, projs, nf = alg_get_center(al);
    5389             :   long i, np;
    5390             : 
    5391        2050 :   pdec = idealprimedec(nf, p);
    5392        2050 :   settyp(pdec, t_COL);
    5393        2050 :   np = lg(pdec)-1;
    5394        2050 :   fa = mkmat2(pdec, const_col(np, gen_1));
    5395        2050 :   if (dvdii(nf_get_disc(nf), p))
    5396         336 :     nfprad = idealprodprime(nf, pdec);
    5397             :   else
    5398        1714 :     nfprad = scalarmat_shallow(p, nf_get_degree(nf));
    5399        2050 :   fa = idealchineseinit(nf, fa);
    5400        2050 :   projs = cgetg(np+1, t_VEC);
    5401        4528 :   for (i=1; i<=np; i++) gel(projs, i) = idealchinese(nf, fa, vec_ei(np,i));
    5402        2050 :   return mkvec2(nfprad, projs);
    5403             : }
    5404             : 
    5405             : static GEN
    5406        3865 : algcenter_prad(GEN al, GEN p, GEN pre)
    5407             : {
    5408             :   GEN nfprad, zprad, mtprad;
    5409             :   long i;
    5410        3865 :   nfprad = gel(pre,1);
    5411        3865 :   zprad = algfromcenterhnf(al, nfprad);
    5412        3865 :   zprad = FpM_image(zprad, p);
    5413        3865 :   mtprad = cgetg(lg(zprad), t_VEC);
    5414        5409 :   for (i=1; i<lg(zprad); i++) gel(mtprad, i) = algbasismultable(al, gel(zprad,i));
    5415        3865 :   mtprad = shallowmatconcat(mtprad);
    5416        3865 :   zprad = FpM_image(mtprad, p);
    5417        3865 :   return zprad;
    5418             : }
    5419             : 
    5420             : static GEN
    5421        3865 : algcenter_p_projs(GEN al, GEN p, GEN pre)
    5422             : {
    5423             :   GEN projs, zprojs;
    5424             :   long i;
    5425        3865 :   projs = gel(pre,2);
    5426        3865 :   zprojs = cgetg(lg(projs), t_VEC);
    5427        8674 :   for (i=1; i<lg(projs); i++) gel(zprojs,i) = FpC_red(algfromcenter(al, gel(projs,i)),p);
    5428        3865 :   return zprojs;
    5429             : }
    5430             : 
    5431             : /* al is assumed to be simple */
    5432             : static GEN
    5433        2050 : alg_pmaximal(GEN al, GEN p)
    5434             : {
    5435             :   pari_sp av;
    5436        2050 :   long n = alg_get_absdim(al);
    5437        2050 :   GEN id = matid(n), al2 = al, prad, lord = gen_0, dec, zprad, projs, pre;
    5438             : 
    5439        2050 :   dbg_printf(0)("Round 2 (noncommutative) at p=%Ps, dim=%d\n", p, n);
    5440        2050 :   pre = algcenter_precompute(al,p); av = avma;
    5441             :   while (1) {
    5442        2984 :     zprad = algcenter_prad(al2, p, pre);
    5443        2984 :     projs = algcenter_p_projs(al2, p, pre);
    5444        2984 :     if (lg(projs) == 2) projs = NULL;
    5445        2984 :     prad = algpradical_i(al2,p,zprad,projs);
    5446        2984 :     if (typ(prad) == t_INT) break;
    5447        2956 :     lord = algleftordermodp(al2,prad,p);
    5448        2956 :     if (!cmp_universal(lord,id)) break;
    5449         934 :     al2 = gerepilecopy(av, alg_change_overorder_shallow(al2,lord));
    5450             :   }
    5451             : 
    5452        2050 :   dec = algpdecompose0(al2,prad,p,projs); av = avma;
    5453        2931 :   while (lg(dec) > 2) {
    5454             :     long i;
    5455        2098 :     for (i = 1; i < lg(dec); i++) {
    5456        1819 :       GEN I = gel(dec,i);
    5457        1819 :       lord = algleftordermodp(al2,I,p);
    5458        1819 :       if (cmp_universal(lord,id)) break;
    5459             :     }
    5460        1160 :     if (i==lg(dec)) break;
    5461         881 :     al2 = gerepilecopy(av, alg_change_overorder_shallow(al2,lord));
    5462         881 :     zprad = algcenter_prad(al2, p, pre);
    5463         881 :     projs = algcenter_p_projs(al2, p, pre);
    5464         881 :     if (lg(projs) == 2) projs = NULL;
    5465         881 :     dec = algpdecompose_i(al2,p,zprad,projs);
    5466             :   }
    5467        2050 :   return al2;
    5468             : }
    5469             : 
    5470             : static GEN
    5471        6412 : algtracematrix(GEN al)
    5472             : {
    5473             :   GEN M, mt;
    5474             :   long n, i, j;
    5475        6412 :   n = alg_get_absdim(al);
    5476        6412 :   mt = alg_get_multable(al);
    5477        6412 :   M = cgetg(n+1, t_MAT);
    5478       48239 :   for (i=1; i<=n; i++)
    5479             :   {
    5480       41827 :     gel(M,i) = cgetg(n+1,t_MAT);
    5481      287624 :     for (j=1; j<=i; j++)
    5482      245797 :       gcoeff(M,j,i) = gcoeff(M,i,j) = algabstrace(al,gmael(mt,i,j));
    5483             :   }
    5484        6412 :   return M;
    5485             : }
    5486             : static GEN
    5487         161 : algdisc_i(GEN al) { return ZM_det(algtracematrix(al)); }
    5488             : GEN
    5489          49 : algdisc(GEN al)
    5490             : {
    5491          49 :   pari_sp av = avma;
    5492          49 :   checkalg(al);
    5493          49 :   if (alg_type(al) == al_REAL) pari_err_TYPE("algdisc [real algebra]", al);
    5494          28 :   return gerepileuptoint(av, algdisc_i(al));
    5495             : }
    5496             : static GEN
    5497         133 : alg_maximal(GEN al)
    5498             : {
    5499         133 :   GEN fa = absZ_factor(algdisc_i(al));
    5500         133 :   return alg_maximal_primes(al, gel(fa,1));
    5501             : }
    5502             : 
    5503             : /** LATTICES **/
    5504             : 
    5505             : /*
    5506             :  Convention: lattice = [I,t] representing t*I, where
    5507             :  - I integral nonsingular upper-triangular matrix representing a lattice over
    5508             :    the integral basis of the algebra, and
    5509             :  - t>0 either an integer or a rational number.
    5510             : 
    5511             :  Recommended and returned by the functions below:
    5512             :  - I HNF and primitive
    5513             : */
    5514             : 
    5515             : /* TODO use hnfmodid whenever possible using a*O <= I <= O
    5516             :  * for instance a = ZM_det_triangular(I) */
    5517             : 
    5518             : static GEN
    5519       63343 : primlat(GEN lat)
    5520             : {
    5521             :   GEN m, t, c;
    5522       63343 :   m = alglat_get_primbasis(lat);
    5523       63343 :   t = alglat_get_scalar(lat);
    5524       63343 :   m = Q_primitive_part(m,&c);
    5525       63343 :   if (c) return mkvec2(m,gmul(t,c));
    5526       53809 :   return lat;
    5527             : }
    5528             : 
    5529             : /* assumes the lattice contains d * integral basis, d=0 allowed */
    5530             : GEN
    5531       51072 : alglathnf(GEN al, GEN m, GEN d)
    5532             : {
    5533       51072 :   pari_sp av = avma;
    5534             :   long N,i,j;
    5535             :   GEN m2, c;
    5536       51072 :   checkalg(al);
    5537       51072 :   if (alg_type(al) == al_REAL) pari_err_TYPE("alglathnf [real algebra]", al);
    5538       51065 :   N = alg_get_absdim(al);
    5539       51065 :   if (!d) d = gen_0;
    5540       51065 :   if (typ(m) == t_VEC) m = matconcat(m);
    5541       51065 :   if (typ(m) == t_COL) m = algleftmultable(al,m);
    5542       51065 :   if (typ(m) != t_MAT) pari_err_TYPE("alglathnf",m);
    5543       51058 :   if (typ(d) != t_FRAC && typ(d) != t_INT) pari_err_TYPE("alglathnf",d);
    5544       51058 :   if (lg(m)-1 < N || lg(gel(m,1))-1 != N) pari_err_DIM("alglathnf");
    5545      459242 :   for (i=1; i<=N; i++)
    5546     6820758 :     for (j=1; j<lg(m); j++)
    5547     6412546 :       if (typ(gcoeff(m,i,j)) != t_FRAC && typ(gcoeff(m,i,j)) != t_INT)
    5548           7 :         pari_err_TYPE("alglathnf", gcoeff(m,i,j));
    5549       51023 :   m2 = Q_primitive_part(m,&c);
    5550       51023 :   if (!c) c = gen_1;
    5551       51023 :   if (!signe(d)) d = detint(m2);
    5552       45593 :   else           d = gdiv(d,c); /* should be an integer */
    5553       51023 :   if (!signe(d)) pari_err_INV("alglathnf [m does not have full rank]", m2);
    5554       51009 :   m2 = ZM_hnfmodid(m2,d);
    5555       51009 :   return gerepilecopy(av, mkvec2(m2,c));
    5556             : }
    5557             : 
    5558             : static GEN
    5559       10689 : prepare_multipliers(GEN *a, GEN *b)
    5560             : {
    5561             :   GEN na, nb, da, db, d;
    5562       10689 :   na = numer_i(*a); da = denom_i(*a);
    5563       10689 :   nb = numer_i(*b); db = denom_i(*b);
    5564       10689 :   na = mulii(na,db);
    5565       10689 :   nb = mulii(nb,da);
    5566       10689 :   d = gcdii(na,nb);
    5567       10689 :   *a = diviiexact(na,d);
    5568       10689 :   *b = diviiexact(nb,d);
    5569       10689 :   return gdiv(d, mulii(da,db));
    5570             : }
    5571             : 
    5572             : static GEN
    5573       10689 : prepare_lat(GEN m1, GEN t1, GEN m2, GEN t2)
    5574             : {
    5575       10689 :   GEN d = prepare_multipliers(&t1, &t2);
    5576       10689 :   m1 = ZM_Z_mul(m1,t1);
    5577       10689 :   m2 = ZM_Z_mul(m2,t2);
    5578       10689 :   return mkvec3(m1,m2,d);
    5579             : }
    5580             : 
    5581             : static GEN
    5582       10703 : alglataddinter(GEN al, GEN lat1, GEN lat2, GEN *sum, GEN *inter)
    5583             : {
    5584             :   GEN d, m1, m2, t1, t2, M, prep, d1, d2, ds, di, K;
    5585       10703 :   checkalg(al);
    5586       10703 :   if (alg_type(al) == al_REAL)
    5587          14 :     pari_err_TYPE("alglataddinter [real algebra]", al);
    5588       10689 :   checklat(al,lat1);
    5589       10689 :   checklat(al,lat2);
    5590             : 
    5591       10689 :   m1 = alglat_get_primbasis(lat1);
    5592       10689 :   t1 = alglat_get_scalar(lat1);
    5593       10689 :   m2 = alglat_get_primbasis(lat2);
    5594       10689 :   t2 = alglat_get_scalar(lat2);
    5595       10689 :   prep = prepare_lat(m1, t1, m2, t2);
    5596       10689 :   m1 = gel(prep,1);
    5597       10689 :   m2 = gel(prep,2);
    5598       10689 :   d = gel(prep,3);
    5599       10689 :   M = matconcat(mkvec2(m1,m2));
    5600       10689 :   d1 = ZM_det_triangular(m1);
    5601       10689 :   d2 = ZM_det_triangular(m2);
    5602       10689 :   ds = gcdii(d1,d2);
    5603       10689 :   if (inter)
    5604             :   {
    5605        7112 :     di = diviiexact(mulii(d1,d2),ds);
    5606        7112 :     K = matkermod(M,di,sum);
    5607        7112 :     K = rowslice(K,1,lg(m1));
    5608        7112 :     *inter = hnfmodid(FpM_mul(m1,K,di),di);
    5609        7112 :     if (sum) *sum = hnfmodid(*sum,ds);
    5610             :   }
    5611        3577 :   else *sum = hnfmodid(M,ds);
    5612       10689 :   return d;
    5613             : }
    5614             : 
    5615             : GEN
    5616        3605 : alglatinter(GEN al, GEN lat1, GEN lat2, GEN* psum)
    5617             : {
    5618        3605 :   pari_sp av = avma;
    5619             :   GEN inter, d;
    5620        3605 :   d = alglataddinter(al, lat1, lat2, psum, &inter);
    5621        3598 :   inter = primlat(mkvec2(inter, d));
    5622        3598 :   if (!psum) return gerepilecopy(av, inter);
    5623          14 :   *psum = primlat(mkvec2(*psum,d));
    5624          14 :   return gc_all(av, 2, &inter, psum);
    5625             : }
    5626             : 
    5627             : GEN
    5628        7098 : alglatadd(GEN al, GEN lat1, GEN lat2, GEN* pinter)
    5629             : {
    5630        7098 :   pari_sp av = avma;
    5631             :   GEN sum, d;
    5632        7098 :   d = alglataddinter(al, lat1, lat2, &sum, pinter);
    5633        7091 :   sum = primlat(mkvec2(sum, d));
    5634        7091 :   if (!pinter) return gerepilecopy(av, sum);
    5635        3514 :   *pinter = primlat(mkvec2(*pinter,d));
    5636        3514 :   return gc_all(av, 2, &sum, pinter);
    5637             : }
    5638             : 
    5639             : /* TODO version that returns the quotient as abelian group? */
    5640             : /* return matrices to convert coordinates from one to other? */
    5641             : int
    5642       31556 : alglatsubset(GEN al, GEN lat1, GEN lat2, GEN* pindex)
    5643             : {
    5644       31556 :   pari_sp av = avma;
    5645             :   int res;
    5646             :   GEN m1, m2, m2i, m, t;
    5647       31556 :   checkalg(al);
    5648       31556 :   if (alg_type(al) == al_REAL) pari_err_TYPE("alglatsubset [real algebra]", al);
    5649       31549 :   checklat(al,lat1);
    5650       31549 :   checklat(al,lat2);
    5651       31549 :   m1 = alglat_get_primbasis(lat1);
    5652       31549 :   m2 = alglat_get_primbasis(lat2);
    5653       31549 :   m2i = RgM_inv_upper(m2);
    5654       31549 :   t = gdiv(alglat_get_scalar(lat1), alglat_get_scalar(lat2));
    5655       31549 :   m = RgM_Rg_mul(RgM_mul(m2i,m1), t);
    5656       31549 :   res = RgM_is_ZM(m);
    5657       31549 :   if (!res || !pindex) return gc_int(av, res);
    5658        1757 :   *pindex = gerepileuptoint(av, mpabs(ZM_det_triangular(m)));
    5659        1757 :   return 1;
    5660             : }
    5661             : 
    5662             : GEN
    5663        5271 : alglatindex(GEN al, GEN lat1, GEN lat2)
    5664             : {
    5665        5271 :   pari_sp av = avma;
    5666             :   long N;
    5667             :   GEN res;
    5668        5271 :   checkalg(al);
    5669        5271 :   if (alg_type(al) == al_REAL) pari_err_TYPE("alglatindex [real algebra]", al);
    5670        5264 :   checklat(al,lat1);
    5671        5264 :   checklat(al,lat2);
    5672        5264 :   N = alg_get_absdim(al);
    5673        5264 :   res = alglat_get_scalar(lat1);
    5674        5264 :   res = gdiv(res, alglat_get_scalar(lat2));
    5675        5264 :   res = gpowgs(res, N);
    5676        5264 :   res = gmul(res,RgM_det_triangular(alglat_get_primbasis(lat1)));
    5677        5264 :   res = gdiv(res, RgM_det_triangular(alglat_get_primbasis(lat2)));
    5678        5264 :   res = gabs(res,0);
    5679        5264 :   return gerepilecopy(av, res);
    5680             : }
    5681             : 
    5682             : GEN
    5683       45612 : alglatmul(GEN al, GEN lat1, GEN lat2)
    5684             : {
    5685       45612 :   pari_sp av = avma;
    5686             :   long N,i;
    5687             :   GEN m1, m2, m, V, lat, t, d, dp;
    5688       45612 :   checkalg(al);
    5689       45612 :   if (alg_type(al) == al_REAL) pari_err_TYPE("alglatmul [real algebra]", al);
    5690       45605 :   if (typ(lat1)==t_COL)
    5691             :   {
    5692       19292 :     if (typ(lat2)==t_COL)
    5693           7 :       pari_err_TYPE("alglatmul [one of lat1, lat2 has to be a lattice]", lat2);
    5694       19285 :     checklat(al,lat2);
    5695       19285 :     lat1 = Q_remove_denom(lat1,&d);
    5696       19285 :     m = algbasismultable(al,lat1);
    5697       19285 :     m2 = alglat_get_primbasis(lat2);
    5698       19285 :     dp = mulii(detint(m),ZM_det_triangular(m2));
    5699       19285 :     m = ZM_mul(m,m2);
    5700       19285 :     t = alglat_get_scalar(lat2);
    5701       19285 :     if (d) t = gdiv(t,d);
    5702             :   }
    5703             :   else /* typ(lat1)!=t_COL */
    5704             :   {
    5705       26313 :     checklat(al,lat1);
    5706       26313 :     if (typ(lat2)==t_COL)
    5707             :     {
    5708       19285 :       lat2 = Q_remove_denom(lat2,&d);
    5709       19285 :       m = algbasisrightmultable(al,lat2);
    5710       19285 :       m1 = alglat_get_primbasis(lat1);
    5711       19285 :       dp = mulii(detint(m),ZM_det_triangular(m1));
    5712       19285 :       m = ZM_mul(m,m1);
    5713       19285 :       t = alglat_get_scalar(lat1);
    5714       19285 :       if (d) t = gdiv(t,d);
    5715             :     }
    5716             :     else /* typ(lat2)!=t_COL */
    5717             :     {
    5718        7028 :       checklat(al,lat2);
    5719        7021 :       N = alg_get_absdim(al);
    5720        7021 :       m1 = alglat_get_primbasis(lat1);
    5721        7021 :       m2 = alglat_get_primbasis(lat2);
    5722        7021 :       dp = mulii(ZM_det_triangular(m1), ZM_det_triangular(m2));
    5723        7021 :       V = cgetg(N+1,t_VEC);
    5724       63189 :       for (i=1; i<=N; i++) {
    5725       56168 :         gel(V,i) = algbasismultable(al,gel(m1,i));
    5726       56168 :         gel(V,i) = ZM_mul(gel(V,i),m2);
    5727             :       }
    5728        7021 :       m = matconcat(V);
    5729        7021 :       t = gmul(alglat_get_scalar(lat1), alglat_get_scalar(lat2));
    5730             :     }
    5731             :   }
    5732             : 
    5733       45591 :   lat = alglathnf(al,m,dp);
    5734       45591 :   gel(lat,2) = gmul(alglat_get_scalar(lat), t);
    5735       45591 :   lat = primlat(lat);
    5736       45591 :   return gerepilecopy(av, lat);
    5737             : }
    5738             : 
    5739             : int
    5740       17528 : alglatcontains(GEN al, GEN lat, GEN x, GEN *ptc)
    5741             : {
    5742       17528 :   pari_sp av = avma;
    5743             :   GEN m, t, sol;
    5744       17528 :   checkalg(al);
    5745       17528 :   if (alg_type(al) == al_REAL)
    5746           7 :     pari_err_TYPE("alglatcontains [real algebra]", al);
    5747       17521 :   checklat(al,lat);
    5748       17521 :   m = alglat_get_primbasis(lat);
    5749       17521 :   t = alglat_get_scalar(lat);
    5750       17521 :   x = RgC_Rg_div(x,t);
    5751       17521 :   if (!RgV_is_ZV(x)) return gc_bool(av,0);
    5752       17521 :   sol = hnf_solve(m,x);
    5753       17521 :   if (!sol) return gc_bool(av,0);
    5754        8771 :   if (!ptc) return gc_bool(av,1);
    5755        8764 :   *ptc = gerepilecopy(av, sol); return 1;
    5756             : }
    5757             : 
    5758             : GEN
    5759        8778 : alglatelement(GEN al, GEN lat, GEN c)
    5760             : {
    5761        8778 :   pari_sp av = avma;
    5762             :   GEN res;
    5763        8778 :   checkalg(al);
    5764        8778 :   if (alg_type(al) == al_REAL)
    5765           7 :     pari_err_TYPE("alglatelement [real algebra]", al);
    5766        8771 :   checklat(al,lat);
    5767        8771 :   if (typ(c)!=t_COL) pari_err_TYPE("alglatelement", c);
    5768        8764 :   res = ZM_ZC_mul(alglat_get_primbasis(lat),c);
    5769        8764 :   res = RgC_Rg_mul(res, alglat_get_scalar(lat));
    5770        8764 :   return gerepilecopy(av,res);
    5771             : }
    5772             : 
    5773             : /* idem QM_invimZ, knowing result is contained in 1/c*Z^n */
    5774             : static GEN
    5775        3535 : QM_invimZ_mod(GEN m, GEN c)
    5776             : {
    5777             :   GEN d, m0, K;
    5778        3535 :   m0 = Q_remove_denom(m, &d);
    5779        3535 :   if (d)    d = mulii(d,c);
    5780          35 :   else      d = c;
    5781        3535 :   K = matkermod(m0, d, NULL);
    5782        3535 :   if (lg(K)==1) K = scalarmat(d, lg(m)-1);
    5783        3493 :   else          K = hnfmodid(K, d);
    5784        3535 :   return RgM_Rg_div(K,c);
    5785             : }
    5786             : 
    5787             : /* If m is injective, computes a Z-basis of the submodule of elements whose
    5788             :  * image under m is integral */
    5789             : static GEN
    5790          14 : QM_invimZ(GEN m)
    5791             : {
    5792          14 :   return RgM_invimage(m, QM_ImQ_hnf(m));
    5793             : }
    5794             : 
    5795             : /* An isomorphism of R-modules M_{m,n}(R) -> R^{m*n} */
    5796             : static GEN
    5797       28322 : mat2col(GEN M, long m, long n)
    5798             : {
    5799             :   long i,j,k,p;
    5800             :   GEN C;
    5801       28322 :   p = m*n;
    5802       28322 :   C = cgetg(p+1,t_COL);
    5803      254702 :   for (i=1,k=1;i<=m;i++)
    5804     2036804 :     for (j=1;j<=n;j++,k++)
    5805     1810424 :       gel(C,k) = gcoeff(M,i,j);
    5806       28322 :   return C;
    5807             : }
    5808             : 
    5809             : static GEN
    5810        3535 : alglattransporter_i(GEN al, GEN lat1, GEN lat2, long right)
    5811             : {
    5812             :   GEN m1, m2, m2i, M, MT, mt, t1, t2, T, c;
    5813             :   long N, i;
    5814        3535 :   N = alg_get_absdim(al);
    5815        3535 :   m1 = alglat_get_primbasis(lat1);
    5816        3535 :   m2 = alglat_get_primbasis(lat2);
    5817        3535 :   m2i = RgM_inv_upper(m2);
    5818        3535 :   c = detint(m1);
    5819        3535 :   t1 = alglat_get_scalar(lat1);
    5820        3535 :   m1 = RgM_Rg_mul(m1,t1);
    5821        3535 :   t2 = alglat_get_scalar(lat2);
    5822        3535 :   m2i = RgM_Rg_div(m2i,t2);
    5823             : 
    5824        3535 :   MT = right? NULL: alg_get_multable(al);
    5825        3535 :   M = cgetg(N+1, t_MAT);
    5826       31815 :   for (i=1; i<=N; i++) {
    5827       28280 :     if (right) mt = algbasisrightmultable(al, vec_ei(N,i));
    5828       14168 :     else       mt = gel(MT,i);
    5829       28280 :     mt = RgM_mul(m2i,mt);
    5830       28280 :     mt = RgM_mul(mt,m1);
    5831       28280 :     gel(M,i) = mat2col(mt, N, N);
    5832             :   }
    5833             : 
    5834        3535 :   c = gdiv(t2,gmul(c,t1));
    5835        3535 :   c = denom_i(c);
    5836        3535 :   T = QM_invimZ_mod(M,c);
    5837        3535 :   return primlat(mkvec2(T,gen_1));
    5838             : }
    5839             : 
    5840             : /*
    5841             :    { x in al | x*lat1 subset lat2}
    5842             : */
    5843             : GEN
    5844        1778 : alglatlefttransporter(GEN al, GEN lat1, GEN lat2)
    5845             : {
    5846        1778 :   pari_sp av = avma;
    5847        1778 :   checkalg(al);
    5848        1778 :   if (alg_type(al) == al_REAL)
    5849           7 :     pari_err_TYPE("alglatlefttransporter [real algebra]", al);
    5850        1771 :   checklat(al,lat1);
    5851        1771 :   checklat(al,lat2);
    5852        1771 :   return gerepilecopy(av, alglattransporter_i(al,lat1,lat2,0));
    5853             : }
    5854             : 
    5855             : /*
    5856             :    { x in al | lat1*x subset lat2}
    5857             : */
    5858             : GEN
    5859        1771 : alglatrighttransporter(GEN al, GEN lat1, GEN lat2)
    5860             : {
    5861        1771 :   pari_sp av = avma;
    5862        1771 :   checkalg(al);
    5863        1771 :   if (alg_type(al) == al_REAL)
    5864           7 :     pari_err_TYPE("alglatrighttransporter [real algebra]", al);
    5865        1764 :   checklat(al,lat1);
    5866        1764 :   checklat(al,lat2);
    5867        1764 :   return gerepilecopy(av, alglattransporter_i(al,lat1,lat2,1));
    5868             : }
    5869             : 
    5870             : GEN
    5871          42 : algmakeintegral(GEN mt0, long maps)
    5872             : {
    5873          42 :   pari_sp av = avma;
    5874             :   long n,i;
    5875             :   GEN m,P,Pi,mt2,mt;
    5876          42 :   n = lg(mt0)-1;
    5877          42 :   mt = check_mt(mt0,NULL);
    5878          42 :   if (!mt) pari_err_TYPE("algmakeintegral", mt0);
    5879          21 :   if (isint1(Q_denom(mt0))) {
    5880           7 :     if (maps) mt = mkvec3(mt,matid(n),matid(n));
    5881           7 :     return gerepilecopy(av,mt);
    5882             :   }
    5883          14 :   dbg_printf(2)(" algmakeintegral: dim=%d, denom=%Ps\n", n, Q_denom(mt0));
    5884          14 :   m = cgetg(n+1,t_MAT);
    5885          56 :   for (i=1;i<=n;i++)
    5886          42 :     gel(m,i) = mat2col(gel(mt,i),n,n);
    5887          14 :   dbg_printf(2)(" computing order, dims m = %d x %d...\n", nbrows(m), lg(m)-1);
    5888          14 :   P = QM_invimZ(m);
    5889          14 :   dbg_printf(2)(" ...done.\n");
    5890          14 :   P = shallowmatconcat(mkvec2(col_ei(n,1),P));
    5891          14 :   P = hnf(P);
    5892          14 :   Pi = RgM_inv(P);
    5893          14 :   mt2 = change_Rgmultable(mt,P,Pi);
    5894          14 :   if (maps) mt2 = mkvec3(mt2,Pi,P); /* mt2, mt->mt2, mt2->mt */
    5895          14 :   return gerepilecopy(av,mt2);
    5896             : }
    5897             : 
    5898             : /** ORDERS **/
    5899             : 
    5900             : /** IDEALS **/
    5901             : 

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