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 :
15 : /*******************************************************************/
16 : /* */
17 : /* BASIC NF OPERATIONS */
18 : /* */
19 : /*******************************************************************/
20 : #include "pari.h"
21 : #include "paripriv.h"
22 :
23 : #define DEBUGLEVEL DEBUGLEVEL_nf
24 :
25 : /*******************************************************************/
26 : /* */
27 : /* OPERATIONS OVER NUMBER FIELD ELEMENTS. */
28 : /* represented as column vectors over the integral basis */
29 : /* */
30 : /*******************************************************************/
31 : static GEN
32 40374135 : get_tab(GEN nf, long *N)
33 : {
34 40374135 : GEN tab = (typ(nf) == t_MAT)? nf: gel(nf,9);
35 40374135 : *N = nbrows(tab); return tab;
36 : }
37 :
38 : /* x != 0, y t_INT. Return x * y (not memory clean if x = 1) */
39 : static GEN
40 1145605780 : _mulii(GEN x, GEN y) {
41 1844922526 : return is_pm1(x)? (signe(x) < 0)? negi(y): y
42 1844745720 : : mulii(x, y);
43 : }
44 :
45 : GEN
46 5834 : tablemul_ei_ej(GEN M, long i, long j)
47 : {
48 : long N;
49 5834 : GEN tab = get_tab(M, &N);
50 5834 : tab += (i-1)*N; return gel(tab,j);
51 : }
52 :
53 : /* Outputs x.ei, where ei is the i-th elt of the algebra basis.
54 : * x an RgV of correct length and arbitrary content (polynomials, scalars...).
55 : * M is the multiplication table ei ej = sum_k M_k^(i,j) ek */
56 : GEN
57 4158 : tablemul_ei(GEN M, GEN x, long i)
58 : {
59 : long j, k, N;
60 : GEN v, tab;
61 :
62 4158 : if (i==1) return gcopy(x);
63 4158 : tab = get_tab(M, &N);
64 4158 : if (typ(x) != t_COL) { v = zerocol(N); gel(v,i) = gcopy(x); return v; }
65 4158 : tab += (i-1)*N; v = cgetg(N+1,t_COL);
66 : /* wi . x = [ sum_j tab[k,j] x[j] ]_k */
67 27020 : for (k=1; k<=N; k++)
68 : {
69 22862 : pari_sp av = avma;
70 22862 : GEN s = gen_0;
71 165396 : for (j=1; j<=N; j++)
72 : {
73 142534 : GEN c = gcoeff(tab,k,j);
74 142534 : if (!gequal0(c)) s = gadd(s, gmul(c, gel(x,j)));
75 : }
76 22862 : gel(v,k) = gc_upto(av,s);
77 : }
78 4158 : return v;
79 : }
80 : /* as tablemul_ei, assume x a ZV of correct length */
81 : GEN
82 24730079 : zk_ei_mul(GEN nf, GEN x, long i)
83 : {
84 : long j, k, N;
85 : GEN v, tab;
86 :
87 24730079 : if (i==1) return ZC_copy(x);
88 24730079 : tab = get_tab(nf, &N); tab += (i-1)*N;
89 24730227 : v = cgetg(N+1,t_COL);
90 174356999 : for (k=1; k<=N; k++)
91 : {
92 149630806 : pari_sp av = avma;
93 149630806 : GEN s = gen_0;
94 2237644121 : for (j=1; j<=N; j++)
95 : {
96 2088154613 : GEN c = gcoeff(tab,k,j);
97 2088154613 : if (signe(c)) s = addii(s, _mulii(c, gel(x,j)));
98 : }
99 149489508 : gel(v,k) = gc_INT(av, s);
100 : }
101 24726193 : return v;
102 : }
103 :
104 : /* table of multiplication by wi in R[w1,..., wN] */
105 : GEN
106 40121 : ei_multable(GEN TAB, long i)
107 : {
108 : long k,N;
109 40121 : GEN m, tab = get_tab(TAB, &N);
110 40121 : tab += (i-1)*N;
111 40121 : m = cgetg(N+1,t_MAT);
112 158521 : for (k=1; k<=N; k++) gel(m,k) = gel(tab,k);
113 40121 : return m;
114 : }
115 :
116 : GEN
117 11379310 : zk_multable(GEN nf, GEN x)
118 : {
119 11379310 : long i, l = lg(x);
120 11379310 : GEN mul = cgetg(l,t_MAT);
121 11379292 : gel(mul,1) = x; /* assume w_1 = 1 */
122 35738886 : for (i=2; i<l; i++) gel(mul,i) = zk_ei_mul(nf,x,i);
123 11375708 : return mul;
124 : }
125 : GEN
126 2058 : multable(GEN M, GEN x)
127 : {
128 : long i, N;
129 : GEN mul;
130 2058 : if (typ(x) == t_MAT) return x;
131 0 : M = get_tab(M, &N);
132 0 : if (typ(x) != t_COL) return scalarmat(x, N);
133 0 : mul = cgetg(N+1,t_MAT);
134 0 : gel(mul,1) = x; /* assume w_1 = 1 */
135 0 : for (i=2; i<=N; i++) gel(mul,i) = tablemul_ei(M,x,i);
136 0 : return mul;
137 : }
138 :
139 : /* x integral in nf; table of multiplication by x in ZK = Z[w1,..., wN].
140 : * Return a t_INT if x is scalar, and a ZM otherwise */
141 : GEN
142 5358550 : zk_scalar_or_multable(GEN nf, GEN x)
143 : {
144 5358550 : long tx = typ(x);
145 5358550 : if (tx == t_MAT || tx == t_INT) return x;
146 5192657 : x = nf_to_scalar_or_basis(nf, x);
147 5192642 : return (typ(x) == t_COL)? zk_multable(nf, x): x;
148 : }
149 :
150 : GEN
151 22497 : nftrace(GEN nf, GEN x)
152 : {
153 22497 : pari_sp av = avma;
154 22497 : nf = checknf(nf);
155 22497 : x = nf_to_scalar_or_basis(nf, x);
156 22479 : x = (typ(x) == t_COL)? RgV_dotproduct(x, gel(nf_get_Tr(nf),1))
157 22500 : : gmulgu(x, nf_get_degree(nf));
158 22500 : return gc_upto(av, x);
159 : }
160 : GEN
161 1050 : rnfelttrace(GEN rnf, GEN x)
162 : {
163 1050 : pari_sp av = avma;
164 1050 : checkrnf(rnf);
165 : /* avoid rnfabstorel special t_POL case misinterpretation */
166 1043 : if (typ(x) == t_POL && varn(x) == rnf_get_varn(rnf))
167 63 : x = gmodulo(x, rnf_get_pol(rnf));
168 1043 : x = rnfeltabstorel(rnf, x);
169 728 : x = (typ(x) == t_POLMOD)? rnfeltdown(rnf, gtrace(x))
170 833 : : gmulgu(x, rnf_get_degree(rnf));
171 833 : return gc_upto(av, x);
172 : }
173 :
174 : static GEN
175 35 : famatQ_to_famatZ(GEN fa)
176 : {
177 35 : GEN E, F, Q, P = gel(fa,1);
178 35 : long i, j, l = lg(P);
179 35 : if (l == 1 || RgV_is_ZV(P)) return fa;
180 7 : Q = cgetg(2*l, t_COL);
181 7 : F = cgetg(2*l, t_COL); E = gel(fa, 2);
182 35 : for (i = j = 1; i < l; i++)
183 : {
184 28 : GEN p = gel(P,i);
185 28 : if (typ(p) == t_INT)
186 14 : { gel(Q, j) = p; gel(F, j) = gel(E, i); j++; }
187 : else
188 : {
189 14 : gel(Q, j) = gel(p,1); gel(F, j) = gel(E, i); j++;
190 14 : gel(Q, j) = gel(p,2); gel(F, j) = negi(gel(E, i)); j++;
191 : }
192 : }
193 7 : setlg(Q, j); setlg(F, j); return mkmat2(Q, F);
194 : }
195 : static GEN
196 35 : famat_cba(GEN fa)
197 : {
198 35 : GEN Q, F, P = gel(fa, 1), E = gel(fa, 2);
199 35 : long i, j, lQ, l = lg(P);
200 35 : if (l == 1) return fa;
201 28 : Q = ZV_cba(P); lQ = lg(Q); settyp(Q, t_COL);
202 28 : F = cgetg(lQ, t_COL);
203 77 : for (j = 1; j < lQ; j++)
204 : {
205 49 : GEN v = gen_0, q = gel(Q,j);
206 49 : if (!equali1(q))
207 203 : for (i = 1; i < l; i++)
208 : {
209 161 : long e = Z_pval(gel(P,i), q);
210 161 : if (e) v = addii(v, muliu(gel(E,i), e));
211 : }
212 49 : gel(F, j) = v;
213 : }
214 28 : return mkmat2(Q, F);
215 : }
216 : static long
217 35 : famat_sign(GEN fa)
218 : {
219 35 : GEN P = gel(fa,1), E = gel(fa,2);
220 35 : long i, l = lg(P), s = 1;
221 126 : for (i = 1; i < l; i++)
222 91 : if (signe(gel(P,i)) < 0 && mpodd(gel(E,i))) s = -s;
223 35 : return s;
224 : }
225 : static GEN
226 35 : famat_abs(GEN fa)
227 : {
228 35 : GEN Q, P = gel(fa,1);
229 : long i, l;
230 35 : Q = cgetg_copy(P, &l);
231 126 : for (i = 1; i < l; i++) gel(Q,i) = absi_shallow(gel(P,i));
232 35 : return mkmat2(Q, gel(fa,2));
233 : }
234 :
235 : /* assume nf is a genuine nf, fa a famat */
236 : static GEN
237 35 : famat_norm(GEN nf, GEN fa)
238 : {
239 35 : pari_sp av = avma;
240 35 : GEN G, g = gel(fa,1);
241 : long i, l, s;
242 :
243 35 : G = cgetg_copy(g, &l);
244 112 : for (i = 1; i < l; i++) gel(G,i) = nfnorm(nf, gel(g,i));
245 35 : fa = mkmat2(G, gel(fa,2));
246 35 : fa = famatQ_to_famatZ(fa);
247 35 : s = famat_sign(fa);
248 35 : fa = famat_reduce(famat_abs(fa));
249 35 : fa = famat_cba(fa);
250 35 : g = factorback(fa);
251 35 : return gc_upto(av, s < 0? gneg(g): g);
252 : }
253 : GEN
254 216424 : nfnorm(GEN nf, GEN x)
255 : {
256 216424 : pari_sp av = avma;
257 : GEN c, den;
258 : long n;
259 216424 : nf = checknf(nf);
260 216424 : n = nf_get_degree(nf);
261 216424 : if (typ(x) == t_MAT) return famat_norm(nf, x);
262 216389 : x = nf_to_scalar_or_basis(nf, x);
263 216386 : if (typ(x)!=t_COL)
264 123495 : return gc_upto(av, gpowgs(x, n));
265 92891 : x = nf_to_scalar_or_alg(nf, Q_primitive_part(x, &c));
266 92893 : x = Q_remove_denom(x, &den);
267 92893 : x = ZX_resultant_all(nf_get_pol(nf), x, den, 0);
268 92893 : return gc_upto(av, c ? gmul(x, gpowgs(c, n)): x);
269 : }
270 :
271 : static GEN
272 119 : to_RgX(GEN P, long vx)
273 : {
274 119 : return varn(P) == vx ? P: scalarpol_shallow(P, vx);
275 : }
276 :
277 : GEN
278 462 : rnfeltnorm(GEN rnf, GEN x)
279 : {
280 462 : pari_sp av = avma;
281 : GEN nf, pol;
282 : long v;
283 462 : checkrnf(rnf);
284 455 : v = rnf_get_varn(rnf);
285 : /* avoid rnfabstorel special t_POL case misinterpretation */
286 455 : if (typ(x) == t_POL && varn(x) == v) x = gmodulo(x, rnf_get_pol(rnf));
287 455 : x = liftpol_shallow(rnfeltabstorel(rnf, x));
288 245 : nf = rnf_get_nf(rnf); pol = rnf_get_pol(rnf);
289 490 : x = (typ(x) == t_POL)
290 119 : ? rnfeltdown(rnf, nfX_resultant(nf,pol,to_RgX(x,v)))
291 245 : : gpowgs(x, rnf_get_degree(rnf));
292 245 : return gc_upto(av, x);
293 : }
294 :
295 : /* x + y in nf */
296 : GEN
297 14282389 : nfadd(GEN nf, GEN x, GEN y)
298 : {
299 14282389 : pari_sp av = avma;
300 : GEN z;
301 :
302 14282389 : nf = checknf(nf);
303 14282389 : x = nf_to_scalar_or_basis(nf, x);
304 14282389 : y = nf_to_scalar_or_basis(nf, y);
305 14282389 : if (typ(x) != t_COL)
306 11157281 : { z = (typ(y) == t_COL)? RgC_Rg_add(y, x): gadd(x,y); }
307 : else
308 3125108 : { z = (typ(y) == t_COL)? RgC_add(x, y): RgC_Rg_add(x, y); }
309 14282388 : return gc_upto(av, z);
310 : }
311 : /* x - y in nf */
312 : GEN
313 1432749 : nfsub(GEN nf, GEN x, GEN y)
314 : {
315 1432749 : pari_sp av = avma;
316 : GEN z;
317 :
318 1432749 : nf = checknf(nf);
319 1432749 : x = nf_to_scalar_or_basis(nf, x);
320 1432749 : y = nf_to_scalar_or_basis(nf, y);
321 1432749 : if (typ(x) != t_COL)
322 1172514 : { z = (typ(y) == t_COL)? Rg_RgC_sub(x,y): gsub(x,y); }
323 : else
324 260235 : { z = (typ(y) == t_COL)? RgC_sub(x,y): RgC_Rg_sub(x,y); }
325 1432749 : return gc_upto(av, z);
326 : }
327 :
328 : /* product of ZC x,y in (true) nf; ( sum_i x_i sum_j y_j m^{i,j}_k )_k */
329 : static GEN
330 8246987 : nfmuli_ZC(GEN nf, GEN x, GEN y)
331 : {
332 : long i, j, k, N;
333 8246987 : GEN TAB = get_tab(nf, &N), v = cgetg(N+1,t_COL);
334 :
335 41449441 : for (k = 1; k <= N; k++)
336 : {
337 33202541 : pari_sp av = avma;
338 33202541 : GEN s, TABi = TAB;
339 33202541 : if (k == 1)
340 8246981 : s = mulii(gel(x,1),gel(y,1));
341 : else
342 24955379 : s = addii(mulii(gel(x,1),gel(y,k)),
343 24955560 : mulii(gel(x,k),gel(y,1)));
344 223282281 : for (i=2; i<=N; i++)
345 : {
346 190083531 : GEN t, xi = gel(x,i);
347 190083531 : TABi += N;
348 190083531 : if (!signe(xi)) continue;
349 :
350 94603131 : t = NULL;
351 1086783280 : for (j=2; j<=N; j++)
352 : {
353 992181897 : GEN p1, c = gcoeff(TABi, k, j); /* m^{i,j}_k */
354 992181897 : if (!signe(c)) continue;
355 294239916 : p1 = _mulii(c, gel(y,j));
356 294244088 : t = t? addii(t, p1): p1;
357 : }
358 94601383 : if (t) s = addii(s, mulii(xi, t));
359 : }
360 33198750 : gel(v,k) = gc_INT(av,s);
361 : }
362 8246900 : return v;
363 : }
364 : static int
365 54069715 : is_famat(GEN x) { return typ(x) == t_MAT && lg(x) == 3; }
366 : /* product of x and y in nf */
367 : GEN
368 24580799 : nfmul(GEN nf, GEN x, GEN y)
369 : {
370 : GEN z;
371 24580799 : pari_sp av = avma;
372 :
373 24580799 : if (x == y) return nfsqr(nf,x);
374 :
375 23325801 : nf = checknf(nf);
376 23325801 : if (is_famat(x) || is_famat(y)) return famat_mul(x, y);
377 23325492 : x = nf_to_scalar_or_basis(nf, x);
378 23325492 : y = nf_to_scalar_or_basis(nf, y);
379 23325491 : if (typ(x) != t_COL)
380 : {
381 16514608 : if (isintzero(x)) return gen_0;
382 13758924 : z = (typ(y) == t_COL)? RgC_Rg_mul(y, x): gmul(x,y); }
383 : else
384 : {
385 6810883 : if (typ(y) != t_COL)
386 : {
387 1892794 : if (isintzero(y)) return gen_0;
388 1047814 : z = RgC_Rg_mul(x, y);
389 : }
390 : else
391 : {
392 : GEN dx, dy;
393 4918089 : x = Q_remove_denom(x, &dx);
394 4918090 : y = Q_remove_denom(y, &dy);
395 4918090 : z = nfmuli_ZC(nf,x,y);
396 4918089 : dx = mul_denom(dx,dy);
397 4918089 : if (dx) z = ZC_Z_div(z, dx);
398 : }
399 : }
400 19724821 : return gc_upto(av, z);
401 : }
402 : /* square of ZC x in nf */
403 : static GEN
404 7349760 : nfsqri_ZC(GEN nf, GEN x)
405 : {
406 : long i, j, k, N;
407 7349760 : GEN TAB = get_tab(nf, &N), v = cgetg(N+1,t_COL);
408 :
409 39772681 : for (k = 1; k <= N; k++)
410 : {
411 32422942 : pari_sp av = avma;
412 32422942 : GEN s, TABi = TAB;
413 32422942 : if (k == 1)
414 7349964 : s = sqri(gel(x,1));
415 : else
416 25072978 : s = shifti(mulii(gel(x,1),gel(x,k)), 1);
417 256784431 : for (i=2; i<=N; i++)
418 : {
419 224381098 : GEN p1, c, t, xi = gel(x,i);
420 224381098 : TABi += N;
421 224381098 : if (!signe(xi)) continue;
422 :
423 80586862 : c = gcoeff(TABi, k, i);
424 80586862 : t = signe(c)? _mulii(c,xi): NULL;
425 679252466 : for (j=i+1; j<=N; j++)
426 : {
427 598667082 : c = gcoeff(TABi, k, j);
428 598667082 : if (!signe(c)) continue;
429 233968508 : p1 = _mulii(c, shifti(gel(x,j),1));
430 233974124 : t = t? addii(t, p1): p1;
431 : }
432 80585384 : if (t) s = addii(s, mulii(xi, t));
433 : }
434 32403333 : gel(v,k) = gc_INT(av,s);
435 : }
436 7349739 : return v;
437 : }
438 : /* square of x in nf */
439 : GEN
440 6108660 : nfsqr(GEN nf, GEN x)
441 : {
442 6108660 : pari_sp av = avma;
443 : GEN z;
444 :
445 6108660 : nf = checknf(nf);
446 6108661 : if (is_famat(x)) return famat_sqr(x);
447 6108663 : x = nf_to_scalar_or_basis(nf, x);
448 6108663 : if (typ(x) != t_COL) z = gsqr(x);
449 : else
450 : {
451 : GEN dx;
452 2644099 : x = Q_remove_denom(x, &dx);
453 2644100 : z = nfsqri_ZC(nf,x);
454 2644092 : if (dx) z = RgC_Rg_div(z, sqri(dx));
455 : }
456 6108656 : return gc_upto(av, z);
457 : }
458 :
459 : /* x a ZC, v a t_COL of ZC/Z */
460 : GEN
461 194459 : zkC_multable_mul(GEN v, GEN x)
462 : {
463 194459 : long i, l = lg(v);
464 194459 : GEN y = cgetg(l, t_COL);
465 761671 : for (i = 1; i < l; i++)
466 : {
467 567212 : GEN c = gel(v,i);
468 567212 : if (typ(c)!=t_COL) {
469 0 : if (!isintzero(c)) c = ZC_Z_mul(gel(x,1), c);
470 : } else {
471 567212 : c = ZM_ZC_mul(x,c);
472 567212 : if (ZV_isscalar(c)) c = gel(c,1);
473 : }
474 567212 : gel(y,i) = c;
475 : }
476 194459 : return y;
477 : }
478 :
479 : GEN
480 31375 : nfC_multable_mul(GEN v, GEN x)
481 : {
482 31375 : long i, l = lg(v);
483 31375 : GEN y = cgetg(l, t_COL);
484 210577 : for (i = 1; i < l; i++)
485 : {
486 179202 : GEN c = gel(v,i);
487 179202 : if (typ(c)!=t_COL) {
488 141601 : if (!isintzero(c)) c = RgC_Rg_mul(gel(x,1), c);
489 : } else {
490 37601 : c = RgM_RgC_mul(x,c);
491 37601 : if (QV_isscalar(c)) c = gel(c,1);
492 : }
493 179202 : gel(y,i) = c;
494 : }
495 31375 : return y;
496 : }
497 :
498 : GEN
499 114287 : nfC_nf_mul(GEN nf, GEN v, GEN x)
500 : {
501 : long tx;
502 : GEN y;
503 :
504 114287 : x = nf_to_scalar_or_basis(nf, x);
505 114287 : tx = typ(x);
506 114287 : if (tx != t_COL)
507 : {
508 : long l, i;
509 85240 : if (tx == t_INT)
510 : {
511 81419 : long s = signe(x);
512 81419 : if (!s) return zerocol(lg(v)-1);
513 76479 : if (is_pm1(x)) return s > 0? leafcopy(v): RgC_neg(v);
514 : }
515 27215 : l = lg(v); y = cgetg(l, t_COL);
516 199154 : for (i=1; i < l; i++)
517 : {
518 171939 : GEN c = gel(v,i);
519 171939 : if (typ(c) != t_COL) c = gmul(c, x); else c = RgC_Rg_mul(c, x);
520 171939 : gel(y,i) = c;
521 : }
522 27215 : return y;
523 : }
524 : else
525 : {
526 : GEN dx;
527 29047 : x = zk_multable(nf, Q_remove_denom(x,&dx));
528 29047 : y = nfC_multable_mul(v, x);
529 29047 : return dx? RgC_Rg_div(y, dx): y;
530 : }
531 : }
532 : static GEN
533 7300 : mulbytab(GEN M, GEN c)
534 7300 : { return typ(c) == t_COL? RgM_RgC_mul(M,c): RgC_Rg_mul(gel(M,1), c); }
535 : GEN
536 2058 : tablemulvec(GEN M, GEN x, GEN v)
537 : {
538 : long l, i;
539 : GEN y;
540 :
541 2058 : if (typ(x) == t_COL && RgV_isscalar(x))
542 : {
543 0 : x = gel(x,1);
544 0 : return typ(v) == t_POL? RgX_Rg_mul(v,x): RgV_Rg_mul(v,x);
545 : }
546 2058 : x = multable(M, x); /* multiplication table by x */
547 2058 : y = cgetg_copy(v, &l);
548 2058 : if (typ(v) == t_POL)
549 : {
550 2058 : y[1] = v[1];
551 9358 : for (i=2; i < l; i++) gel(y,i) = mulbytab(x, gel(v,i));
552 2058 : y = normalizepol(y);
553 : }
554 : else
555 : {
556 0 : for (i=1; i < l; i++) gel(y,i) = mulbytab(x, gel(v,i));
557 : }
558 2058 : return y;
559 : }
560 :
561 : GEN
562 1330594 : zkmultable_capZ(GEN mx) { return Q_denom(zkmultable_inv(mx)); }
563 : GEN
564 1639928 : zkmultable_inv(GEN mx) { return ZM_gauss(mx, col_ei(lg(mx)-1,1)); }
565 : /* nf a true nf, x a ZC */
566 : GEN
567 309335 : zk_inv(GEN nf, GEN x) { return zkmultable_inv(zk_multable(nf,x)); }
568 :
569 : /* inverse of x in nf */
570 : GEN
571 188896 : nfinv(GEN nf, GEN x)
572 : {
573 188896 : pari_sp av = avma;
574 : GEN z;
575 :
576 188896 : nf = checknf(nf);
577 188896 : if (is_famat(x)) return famat_inv(x);
578 188896 : x = nf_to_scalar_or_basis(nf, x);
579 188896 : if (typ(x) == t_COL)
580 : {
581 : GEN d;
582 173232 : x = Q_remove_denom(x, &d);
583 173232 : z = zk_inv(nf, x);
584 173232 : if (d) z = RgC_Rg_mul(z, d);
585 : }
586 : else
587 15664 : z = ginv(x);
588 188896 : return gc_upto(av, z);
589 : }
590 :
591 : /* quotient of x and y in nf */
592 : GEN
593 40151 : nfdiv(GEN nf, GEN x, GEN y)
594 : {
595 40151 : pari_sp av = avma;
596 : GEN z;
597 :
598 40151 : nf = checknf(nf);
599 40151 : if (is_famat(x) || is_famat(y)) return famat_div(x,y);
600 40060 : y = nf_to_scalar_or_basis(nf, y);
601 40060 : if (typ(y) != t_COL)
602 : {
603 19887 : x = nf_to_scalar_or_basis(nf, x);
604 19887 : z = (typ(x) == t_COL)? RgC_Rg_div(x, y): gdiv(x,y);
605 : }
606 : else
607 : {
608 : GEN d;
609 20173 : y = Q_remove_denom(y, &d);
610 20173 : z = nfmul(nf, x, zk_inv(nf,y));
611 20173 : if (d) z = typ(z) == t_COL? RgC_Rg_mul(z, d): gmul(z, d);
612 : }
613 40060 : return gc_upto(av, z);
614 : }
615 :
616 : /* product of INTEGERS (t_INT or ZC) x and y in (true) nf */
617 : GEN
618 4817722 : nfmuli(GEN nf, GEN x, GEN y)
619 : {
620 4817722 : if (typ(x) == t_INT) return (typ(y) == t_COL)? ZC_Z_mul(y, x): mulii(x,y);
621 3584514 : if (typ(y) == t_INT) return ZC_Z_mul(x, y);
622 3328840 : return nfmuli_ZC(nf, x, y);
623 : }
624 : GEN
625 4705714 : nfsqri(GEN nf, GEN x)
626 4705714 : { return (typ(x) == t_INT)? sqri(x): nfsqri_ZC(nf, x); }
627 :
628 : /* both x and y are RgV */
629 : GEN
630 0 : tablemul(GEN TAB, GEN x, GEN y)
631 : {
632 : long i, j, k, N;
633 : GEN s, v;
634 0 : if (typ(x) != t_COL) return gmul(x, y);
635 0 : if (typ(y) != t_COL) return gmul(y, x);
636 0 : N = lg(x)-1;
637 0 : v = cgetg(N+1,t_COL);
638 0 : for (k=1; k<=N; k++)
639 : {
640 0 : pari_sp av = avma;
641 0 : GEN TABi = TAB;
642 0 : if (k == 1)
643 0 : s = gmul(gel(x,1),gel(y,1));
644 : else
645 0 : s = gadd(gmul(gel(x,1),gel(y,k)),
646 0 : gmul(gel(x,k),gel(y,1)));
647 0 : for (i=2; i<=N; i++)
648 : {
649 0 : GEN t, xi = gel(x,i);
650 0 : TABi += N;
651 0 : if (gequal0(xi)) continue;
652 :
653 0 : t = NULL;
654 0 : for (j=2; j<=N; j++)
655 : {
656 0 : GEN p1, c = gcoeff(TABi, k, j); /* m^{i,j}_k */
657 0 : if (gequal0(c)) continue;
658 0 : p1 = gmul(c, gel(y,j));
659 0 : t = t? gadd(t, p1): p1;
660 : }
661 0 : if (t) s = gadd(s, gmul(xi, t));
662 : }
663 0 : gel(v,k) = gc_upto(av,s);
664 : }
665 0 : return v;
666 : }
667 : GEN
668 10356 : tablesqr(GEN TAB, GEN x)
669 : {
670 : long i, j, k, N;
671 : GEN s, v;
672 :
673 10356 : if (typ(x) != t_COL) return gsqr(x);
674 10356 : N = lg(x)-1;
675 10356 : v = cgetg(N+1,t_COL);
676 :
677 83106 : for (k=1; k<=N; k++)
678 : {
679 72750 : pari_sp av = avma;
680 72750 : GEN TABi = TAB;
681 72750 : if (k == 1)
682 10356 : s = gsqr(gel(x,1));
683 : else
684 62394 : s = gmul2n(gmul(gel(x,1),gel(x,k)), 1);
685 558276 : for (i=2; i<=N; i++)
686 : {
687 485526 : GEN p1, c, t, xi = gel(x,i);
688 485526 : TABi += N;
689 485526 : if (gequal0(xi)) continue;
690 :
691 140138 : c = gcoeff(TABi, k, i);
692 140138 : t = !gequal0(c)? gmul(c,xi): NULL;
693 652832 : for (j=i+1; j<=N; j++)
694 : {
695 512694 : c = gcoeff(TABi, k, j);
696 512694 : if (gequal0(c)) continue;
697 248752 : p1 = gmul(gmul2n(c,1), gel(x,j));
698 248752 : t = t? gadd(t, p1): p1;
699 : }
700 140138 : if (t) s = gadd(s, gmul(xi, t));
701 : }
702 72750 : gel(v,k) = gc_upto(av,s);
703 : }
704 10356 : return v;
705 : }
706 :
707 : static GEN
708 398768 : _mul(void *data, GEN x, GEN y) { return nfmuli((GEN)data,x,y); }
709 : static GEN
710 1061744 : _sqr(void *data, GEN x) { return nfsqri((GEN)data,x); }
711 :
712 : /* Compute z^n in nf, left-shift binary powering */
713 : GEN
714 1025577 : nfpow(GEN nf, GEN z, GEN n)
715 : {
716 1025577 : pari_sp av = avma;
717 : long s;
718 : GEN x, cx;
719 :
720 1025577 : if (typ(n)!=t_INT) pari_err_TYPE("nfpow",n);
721 1025577 : nf = checknf(nf);
722 1025576 : s = signe(n); if (!s) return gen_1;
723 1025576 : if (is_famat(z)) return famat_pow(z, n);
724 964935 : x = nf_to_scalar_or_basis(nf, z);
725 964942 : if (typ(x) != t_COL) return powgi(x,n);
726 838713 : if (s < 0)
727 : { /* simplified nfinv */
728 : GEN d;
729 46705 : x = Q_remove_denom(x, &d);
730 46705 : x = zk_inv(nf, x);
731 46705 : x = primitive_part(x, &cx);
732 46705 : cx = mul_content(cx, d);
733 46705 : n = negi(n);
734 : }
735 : else
736 792008 : x = primitive_part(x, &cx);
737 838691 : x = gen_pow_i(x, n, (void*)nf, _sqr, _mul);
738 838706 : if (cx)
739 48130 : x = gc_upto(av, gmul(x, powgi(cx, n)));
740 : else
741 790576 : x = gc_GEN(av, x);
742 838714 : return x;
743 : }
744 : /* Compute z^n in nf, left-shift binary powering */
745 : GEN
746 370600 : nfpow_u(GEN nf, GEN z, ulong n)
747 : {
748 370600 : pari_sp av = avma;
749 : GEN x, cx;
750 :
751 370600 : if (!n) return gen_1;
752 370600 : x = nf_to_scalar_or_basis(nf, z);
753 370600 : if (typ(x) != t_COL) return gpowgs(x,n);
754 329777 : x = primitive_part(x, &cx);
755 329778 : x = gen_powu_i(x, n, (void*)nf, _sqr, _mul);
756 329777 : if (cx)
757 : {
758 113497 : x = gmul(x, powgi(cx, utoipos(n)));
759 113497 : return gc_upto(av,x);
760 : }
761 216280 : return gc_GEN(av, x);
762 : }
763 :
764 : long
765 2163 : nfissquare(GEN nf, GEN z, GEN *px)
766 : {
767 2163 : pari_sp av = avma;
768 2163 : long v = fetch_var_higher();
769 : GEN R;
770 2163 : nf = checknf(nf);
771 2163 : if (nf_get_degree(nf) == 1)
772 : {
773 210 : z = algtobasis(nf, z);
774 210 : if (!issquareall(gel(z,1), px)) return gc_long(av, 0);
775 21 : if (px) *px = gc_upto(av, *px); else set_avma(av);
776 21 : return 1;
777 : }
778 1953 : z = nf_to_scalar_or_alg(nf, z);
779 1953 : R = nfroots(nf, deg2pol_shallow(gen_m1, gen_0, z, v));
780 1953 : delete_var(); if (lg(R) == 1) return gc_long(av, 0);
781 1330 : if (px) *px = gc_GEN(av, nf_to_scalar_or_basis(nf, gel(R,1)));
782 14 : else set_avma(av);
783 1330 : return 1;
784 : }
785 :
786 : long
787 11702 : nfispower(GEN nf, GEN z, long n, GEN *px)
788 : {
789 11702 : pari_sp av = avma;
790 11702 : long v = fetch_var_higher();
791 : GEN R;
792 11702 : nf = checknf(nf);
793 11702 : if (nf_get_degree(nf) == 1)
794 : {
795 329 : z = algtobasis(nf, z);
796 329 : if (!ispower(gel(z,1), stoi(n), px)) return gc_long(av, 0);
797 147 : if (px) *px = gc_upto(av, *px); else set_avma(av);
798 147 : return 1;
799 : }
800 11373 : if (n <= 0)
801 0 : pari_err_DOMAIN("nfeltispower","exponent","<=",gen_0,stoi(n));
802 11373 : z = nf_to_scalar_or_alg(nf, z);
803 11373 : if (n==1)
804 : {
805 0 : if (px) *px = gc_GEN(av, z);
806 0 : return 1;
807 : }
808 11373 : R = nfroots(nf, gsub(pol_xn(n, v), z));
809 11373 : delete_var(); if (lg(R) == 1) return gc_long(av, 0);
810 3780 : if (px) *px = gc_GEN(av, nf_to_scalar_or_basis(nf, gel(R,1)));
811 3766 : else set_avma(av);
812 3780 : return 1;
813 : }
814 :
815 : static GEN
816 56 : idmulred(void *nf, GEN x, GEN y) { return idealmulred((GEN) nf, x, y); }
817 : static GEN
818 413 : idpowred(void *nf, GEN x, GEN n) { return idealpowred((GEN) nf, x, n); }
819 : static GEN
820 72020 : idmul(void *nf, GEN x, GEN y) { return idealmul((GEN) nf, x, y); }
821 : static GEN
822 87971 : idpow(void *nf, GEN x, GEN n) { return idealpow((GEN) nf, x, n); }
823 : GEN
824 94611 : idealfactorback(GEN nf, GEN L, GEN e, long red)
825 : {
826 94611 : nf = checknf(nf);
827 94611 : if (red) return gen_factorback(L, e, (void*)nf, &idmulred, &idpowred, NULL);
828 94254 : if (!e && typ(L) == t_MAT && lg(L) == 3) { e = gel(L,2); L = gel(L,1); }
829 94254 : if (is_vec_t(typ(L)) && RgV_is_prV(L))
830 : { /* don't use gen_factorback since *= pr^v can be done more efficiently */
831 73620 : pari_sp av = avma;
832 73620 : long i, l = lg(L);
833 : GEN a;
834 73620 : if (!e) e = const_vec(l-1, gen_1);
835 70757 : else switch(typ(e))
836 : {
837 7742 : case t_VECSMALL: e = zv_to_ZV(e); break;
838 63015 : case t_VEC: case t_COL:
839 63015 : if (!RgV_is_ZV(e))
840 0 : pari_err_TYPE("factorback [not an exponent vector]", e);
841 63015 : break;
842 0 : default: pari_err_TYPE("idealfactorback", e);
843 : }
844 73620 : if (l != lg(e))
845 0 : pari_err_TYPE("factorback [not an exponent vector]", e);
846 73620 : if (l == 1 || ZV_equal0(e)) return gc_const(av, gen_1);
847 23072 : a = idealpow(nf, gel(L,1), gel(e,1));
848 250469 : for (i = 2; i < l; i++)
849 227397 : if (signe(gel(e,i))) a = idealmulpowprime(nf, a, gel(L,i), gel(e,i));
850 23072 : return gc_upto(av, a);
851 : }
852 20634 : return gen_factorback(L, e, (void*)nf, &idmul, &idpow, NULL);
853 : }
854 : static GEN
855 350813 : eltmul(void *nf, GEN x, GEN y) { return nfmul((GEN) nf, x, y); }
856 : static GEN
857 499812 : eltpow(void *nf, GEN x, GEN n) { return nfpow((GEN) nf, x, n); }
858 : GEN
859 281075 : nffactorback(GEN nf, GEN L, GEN e)
860 281075 : { return gen_factorback(L, e, (void*)checknf(nf), &eltmul, &eltpow, NULL); }
861 :
862 : static GEN
863 778543 : _nf_red(void *E, GEN x) { (void)E; return gcopy(x); }
864 :
865 : static GEN
866 3861892 : _nf_add(void *E, GEN x, GEN y) { return nfadd((GEN)E,x,y); }
867 :
868 : static GEN
869 204392 : _nf_neg(void *E, GEN x) { (void)E; return gneg(x); }
870 :
871 : static GEN
872 4502050 : _nf_mul(void *E, GEN x, GEN y) { return nfmul((GEN)E,x,y); }
873 :
874 : static GEN
875 13727 : _nf_inv(void *E, GEN x) { return nfinv((GEN)E,x); }
876 :
877 : static GEN
878 3353 : _nf_s(void *E, long x) { (void)E; return stoi(x); }
879 :
880 : static const struct bb_field nf_field={_nf_red,_nf_add,_nf_mul,_nf_neg,
881 : _nf_inv,&gequal0,_nf_s };
882 :
883 54830 : const struct bb_field *get_nf_field(void **E, GEN nf)
884 54830 : { *E = (void*)nf; return &nf_field; }
885 :
886 : GEN
887 14 : nfM_det(GEN nf, GEN M)
888 : {
889 : void *E;
890 14 : const struct bb_field *S = get_nf_field(&E, nf);
891 14 : return gen_det(M, E, S);
892 : }
893 : GEN
894 3339 : nfM_inv(GEN nf, GEN M)
895 : {
896 : void *E;
897 3339 : const struct bb_field *S = get_nf_field(&E, nf);
898 3339 : return gen_Gauss(M, matid(lg(M)-1), E, S);
899 : }
900 :
901 : GEN
902 0 : nfM_ker(GEN nf, GEN M)
903 : {
904 : void *E;
905 0 : const struct bb_field *S = get_nf_field(&E, nf);
906 0 : return gen_ker(M, 0, E, S);
907 : }
908 :
909 : GEN
910 2948 : nfM_mul(GEN nf, GEN A, GEN B)
911 : {
912 : void *E;
913 2948 : const struct bb_field *S = get_nf_field(&E, nf);
914 2948 : return gen_matmul(A, B, E, S);
915 : }
916 : GEN
917 48529 : nfM_nfC_mul(GEN nf, GEN A, GEN B)
918 : {
919 : void *E;
920 48529 : const struct bb_field *S = get_nf_field(&E, nf);
921 48529 : return gen_matcolmul(A, B, E, S);
922 : }
923 :
924 : /* valuation of integral x (ZV), with resp. to prime ideal pr */
925 : long
926 24763706 : ZC_nfvalrem(GEN x, GEN pr, GEN *newx)
927 : {
928 24763706 : pari_sp av = avma;
929 : long i, v, l;
930 24763706 : GEN r, y, p = pr_get_p(pr), mul = pr_get_tau(pr);
931 :
932 : /* p inert */
933 24763730 : if (typ(mul) == t_INT) return newx? ZV_pvalrem(x, p, newx):ZV_pval(x, p);
934 23761582 : y = cgetg_copy(x, &l); /* will hold the new x */
935 23761981 : x = leafcopy(x);
936 38519315 : for(v=0;; v++)
937 : {
938 146389090 : for (i=1; i<l; i++)
939 : { /* is (x.b)[i] divisible by p ? */
940 131626495 : gel(y,i) = dvmdii(ZMrow_ZC_mul(mul,x,i),p,&r);
941 131629783 : if (r != gen_0) { if (newx) *newx = x; return v; }
942 : }
943 14762595 : swap(x, y);
944 14762595 : if (!newx && (v & 0xf) == 0xf) v += pr_get_e(pr) * ZV_pvalrem(x, p, &x);
945 14762595 : if (gc_needed(av,1))
946 : {
947 0 : if(DEBUGMEM>1) pari_warn(warnmem,"ZC_nfvalrem, v >= %ld", v);
948 0 : (void)gc_all(av, 2, &x, &y);
949 : }
950 : }
951 : }
952 : long
953 20261500 : ZC_nfval(GEN x, GEN P)
954 20261500 : { return ZC_nfvalrem(x, P, NULL); }
955 :
956 : /* v_P(x) != 0, x a ZV. Simpler version of ZC_nfvalrem */
957 : int
958 1294475 : ZC_prdvd(GEN x, GEN P)
959 : {
960 1294475 : pari_sp av = avma;
961 : long i, l;
962 1294475 : GEN p = pr_get_p(P), mul = pr_get_tau(P);
963 1294492 : if (typ(mul) == t_INT) return ZV_Z_dvd(x, p);
964 1293925 : l = lg(x);
965 5204535 : for (i=1; i<l; i++)
966 4663341 : if (!dvdii(ZMrow_ZC_mul(mul,x,i), p)) return gc_bool(av,0);
967 541194 : return gc_bool(av,1);
968 : }
969 :
970 : int
971 357 : pr_equal(GEN P, GEN Q)
972 : {
973 357 : GEN gQ, p = pr_get_p(P);
974 357 : long e = pr_get_e(P), f = pr_get_f(P), n;
975 357 : if (!equalii(p, pr_get_p(Q)) || e != pr_get_e(Q) || f != pr_get_f(Q))
976 336 : return 0;
977 21 : gQ = pr_get_gen(Q); n = lg(gQ)-1;
978 21 : if (2*e*f > n) return 1; /* room for only one such pr */
979 14 : return ZV_equal(pr_get_gen(P), gQ) || ZC_prdvd(gQ, P);
980 : }
981 :
982 : GEN
983 422947 : famat_nfvalrem(GEN nf, GEN x, GEN pr, GEN *py)
984 : {
985 422947 : pari_sp av = avma;
986 422947 : GEN P = gel(x,1), E = gel(x,2), V = gen_0, y = NULL;
987 422947 : long l = lg(P), simplify = 0, i;
988 422947 : if (py) { *py = gen_1; y = cgetg(l, t_COL); }
989 :
990 2265071 : for (i = 1; i < l; i++)
991 : {
992 1842124 : GEN e = gel(E,i);
993 : long v;
994 1842124 : if (!signe(e))
995 : {
996 7 : if (py) gel(y,i) = gen_1;
997 7 : simplify = 1; continue;
998 : }
999 1842117 : v = nfvalrem(nf, gel(P,i), pr, py? &gel(y,i): NULL);
1000 1842117 : if (v == LONG_MAX) { set_avma(av); if (py) *py = gen_0; return mkoo(); }
1001 1842117 : V = addmulii(V, stoi(v), e);
1002 : }
1003 422947 : if (!py) V = gc_INT(av, V);
1004 : else
1005 : {
1006 56 : y = mkmat2(y, gel(x,2));
1007 56 : if (simplify) y = famat_remove_trivial(y);
1008 56 : (void)gc_all(av, 2, &V, &y); *py = y;
1009 : }
1010 422947 : return V;
1011 : }
1012 : long
1013 5614563 : nfval(GEN nf, GEN x, GEN pr)
1014 : {
1015 5614563 : pari_sp av = avma;
1016 : long w, e;
1017 : GEN cx, p;
1018 :
1019 5614563 : if (gequal0(x)) return LONG_MAX;
1020 5601723 : nf = checknf(nf);
1021 5601722 : checkprid(pr);
1022 5601713 : p = pr_get_p(pr);
1023 5601710 : e = pr_get_e(pr);
1024 5601707 : x = nf_to_scalar_or_basis(nf, x);
1025 5601664 : if (typ(x) != t_COL) return e*Q_pval(x,p);
1026 2377985 : x = Q_primitive_part(x, &cx);
1027 2377999 : w = ZC_nfval(x,pr);
1028 2377940 : if (cx) w += e*Q_pval(cx,p);
1029 2377941 : return gc_long(av,w);
1030 : }
1031 :
1032 : /* want to write p^v = uniformizer^(e*v) * z^v, z coprime to pr */
1033 : /* z := tau^e / p^(e-1), algebraic integer coprime to pr; return z^v */
1034 : static GEN
1035 973434 : powp(GEN nf, GEN pr, long v)
1036 : {
1037 : GEN b, z;
1038 : long e;
1039 973434 : if (!v) return gen_1;
1040 446824 : b = pr_get_tau(pr);
1041 446824 : if (typ(b) == t_INT) return gen_1;
1042 131299 : e = pr_get_e(pr);
1043 131299 : z = gel(b,1);
1044 131299 : if (e != 1) z = gdiv(nfpow_u(nf, z, e), powiu(pr_get_p(pr),e-1));
1045 131299 : if (v < 0) { v = -v; z = nfinv(nf, z); }
1046 131299 : if (v != 1) z = nfpow_u(nf, z, v);
1047 131299 : return z;
1048 : }
1049 : long
1050 3666399 : nfvalrem(GEN nf, GEN x, GEN pr, GEN *py)
1051 : {
1052 3666399 : pari_sp av = avma;
1053 : long w, e;
1054 : GEN cx, p, t;
1055 :
1056 3666399 : if (!py) return nfval(nf,x,pr);
1057 1810919 : if (gequal0(x)) { *py = gen_0; return LONG_MAX; }
1058 1810863 : nf = checknf(nf);
1059 1810861 : checkprid(pr);
1060 1810861 : p = pr_get_p(pr);
1061 1810859 : e = pr_get_e(pr);
1062 1810859 : x = nf_to_scalar_or_basis(nf, x);
1063 1810858 : if (typ(x) != t_COL) {
1064 557858 : w = Q_pvalrem(x,p, py);
1065 557858 : if (!w) { *py = gc_GEN(av, x); return 0; }
1066 349279 : *py = gc_upto(av, gmul(powp(nf, pr, w), *py));
1067 349279 : return e*w;
1068 : }
1069 1253000 : x = Q_primitive_part(x, &cx);
1070 1252994 : w = ZC_nfvalrem(x,pr, py);
1071 1252991 : if (cx)
1072 : {
1073 624155 : long v = Q_pvalrem(cx,p, &t);
1074 624155 : *py = nfmul(nf, *py, gmul(powp(nf,pr,v), t));
1075 624155 : *py = gc_upto(av, *py);
1076 624155 : w += e*v;
1077 : }
1078 : else
1079 628836 : *py = gc_GEN(av, *py);
1080 1253005 : return w;
1081 : }
1082 : GEN
1083 15015 : gpnfvalrem(GEN nf, GEN x, GEN pr, GEN *py)
1084 : {
1085 : long v;
1086 15015 : if (is_famat(x)) return famat_nfvalrem(nf, x, pr, py);
1087 15008 : v = nfvalrem(nf,x,pr,py);
1088 15008 : return v == LONG_MAX? mkoo(): stoi(v);
1089 : }
1090 :
1091 : GEN
1092 226047 : basistoalg(GEN nf, GEN x)
1093 : {
1094 : GEN T;
1095 :
1096 226047 : nf = checknf(nf);
1097 226047 : switch(typ(x))
1098 : {
1099 142578 : case t_COL: {
1100 142578 : pari_sp av = avma; x = nf_to_scalar_or_alg(nf, x);
1101 142571 : return gc_GEN(av, mkpolmod(x, nf_get_pol(nf)));
1102 : }
1103 45563 : case t_POLMOD:
1104 45563 : T = nf_get_pol(nf);
1105 45563 : if (!RgX_equal_var(T,gel(x,1)))
1106 0 : pari_err_MODULUS("basistoalg", T,gel(x,1));
1107 45563 : return gcopy(x);
1108 4865 : case t_POL:
1109 4865 : T = nf_get_pol(nf);
1110 4865 : if (varn(T) != varn(x)) pari_err_VAR("basistoalg",x,T);
1111 4858 : retmkpolmod(RgX_rem(x, T), ZX_copy(T));
1112 33041 : case t_INT:
1113 : case t_FRAC:
1114 33041 : T = nf_get_pol(nf);
1115 33041 : retmkpolmod(gcopy(x), ZX_copy(T));
1116 0 : default:
1117 0 : pari_err_TYPE("basistoalg",x);
1118 : return NULL; /* LCOV_EXCL_LINE */
1119 : }
1120 : }
1121 :
1122 : /* true nf, x a t_POL */
1123 : static GEN
1124 3480123 : pol_to_scalar_or_basis(GEN nf, GEN x)
1125 : {
1126 3480123 : GEN T = nf_get_pol(nf);
1127 3480121 : long l = lg(x);
1128 3480121 : if (varn(x) != varn(T)) pari_err_VAR("nf_to_scalar_or_basis", x,T);
1129 3480016 : if (l >= lg(T)) { x = RgX_rem(x, T); l = lg(x); }
1130 3480016 : if (l == 2) return gen_0;
1131 3016749 : if (l == 3)
1132 : {
1133 677516 : x = gel(x,2);
1134 677516 : if (!is_rational_t(typ(x))) pari_err_TYPE("nf_to_scalar_or_basis",x);
1135 677509 : return x;
1136 : }
1137 2339233 : return poltobasis(nf,x);
1138 : }
1139 : /* Assume nf is a genuine nf. */
1140 : GEN
1141 123900087 : nf_to_scalar_or_basis(GEN nf, GEN x)
1142 : {
1143 123900087 : switch(typ(x))
1144 : {
1145 71140589 : case t_INT: case t_FRAC:
1146 71140589 : return x;
1147 525473 : case t_POLMOD:
1148 525473 : x = checknfelt_mod(nf,x,"nf_to_scalar_or_basis");
1149 525342 : switch(typ(x))
1150 : {
1151 38605 : case t_INT: case t_FRAC: return x;
1152 486737 : case t_POL: return pol_to_scalar_or_basis(nf,x);
1153 : }
1154 0 : break;
1155 2993387 : case t_POL: return pol_to_scalar_or_basis(nf,x);
1156 49245339 : case t_COL:
1157 49245339 : if (lg(x)-1 != nf_get_degree(nf)) break;
1158 49245179 : return QV_isscalar(x)? gel(x,1): x;
1159 : }
1160 96 : pari_err_TYPE("nf_to_scalar_or_basis",x);
1161 : return NULL; /* LCOV_EXCL_LINE */
1162 : }
1163 : /* Let x be a polynomial with coefficients in Q or nf. Return the same
1164 : * polynomial with coefficients expressed as vectors (on the integral basis).
1165 : * No consistency checks, not memory-clean. */
1166 : GEN
1167 33750 : RgX_to_nfX(GEN nf, GEN x)
1168 258721 : { pari_APPLY_pol_normalized(nf_to_scalar_or_basis(nf, gel(x,i))); }
1169 :
1170 : /* Assume nf is a genuine nf. */
1171 : GEN
1172 7169074 : nf_to_scalar_or_alg(GEN nf, GEN x)
1173 : {
1174 7169074 : switch(typ(x))
1175 : {
1176 1921555 : case t_INT: case t_FRAC:
1177 1921555 : return x;
1178 94689 : case t_POLMOD:
1179 94689 : x = checknfelt_mod(nf,x,"nf_to_scalar_or_alg");
1180 94689 : if (typ(x) != t_POL) return x;
1181 : /* fall through */
1182 : case t_POL:
1183 : {
1184 100443 : GEN T = nf_get_pol(nf);
1185 100443 : long l = lg(x);
1186 100443 : if (varn(x) != varn(T)) pari_err_VAR("nf_to_scalar_or_alg", x,T);
1187 100443 : if (l >= lg(T)) { x = RgX_rem(x, T); l = lg(x); }
1188 100443 : if (l == 2) return gen_0;
1189 100380 : if (l == 3) return gel(x,2);
1190 98665 : return x;
1191 : }
1192 5146091 : case t_COL:
1193 : {
1194 : GEN dx;
1195 5146091 : if (lg(x)-1 != nf_get_degree(nf)) break;
1196 10191007 : if (QV_isscalar(x)) return gel(x,1);
1197 5044743 : x = Q_remove_denom(x, &dx);
1198 5044797 : x = RgV_RgC_mul(nf_get_zkprimpart(nf), x);
1199 5044915 : dx = mul_denom(dx, nf_get_zkden(nf));
1200 5044898 : return gdiv(x,dx);
1201 : }
1202 : }
1203 54 : pari_err_TYPE("nf_to_scalar_or_alg",x);
1204 : return NULL; /* LCOV_EXCL_LINE */
1205 : }
1206 :
1207 : /* Assume nf is a genuine nf. */
1208 : GEN
1209 2134874 : nf_to_scalar_or_polmod(GEN nf, GEN x)
1210 : {
1211 2134874 : x = nf_to_scalar_or_alg(nf, x);
1212 2134874 : if (typ(x) == t_POL && varn(x) == nf_get_varn(nf))
1213 281358 : x = mkpolmod(x, nf_get_pol(nf));
1214 2134874 : return x;
1215 : }
1216 :
1217 : /* gmul(A, RgX_to_RgC(x)), A t_MAT of compatible dimensions */
1218 : GEN
1219 1365 : RgM_RgX_mul(GEN A, GEN x)
1220 : {
1221 1365 : long i, l = lg(x)-1;
1222 : GEN z;
1223 1365 : if (l == 1) return zerocol(nbrows(A));
1224 1351 : z = gmul(gel(x,2), gel(A,1));
1225 2555 : for (i = 2; i < l; i++)
1226 1204 : if (!gequal0(gel(x,i+1))) z = gadd(z, gmul(gel(x,i+1), gel(A,i)));
1227 1351 : return z;
1228 : }
1229 : GEN
1230 10291684 : ZM_ZX_mul(GEN A, GEN x)
1231 : {
1232 10291684 : long i, l = lg(x)-1;
1233 : GEN z;
1234 10291684 : if (l == 1) return zerocol(nbrows(A));
1235 10290354 : z = ZC_Z_mul(gel(A,1), gel(x,2));
1236 33522004 : for (i = 2; i < l ; i++)
1237 23239029 : if (signe(gel(x,i+1))) z = ZC_add(z, ZC_Z_mul(gel(A,i), gel(x,i+1)));
1238 10282975 : return z;
1239 : }
1240 : /* x a t_POL, nf a genuine nf. No garbage collecting. No check. */
1241 : GEN
1242 9652545 : poltobasis(GEN nf, GEN x)
1243 : {
1244 9652545 : GEN d, T = nf_get_pol(nf);
1245 9652519 : if (varn(x) != varn(T)) pari_err_VAR( "poltobasis", x,T);
1246 9652386 : if (degpol(x) >= degpol(T)) x = RgX_rem(x,T);
1247 9652435 : x = Q_remove_denom(x, &d);
1248 9652618 : if (!RgX_is_ZX(x)) pari_err_TYPE("poltobasis",x);
1249 9652563 : x = ZM_ZX_mul(nf_get_invzk(nf), x);
1250 9650390 : if (d) x = RgC_Rg_div(x, d);
1251 9650504 : return x;
1252 : }
1253 :
1254 : GEN
1255 975426 : algtobasis(GEN nf, GEN x)
1256 : {
1257 : pari_sp av;
1258 :
1259 975426 : nf = checknf(nf);
1260 975424 : switch(typ(x))
1261 : {
1262 153854 : case t_POLMOD:
1263 153854 : if (!RgX_equal_var(nf_get_pol(nf),gel(x,1)))
1264 7 : pari_err_MODULUS("algtobasis", nf_get_pol(nf),gel(x,1));
1265 153847 : x = gel(x,2);
1266 153847 : switch(typ(x))
1267 : {
1268 12635 : case t_INT:
1269 12635 : case t_FRAC: return scalarcol(x, nf_get_degree(nf));
1270 141212 : case t_POL:
1271 141212 : av = avma;
1272 141212 : return gc_upto(av,poltobasis(nf,x));
1273 : }
1274 0 : break;
1275 :
1276 252912 : case t_POL:
1277 252912 : av = avma;
1278 252912 : return gc_upto(av,poltobasis(nf,x));
1279 :
1280 85750 : case t_COL:
1281 85750 : if (!RgV_is_QV(x)) pari_err_TYPE("nfalgtobasis",x);
1282 85744 : if (lg(x)-1 != nf_get_degree(nf)) pari_err_DIM("nfalgtobasis");
1283 85744 : return gcopy(x);
1284 :
1285 482910 : case t_INT:
1286 482910 : case t_FRAC: return scalarcol(x, nf_get_degree(nf));
1287 : }
1288 0 : pari_err_TYPE("algtobasis",x);
1289 : return NULL; /* LCOV_EXCL_LINE */
1290 : }
1291 :
1292 : GEN
1293 61264 : rnfbasistoalg(GEN rnf,GEN x)
1294 : {
1295 61264 : const char *f = "rnfbasistoalg";
1296 : long lx, i;
1297 61264 : pari_sp av = avma;
1298 : GEN z, nf, R, T;
1299 :
1300 61264 : checkrnf(rnf);
1301 61264 : nf = rnf_get_nf(rnf);
1302 61264 : T = nf_get_pol(nf);
1303 61264 : R = QXQX_to_mod_shallow(rnf_get_pol(rnf), T);
1304 61264 : switch(typ(x))
1305 : {
1306 875 : case t_COL:
1307 875 : z = cgetg_copy(x, &lx);
1308 2597 : for (i=1; i<lx; i++)
1309 : {
1310 1778 : GEN c = nf_to_scalar_or_alg(nf, gel(x,i));
1311 1722 : if (typ(c) == t_POL) c = mkpolmod(c,T);
1312 1722 : gel(z,i) = c;
1313 : }
1314 819 : z = RgV_RgC_mul(gel(rnf_get_zk(rnf),1), z);
1315 735 : return gc_upto(av, gmodulo(z,R));
1316 :
1317 37555 : case t_POLMOD:
1318 37555 : x = polmod_nffix(f, rnf, x, 0);
1319 37282 : if (typ(x) != t_POL) break;
1320 17549 : retmkpolmod(RgX_copy(x), RgX_copy(R));
1321 1841 : case t_POL:
1322 1841 : if (varn(x) == varn(T)) { RgX_check_QX(x,f); x = gmodulo(x,T); break; }
1323 1596 : if (varn(x) == varn(R))
1324 : {
1325 1540 : x = RgX_nffix(f,nf_get_pol(nf),x,0);
1326 1540 : return gmodulo(x, R);
1327 : }
1328 56 : pari_err_VAR(f, x,R);
1329 : }
1330 40915 : retmkpolmod(scalarpol(x, varn(R)), RgX_copy(R));
1331 : }
1332 :
1333 : GEN
1334 3220 : matbasistoalg(GEN nf,GEN x)
1335 : {
1336 : long i, j, li, lx;
1337 3220 : GEN z = cgetg_copy(x, &lx);
1338 :
1339 3220 : if (lx == 1) return z;
1340 3213 : switch(typ(x))
1341 : {
1342 91 : case t_VEC: case t_COL:
1343 371 : for (i=1; i<lx; i++) gel(z,i) = basistoalg(nf, gel(x,i));
1344 91 : return z;
1345 3122 : case t_MAT: break;
1346 0 : default: pari_err_TYPE("matbasistoalg",x);
1347 : }
1348 3122 : li = lgcols(x);
1349 11018 : for (j=1; j<lx; j++)
1350 : {
1351 7896 : GEN c = cgetg(li,t_COL), xj = gel(x,j);
1352 7896 : gel(z,j) = c;
1353 33943 : for (i=1; i<li; i++) gel(c,i) = basistoalg(nf, gel(xj,i));
1354 : }
1355 3122 : return z;
1356 : }
1357 :
1358 : GEN
1359 33258 : matalgtobasis(GEN nf,GEN x)
1360 : {
1361 : long i, j, li, lx;
1362 33258 : GEN z = cgetg_copy(x, &lx);
1363 :
1364 33258 : if (lx == 1) return z;
1365 32789 : switch(typ(x))
1366 : {
1367 32782 : case t_VEC: case t_COL:
1368 86240 : for (i=1; i<lx; i++) gel(z,i) = algtobasis(nf, gel(x,i));
1369 32782 : return z;
1370 7 : case t_MAT: break;
1371 0 : default: pari_err_TYPE("matalgtobasis",x);
1372 : }
1373 7 : li = lgcols(x);
1374 14 : for (j=1; j<lx; j++)
1375 : {
1376 7 : GEN c = cgetg(li,t_COL), xj = gel(x,j);
1377 7 : gel(z,j) = c;
1378 21 : for (i=1; i<li; i++) gel(c,i) = algtobasis(nf, gel(xj,i));
1379 : }
1380 7 : return z;
1381 : }
1382 : GEN
1383 6800 : RgM_to_nfM(GEN nf,GEN x)
1384 : {
1385 : long i, j, li, lx;
1386 6800 : GEN z = cgetg_copy(x, &lx);
1387 :
1388 6800 : if (lx == 1) return z;
1389 6800 : li = lgcols(x);
1390 43755 : for (j=1; j<lx; j++)
1391 : {
1392 36955 : GEN c = cgetg(li,t_COL), xj = gel(x,j);
1393 36955 : gel(z,j) = c;
1394 228983 : for (i=1; i<li; i++) gel(c,i) = nf_to_scalar_or_basis(nf, gel(xj,i));
1395 : }
1396 6800 : return z;
1397 : }
1398 : GEN
1399 112896 : RgC_to_nfC(GEN nf, GEN x)
1400 661480 : { pari_APPLY_type(t_COL, nf_to_scalar_or_basis(nf, gel(x,i))) }
1401 :
1402 : /* x a t_POLMOD, supposedly in rnf = K[z]/(T), K = Q[y]/(Tnf) */
1403 : GEN
1404 181329 : polmod_nffix(const char *f, GEN rnf, GEN x, int lift)
1405 181329 : { return polmod_nffix2(f, rnf_get_nfpol(rnf), rnf_get_pol(rnf), x,lift); }
1406 : GEN
1407 181420 : polmod_nffix2(const char *f, GEN T, GEN R, GEN x, int lift)
1408 : {
1409 181420 : if (RgX_equal_var(gel(x,1), R))
1410 : {
1411 149408 : x = gel(x,2);
1412 149408 : if (typ(x) == t_POL && varn(x) == varn(R))
1413 : {
1414 115887 : x = RgX_nffix(f, T, x, lift);
1415 115887 : switch(lg(x))
1416 : {
1417 5691 : case 2: return gen_0;
1418 18786 : case 3: return gel(x,2);
1419 : }
1420 91410 : return x;
1421 : }
1422 : }
1423 65533 : return Rg_nffix(f, T, x, lift);
1424 : }
1425 : GEN
1426 1428 : rnfalgtobasis(GEN rnf,GEN x)
1427 : {
1428 1428 : const char *f = "rnfalgtobasis";
1429 1428 : pari_sp av = avma;
1430 : GEN T, R;
1431 :
1432 1428 : checkrnf(rnf);
1433 1428 : R = rnf_get_pol(rnf);
1434 1428 : T = rnf_get_nfpol(rnf);
1435 1428 : switch(typ(x))
1436 : {
1437 98 : case t_COL:
1438 98 : if (lg(x)-1 != rnf_get_degree(rnf)) pari_err_DIM(f);
1439 49 : x = RgV_nffix(f, T, x, 0);
1440 42 : return gc_GEN(av, x);
1441 :
1442 1162 : case t_POLMOD:
1443 1162 : x = polmod_nffix(f, rnf, x, 0);
1444 1057 : if (typ(x) != t_POL) break;
1445 714 : return gc_upto(av, RgM_RgX_mul(rnf_get_invzk(rnf), x));
1446 112 : case t_POL:
1447 112 : if (varn(x) == varn(T))
1448 : {
1449 42 : RgX_check_QX(x,f);
1450 28 : if (degpol(x) >= degpol(T)) x = RgX_rem(x,T);
1451 28 : x = mkpolmod(x,T); break;
1452 : }
1453 70 : x = RgX_nffix(f, T, x, 0);
1454 56 : if (degpol(x) >= degpol(R)) x = RgX_rem(x, R);
1455 56 : return gc_upto(av, RgM_RgX_mul(rnf_get_invzk(rnf), x));
1456 : }
1457 427 : return gc_upto(av, scalarcol(x, rnf_get_degree(rnf)));
1458 : }
1459 :
1460 : /* Given a and b in nf, gives an algebraic integer y in nf such that a-b.y
1461 : * is "small" */
1462 : GEN
1463 259 : nfdiveuc(GEN nf, GEN a, GEN b)
1464 : {
1465 259 : pari_sp av = avma;
1466 259 : a = nfdiv(nf,a,b);
1467 259 : return gc_upto(av, ground(a));
1468 : }
1469 :
1470 : /* Given a and b in nf, gives a "small" algebraic integer r in nf
1471 : * of the form a-b.y */
1472 : GEN
1473 259 : nfmod(GEN nf, GEN a, GEN b)
1474 : {
1475 259 : pari_sp av = avma;
1476 259 : GEN p1 = gneg_i(nfmul(nf,b,ground(nfdiv(nf,a,b))));
1477 259 : return gc_upto(av, nfadd(nf,a,p1));
1478 : }
1479 :
1480 : /* Given a and b in nf, gives a two-component vector [y,r] in nf such
1481 : * that r=a-b.y is "small". */
1482 : GEN
1483 259 : nfdivrem(GEN nf, GEN a, GEN b)
1484 : {
1485 259 : pari_sp av = avma;
1486 259 : GEN p1,z, y = ground(nfdiv(nf,a,b));
1487 :
1488 259 : p1 = gneg_i(nfmul(nf,b,y));
1489 259 : z = cgetg(3,t_VEC);
1490 259 : gel(z,1) = gcopy(y);
1491 259 : gel(z,2) = nfadd(nf,a,p1); return gc_upto(av, z);
1492 : }
1493 :
1494 : /*************************************************************************/
1495 : /** **/
1496 : /** LOGARITHMIC EMBEDDINGS **/
1497 : /** **/
1498 : /*************************************************************************/
1499 :
1500 : static int
1501 4744911 : low_prec(GEN x)
1502 : {
1503 4744911 : switch(typ(x))
1504 : {
1505 0 : case t_INT: return !signe(x);
1506 4744911 : case t_REAL: return !signe(x) || realprec(x) <= DEFAULTPREC;
1507 0 : default: return 0;
1508 : }
1509 : }
1510 :
1511 : static GEN
1512 23449 : cxlog_1(GEN nf) { return zerocol(lg(nf_get_roots(nf))-1); }
1513 : static GEN
1514 541 : cxlog_m1(GEN nf, long prec)
1515 : {
1516 541 : long i, l = lg(nf_get_roots(nf)), r1 = nf_get_r1(nf);
1517 541 : GEN v = cgetg(l, t_COL), p, P;
1518 541 : p = mppi(prec); P = mkcomplex(gen_0, p);
1519 1258 : for (i = 1; i <= r1; i++) gel(v,i) = P; /* IPi*/
1520 541 : if (i < l) P = gmul2n(P,1);
1521 1133 : for ( ; i < l; i++) gel(v,i) = P; /* 2IPi */
1522 541 : return v;
1523 : }
1524 : static GEN
1525 1768971 : ZC_cxlog(GEN nf, GEN x, long prec)
1526 : {
1527 : long i, l, r1;
1528 : GEN v;
1529 1768971 : x = RgM_RgC_mul(nf_get_M(nf), Q_primpart(x));
1530 1768974 : l = lg(x); r1 = nf_get_r1(nf);
1531 4496239 : for (i = 1; i <= r1; i++)
1532 2727265 : if (low_prec(gel(x,i))) return NULL;
1533 3572540 : for ( ; i < l; i++)
1534 1803567 : if (low_prec(gnorm(gel(x,i)))) return NULL;
1535 1768973 : v = cgetg(l,t_COL);
1536 4496238 : for (i = 1; i <= r1; i++) gel(v,i) = glog(gel(x,i),prec);
1537 3572539 : for ( ; i < l; i++) gel(v,i) = gmul2n(glog(gel(x,i),prec),1);
1538 1768974 : return v;
1539 : }
1540 : static GEN
1541 227519 : famat_cxlog(GEN nf, GEN fa, long prec)
1542 : {
1543 227519 : GEN G, E, y = NULL;
1544 : long i, l;
1545 :
1546 227519 : if (typ(fa) != t_MAT) pari_err_TYPE("famat_cxlog",fa);
1547 227519 : if (lg(fa) == 1) return cxlog_1(nf);
1548 227519 : G = gel(fa,1);
1549 227519 : E = gel(fa,2); l = lg(E);
1550 1133078 : for (i = 1; i < l; i++)
1551 : {
1552 905559 : GEN t, e = gel(E,i), x = nf_to_scalar_or_basis(nf, gel(G,i));
1553 : /* multiplicative arch would be better (save logs), but exponents overflow
1554 : * [ could keep track of expo separately, but not worth it ] */
1555 905559 : switch(typ(x))
1556 : { /* ignore positive rationals */
1557 16411 : case t_FRAC: x = gel(x,1); /* fall through */
1558 271095 : case t_INT: if (signe(x) > 0) continue;
1559 86 : if (!mpodd(e)) continue;
1560 30 : t = cxlog_m1(nf, prec); /* we probably should not reach this line */
1561 30 : break;
1562 634464 : default: /* t_COL */
1563 634464 : t = ZC_cxlog(nf,x,prec); if (!t) return NULL;
1564 634464 : t = RgC_Rg_mul(t, e);
1565 : }
1566 634494 : y = y? RgV_add(y,t): t;
1567 : }
1568 227519 : return y ? y: cxlog_1(nf);
1569 : }
1570 : /* Archimedean components: [e_i Log( sigma_i(X) )], where X = primpart(x),
1571 : * and e_i = 1 (resp 2.) for i <= R1 (resp. > R1) */
1572 : GEN
1573 1363181 : nf_cxlog(GEN nf, GEN x, long prec)
1574 : {
1575 1363181 : if (typ(x) == t_MAT) return famat_cxlog(nf,x,prec);
1576 1135662 : x = nf_to_scalar_or_basis(nf,x);
1577 1135662 : switch(typ(x))
1578 : {
1579 0 : case t_FRAC: x = gel(x,1); /* fall through */
1580 1155 : case t_INT:
1581 1155 : return signe(x) > 0? cxlog_1(nf): cxlog_m1(nf, prec);
1582 1134507 : default:
1583 1134507 : return ZC_cxlog(nf, x, prec);
1584 : }
1585 : }
1586 : GEN
1587 97 : nfV_cxlog(GEN nf, GEN x, long prec)
1588 : {
1589 : long i, l;
1590 97 : GEN v = cgetg_copy(x, &l);
1591 167 : for (i = 1; i < l; i++)
1592 70 : if (!(gel(v,i) = nf_cxlog(nf, gel(x,i), prec))) return NULL;
1593 97 : return v;
1594 : }
1595 :
1596 : static GEN
1597 15820 : scalar_logembed(GEN nf, GEN u, GEN *emb)
1598 : {
1599 : GEN v, logu;
1600 15820 : long i, s = signe(u), RU = lg(nf_get_roots(nf))-1, R1 = nf_get_r1(nf);
1601 :
1602 15820 : if (!s) pari_err_DOMAIN("nflogembed","argument","=",gen_0,u);
1603 15820 : v = cgetg(RU+1, t_COL); logu = logr_abs(u);
1604 18977 : for (i = 1; i <= R1; i++) gel(v,i) = logu;
1605 15820 : if (i <= RU)
1606 : {
1607 14350 : GEN logu2 = shiftr(logu,1);
1608 55839 : for ( ; i <= RU; i++) gel(v,i) = logu2;
1609 : }
1610 15820 : if (emb) *emb = const_col(RU, u);
1611 15820 : return v;
1612 : }
1613 :
1614 : static GEN
1615 1309 : famat_logembed(GEN nf,GEN x,GEN *emb,long prec)
1616 : {
1617 1309 : GEN A, M, T, a, t, g = gel(x,1), e = gel(x,2);
1618 1309 : long i, l = lg(e);
1619 :
1620 1309 : if (l == 1) return scalar_logembed(nf, real_1(prec), emb);
1621 1309 : A = NULL; T = emb? cgetg(l, t_COL): NULL;
1622 1309 : if (emb) *emb = M = mkmat2(T, e);
1623 62132 : for (i = 1; i < l; i++)
1624 : {
1625 60823 : a = nflogembed(nf, gel(g,i), &t, prec);
1626 60823 : if (!a) return NULL;
1627 60823 : a = RgC_Rg_mul(a, gel(e,i));
1628 60823 : A = A? RgC_add(A, a): a;
1629 60823 : if (emb) gel(T,i) = t;
1630 : }
1631 1309 : return A;
1632 : }
1633 :
1634 : /* Get archimedean components: [e_i log( | sigma_i(x) | )], with e_i = 1
1635 : * (resp 2.) for i <= R1 (resp. > R1) and set emb to the embeddings of x.
1636 : * Return NULL if precision problem */
1637 : GEN
1638 107642 : nflogembed(GEN nf, GEN x, GEN *emb, long prec)
1639 : {
1640 : long i, l, r1;
1641 : GEN v, t;
1642 :
1643 107642 : if (typ(x) == t_MAT) return famat_logembed(nf,x,emb,prec);
1644 106333 : x = nf_to_scalar_or_basis(nf,x);
1645 106333 : if (typ(x) != t_COL) return scalar_logembed(nf, gtofp(x,prec), emb);
1646 90513 : x = RgM_RgC_mul(nf_get_M(nf), x);
1647 90517 : l = lg(x); r1 = nf_get_r1(nf); v = cgetg(l,t_COL);
1648 134610 : for (i = 1; i <= r1; i++)
1649 : {
1650 44093 : t = gabs(gel(x,i),prec); if (low_prec(t)) return NULL;
1651 44093 : gel(v,i) = glog(t,prec);
1652 : }
1653 260505 : for ( ; i < l; i++)
1654 : {
1655 169988 : t = gnorm(gel(x,i)); if (low_prec(t)) return NULL;
1656 169988 : gel(v,i) = glog(t,prec);
1657 : }
1658 90517 : if (emb) *emb = x;
1659 90517 : return v;
1660 : }
1661 :
1662 : /*************************************************************************/
1663 : /** **/
1664 : /** REAL EMBEDDINGS **/
1665 : /** **/
1666 : /*************************************************************************/
1667 : static GEN
1668 512131 : sarch_get_cyc(GEN sarch) { return gel(sarch,1); }
1669 : static GEN
1670 1863628 : sarch_get_archp(GEN sarch) { return gel(sarch,2); }
1671 : static GEN
1672 730425 : sarch_get_MI(GEN sarch) { return gel(sarch,3); }
1673 : static GEN
1674 730424 : sarch_get_lambda(GEN sarch) { return gel(sarch,4); }
1675 : static GEN
1676 730424 : sarch_get_F(GEN sarch) { return gel(sarch,5); }
1677 :
1678 : /* true nf, x non-zero algebraic integer; return number of positive real roots
1679 : * of char_x */
1680 : static long
1681 1163999 : num_positive(GEN nf, GEN x)
1682 : {
1683 1163999 : GEN T = nf_get_pol(nf), B, charx;
1684 1163998 : long dnf, vnf, N, r1 = nf_get_r1(nf);
1685 1164000 : x = nf_to_scalar_or_alg(nf, x);
1686 1163998 : if (typ(x) != t_POL) return (signe(x) < 0)? 0: degpol(T);
1687 : /* x not a scalar */
1688 1158530 : if (r1 == 1)
1689 : {
1690 31402 : long s = signe(ZX_resultant(T, Q_primpart(x)));
1691 31402 : return s > 0? 1: 0;
1692 : }
1693 1127128 : charx = ZXQ_charpoly(x, T, 0);
1694 1127136 : charx = ZX_radical(charx);
1695 1127129 : N = degpol(T) / degpol(charx);
1696 : /* real places are unramified ? */
1697 1127128 : if (N == 1 || ZX_sturm(charx) * N == r1)
1698 1126518 : return ZX_sturmpart(charx, mkvec2(gen_0,mkoo())) * N;
1699 : /* painful case, multiply by random square until primitive */
1700 610 : dnf = nf_get_degree(nf);
1701 610 : vnf = varn(T);
1702 610 : B = int2n(10);
1703 : for(;;)
1704 0 : {
1705 610 : GEN y = RgXQ_sqr(random_FpX(dnf, vnf, B), T);
1706 610 : y = RgXQ_mul(x, y, T);
1707 610 : charx = ZXQ_charpoly(y, T, 0);
1708 610 : if (ZX_is_squarefree(charx))
1709 610 : return ZX_sturmpart(charx, mkvec2(gen_0,mkoo()));
1710 : }
1711 : }
1712 :
1713 : /* x a QC: return sigma_k(x) where 1 <= k <= r1+r2; correct but inefficient
1714 : * if x in Q. M = nf_get_M(nf) */
1715 : static GEN
1716 2147 : nfembed_i(GEN M, GEN x, long k)
1717 : {
1718 2147 : long i, l = lg(M);
1719 2147 : GEN z = gel(x,1);
1720 24394 : for (i = 2; i < l; i++) z = gadd(z, gmul(gcoeff(M,k,i), gel(x,i)));
1721 2147 : return z;
1722 : }
1723 : GEN
1724 0 : nfembed(GEN nf, GEN x, long k)
1725 : {
1726 0 : pari_sp av = avma;
1727 0 : nf = checknf(nf);
1728 0 : x = nf_to_scalar_or_basis(nf,x);
1729 0 : if (typ(x) != t_COL) return gc_GEN(av, x);
1730 0 : return gc_upto(av, nfembed_i(nf_get_M(nf),x,k));
1731 : }
1732 :
1733 : /* x a ZC */
1734 : static GEN
1735 104390 : zk_embed(GEN M, GEN x, long k)
1736 : {
1737 104390 : long i, l = lg(x);
1738 : GEN z; /* times M[k,1], which is 1 */
1739 104390 : if (l == 2) return gel(x,1); /* K = Q */
1740 104390 : z = addir(gel(x,1), mulri(gcoeff(M,k,2), gel(x,2)));
1741 154111 : for (i = 3; i < l; i++) z = addrr(z, mulri(gcoeff(M,k,i), gel(x,i)));
1742 104389 : return z;
1743 : }
1744 :
1745 : /* check that signs[i..#signs] == s; signs = NULL encodes "totally positive" */
1746 : static int
1747 34237 : oksigns(long l, GEN signs, long i, long s)
1748 : {
1749 34237 : if (!signs) return s == 0;
1750 40492 : for (; i < l; i++)
1751 30493 : if (signs[i] != s) return 0;
1752 9999 : return 1;
1753 : }
1754 :
1755 : /* true nf, x a ZC (primitive for efficiency) which is not a scalar */
1756 : static int
1757 105795 : nfchecksigns_i(GEN nf, GEN x, GEN signs, GEN archp)
1758 : {
1759 105795 : long i, np, npc, l = lg(archp), r1 = nf_get_r1(nf);
1760 : GEN sarch;
1761 :
1762 105795 : if (r1 == 0) return 1;
1763 105403 : np = num_positive(nf, x);
1764 105403 : if (np == 0) return oksigns(l, signs, 1, 1);
1765 89833 : if (np == r1) return oksigns(l, signs, 1, 0);
1766 71166 : sarch = nfarchstar(nf, NULL, identity_perm(r1));
1767 80346 : for (i = 1, npc = 0; i < l; i++)
1768 : {
1769 80096 : GEN xi = set_sign_mod_divisor(nf, vecsmall_ei(r1, archp[i]), gen_1, sarch);
1770 : long ni, s;
1771 80095 : xi = Q_primpart(xi);
1772 80096 : ni = num_positive(nf, nfmuli(nf,x,xi));
1773 80096 : s = ni < np? 0: 1;
1774 80096 : if (s != (signs? signs[i]: 0)) return 0;
1775 34822 : if (!s) npc++; /* found a positive root */
1776 34822 : if (npc == np)
1777 : { /* found all positive roots */
1778 24979 : if (!signs) return i == l-1;
1779 18096 : for (i++; i < l; i++)
1780 8827 : if (signs[i] != 1) return 0;
1781 9269 : return 1;
1782 : }
1783 9843 : if (i - npc == r1 - np)
1784 : { /* found all negative roots */
1785 663 : if (!signs) return 1;
1786 719 : for (i++; i < l; i++)
1787 77 : if (signs[i]) return 0;
1788 642 : return 1;
1789 : }
1790 : }
1791 250 : return 1;
1792 : }
1793 : static void
1794 1220 : pl_convert(GEN pl, GEN *psigns, GEN *parchp)
1795 : {
1796 1220 : long i, j, l = lg(pl);
1797 1220 : GEN signs = cgetg(l, t_VECSMALL);
1798 1220 : GEN archp = cgetg(l, t_VECSMALL);
1799 3928 : for (i = j = 1; i < l; i++)
1800 : {
1801 2708 : if (!pl[i]) continue;
1802 1926 : archp[j] = i;
1803 1926 : signs[j] = (pl[i] < 0)? 1: 0;
1804 1926 : j++;
1805 : }
1806 1220 : setlg(archp, j); *parchp = archp;
1807 1220 : setlg(signs, j); *psigns = signs;
1808 1220 : }
1809 : /* pl : requested signs for real embeddings, 0 = no sign constraint */
1810 : int
1811 14506 : nfchecksigns(GEN nf, GEN x, GEN pl)
1812 : {
1813 14506 : pari_sp av = avma;
1814 : GEN signs, archp;
1815 14506 : nf = checknf(nf);
1816 14506 : x = nf_to_scalar_or_basis(nf,x);
1817 14506 : if (typ(x) != t_COL)
1818 : {
1819 13286 : long i, l = lg(pl), s = gsigne(x);
1820 24332 : for (i = 1; i < l; i++)
1821 13349 : if (pl[i] && pl[i] != s) return gc_bool(av,0);
1822 10983 : return gc_bool(av,1);
1823 : }
1824 1220 : pl_convert(pl, &signs, &archp);
1825 1220 : return gc_bool(av, nfchecksigns_i(nf, x, signs, archp));
1826 : }
1827 :
1828 : /* signs = NULL: totally positive, else sign[i] = 0 (+) or 1 (-) */
1829 : static GEN
1830 730426 : get_C(GEN lambda, long l, GEN signs)
1831 : {
1832 : long i;
1833 : GEN C, mlambda;
1834 730426 : if (!signs) return const_vec(l-1, lambda);
1835 700676 : C = cgetg(l, t_COL); mlambda = gneg(lambda);
1836 2716795 : for (i = 1; i < l; i++) gel(C,i) = signs[i]? mlambda: lambda;
1837 700675 : return C;
1838 : }
1839 : /* signs = NULL: totally positive at archp.
1840 : * Assume that a t_COL x is not a scalar */
1841 : static GEN
1842 864208 : nfsetsigns(GEN nf, GEN signs, GEN x, GEN sarch)
1843 : {
1844 864208 : long i, l = lg(sarch_get_archp(sarch));
1845 864208 : GEN ex = NULL;
1846 : /* Is signature already correct ? */
1847 864208 : if (typ(x) != t_COL)
1848 : {
1849 759636 : long s = gsigne(x);
1850 759635 : if (!s) i = 1;
1851 759614 : else if (!signs)
1852 7399 : i = (s < 0)? 1: l;
1853 : else
1854 : {
1855 752215 : s = s < 0? 1: 0;
1856 1285944 : for (i = 1; i < l; i++)
1857 1194668 : if (signs[i] != s) break;
1858 : }
1859 759635 : if (i < l) ex = const_col(l-1, x);
1860 : }
1861 : else
1862 : { /* inefficient if x scalar, wrong if x = 0 */
1863 104572 : pari_sp av = avma;
1864 104572 : GEN cex, M = nf_get_M(nf), archp = sarch_get_archp(sarch);
1865 104575 : GEN xp = Q_primitive_part(x,&cex);
1866 104575 : if (nfchecksigns_i(nf, xp, signs, archp)) set_avma(av);
1867 : else
1868 : {
1869 69325 : ex = cgetg(l,t_COL);
1870 173714 : for (i = 1; i < l; i++) gel(ex,i) = zk_embed(M,xp,archp[i]);
1871 69324 : if (cex) ex = RgC_Rg_mul(ex, cex); /* put back content */
1872 : }
1873 : }
1874 864201 : if (ex)
1875 : { /* If no, fix it */
1876 730425 : GEN MI = sarch_get_MI(sarch), F = sarch_get_F(sarch);
1877 730424 : GEN lambda = sarch_get_lambda(sarch);
1878 730425 : GEN t = RgC_sub(get_C(lambda, l, signs), ex);
1879 730421 : t = grndtoi(RgM_RgC_mul(MI,t), NULL);
1880 730407 : if (lg(F) != 1) t = ZM_ZC_mul(F, t);
1881 730407 : x = typ(x) == t_COL? RgC_add(t, x): RgC_Rg_add(t, x);
1882 : }
1883 864180 : return x;
1884 : }
1885 : /* - true nf
1886 : * - sarch = nfarchstar(nf, F);
1887 : * - x encodes a vector of signs at arch.archp: either a t_VECSMALL
1888 : * (vector of signs as {0,1}-vector), NULL (totally positive at archp),
1889 : * or a nonzero number field element (replaced by its signature at archp);
1890 : * - y is a nonzero number field element
1891 : * Return z = y (mod F) with signs(y, archp) = signs(x) (a {0,1}-vector).
1892 : * Not stack-clean */
1893 : GEN
1894 894852 : set_sign_mod_divisor(GEN nf, GEN x, GEN y, GEN sarch)
1895 : {
1896 894852 : GEN archp = sarch_get_archp(sarch);
1897 894851 : if (lg(archp) == 1) return y;
1898 861958 : if (x && typ(x) != t_VECSMALL) x = nfsign_arch(nf, x, archp);
1899 861958 : return nfsetsigns(nf, x, nf_to_scalar_or_basis(nf,y), sarch);
1900 : }
1901 :
1902 : static GEN
1903 480624 : setsigns_init(GEN nf, GEN archp, GEN F, GEN DATA)
1904 : {
1905 480624 : GEN lambda, Mr = rowpermute(nf_get_M(nf), archp), MI = F? RgM_mul(Mr,F): Mr;
1906 480635 : lambda = gmul2n(matrixnorm(MI,DEFAULTPREC), -1);
1907 480629 : if (typ(lambda) != t_REAL) lambda = gmul(lambda, uutoQ(1001,1000));
1908 480634 : if (lg(archp) < lg(MI))
1909 : {
1910 81853 : GEN perm = gel(indexrank(MI), 2);
1911 81852 : if (!F) F = matid(nf_get_degree(nf));
1912 81852 : MI = vecpermute(MI, perm);
1913 81852 : F = vecpermute(F, perm);
1914 : }
1915 480632 : if (!F) F = cgetg(1,t_MAT);
1916 480631 : MI = RgM_inv(MI);
1917 480635 : return mkvec5(DATA, archp, MI, lambda, F);
1918 : }
1919 : /* F nonzero integral ideal in HNF (or NULL: Z_K), compute elements in 1+F
1920 : * whose sign matrix at archp is identity; archp in 'indices' format */
1921 : GEN
1922 658941 : nfarchstar(GEN nf, GEN F, GEN archp)
1923 : {
1924 658941 : long nba = lg(archp) - 1;
1925 658941 : if (!nba) return mkvec2(cgetg(1,t_VEC), archp);
1926 478387 : if (F && equali1(gcoeff(F,1,1))) F = NULL;
1927 478388 : if (F) F = idealpseudored(F, nf_get_roundG(nf));
1928 478379 : return setsigns_init(nf, archp, F, const_vec(nba, gen_2));
1929 : }
1930 :
1931 : /*************************************************************************/
1932 : /** **/
1933 : /** IDEALCHINESE **/
1934 : /** **/
1935 : /*************************************************************************/
1936 : static int
1937 6218 : isprfact(GEN x)
1938 : {
1939 : long i, l;
1940 : GEN L, E;
1941 6218 : if (typ(x) != t_MAT || lg(x) != 3) return 0;
1942 6218 : L = gel(x,1); l = lg(L);
1943 6218 : E = gel(x,2);
1944 19015 : for(i=1; i<l; i++)
1945 : {
1946 12797 : checkprid(gel(L,i));
1947 12797 : if (typ(gel(E,i)) != t_INT) return 0;
1948 : }
1949 6218 : return 1;
1950 : }
1951 :
1952 : /* initialize projectors mod pr[i]^e[i] for idealchinese */
1953 : static GEN
1954 6218 : pr_init(GEN nf, GEN fa, GEN w, GEN dw)
1955 : {
1956 6218 : GEN U, E, F, FZ, L = gel(fa,1), E0 = gel(fa,2);
1957 6218 : long i, r = lg(L);
1958 :
1959 6218 : if (w && lg(w) != r) pari_err_TYPE("idealchinese", w);
1960 6218 : if (r == 1 && !dw) return cgetg(1,t_VEC);
1961 6204 : E = leafcopy(E0); /* do not destroy fa[2] */
1962 19001 : for (i = 1; i < r; i++)
1963 12797 : if (signe(gel(E,i)) < 0) gel(E,i) = gen_0;
1964 6204 : F = factorbackprime(nf, L, E);
1965 6204 : if (dw)
1966 : {
1967 693 : F = ZM_Z_mul(F, dw);
1968 1596 : for (i = 1; i < r; i++)
1969 : {
1970 903 : GEN pr = gel(L,i);
1971 903 : long e = itos(gel(E0,i)), v = idealval(nf, dw, pr);
1972 903 : if (e >= 0)
1973 896 : gel(E,i) = addiu(gel(E,i), v);
1974 7 : else if (v + e <= 0)
1975 0 : F = idealmulpowprime(nf, F, pr, stoi(-v)); /* coprime to pr */
1976 : else
1977 : {
1978 7 : F = idealmulpowprime(nf, F, pr, stoi(e));
1979 7 : gel(E,i) = stoi(v + e);
1980 : }
1981 : }
1982 : }
1983 6204 : U = cgetg(r, t_VEC);
1984 19001 : for (i = 1; i < r; i++)
1985 : {
1986 : GEN u;
1987 12797 : if (w && gequal0(gel(w,i))) u = gen_0; /* unused */
1988 : else
1989 : {
1990 12720 : GEN pr = gel(L,i), e = gel(E,i), t;
1991 12720 : t = idealdivpowprime(nf,F, pr, e);
1992 12720 : u = hnfmerge_get_1(t, idealpow(nf, pr, e));
1993 12720 : if (!u) pari_err_COPRIME("idealchinese", t,pr);
1994 : }
1995 12797 : gel(U,i) = u;
1996 : }
1997 6204 : FZ = gcoeff(F, 1, 1);
1998 6204 : F = idealpseudored(F, nf_get_roundG(nf));
1999 6204 : return mkvec2(mkvec2(F, FZ), U);
2000 : }
2001 :
2002 : static GEN
2003 2996 : pl_normalize(GEN nf, GEN pl)
2004 : {
2005 2996 : const char *fun = "idealchinese";
2006 2996 : if (lg(pl)-1 != nf_get_r1(nf)) pari_err_TYPE(fun,pl);
2007 2996 : switch(typ(pl))
2008 : {
2009 707 : case t_VEC: RgV_check_ZV(pl,fun); pl = ZV_to_zv(pl);
2010 : /* fall through */
2011 2996 : case t_VECSMALL: break;
2012 0 : default: pari_err_TYPE(fun,pl);
2013 : }
2014 2996 : return pl;
2015 : }
2016 :
2017 : static int
2018 13274 : is_chineseinit(GEN x)
2019 : {
2020 : GEN fa, pl;
2021 : long l;
2022 13274 : if (typ(x) != t_VEC || lg(x)!=3) return 0;
2023 10712 : fa = gel(x,1);
2024 10712 : pl = gel(x,2);
2025 10712 : if (typ(fa) != t_VEC || typ(pl) != t_VEC) return 0;
2026 6568 : l = lg(fa);
2027 6568 : if (l != 1)
2028 : {
2029 : GEN z;
2030 6526 : if (l != 3) return 0;
2031 6526 : z = gel(fa, 1);
2032 6526 : if (typ(z) != t_VEC || lg(z) != 3 || typ(gel(z,1)) != t_MAT
2033 6519 : || typ(gel(z,2)) != t_INT
2034 6519 : || typ(gel(fa,2)) != t_VEC)
2035 7 : return 0;
2036 : }
2037 6561 : l = lg(pl);
2038 6561 : if (l != 1)
2039 : {
2040 1136 : if (l != 6 || typ(gel(pl,3)) != t_MAT || typ(gel(pl,1)) != t_VECSMALL
2041 1136 : || typ(gel(pl,2)) != t_VECSMALL)
2042 0 : return 0;
2043 : }
2044 6561 : return 1;
2045 : }
2046 :
2047 : /* nf a true 'nf' */
2048 : static GEN
2049 6687 : chineseinit_i(GEN nf, GEN fa, GEN w, GEN dw)
2050 : {
2051 6687 : const char *fun = "idealchineseinit";
2052 6687 : GEN archp = NULL, pl = NULL;
2053 6687 : switch(typ(fa))
2054 : {
2055 2996 : case t_VEC:
2056 2996 : if (is_chineseinit(fa))
2057 : {
2058 0 : if (dw) pari_err_DOMAIN(fun, "denom(y)", "!=", gen_1, w);
2059 0 : return fa;
2060 : }
2061 2996 : if (lg(fa) != 3) pari_err_TYPE(fun, fa);
2062 : /* of the form [x,s] */
2063 2996 : pl = pl_normalize(nf, gel(fa,2));
2064 2996 : fa = gel(fa,1);
2065 2996 : archp = vecsmall01_to_indices(pl);
2066 : /* keep pr_init, reset pl */
2067 2996 : if (is_chineseinit(fa)) { fa = gel(fa,1); break; }
2068 : /* fall through */
2069 : case t_MAT: /* factorization? */
2070 6218 : if (isprfact(fa)) { fa = pr_init(nf, fa, w, dw); break; }
2071 0 : default: pari_err_TYPE(fun,fa);
2072 : }
2073 :
2074 6687 : if (!pl) pl = cgetg(1,t_VEC);
2075 : else
2076 : {
2077 2996 : long r = lg(archp);
2078 2996 : if (r == 1) pl = cgetg(1, t_VEC);
2079 : else
2080 : {
2081 2242 : GEN F = (lg(fa) == 1)? NULL: gmael(fa,1,1), signs = cgetg(r, t_VECSMALL);
2082 : long i;
2083 6234 : for (i = 1; i < r; i++) signs[i] = (pl[archp[i]] < 0)? 1: 0;
2084 2242 : pl = setsigns_init(nf, archp, F, signs);
2085 : }
2086 : }
2087 6687 : return mkvec2(fa, pl);
2088 : }
2089 :
2090 : /* Given a prime ideal factorization x, possibly with 0 or negative exponents,
2091 : * and a vector w of elements of nf, gives b such that
2092 : * v_p(b-w_p)>=v_p(x) for all prime ideals p in the ideal factorization
2093 : * and v_p(b)>=0 for all other p, using the standard proof given in GTM 138. */
2094 : GEN
2095 12779 : idealchinese(GEN nf, GEN x0, GEN w)
2096 : {
2097 12779 : const char *fun = "idealchinese";
2098 12779 : pari_sp av = avma;
2099 12779 : GEN x = x0, x1, x2, s, dw, F;
2100 :
2101 12779 : nf = checknf(nf);
2102 12779 : if (!w) return gc_GEN(av, chineseinit_i(nf,x,NULL,NULL));
2103 :
2104 7282 : if (typ(w) != t_VEC) pari_err_TYPE(fun,w);
2105 7282 : w = Q_remove_denom(matalgtobasis(nf,w), &dw);
2106 7282 : if (!is_chineseinit(x)) x = chineseinit_i(nf,x,w,dw);
2107 : /* x is a 'chineseinit' */
2108 7282 : x1 = gel(x,1); s = NULL;
2109 7282 : x2 = gel(x,2);
2110 7282 : if (lg(x1) == 1) { F = NULL; dw = NULL; }
2111 : else
2112 : {
2113 7240 : GEN U = gel(x1,2), FZ;
2114 7240 : long i, r = lg(w);
2115 7240 : F = gmael(x1,1,1); FZ = gmael(x1,1,2);
2116 23726 : for (i=1; i<r; i++)
2117 16486 : if (!ZV_equal0(gel(w,i)))
2118 : {
2119 12363 : GEN t = nfmuli(nf, gel(U,i), gel(w,i));
2120 12363 : s = s? ZC_add(s,t): t;
2121 : }
2122 7240 : if (s)
2123 : {
2124 7219 : s = ZC_reducemodmatrix(s, F);
2125 7219 : if (dw && x == x0) /* input was a chineseinit */
2126 : {
2127 7 : dw = modii(dw, FZ);
2128 7 : s = FpC_Fp_mul(s, Fp_inv(dw, FZ), FZ);
2129 7 : dw = NULL;
2130 : }
2131 7219 : if (ZV_isscalar(s)) s = icopy(gel(s,1));
2132 : }
2133 : }
2134 7282 : if (lg(x2) != 1)
2135 : {
2136 2249 : s = nfsetsigns(nf, gel(x2,1), s? s: gen_0, x2);
2137 2249 : if (typ(s) == t_COL && QV_isscalar(s))
2138 : {
2139 434 : s = gel(s,1); if (!dw) s = gcopy(s);
2140 : }
2141 : }
2142 5033 : else if (!s) return gc_const(av, gen_0);
2143 7233 : return gc_upto(av, dw? gdiv(s, dw): s);
2144 : }
2145 :
2146 : /*************************************************************************/
2147 : /** **/
2148 : /** (Z_K/I)^* **/
2149 : /** **/
2150 : /*************************************************************************/
2151 : GEN
2152 2996 : vecsmall01_to_indices(GEN v)
2153 : {
2154 2996 : long i, k, l = lg(v);
2155 2996 : GEN p = new_chunk(l) + l;
2156 8449 : for (k=1, i=l-1; i; i--)
2157 5453 : if (v[i]) { *--p = i; k++; }
2158 2996 : *--p = _evallg(k) | evaltyp(t_VECSMALL);
2159 2996 : set_avma((pari_sp)p); return p;
2160 : }
2161 : GEN
2162 1330886 : vec01_to_indices(GEN v)
2163 : {
2164 : long i, k, l;
2165 : GEN p;
2166 :
2167 1330886 : switch (typ(v))
2168 : {
2169 1271064 : case t_VECSMALL: return v;
2170 59822 : case t_VEC: break;
2171 0 : default: pari_err_TYPE("vec01_to_indices",v);
2172 : }
2173 59822 : l = lg(v);
2174 59822 : p = new_chunk(l) + l;
2175 180145 : for (k=1, i=l-1; i; i--)
2176 120323 : if (signe(gel(v,i))) { *--p = i; k++; }
2177 59822 : *--p = _evallg(k) | evaltyp(t_VECSMALL);
2178 59822 : set_avma((pari_sp)p); return p;
2179 : }
2180 : GEN
2181 143702 : indices_to_vec01(GEN p, long r)
2182 : {
2183 143702 : long i, l = lg(p);
2184 143702 : GEN v = zerovec(r);
2185 217823 : for (i = 1; i < l; i++) gel(v, p[i]) = gen_1;
2186 143702 : return v;
2187 : }
2188 :
2189 : /* return (column) vector of R1 signatures of x (0 or 1) */
2190 : GEN
2191 1271064 : nfsign_arch(GEN nf, GEN x, GEN arch)
2192 : {
2193 1271064 : GEN sarch, V, archp = vec01_to_indices(arch);
2194 1271064 : long i, s, np, npc, r1, n = lg(archp)-1;
2195 : pari_sp av;
2196 :
2197 1271064 : if (!n) return cgetg(1,t_VECSMALL);
2198 1064821 : if (typ(x) == t_MAT)
2199 : { /* factorisation */
2200 337495 : GEN g = gel(x,1), e = gel(x,2);
2201 337495 : long l = lg(g);
2202 337495 : V = zero_zv(n);
2203 954127 : for (i = 1; i < l; i++)
2204 616630 : if (mpodd(gel(e,i)))
2205 504705 : Flv_add_inplace(V, nfsign_arch(nf,gel(g,i),archp), 2);
2206 337497 : set_avma((pari_sp)V); return V;
2207 : }
2208 727326 : av = avma; V = cgetg(n+1,t_VECSMALL);
2209 727326 : x = nf_to_scalar_or_basis(nf, x);
2210 727326 : switch(typ(x))
2211 : {
2212 202822 : case t_INT:
2213 202822 : s = signe(x);
2214 202822 : if (!s) pari_err_DOMAIN("nfsign_arch","element","=",gen_0,x);
2215 202822 : set_avma(av); return const_vecsmall(n, (s < 0)? 1: 0);
2216 644 : case t_FRAC:
2217 644 : s = signe(gel(x,1));
2218 644 : set_avma(av); return const_vecsmall(n, (s < 0)? 1: 0);
2219 : }
2220 523860 : r1 = nf_get_r1(nf); x = Q_primpart(x); np = num_positive(nf, x);
2221 523861 : if (np == 0) { set_avma(av); return const_vecsmall(n, 1); }
2222 461840 : if (np == r1){ set_avma(av); return const_vecsmall(n, 0); }
2223 315783 : sarch = nfarchstar(nf, NULL, identity_perm(r1));
2224 456010 : for (i = 1, npc = 0; i <= n; i++)
2225 : {
2226 454650 : GEN xi = set_sign_mod_divisor(nf, vecsmall_ei(r1, archp[i]), gen_1, sarch);
2227 : long ni;
2228 454643 : xi = Q_primpart(xi);
2229 454645 : ni = num_positive(nf, nfmuli(nf,x,xi));
2230 454650 : V[i] = ni < np? 0: 1;
2231 454650 : if (!V[i]) npc++; /* found a positive root */
2232 454650 : if (npc == np)
2233 : { /* found all positive roots */
2234 323307 : for (i++; i <= n; i++) V[i] = 1;
2235 178720 : break;
2236 : }
2237 275930 : if (i - npc == r1 - np)
2238 : { /* found all negative roots */
2239 212841 : for (i++; i <= n; i++) V[i] = 0;
2240 135703 : break;
2241 : }
2242 : }
2243 315783 : set_avma((pari_sp)V); return V;
2244 : }
2245 : static void
2246 37121 : chk_ind(const char *s, long i, long r1)
2247 : {
2248 37121 : if (i <= 0) pari_err_DOMAIN(s, "index", "<=", gen_0, stoi(i));
2249 37107 : if (i > r1) pari_err_DOMAIN(s, "index", ">", utoi(r1), utoi(i));
2250 37072 : }
2251 : static GEN
2252 129066 : parse_embed(GEN ind, long r, const char *f)
2253 : {
2254 : long l, i;
2255 129066 : if (!ind) return identity_perm(r);
2256 34755 : switch(typ(ind))
2257 : {
2258 70 : case t_INT: ind = mkvecsmall(itos(ind)); break;
2259 84 : case t_VEC: case t_COL: ind = vec_to_vecsmall(ind); break;
2260 34601 : case t_VECSMALL: break;
2261 0 : default: pari_err_TYPE(f, ind);
2262 : }
2263 34755 : l = lg(ind);
2264 71827 : for (i = 1; i < l; i++) chk_ind(f, ind[i], r);
2265 34706 : return ind;
2266 : }
2267 : GEN
2268 126175 : nfeltsign(GEN nf, GEN x, GEN ind0)
2269 : {
2270 126175 : pari_sp av = avma;
2271 : long i, l;
2272 : GEN v, ind;
2273 126175 : nf = checknf(nf);
2274 126175 : ind = parse_embed(ind0, nf_get_r1(nf), "nfeltsign");
2275 126154 : l = lg(ind);
2276 126154 : if (is_rational_t(typ(x)))
2277 : { /* nfsign_arch would test this, but avoid converting t_VECSMALL -> t_VEC */
2278 : GEN s;
2279 31913 : switch(gsigne(x))
2280 : {
2281 16625 : case -1:s = gen_m1; break;
2282 15281 : case 1: s = gen_1; break;
2283 7 : default: s = gen_0; break;
2284 : }
2285 31913 : set_avma(av);
2286 31913 : return (ind0 && typ(ind0) == t_INT)? s: const_vec(l-1, s);
2287 : }
2288 94241 : v = nfsign_arch(nf, x, ind);
2289 94241 : if (ind0 && typ(ind0) == t_INT) { set_avma(av); return v[1]? gen_m1: gen_1; }
2290 94227 : settyp(v, t_VEC);
2291 264264 : for (i = 1; i < l; i++) gel(v,i) = v[i]? gen_m1: gen_1;
2292 94227 : return gc_upto(av, v);
2293 : }
2294 :
2295 : /* true nf */
2296 : GEN
2297 735 : nfeltembed_i(GEN *pnf, GEN x, GEN ind0, long prec0)
2298 : {
2299 : long i, e, l, r1, r2, prec, prec1;
2300 735 : GEN v, ind, cx, nf = *pnf;
2301 735 : nf_get_sign(nf,&r1,&r2);
2302 735 : x = nf_to_scalar_or_basis(nf, x);
2303 728 : ind = parse_embed(ind0, r1+r2, "nfeltembed");
2304 721 : l = lg(ind);
2305 721 : if (typ(x) != t_COL)
2306 : {
2307 224 : if (!(ind0 && typ(ind0) == t_INT)) x = const_vec(l-1, x);
2308 224 : return x;
2309 : }
2310 497 : x = Q_primitive_part(x, &cx);
2311 497 : prec1 = prec0; e = gexpo(x);
2312 497 : if (e > 8) prec1 += nbits2extraprec(e);
2313 497 : prec = prec1;
2314 497 : if (nf_get_prec(nf) < prec) nf = nfnewprec_shallow(nf, prec);
2315 497 : v = cgetg(l, t_VEC);
2316 : for(;;)
2317 138 : {
2318 635 : GEN M = nf_get_M(nf);
2319 2644 : for (i = 1; i < l; i++)
2320 : {
2321 2147 : GEN t = nfembed_i(M, x, ind[i]);
2322 2147 : long e = gexpo(t);
2323 2147 : if (gequal0(t) || precision(t) < prec0
2324 2147 : || (e < 0 && prec < prec1 + nbits2extraprec(-e)) ) break;
2325 2009 : if (cx) t = gmul(t, cx);
2326 2009 : gel(v,i) = t;
2327 : }
2328 635 : if (i == l) break;
2329 138 : prec = precdbl(prec);
2330 138 : if (DEBUGLEVEL>1) pari_warn(warnprec,"eltnfembed", prec);
2331 138 : *pnf = nf = nfnewprec_shallow(nf, prec);
2332 : }
2333 497 : if (ind0 && typ(ind0) == t_INT) v = gel(v,1);
2334 497 : return v;
2335 : }
2336 : GEN
2337 735 : nfeltembed(GEN nf, GEN x, GEN ind0, long prec0)
2338 : {
2339 735 : pari_sp av = avma; nf = checknf(nf);
2340 735 : return gc_GEN(av, nfeltembed_i(&nf, x, ind0, prec0));
2341 : }
2342 :
2343 : /* number of distinct roots of sigma(f) */
2344 : GEN
2345 2163 : nfpolsturm(GEN nf, GEN f, GEN ind0)
2346 : {
2347 2163 : pari_sp av = avma;
2348 : long d, l, r1, single;
2349 : GEN ind, u, v, vr1, T, s, t;
2350 :
2351 2163 : nf = checknf(nf); T = nf_get_pol(nf); r1 = nf_get_r1(nf);
2352 2163 : ind = parse_embed(ind0, r1, "nfpolsturm");
2353 2142 : single = ind0 && typ(ind0) == t_INT;
2354 2142 : l = lg(ind);
2355 :
2356 2142 : if (gequal0(f)) pari_err_ROOTS0("nfpolsturm");
2357 2135 : if (typ(f) == t_POL && varn(f) != varn(T))
2358 : {
2359 2114 : f = RgX_nffix("nfpolsturm", T, f,1);
2360 2114 : if (lg(f) == 3) f = NULL;
2361 : }
2362 : else
2363 : {
2364 21 : (void)Rg_nffix("nfpolsturm", T, f, 0);
2365 21 : f = NULL;
2366 : }
2367 2135 : if (!f) { set_avma(av); return single? gen_0: zerovec(l-1); }
2368 2114 : d = degpol(f);
2369 2114 : if (d == 1) { set_avma(av); return single? gen_1: const_vec(l-1,gen_1); }
2370 :
2371 2023 : vr1 = const_vecsmall(l-1, 1);
2372 2023 : u = Q_primpart(f); s = ZV_to_zv(nfeltsign(nf, gel(u,d+2), ind));
2373 2023 : v = RgX_deriv(u); t = odd(d)? leafcopy(s): zv_neg(s);
2374 : for(;;)
2375 301 : {
2376 2324 : GEN r = RgX_neg( Q_primpart(RgX_pseudorem(u, v)) ), sr;
2377 2324 : long i, dr = degpol(r);
2378 2324 : if (dr < 0) break;
2379 2324 : sr = ZV_to_zv(nfeltsign(nf, gel(r,dr+2), ind));
2380 5579 : for (i = 1; i < l; i++)
2381 3255 : if (sr[i] != s[i]) { s[i] = sr[i], vr1[i]--; }
2382 2324 : if (odd(dr)) sr = zv_neg(sr);
2383 5579 : for (i = 1; i < l; i++)
2384 3255 : if (sr[i] != t[i]) { t[i] = sr[i], vr1[i]++; }
2385 2324 : if (!dr) break;
2386 301 : u = v; v = r;
2387 : }
2388 2023 : if (single) return gc_stoi(av,vr1[1]);
2389 2016 : return gc_upto(av, zv_to_ZV(vr1));
2390 : }
2391 :
2392 : /* True nf; return the vector of signs of x; the matrix of such if x is a vector
2393 : * of nf elements */
2394 : GEN
2395 53914 : nfsign(GEN nf, GEN x)
2396 : {
2397 : long i, l;
2398 : GEN archp, S;
2399 :
2400 53914 : archp = identity_perm( nf_get_r1(nf) );
2401 53914 : if (typ(x) != t_VEC) return nfsign_arch(nf, x, archp);
2402 45507 : l = lg(x); S = cgetg(l, t_MAT);
2403 196363 : for (i=1; i<l; i++) gel(S,i) = nfsign_arch(nf, gel(x,i), archp);
2404 45508 : return S;
2405 : }
2406 :
2407 : /* x integral elt, A integral ideal in HNF; reduce x mod A */
2408 : static GEN
2409 8142385 : zk_modHNF(GEN x, GEN A)
2410 8142385 : { return (typ(x) == t_COL)? ZC_hnfrem(x, A): modii(x, gcoeff(A,1,1)); }
2411 :
2412 : /* given an element x in Z_K and an integral ideal y in HNF, coprime with x,
2413 : outputs an element inverse of x modulo y */
2414 : GEN
2415 175 : nfinvmodideal(GEN nf, GEN x, GEN y)
2416 : {
2417 175 : pari_sp av = avma;
2418 175 : GEN a, yZ = gcoeff(y,1,1);
2419 :
2420 175 : if (equali1(yZ)) return gen_0;
2421 175 : x = nf_to_scalar_or_basis(nf, x);
2422 175 : if (typ(x) == t_INT) return gc_upto(av, Fp_inv(x, yZ));
2423 :
2424 79 : a = hnfmerge_get_1(idealhnf_principal(nf,x), y);
2425 79 : if (!a) pari_err_INV("nfinvmodideal", x);
2426 79 : return gc_upto(av, zk_modHNF(nfdiv(nf,a,x), y));
2427 : }
2428 :
2429 : static GEN
2430 2813417 : nfsqrmodideal(GEN nf, GEN x, GEN id)
2431 2813417 : { return zk_modHNF(nfsqri(nf,x), id); }
2432 : static GEN
2433 7852469 : nfmulmodideal(GEN nf, GEN x, GEN y, GEN id)
2434 7852469 : { return x? zk_modHNF(nfmuli(nf,x,y), id): y; }
2435 : /* assume x integral, k integer, A in HNF */
2436 : GEN
2437 6490086 : nfpowmodideal(GEN nf,GEN x,GEN k,GEN A)
2438 : {
2439 6490086 : long s = signe(k);
2440 : pari_sp av;
2441 : GEN y;
2442 :
2443 6490086 : if (!s) return gen_1;
2444 6490086 : av = avma;
2445 6490086 : x = nf_to_scalar_or_basis(nf, x);
2446 6490343 : if (typ(x) != t_COL) return Fp_pow(x, k, gcoeff(A,1,1));
2447 2807149 : if (s < 0) { k = negi(k); x = nfinvmodideal(nf, x,A); }
2448 2807149 : if (equali1(k)) return gc_upto(av, s > 0? zk_modHNF(x, A): x);
2449 1274805 : for(y = NULL;;)
2450 : {
2451 4088438 : if (mpodd(k)) y = nfmulmodideal(nf,y,x,A);
2452 4088400 : k = shifti(k,-1); if (!signe(k)) break;
2453 2813078 : x = nfsqrmodideal(nf,x,A);
2454 : }
2455 1274799 : return gc_upto(av, y);
2456 : }
2457 :
2458 : /* a * g^n mod id */
2459 : static GEN
2460 5090117 : nfmulpowmodideal(GEN nf, GEN a, GEN g, GEN n, GEN id)
2461 : {
2462 5090117 : return nfmulmodideal(nf, a, nfpowmodideal(nf,g,n,id), id);
2463 : }
2464 :
2465 : /* assume (num(g[i]), id) = 1 for all i. Return prod g[i]^e[i] mod id.
2466 : * EX = multiple of exponent of (O_K/id)^* */
2467 : GEN
2468 2927839 : famat_to_nf_modideal_coprime(GEN nf, GEN g, GEN e, GEN id, GEN EX)
2469 : {
2470 2927839 : GEN EXo2, plus = NULL, minus = NULL, idZ = gcoeff(id,1,1);
2471 2927839 : long i, lx = lg(g);
2472 :
2473 2927839 : if (equali1(idZ)) return gen_1; /* id = Z_K */
2474 2927393 : EXo2 = (expi(EX) > 10)? shifti(EX,-1): NULL;
2475 9193877 : for (i = 1; i < lx; i++)
2476 : {
2477 6266537 : GEN h, n = centermodii(gel(e,i), EX, EXo2);
2478 6266043 : long sn = signe(n);
2479 6266043 : if (!sn) continue;
2480 :
2481 4356312 : h = nf_to_scalar_or_basis(nf, gel(g,i));
2482 4356710 : switch(typ(h))
2483 : {
2484 2650305 : case t_INT: break;
2485 0 : case t_FRAC:
2486 0 : h = Fp_div(gel(h,1), gel(h,2), idZ); break;
2487 1706405 : default:
2488 : {
2489 : GEN dh;
2490 1706405 : h = Q_remove_denom(h, &dh);
2491 1706556 : if (dh) h = FpC_Fp_mul(h, Fp_inv(dh,idZ), idZ);
2492 : }
2493 : }
2494 4356769 : if (sn > 0)
2495 4354933 : plus = nfmulpowmodideal(nf, plus, h, n, id);
2496 : else /* sn < 0 */
2497 1836 : minus = nfmulpowmodideal(nf, minus, h, negi(n), id);
2498 : }
2499 2927340 : if (minus) plus = nfmulmodideal(nf, plus, nfinvmodideal(nf,minus,id), id);
2500 2927405 : return plus? plus: gen_1;
2501 : }
2502 :
2503 : /* given 2 integral ideals x, y in HNF s.t x | y | x^2, compute (1+x)/(1+y) in
2504 : * the form [[cyc],[gen], U], where U := ux^-1 as a pair [ZM, denom(U)] */
2505 : static GEN
2506 275318 : zidealij(GEN x, GEN y)
2507 : {
2508 275318 : GEN U, G, cyc, xp = gcoeff(x,1,1), xi = hnf_invscale(x, xp);
2509 : long j, N;
2510 :
2511 : /* x^(-1) y = relations between the 1 + x_i (HNF) */
2512 275314 : cyc = ZM_snf_group(ZM_Z_divexact(ZM_mul(xi, y), xp), &U, &G);
2513 275309 : N = lg(cyc); G = ZM_mul(x,G); settyp(G, t_VEC); /* new generators */
2514 652063 : for (j=1; j<N; j++)
2515 : {
2516 376766 : GEN c = gel(G,j);
2517 376766 : gel(c,1) = addiu(gel(c,1), 1); /* 1 + g_j */
2518 376751 : if (ZV_isscalar(c)) gel(G,j) = gel(c,1);
2519 : }
2520 275297 : return mkvec4(cyc, G, ZM_mul(U,xi), xp);
2521 : }
2522 :
2523 : /* lg(x) > 1, x + 1; shallow */
2524 : static GEN
2525 204721 : ZC_add1(GEN x)
2526 : {
2527 204721 : long i, l = lg(x);
2528 204721 : GEN y = cgetg(l, t_COL);
2529 471434 : for (i = 2; i < l; i++) gel(y,i) = gel(x,i);
2530 204724 : gel(y,1) = addiu(gel(x,1), 1); return y;
2531 : }
2532 : /* lg(x) > 1, x - 1; shallow */
2533 : static GEN
2534 71777 : ZC_sub1(GEN x)
2535 : {
2536 71777 : long i, l = lg(x);
2537 71777 : GEN y = cgetg(l, t_COL);
2538 182047 : for (i = 2; i < l; i++) gel(y,i) = gel(x,i);
2539 71777 : gel(y,1) = subiu(gel(x,1), 1); return y;
2540 : }
2541 :
2542 : /* x,y are t_INT or ZC */
2543 : static GEN
2544 0 : zkadd(GEN x, GEN y)
2545 : {
2546 0 : long tx = typ(x);
2547 0 : if (tx == typ(y))
2548 0 : return tx == t_INT? addii(x,y): ZC_add(x,y);
2549 : else
2550 0 : return tx == t_INT? ZC_Z_add(y,x): ZC_Z_add(x,y);
2551 : }
2552 : /* x a t_INT or ZC, x+1; shallow */
2553 : static GEN
2554 301813 : zkadd1(GEN x)
2555 : {
2556 301813 : long tx = typ(x);
2557 301813 : return tx == t_INT? addiu(x,1): ZC_add1(x);
2558 : }
2559 : /* x a t_INT or ZC, x-1; shallow */
2560 : static GEN
2561 301847 : zksub1(GEN x)
2562 : {
2563 301847 : long tx = typ(x);
2564 301847 : return tx == t_INT? subiu(x,1): ZC_sub1(x);
2565 : }
2566 : /* x,y are t_INT or ZC; x - y */
2567 : static GEN
2568 0 : zksub(GEN x, GEN y)
2569 : {
2570 0 : long tx = typ(x), ty = typ(y);
2571 0 : if (tx == ty)
2572 0 : return tx == t_INT? subii(x,y): ZC_sub(x,y);
2573 : else
2574 0 : return tx == t_INT? Z_ZC_sub(x,y): ZC_Z_sub(x,y);
2575 : }
2576 : /* x is t_INT or ZM (mult. map), y is t_INT or ZC; x * y */
2577 : static GEN
2578 301828 : zkmul(GEN x, GEN y)
2579 : {
2580 301828 : long tx = typ(x), ty = typ(y);
2581 301828 : if (ty == t_INT)
2582 230066 : return tx == t_INT? mulii(x,y): ZC_Z_mul(gel(x,1),y);
2583 : else
2584 71762 : return tx == t_INT? ZC_Z_mul(y,x): ZM_ZC_mul(x,y);
2585 : }
2586 :
2587 : /* (U,V) = 1 coprime ideals. Want z = x mod U, = y mod V; namely
2588 : * z =vx + uy = v(x-y) + y, where u + v = 1, u in U, v in V.
2589 : * zkc = [v, UV], v a t_INT or ZM (mult. by v map), UV a ZM (ideal in HNF);
2590 : * shallow */
2591 : GEN
2592 0 : zkchinese(GEN zkc, GEN x, GEN y)
2593 : {
2594 0 : GEN v = gel(zkc,1), UV = gel(zkc,2), z = zkadd(zkmul(v, zksub(x,y)), y);
2595 0 : return zk_modHNF(z, UV);
2596 : }
2597 : /* special case z = x mod U, = 1 mod V; shallow */
2598 : GEN
2599 301847 : zkchinese1(GEN zkc, GEN x)
2600 : {
2601 301847 : GEN v = gel(zkc,1), UV = gel(zkc,2), z = zkadd1(zkmul(v, zksub1(x)));
2602 301829 : return (typ(z) == t_INT)? z: ZC_hnfrem(z, UV);
2603 : }
2604 : static GEN
2605 270569 : zkVchinese1(GEN zkc, GEN v)
2606 : {
2607 : long i, ly;
2608 270569 : GEN y = cgetg_copy(v, &ly);
2609 572386 : for (i=1; i<ly; i++) gel(y,i) = zkchinese1(zkc, gel(v,i));
2610 270539 : return y;
2611 : }
2612 :
2613 : /* prepare to solve z = x (mod A), z = y mod (B) [zkchinese or zkchinese1] */
2614 : GEN
2615 270320 : zkchineseinit(GEN nf, GEN A, GEN B, GEN AB)
2616 : {
2617 270320 : GEN v = idealaddtoone_raw(nf, A, B);
2618 : long e;
2619 270309 : if ((e = gexpo(v)) > 5)
2620 : {
2621 83351 : GEN b = (typ(v) == t_COL)? v: scalarcol_shallow(v, nf_get_degree(nf));
2622 83351 : b= ZC_reducemodlll(b, AB);
2623 83356 : if (gexpo(b) < e) v = b;
2624 : }
2625 270301 : return mkvec2(zk_scalar_or_multable(nf,v), AB);
2626 : }
2627 : /* prepare to solve z = x (mod A), z = 1 mod (B)
2628 : * and then z = 1 (mod A), z = y mod (B) [zkchinese1 twice] */
2629 : static GEN
2630 259 : zkchinese1init2(GEN nf, GEN A, GEN B, GEN AB)
2631 : {
2632 259 : GEN zkc = zkchineseinit(nf, A, B, AB);
2633 259 : GEN mv = gel(zkc,1), mu;
2634 259 : if (typ(mv) == t_INT) return mkvec2(zkc, mkvec2(subui(1,mv),AB));
2635 35 : mu = RgM_Rg_add_shallow(ZM_neg(mv), gen_1);
2636 35 : return mkvec2(mkvec2(mv,AB), mkvec2(mu,AB));
2637 : }
2638 :
2639 : static GEN
2640 2781910 : apply_U(GEN L, GEN a)
2641 : {
2642 2781910 : GEN e, U = gel(L,3), dU = gel(L,4);
2643 2781910 : if (typ(a) == t_INT)
2644 973942 : e = ZC_Z_mul(gel(U,1), subiu(a, 1));
2645 : else
2646 : { /* t_COL */
2647 1807968 : GEN t = shallowcopy(a);
2648 1808012 : gel(t,1) = subiu(gel(t,1), 1); /* t = a - 1 */
2649 1807921 : e = ZM_ZC_mul(U, t);
2650 : }
2651 2781839 : return gdiv(e, dU);
2652 : }
2653 :
2654 : /* true nf; vectors of [[cyc],[g],U.X^-1]. Assume k > 1. */
2655 : static GEN
2656 194100 : principal_units(GEN nf, GEN pr, long k, GEN prk)
2657 : {
2658 : GEN list, prb;
2659 194100 : ulong mask = quadratic_prec_mask(k);
2660 194100 : long a = 1;
2661 :
2662 194100 : prb = pr_hnf(nf,pr);
2663 194100 : list = vectrunc_init(k);
2664 469414 : while (mask > 1)
2665 : {
2666 275321 : GEN pra = prb;
2667 275321 : long b = a << 1;
2668 :
2669 275321 : if (mask & 1) b--;
2670 275321 : mask >>= 1;
2671 : /* compute 1 + pr^a / 1 + pr^b, 2a <= b */
2672 275321 : prb = (b >= k)? prk: idealpows(nf,pr,b);
2673 275320 : vectrunc_append(list, zidealij(pra, prb));
2674 275316 : a = b;
2675 : }
2676 194093 : return list;
2677 : }
2678 : /* a = 1 mod (pr) return log(a) on local-gens of 1+pr/1+pr^k */
2679 : static GEN
2680 1686163 : log_prk1(GEN nf, GEN a, long nh, GEN L2, GEN prk)
2681 : {
2682 1686163 : GEN y = cgetg(nh+1, t_COL);
2683 1686170 : long j, iy, c = lg(L2)-1;
2684 4467990 : for (j = iy = 1; j <= c; j++)
2685 : {
2686 2781902 : GEN L = gel(L2,j), cyc = gel(L,1), gen = gel(L,2), E = apply_U(L,a);
2687 2781874 : long i, nc = lg(cyc)-1;
2688 2781874 : int last = (j == c);
2689 7111992 : for (i = 1; i <= nc; i++, iy++)
2690 : {
2691 4330172 : GEN t, e = gel(E,i);
2692 4330172 : if (typ(e) != t_INT) pari_err_COPRIME("zlog_prk1", a, prk);
2693 4330165 : t = Fp_neg(e, gel(cyc,i));
2694 4330020 : gel(y,iy) = negi(t);
2695 4330104 : if (!last && signe(t)) a = nfmulpowmodideal(nf, a, gel(gen,i), t, prk);
2696 : }
2697 : }
2698 1686088 : return y;
2699 : }
2700 : /* true nf */
2701 : static GEN
2702 69957 : principal_units_relations(GEN nf, GEN L2, GEN prk, long nh)
2703 : {
2704 69957 : GEN h = cgetg(nh+1,t_MAT);
2705 69957 : long ih, j, c = lg(L2)-1;
2706 221141 : for (j = ih = 1; j <= c; j++)
2707 : {
2708 151184 : GEN L = gel(L2,j), F = gel(L,1), G = gel(L,2);
2709 151184 : long k, lG = lg(G);
2710 358918 : for (k = 1; k < lG; k++,ih++)
2711 : { /* log(g^f) mod pr^e */
2712 207734 : GEN a = nfpowmodideal(nf,gel(G,k),gel(F,k),prk);
2713 207734 : gel(h,ih) = ZC_neg(log_prk1(nf, a, nh, L2, prk));
2714 207734 : gcoeff(h,ih,ih) = gel(F,k);
2715 : }
2716 : }
2717 69957 : return h;
2718 : }
2719 : /* true nf; k > 1; multiplicative group (1 + pr) / (1 + pr^k) */
2720 : static GEN
2721 194091 : idealprincipalunits_i(GEN nf, GEN pr, long k, GEN *pU)
2722 : {
2723 194091 : GEN cyc, gen, L2, prk = idealpows(nf, pr, k);
2724 :
2725 194100 : L2 = principal_units(nf, pr, k, prk);
2726 194099 : if (k == 2)
2727 : {
2728 124143 : GEN L = gel(L2,1);
2729 124143 : cyc = gel(L,1);
2730 124143 : gen = gel(L,2);
2731 124143 : if (pU) *pU = matid(lg(gen)-1);
2732 : }
2733 : else
2734 : {
2735 69956 : long c = lg(L2), j;
2736 69956 : GEN EX, h, Ui, vg = cgetg(c, t_VEC);
2737 221141 : for (j = 1; j < c; j++) gel(vg, j) = gmael(L2,j,2);
2738 69957 : vg = shallowconcat1(vg);
2739 69957 : h = principal_units_relations(nf, L2, prk, lg(vg)-1);
2740 69958 : h = ZM_hnfall_i(h, NULL, 0);
2741 69956 : cyc = ZM_snf_group(h, pU, &Ui);
2742 69956 : c = lg(Ui); gen = cgetg(c, t_VEC); EX = cyc_get_expo(cyc);
2743 228835 : for (j = 1; j < c; j++)
2744 158877 : gel(gen,j) = famat_to_nf_modideal_coprime(nf, vg, gel(Ui,j), prk, EX);
2745 : }
2746 194102 : return mkvec4(cyc, gen, prk, L2);
2747 : }
2748 : GEN
2749 196 : idealprincipalunits(GEN nf, GEN pr, long k)
2750 : {
2751 : pari_sp av;
2752 : GEN v;
2753 196 : nf = checknf(nf);
2754 196 : if (k == 1) { checkprid(pr); retmkvec3(gen_1,cgetg(1,t_VEC),cgetg(1,t_VEC)); }
2755 189 : av = avma; v = idealprincipalunits_i(nf, pr, k, NULL);
2756 189 : return gc_GEN(av, mkvec3(powiu(pr_norm(pr), k-1), gel(v,1), gel(v,2)));
2757 : }
2758 :
2759 : /* true nf; given an ideal pr^k dividing an integral ideal x (in HNF form)
2760 : * compute an 'sprk', the structure of G = (Z_K/pr^k)^* [ x = NULL for x=pr^k ]
2761 : * Return a vector with at least 4 components [cyc],[gen],[HNF pr^k,pr,k],ff,
2762 : * where
2763 : * cyc : type of G as abelian group (SNF)
2764 : * gen : generators of G, coprime to x
2765 : * pr^k: in HNF
2766 : * ff : data for log_g in (Z_K/pr)^*
2767 : * Two extra components are present iff k > 1: L2, U
2768 : * L2 : list of data structures to compute local DL in (Z_K/pr)^*,
2769 : * and 1 + pr^a/ 1 + pr^b for various a < b <= min(2a, k)
2770 : * U : base change matrices to convert a vector of local DL to DL wrt gen
2771 : * If MOD is not NULL, initialize G / G^MOD instead */
2772 : static GEN
2773 465698 : sprkinit(GEN nf, GEN pr, long k, GEN x, GEN MOD)
2774 : {
2775 465698 : GEN T, p, Ld, modpr, cyc, gen, g, g0, A, prk, U, L2, ord0 = NULL;
2776 465698 : long f = pr_get_f(pr);
2777 :
2778 465698 : if(DEBUGLEVEL>3) err_printf("treating pr^%ld, pr = %Ps\n",k,pr);
2779 465698 : modpr = nf_to_Fq_init(nf, &pr,&T,&p);
2780 465716 : if (MOD)
2781 : {
2782 418025 : GEN o = subiu(powiu(p,f), 1), d = gcdii(o, MOD), fa = Z_factor(d);
2783 418000 : ord0 = mkvec2(o, fa); /* true order, factorization of order in G/G^MOD */
2784 417991 : Ld = gel(fa,1);
2785 417991 : if (lg(Ld) > 1 && equaliu(gel(Ld,1),2)) Ld = vecslice(Ld,2,lg(Ld)-1);
2786 : }
2787 : /* (Z_K / pr)^* */
2788 465689 : if (f == 1)
2789 : {
2790 375732 : g0 = g = MOD? pgener_Fp_local(p, Ld): pgener_Fp(p);
2791 375754 : if (!ord0) ord0 = get_arith_ZZM(subiu(p,1));
2792 : }
2793 : else
2794 : {
2795 89957 : g0 = g = MOD? gener_FpXQ_local(T, p, Ld): gener_FpXQ(T,p, &ord0);
2796 89957 : g = Fq_to_nf(g, modpr);
2797 89956 : if (typ(g) == t_POL) g = poltobasis(nf, g);
2798 : }
2799 465712 : A = gel(ord0, 1); /* Norm(pr)-1 */
2800 : /* If MOD != NULL, d = gcd(A, MOD): g^(A/d) has order d */
2801 465712 : if (k == 1)
2802 : {
2803 271809 : cyc = mkvec(A);
2804 271808 : gen = mkvec(g);
2805 271803 : prk = pr_hnf(nf,pr);
2806 271816 : L2 = U = NULL;
2807 : }
2808 : else
2809 : { /* local-gens of (1 + pr)/(1 + pr^k) = SNF-gens * U */
2810 : GEN AB, B, u, v, w;
2811 : long j, l;
2812 193903 : w = idealprincipalunits_i(nf, pr, k, &U);
2813 : /* incorporate (Z_K/pr)^*, order A coprime to B = expo(1+pr/1+pr^k)*/
2814 193913 : cyc = leafcopy(gel(w,1)); B = cyc_get_expo(cyc); AB = mulii(A,B);
2815 193901 : gen = leafcopy(gel(w,2));
2816 193895 : prk = gel(w,3);
2817 193895 : g = nfpowmodideal(nf, g, B, prk);
2818 193909 : g0 = Fq_pow(g0, modii(B,A), T, p); /* update primitive root */
2819 193900 : L2 = mkvec3(A, g, gel(w,4));
2820 193908 : gel(cyc,1) = AB;
2821 193908 : gel(gen,1) = nfmulmodideal(nf, gel(gen,1), g, prk);
2822 193905 : u = mulii(Fp_inv(A,B), A);
2823 193894 : v = subui(1, u); l = lg(U);
2824 570064 : for (j = 1; j < l; j++) gcoeff(U,1,j) = Fp_mul(u, gcoeff(U,1,j), AB);
2825 193899 : U = mkvec2(Rg_col_ei(v, lg(gen)-1, 1), U);
2826 : }
2827 : /* local-gens of (Z_K/pr^k)^* = SNF-gens * U */
2828 465718 : if (x)
2829 : {
2830 270065 : GEN uv = zkchineseinit(nf, idealmulpowprime(nf,x,pr,utoineg(k)), prk, x);
2831 270052 : gen = zkVchinese1(uv, gen);
2832 : }
2833 465676 : return mkvecn(U? 6: 4, cyc, gen, prk, mkvec3(modpr,g0,ord0), L2, U);
2834 : }
2835 : GEN
2836 4570942 : sprk_get_cyc(GEN s) { return gel(s,1); }
2837 : GEN
2838 2143762 : sprk_get_expo(GEN s) { return cyc_get_expo(sprk_get_cyc(s)); }
2839 : GEN
2840 370358 : sprk_get_gen(GEN s) { return gel(s,2); }
2841 : GEN
2842 5630544 : sprk_get_prk(GEN s) { return gel(s,3); }
2843 : GEN
2844 3041832 : sprk_get_ff(GEN s) { return gel(s,4); }
2845 : GEN
2846 2820300 : sprk_get_pr(GEN s) { GEN ff = gel(s,4); return modpr_get_pr(gel(ff,1)); }
2847 : /* L2 to 1 + pr / 1 + pr^k */
2848 : static GEN
2849 1552255 : sprk_get_L2(GEN s) { return gmael(s,5,3); }
2850 : /* lift to nf of primitive root of k(pr) */
2851 : static GEN
2852 318214 : sprk_get_gnf(GEN s) { return gmael(s,5,2); }
2853 : /* A = Npr-1, <g> = (Z_K/pr)^*, L2 to 1 + pr / 1 + pr^k */
2854 : void
2855 0 : sprk_get_AgL2(GEN s, GEN *A, GEN *g, GEN *L2)
2856 0 : { GEN v = gel(s,5); *A = gel(v,1); *g = gel(v,2); *L2 = gel(v,3); }
2857 : void
2858 1532126 : sprk_get_U2(GEN s, GEN *U1, GEN *U2)
2859 1532126 : { GEN v = gel(s,6); *U1 = gel(v,1); *U2 = gel(v,2); }
2860 : static int
2861 3041831 : sprk_is_prime(GEN s) { return lg(s) == 5; }
2862 :
2863 : GEN
2864 2143566 : famat_zlog_pr(GEN nf, GEN g, GEN e, GEN sprk, GEN mod)
2865 : {
2866 2143566 : GEN x, expo = sprk_get_expo(sprk);
2867 2143565 : if (mod) expo = gcdii(expo,mod);
2868 2143554 : x = famat_makecoprime(nf, g, e, sprk_get_pr(sprk), sprk_get_prk(sprk), expo);
2869 2143558 : return log_prk(nf, x, sprk, mod);
2870 : }
2871 : /* famat_zlog_pr assuming (g,sprk.pr) = 1 */
2872 : static GEN
2873 196 : famat_zlog_pr_coprime(GEN nf, GEN g, GEN e, GEN sprk, GEN MOD)
2874 : {
2875 196 : GEN x = famat_to_nf_modideal_coprime(nf, g, e, sprk_get_prk(sprk),
2876 : sprk_get_expo(sprk));
2877 196 : return log_prk(nf, x, sprk, MOD);
2878 : }
2879 :
2880 : /* o t_INT, O = [ord,fa] format for multiple of o (for Fq_log);
2881 : * return o in [ord,fa] format */
2882 : static GEN
2883 739833 : order_update(GEN o, GEN O)
2884 : {
2885 739833 : GEN p = gmael(O,2,1), z = o, P, E;
2886 739833 : long i, j, l = lg(p);
2887 739833 : P = cgetg(l, t_COL);
2888 739828 : E = cgetg(l, t_COL);
2889 797033 : for (i = j = 1; i < l; i++)
2890 : {
2891 797033 : long v = Z_pvalrem(z, gel(p,i), &z);
2892 796982 : if (v)
2893 : {
2894 783888 : gel(P,j) = gel(p,i);
2895 783888 : gel(E,j) = utoipos(v); j++;
2896 783900 : if (is_pm1(z)) break;
2897 : }
2898 : }
2899 739790 : setlg(P, j);
2900 739787 : setlg(E, j); return mkvec2(o, mkmat2(P,E));
2901 : }
2902 :
2903 : /* a in Z_K (t_COL or t_INT), pr prime ideal, sprk = sprkinit(nf,pr,k,x),
2904 : * mod positive t_INT or NULL (meaning mod=0).
2905 : * return log(a) modulo mod on SNF-generators of (Z_K/pr^k)^* */
2906 : GEN
2907 3124427 : log_prk(GEN nf, GEN a, GEN sprk, GEN mod)
2908 : {
2909 : GEN e, prk, g, U1, U2, y, ff, O, o, oN, gN, N, T, p, modpr, pr, cyc;
2910 :
2911 3124427 : if (typ(a) == t_MAT) return famat_zlog_pr(nf, gel(a,1), gel(a,2), sprk, mod);
2912 3041820 : N = NULL;
2913 3041820 : ff = sprk_get_ff(sprk);
2914 3041827 : pr = gel(ff,1); /* modpr */
2915 3041827 : g = gN = gel(ff,2);
2916 3041827 : O = gel(ff,3); /* order of g = |Fq^*|, in [ord, fa] format */
2917 3041827 : o = oN = gel(O,1); /* order as a t_INT */
2918 3041827 : prk = sprk_get_prk(sprk);
2919 3041848 : modpr = nf_to_Fq_init(nf, &pr, &T, &p);
2920 3041841 : if (mod)
2921 : {
2922 2523120 : GEN d = gcdii(o,mod);
2923 2522906 : if (!equalii(o, d))
2924 : {
2925 947441 : N = diviiexact(o,d); /* > 1, coprime to p */
2926 947378 : a = nfpowmodideal(nf, a, N, prk);
2927 947558 : oN = d; /* order of g^N mod pr */
2928 : }
2929 : }
2930 3041690 : if (equali1(oN))
2931 684764 : e = gen_0;
2932 : else
2933 : {
2934 2356957 : if (N) { O = order_update(oN, O); gN = Fq_pow(g, N, T, p); }
2935 2356960 : e = Fq_log(nf_to_Fq(nf,a,modpr), gN, O, T, p);
2936 : }
2937 : /* 0 <= e < oN is correct modulo oN */
2938 3041848 : if (sprk_is_prime(sprk)) return mkcol(e); /* k = 1 */
2939 :
2940 1087196 : sprk_get_U2(sprk, &U1,&U2);
2941 1087278 : cyc = sprk_get_cyc(sprk);
2942 1087282 : if (mod)
2943 : {
2944 663686 : cyc = ZV_snf_gcd(cyc, mod);
2945 663673 : if (signe(remii(mod,p))) return ZV_ZV_mod(ZC_Z_mul(U1,e), cyc);
2946 : }
2947 1033549 : if (signe(e))
2948 : {
2949 318214 : GEN E = N? mulii(e, N): e;
2950 318214 : a = nfmulpowmodideal(nf, a, sprk_get_gnf(sprk), Fp_neg(E, o), prk);
2951 : }
2952 : /* a = 1 mod pr */
2953 1033549 : y = log_prk1(nf, a, lg(U2)-1, sprk_get_L2(sprk), prk);
2954 1033581 : if (N)
2955 : { /* from DL(a^N) to DL(a) */
2956 152175 : GEN E = gel(sprk_get_cyc(sprk), 1), q = powiu(p, Z_pval(E, p));
2957 152176 : y = ZC_Z_mul(y, Fp_inv(N, q));
2958 : }
2959 1033581 : y = ZC_lincomb(gen_1, e, ZM_ZC_mul(U2,y), U1);
2960 1033588 : return ZV_ZV_mod(y, cyc);
2961 : }
2962 : /* true nf */
2963 : GEN
2964 95417 : log_prk_init(GEN nf, GEN pr, long k, GEN MOD)
2965 95417 : { return sprkinit(nf,pr,k,NULL,MOD);}
2966 : GEN
2967 497 : veclog_prk(GEN nf, GEN v, GEN sprk)
2968 : {
2969 497 : long l = lg(v), i;
2970 497 : GEN w = cgetg(l, t_MAT);
2971 1232 : for (i = 1; i < l; i++) gel(w,i) = log_prk(nf, gel(v,i), sprk, NULL);
2972 497 : return w;
2973 : }
2974 :
2975 : static GEN
2976 1424660 : famat_zlog(GEN nf, GEN fa, GEN sgn, zlog_S *S)
2977 : {
2978 1424660 : long i, l0, l = lg(S->U);
2979 1424660 : GEN g = gel(fa,1), e = gel(fa,2), y = cgetg(l, t_COL);
2980 1424661 : l0 = lg(S->sprk); /* = l (trivial arch. part), or l-1 */
2981 3065925 : for (i=1; i < l0; i++) gel(y,i) = famat_zlog_pr(nf, g, e, gel(S->sprk,i), S->mod);
2982 1424657 : if (l0 != l)
2983 : {
2984 233420 : if (!sgn) sgn = nfsign_arch(nf, fa, S->archp);
2985 233420 : gel(y,l0) = Flc_to_ZC(sgn);
2986 : }
2987 1424657 : return y;
2988 : }
2989 :
2990 : /* assume that cyclic factors are normalized, in particular != [1] */
2991 : static GEN
2992 270376 : split_U(GEN U, GEN Sprk)
2993 : {
2994 270376 : long t = 0, k, n, l = lg(Sprk);
2995 270376 : GEN vU = cgetg(l+1, t_VEC);
2996 639946 : for (k = 1; k < l; k++)
2997 : {
2998 369574 : n = lg(sprk_get_cyc(gel(Sprk,k))) - 1; /* > 0 */
2999 369575 : gel(vU,k) = vecslice(U, t+1, t+n);
3000 369578 : t += n;
3001 : }
3002 : /* t+1 .. lg(U)-1 */
3003 270372 : n = lg(U) - t - 1; /* can be 0 */
3004 270372 : if (!n) setlg(vU,l); else gel(vU,l) = vecslice(U, t+1, t+n);
3005 270380 : return vU;
3006 : }
3007 :
3008 : static void
3009 2157256 : init_zlog_mod(zlog_S *S, GEN bid, GEN mod)
3010 : {
3011 2157256 : GEN fa2 = bid_get_fact2(bid), MOD = bid_get_MOD(bid);
3012 2157256 : S->U = bid_get_U(bid);
3013 2157253 : S->hU = lg(bid_get_cyc(bid))-1;
3014 2157250 : S->archp = bid_get_archp(bid);
3015 2157250 : S->sprk = bid_get_sprk(bid);
3016 2157249 : S->bid = bid;
3017 2157249 : if (MOD) mod = mod? gcdii(mod, MOD): MOD;
3018 2157161 : S->mod = mod;
3019 2157161 : S->P = gel(fa2,1);
3020 2157161 : S->k = gel(fa2,2);
3021 2157161 : S->no2 = lg(S->P) == lg(gel(bid_get_fact(bid),1));
3022 2157173 : }
3023 : void
3024 395953 : init_zlog(zlog_S *S, GEN bid)
3025 : {
3026 395953 : return init_zlog_mod(S, bid, NULL);
3027 : }
3028 :
3029 : /* a a t_FRAC/t_INT, reduce mod bid */
3030 : static GEN
3031 14 : Q_mod_bid(GEN bid, GEN a)
3032 : {
3033 14 : GEN xZ = gcoeff(bid_get_ideal(bid),1,1);
3034 14 : GEN b = Rg_to_Fp(a, xZ);
3035 14 : if (gsigne(a) < 0) b = subii(b, xZ);
3036 14 : return signe(b)? b: xZ;
3037 : }
3038 : /* Return decomposition of a on the CRT generators blocks attached to the
3039 : * S->sprk and sarch; sgn = sign(a, S->arch), NULL if unknown */
3040 : static GEN
3041 495227 : zlog(GEN nf, GEN a, GEN sgn, zlog_S *S)
3042 : {
3043 : long k, l;
3044 : GEN y;
3045 495227 : a = nf_to_scalar_or_basis(nf, a);
3046 495220 : switch(typ(a))
3047 : {
3048 184968 : case t_INT: break;
3049 14 : case t_FRAC: a = Q_mod_bid(S->bid, a); break;
3050 310238 : default: /* case t_COL: */
3051 : {
3052 : GEN den;
3053 310238 : a = Q_remove_denom(a, &den);
3054 310251 : if (den)
3055 : {
3056 84 : a = mkmat2(mkcol2(a, den), mkcol2(gen_1, gen_m1));
3057 84 : return famat_zlog(nf, a, sgn, S);
3058 : }
3059 : }
3060 : }
3061 495144 : if (sgn)
3062 400812 : sgn = (lg(sgn) == 1)? NULL: leafcopy(sgn);
3063 : else
3064 94332 : sgn = (lg(S->archp) == 1)? NULL: nfsign_arch(nf, a, S->archp);
3065 495141 : l = lg(S->sprk);
3066 495141 : y = cgetg(sgn? l+1: l, t_COL);
3067 1350393 : for (k = 1; k < l; k++)
3068 : {
3069 855295 : GEN sprk = gel(S->sprk,k);
3070 855295 : gel(y,k) = log_prk(nf, a, sprk, S->mod);
3071 : }
3072 495098 : if (sgn) gel(y,l) = Flc_to_ZC(sgn);
3073 495103 : return y;
3074 : }
3075 :
3076 : /* true nf */
3077 : GEN
3078 72247 : pr_basis_perm(GEN nf, GEN pr)
3079 : {
3080 72247 : long f = pr_get_f(pr);
3081 : GEN perm;
3082 72247 : if (f == nf_get_degree(nf)) return identity_perm(f);
3083 62167 : perm = cgetg(f+1, t_VECSMALL);
3084 62167 : perm[1] = 1;
3085 62167 : if (f > 1)
3086 : {
3087 3248 : GEN H = pr_hnf(nf,pr);
3088 : long i, k;
3089 11480 : for (i = k = 2; k <= f; i++)
3090 8232 : if (!equali1(gcoeff(H,i,i))) perm[k++] = i;
3091 : }
3092 62167 : return perm;
3093 : }
3094 :
3095 : /* \sum U[i]*y[i], U[i] ZM, y[i] ZC. We allow lg(y) > lg(U). */
3096 : static GEN
3097 1919860 : ZMV_ZCV_mul(GEN U, GEN y)
3098 : {
3099 1919860 : long i, l = lg(U);
3100 1919860 : GEN z = NULL;
3101 1919860 : if (l == 1) return cgetg(1,t_COL);
3102 4915472 : for (i = 1; i < l; i++)
3103 : {
3104 2995722 : GEN u = ZM_ZC_mul(gel(U,i), gel(y,i));
3105 2995678 : z = z? ZC_add(z, u): u;
3106 : }
3107 1919750 : return z;
3108 : }
3109 : /* A * (x[1], ..., x[d] */
3110 : static GEN
3111 518 : ZM_ZMV_mul(GEN A, GEN x)
3112 1057 : { pari_APPLY_same(ZM_mul(A,gel(x,i))); }
3113 :
3114 : /* a = 1 mod pr, sprk mod pr^e, e >= 1 */
3115 : static GEN
3116 444864 : sprk_log_prk1_2(GEN nf, GEN a, GEN sprk)
3117 : {
3118 444864 : GEN U1, U2, y, L2 = sprk_get_L2(sprk);
3119 444865 : sprk_get_U2(sprk, &U1,&U2);
3120 444865 : y = ZM_ZC_mul(U2, log_prk1(nf, a, lg(U2)-1, L2, sprk_get_prk(sprk)));
3121 444859 : return ZV_ZV_mod(y, sprk_get_cyc(sprk));
3122 : }
3123 : /* true nf; assume e >= 2 */
3124 : GEN
3125 145845 : sprk_log_gen_pr2(GEN nf, GEN sprk, long e)
3126 : {
3127 145845 : GEN M, G, pr = sprk_get_pr(sprk);
3128 : long i, l;
3129 145845 : if (e == 2)
3130 : {
3131 73850 : GEN L2 = sprk_get_L2(sprk), L = gel(L2,1);
3132 73850 : G = gel(L,2); l = lg(G);
3133 : }
3134 : else
3135 : {
3136 71995 : GEN perm = pr_basis_perm(nf,pr), PI = nfpow_u(nf, pr_get_gen(pr), e-1);
3137 71995 : l = lg(perm);
3138 71995 : G = cgetg(l, t_VEC);
3139 71995 : if (typ(PI) == t_INT)
3140 : { /* zk_ei_mul doesn't allow t_INT */
3141 10073 : long N = nf_get_degree(nf);
3142 10073 : gel(G,1) = addiu(PI,1);
3143 13076 : for (i = 2; i < l; i++)
3144 : {
3145 3003 : GEN z = col_ei(N, 1);
3146 3003 : gel(G,i) = z; gel(z, perm[i]) = PI;
3147 : }
3148 : }
3149 : else
3150 : {
3151 61922 : gel(G,1) = nfadd(nf, gen_1, PI);
3152 69041 : for (i = 2; i < l; i++)
3153 7119 : gel(G,i) = nfadd(nf, gen_1, zk_ei_mul(nf, PI, perm[i]));
3154 : }
3155 : }
3156 145845 : M = cgetg(l, t_MAT);
3157 314819 : for (i = 1; i < l; i++) gel(M,i) = sprk_log_prk1_2(nf, gel(G,i), sprk);
3158 145827 : return M;
3159 : }
3160 : /* Log on bid.gen of generators of P_{1,I pr^{e-1}} / P_{1,I pr^e} (I,pr) = 1,
3161 : * defined implicitly via CRT. 'ind' is the index of pr in modulus
3162 : * factorization; true nf */
3163 : GEN
3164 484277 : log_gen_pr(zlog_S *S, long ind, GEN nf, long e)
3165 : {
3166 484277 : GEN Uind = gel(S->U, ind);
3167 484277 : if (e == 1) retmkmat( gel(Uind,1) );
3168 143127 : return ZM_mul(Uind, sprk_log_gen_pr2(nf, gel(S->sprk,ind), e));
3169 : }
3170 : /* true nf */
3171 : GEN
3172 2037 : sprk_log_gen_pr(GEN nf, GEN sprk, long e)
3173 : {
3174 2037 : if (e == 1)
3175 : {
3176 0 : long n = lg(sprk_get_cyc(sprk))-1;
3177 0 : retmkmat(col_ei(n, 1));
3178 : }
3179 2037 : return sprk_log_gen_pr2(nf, sprk, e);
3180 : }
3181 : /* a = 1 mod pr */
3182 : GEN
3183 275872 : sprk_log_prk1(GEN nf, GEN a, GEN sprk)
3184 : {
3185 275872 : if (lg(sprk) == 5) return mkcol(gen_0); /* mod pr */
3186 275872 : return sprk_log_prk1_2(nf, a, sprk);
3187 : }
3188 : /* Log on bid.gen of generator of P_{1,f} / P_{1,f v[index]}
3189 : * v = vector of r1 real places */
3190 : GEN
3191 111991 : log_gen_arch(zlog_S *S, long index) { return gel(veclast(S->U), index); }
3192 :
3193 : /* compute bid.clgp: [h,cyc] or [h,cyc,gen] */
3194 : static GEN
3195 271392 : bid_grp(GEN nf, GEN U, GEN cyc, GEN g, GEN F, GEN sarch)
3196 : {
3197 271392 : GEN G, h = ZV_prod(cyc);
3198 : long c;
3199 271416 : if (!U) return mkvec2(h,cyc);
3200 271059 : c = lg(U);
3201 271059 : G = cgetg(c,t_VEC);
3202 271065 : if (c > 1)
3203 : {
3204 240974 : GEN U0, Uoo, EX = cyc_get_expo(cyc); /* exponent of bid */
3205 240973 : long i, hU = nbrows(U), nba = lg(sarch_get_cyc(sarch))-1; /* #f_oo */
3206 240981 : if (!nba) { U0 = U; Uoo = NULL; }
3207 90583 : else if (nba == hU) { U0 = NULL; Uoo = U; }
3208 : else
3209 : {
3210 81441 : U0 = rowslice(U, 1, hU-nba);
3211 81444 : Uoo = rowslice(U, hU-nba+1, hU);
3212 : }
3213 771488 : for (i = 1; i < c; i++)
3214 : {
3215 530510 : GEN t = gen_1;
3216 530510 : if (U0) t = famat_to_nf_modideal_coprime(nf, g, gel(U0,i), F, EX);
3217 530510 : if (Uoo) t = set_sign_mod_divisor(nf, ZV_to_Flv(gel(Uoo,i),2), t, sarch);
3218 530507 : gel(G,i) = t;
3219 : }
3220 : }
3221 271069 : return mkvec3(h, cyc, G);
3222 : }
3223 :
3224 : /* remove prime ideals of norm 2 with exponent 1 from factorization */
3225 : static GEN
3226 271767 : famat_strip2(GEN fa)
3227 : {
3228 271767 : GEN P = gel(fa,1), E = gel(fa,2), Q, F;
3229 271767 : long l = lg(P), i, j;
3230 271767 : Q = cgetg(l, t_COL);
3231 271764 : F = cgetg(l, t_COL);
3232 681394 : for (i = j = 1; i < l; i++)
3233 : {
3234 409624 : GEN pr = gel(P,i), e = gel(E,i);
3235 409624 : if (!absequaliu(pr_get_p(pr), 2) || itou(e) != 1 || pr_get_f(pr) != 1)
3236 : {
3237 370984 : gel(Q,j) = pr;
3238 370984 : gel(F,j) = e; j++;
3239 : }
3240 : }
3241 271770 : setlg(Q,j);
3242 271769 : setlg(F,j); return mkmat2(Q,F);
3243 : }
3244 : static int
3245 134104 : checkarchp(GEN v, long r1)
3246 : {
3247 134104 : long i, l = lg(v);
3248 134104 : pari_sp av = avma;
3249 : GEN p;
3250 134104 : if (l == 1) return 1;
3251 47167 : if (l == 2) return v[1] > 0 && v[1] <= r1;
3252 22016 : p = zero_zv(r1);
3253 66147 : for (i = 1; i < l; i++)
3254 : {
3255 44161 : long j = v[i];
3256 44161 : if (j <= 0 || j > r1 || p[j]) return gc_long(av, 0);
3257 44126 : p[j] = 1;
3258 : }
3259 21986 : return gc_long(av, 1);
3260 : }
3261 :
3262 : /* True nf. Put ideal to form [[ideal,arch]] and set fa and fa2 to its
3263 : * factorization, archp to the indices of arch places */
3264 : GEN
3265 271761 : check_mod_factored(GEN nf, GEN ideal, GEN *fa_, GEN *fa2_, GEN *archp_, GEN MOD)
3266 : {
3267 : GEN arch, x, fa, fa2, archp;
3268 : long R1;
3269 :
3270 271761 : R1 = nf_get_r1(nf);
3271 271763 : if (typ(ideal) == t_VEC && lg(ideal) == 3)
3272 : {
3273 191560 : arch = gel(ideal,2);
3274 191560 : ideal= gel(ideal,1);
3275 191560 : switch(typ(arch))
3276 : {
3277 57456 : case t_VEC:
3278 57456 : if (lg(arch) != R1+1)
3279 7 : pari_err_TYPE("Idealstar [incorrect archimedean component]",arch);
3280 57449 : archp = vec01_to_indices(arch);
3281 57449 : break;
3282 134104 : case t_VECSMALL:
3283 134104 : if (!checkarchp(arch, R1))
3284 35 : pari_err_TYPE("Idealstar [incorrect archimedean component]",arch);
3285 134071 : archp = arch;
3286 134071 : arch = indices_to_vec01(archp, R1);
3287 134069 : break;
3288 0 : default:
3289 0 : pari_err_TYPE("Idealstar [incorrect archimedean component]",arch);
3290 : return NULL;/*LCOV_EXCL_LINE*/
3291 : }
3292 : }
3293 : else
3294 : {
3295 80203 : arch = zerovec(R1);
3296 80195 : archp = cgetg(1, t_VECSMALL);
3297 : }
3298 271718 : if (MOD)
3299 : {
3300 227086 : if (typ(MOD) != t_INT) pari_err_TYPE("bnrinit [incorrect cycmod]", MOD);
3301 227086 : if (mpodd(MOD) && lg(archp) != 1)
3302 231 : MOD = shifti(MOD, 1); /* ensure elements of G^MOD are >> 0 */
3303 : }
3304 271714 : if (is_nf_factor(ideal))
3305 : {
3306 53172 : fa = ideal;
3307 53172 : x = factorbackprime(nf, gel(fa,1), gel(fa,2));
3308 : }
3309 : else
3310 : {
3311 218544 : fa = idealfactor(nf, ideal);
3312 218551 : x = ideal;
3313 : }
3314 271723 : if (typ(x) != t_MAT) x = idealhnf_shallow(nf, x);
3315 271711 : if (lg(x) == 1) pari_err_DOMAIN("Idealstar", "ideal","=",gen_0,x);
3316 271711 : if (typ(gcoeff(x,1,1)) != t_INT)
3317 7 : pari_err_DOMAIN("Idealstar","denominator(ideal)", "!=",gen_1,x);
3318 :
3319 271704 : fa2 = famat_strip2(fa);
3320 271704 : if (fa_ != NULL) *fa_ = fa;
3321 271704 : if (fa2_ != NULL) *fa2_ = fa2;
3322 271704 : if (fa2_ != NULL) *archp_ = archp;
3323 271704 : return mkvec2(x, arch);
3324 : }
3325 :
3326 : /* True nf. Compute [[ideal,arch], [h,[cyc],[gen]], idealfact, [liste], U]
3327 : flag may include nf_GEN | nf_INIT */
3328 : static GEN
3329 271114 : Idealstarmod_i(GEN nf, GEN ideal, long flag, GEN MOD)
3330 : {
3331 : long i, nbp;
3332 271114 : GEN y, cyc, U, u1 = NULL, fa, fa2, sprk, x_arch, x, arch, archp, E, P, sarch, gen;
3333 :
3334 271114 : x_arch = check_mod_factored(nf, ideal, &fa, &fa2, &archp, MOD);
3335 271063 : x = gel(x_arch, 1);
3336 271063 : arch = gel(x_arch, 2);
3337 :
3338 271063 : sarch = nfarchstar(nf, x, archp);
3339 271061 : P = gel(fa2,1);
3340 271061 : E = gel(fa2,2);
3341 271061 : nbp = lg(P)-1;
3342 271061 : sprk = cgetg(nbp+1,t_VEC);
3343 271070 : if (nbp)
3344 : {
3345 231793 : GEN t = (lg(gel(fa,1))==2)? NULL: x; /* beware fa != fa2 */
3346 231793 : cyc = cgetg(nbp+2,t_VEC);
3347 231786 : gen = cgetg(nbp+1,t_VEC);
3348 602084 : for (i = 1; i <= nbp; i++)
3349 : {
3350 370285 : GEN L = sprkinit(nf, gel(P,i), itou(gel(E,i)), t, MOD);
3351 370298 : gel(sprk,i) = L;
3352 370298 : gel(cyc,i) = sprk_get_cyc(L);
3353 : /* true gens are congruent to those mod x AND positive at archp */
3354 370296 : gel(gen,i) = sprk_get_gen(L);
3355 : }
3356 231799 : gel(cyc,i) = sarch_get_cyc(sarch);
3357 231799 : cyc = shallowconcat1(cyc);
3358 231799 : gen = shallowconcat1(gen);
3359 231803 : cyc = ZV_snf_group(cyc, &U, (flag & nf_GEN)? &u1: NULL);
3360 : }
3361 : else
3362 : {
3363 39277 : cyc = sarch_get_cyc(sarch);
3364 39277 : gen = cgetg(1,t_VEC);
3365 39276 : U = matid(lg(cyc)-1);
3366 39275 : if (flag & nf_GEN) u1 = U;
3367 : }
3368 271052 : if (MOD) cyc = ZV_snf_gcd(cyc, MOD);
3369 271035 : y = bid_grp(nf, u1, cyc, gen, x, sarch);
3370 271070 : if (!(flag & nf_INIT)) return y;
3371 270272 : U = split_U(U, sprk);
3372 540539 : return mkvec5(mkvec2(x, arch), y, mkvec2(fa,fa2),
3373 270269 : MOD? mkvec3(sprk, sarch, MOD): mkvec2(sprk, sarch),
3374 : U);
3375 : }
3376 :
3377 : static long
3378 63 : idealHNF_norm_pval(GEN x, GEN p)
3379 : {
3380 63 : long i, v = 0, l = lg(x);
3381 175 : for (i = 1; i < l; i++) v += Z_pval(gcoeff(x,i,i), p);
3382 63 : return v;
3383 : }
3384 : static long
3385 63 : sprk_get_k(GEN sprk)
3386 : {
3387 : GEN pr, prk;
3388 63 : if (sprk_is_prime(sprk)) return 1;
3389 63 : pr = sprk_get_pr(sprk);
3390 63 : prk = sprk_get_prk(sprk);
3391 63 : return idealHNF_norm_pval(prk, pr_get_p(pr)) / pr_get_f(pr);
3392 : }
3393 : /* true nf, L a sprk */
3394 : GEN
3395 63 : sprk_to_bid(GEN nf, GEN L, long flag)
3396 : {
3397 63 : GEN y, cyc, U, u1 = NULL, fa, fa2, arch, sarch, gen, sprk;
3398 :
3399 63 : arch = zerovec(nf_get_r1(nf));
3400 63 : fa = to_famat_shallow(sprk_get_pr(L), utoipos(sprk_get_k(L)));
3401 63 : sarch = nfarchstar(nf, NULL, cgetg(1, t_VECSMALL));
3402 63 : fa2 = famat_strip2(fa);
3403 63 : sprk = mkvec(L);
3404 63 : cyc = shallowconcat(sprk_get_cyc(L), sarch_get_cyc(sarch));
3405 63 : gen = sprk_get_gen(L);
3406 63 : cyc = ZV_snf_group(cyc, &U, (flag & nf_GEN)? &u1: NULL);
3407 63 : y = bid_grp(nf, u1, cyc, gen, NULL, sarch);
3408 63 : if (!(flag & nf_INIT)) return y;
3409 63 : return mkvec5(mkvec2(sprk_get_prk(L), arch), y, mkvec2(fa,fa2),
3410 : mkvec2(sprk, sarch), split_U(U, sprk));
3411 : }
3412 : GEN
3413 270833 : Idealstarmod(GEN nf, GEN ideal, long flag, GEN MOD)
3414 : {
3415 270833 : pari_sp av = avma;
3416 270833 : nf = nf? checknf(nf): nfinit(pol_x(0), DEFAULTPREC);
3417 270841 : return gc_GEN(av, Idealstarmod_i(nf, ideal, flag, MOD));
3418 : }
3419 : GEN
3420 938 : Idealstar(GEN nf, GEN ideal, long flag) { return Idealstarmod(nf, ideal, flag, NULL); }
3421 : GEN
3422 273 : Idealstarprk(GEN nf, GEN pr, long k, long flag)
3423 : {
3424 273 : pari_sp av = avma;
3425 273 : GEN z = Idealstarmod_i(nf, mkmat2(mkcol(pr),mkcols(k)), flag, NULL);
3426 273 : return gc_GEN(av, z);
3427 : }
3428 :
3429 : /* FIXME: obsolete */
3430 : GEN
3431 0 : zidealstarinitgen(GEN nf, GEN ideal)
3432 0 : { return Idealstar(nf,ideal, nf_INIT|nf_GEN); }
3433 : GEN
3434 0 : zidealstarinit(GEN nf, GEN ideal)
3435 0 : { return Idealstar(nf,ideal, nf_INIT); }
3436 : GEN
3437 0 : zidealstar(GEN nf, GEN ideal)
3438 0 : { return Idealstar(nf,ideal, nf_GEN); }
3439 :
3440 : GEN
3441 112 : idealstarmod(GEN nf, GEN ideal, long flag, GEN MOD)
3442 : {
3443 112 : switch(flag)
3444 : {
3445 0 : case 0: return Idealstarmod(nf,ideal, nf_GEN, MOD);
3446 98 : case 1: return Idealstarmod(nf,ideal, nf_INIT, MOD);
3447 14 : case 2: return Idealstarmod(nf,ideal, nf_INIT|nf_GEN, MOD);
3448 0 : default: pari_err_FLAG("idealstar");
3449 : }
3450 : return NULL; /* LCOV_EXCL_LINE */
3451 : }
3452 : GEN
3453 0 : idealstar0(GEN nf, GEN ideal,long flag) { return idealstarmod(nf, ideal, flag, NULL); }
3454 :
3455 : GEN
3456 2321590 : ZV_snf_gcd(GEN x, GEN mod)
3457 5501777 : { pari_APPLY_same(gcdii(gel(x,i), mod)); }
3458 :
3459 : /* assume a true bnf and bid */
3460 : GEN
3461 239957 : ideallog_units0(GEN bnf, GEN bid, GEN MOD)
3462 : {
3463 239957 : GEN nf = bnf_get_nf(bnf), D, y, C, cyc;
3464 239956 : long j, lU = lg(bnf_get_logfu(bnf)); /* r1+r2 */
3465 : zlog_S S;
3466 239956 : init_zlog_mod(&S, bid, MOD);
3467 239935 : if (!S.hU) return zeromat(0,lU);
3468 239935 : cyc = bid_get_cyc(bid);
3469 239932 : D = nfsign_fu(bnf, bid_get_archp(bid));
3470 239950 : y = cgetg(lU, t_MAT);
3471 239947 : if ((C = bnf_build_cheapfu(bnf)))
3472 400767 : { for (j = 1; j < lU; j++) gel(y,j) = zlog(nf, gel(C,j), gel(D,j), &S); }
3473 : else
3474 : {
3475 49 : long i, l = lg(S.U), l0 = lg(S.sprk);
3476 : GEN X, U;
3477 49 : if (!(C = bnf_compactfu_mat(bnf))) bnf_build_units(bnf); /* error */
3478 49 : X = gel(C,1); U = gel(C,2);
3479 147 : for (j = 1; j < lU; j++) gel(y,j) = cgetg(l, t_COL);
3480 126 : for (i = 1; i < l0; i++)
3481 : {
3482 77 : GEN sprk = gel(S.sprk, i);
3483 77 : GEN Xi = sunits_makecoprime(X, sprk_get_pr(sprk), sprk_get_prk(sprk));
3484 231 : for (j = 1; j < lU; j++)
3485 154 : gcoeff(y,i,j) = famat_zlog_pr_coprime(nf, Xi, gel(U,j), sprk, MOD);
3486 : }
3487 49 : if (l0 != l)
3488 56 : for (j = 1; j < lU; j++) gcoeff(y,l0,j) = Flc_to_ZC(gel(D,j));
3489 : }
3490 239946 : y = vec_prepend(y, zlog(nf, bnf_get_tuU(bnf), nfsign_tu(bnf, S.archp), &S));
3491 640818 : for (j = 1; j <= lU; j++)
3492 400889 : gel(y,j) = ZV_ZV_mod(ZMV_ZCV_mul(S.U, gel(y,j)), cyc);
3493 239929 : return y;
3494 : }
3495 : GEN
3496 84 : ideallog_units(GEN bnf, GEN bid)
3497 84 : { return ideallog_units0(bnf, bid, NULL); }
3498 : GEN
3499 518 : log_prk_units(GEN nf, GEN D, GEN sprk)
3500 : {
3501 518 : GEN L, Ltu = log_prk(nf, gel(D,1), sprk, NULL);
3502 518 : D = gel(D,2);
3503 518 : if (lg(D) != 3 || typ(gel(D,2)) != t_MAT) L = veclog_prk(nf, D, sprk);
3504 : else
3505 : {
3506 21 : GEN X = gel(D,1), U = gel(D,2);
3507 21 : long j, lU = lg(U);
3508 21 : X = sunits_makecoprime(X, sprk_get_pr(sprk), sprk_get_prk(sprk));
3509 21 : L = cgetg(lU, t_MAT);
3510 63 : for (j = 1; j < lU; j++)
3511 42 : gel(L,j) = famat_zlog_pr_coprime(nf, X, gel(U,j), sprk, NULL);
3512 : }
3513 518 : return vec_prepend(L, Ltu);
3514 : }
3515 :
3516 : static GEN
3517 1521357 : ideallog_i(GEN nf, GEN x, zlog_S *S)
3518 : {
3519 1521357 : pari_sp av = avma;
3520 : GEN y;
3521 1521357 : if (!S->hU) return cgetg(1, t_COL);
3522 1518991 : if (typ(x) == t_MAT)
3523 1424575 : y = famat_zlog(nf, x, NULL, S);
3524 : else
3525 94416 : y = zlog(nf, x, NULL, S);
3526 1518984 : y = ZMV_ZCV_mul(S->U, y);
3527 1518986 : return gc_upto(av, ZV_ZV_mod(y, bid_get_cyc(S->bid)));
3528 : }
3529 : GEN
3530 1528039 : ideallogmod(GEN nf, GEN x, GEN bid, GEN mod)
3531 : {
3532 : zlog_S S;
3533 1528039 : if (!nf)
3534 : {
3535 6671 : if (mod) pari_err_IMPL("Zideallogmod");
3536 6671 : return Zideallog(bid, x);
3537 : }
3538 1521368 : checkbid(bid); init_zlog_mod(&S, bid, mod);
3539 1521358 : return ideallog_i(checknf(nf), x, &S);
3540 : }
3541 : GEN
3542 13769 : ideallog(GEN nf, GEN x, GEN bid) { return ideallogmod(nf, x, bid, NULL); }
3543 :
3544 : /*************************************************************************/
3545 : /** **/
3546 : /** JOIN BID STRUCTURES, IDEAL LISTS **/
3547 : /** **/
3548 : /*************************************************************************/
3549 : /* bid1, bid2: for coprime modules m1 and m2 (without arch. part).
3550 : * Output: bid for m1 m2 */
3551 : static GEN
3552 469 : join_bid(GEN nf, GEN bid1, GEN bid2)
3553 : {
3554 469 : pari_sp av = avma;
3555 : long nbgen, l1,l2;
3556 : GEN I1,I2, G1,G2, sprk1,sprk2, cyc1,cyc2, sarch;
3557 469 : GEN sprk, fa,fa2, U, cyc, y, u1 = NULL, x, gen;
3558 :
3559 469 : I1 = bid_get_ideal(bid1);
3560 469 : I2 = bid_get_ideal(bid2);
3561 469 : if (gequal1(gcoeff(I1,1,1))) return bid2; /* frequent trivial case */
3562 259 : G1 = bid_get_grp(bid1);
3563 259 : G2 = bid_get_grp(bid2);
3564 259 : x = idealmul(nf, I1,I2);
3565 259 : fa = famat_mul_shallow(bid_get_fact(bid1), bid_get_fact(bid2));
3566 259 : fa2= famat_mul_shallow(bid_get_fact2(bid1), bid_get_fact2(bid2));
3567 259 : sprk1 = bid_get_sprk(bid1);
3568 259 : sprk2 = bid_get_sprk(bid2);
3569 259 : sprk = shallowconcat(sprk1, sprk2);
3570 :
3571 259 : cyc1 = abgrp_get_cyc(G1); l1 = lg(cyc1);
3572 259 : cyc2 = abgrp_get_cyc(G2); l2 = lg(cyc2);
3573 259 : gen = (lg(G1)>3 && lg(G2)>3)? gen_1: NULL;
3574 259 : nbgen = l1+l2-2;
3575 259 : cyc = ZV_snf_group(shallowconcat(cyc1,cyc2), &U, gen? &u1: NULL);
3576 259 : if (nbgen)
3577 : {
3578 259 : GEN U1 = bid_get_U(bid1), U2 = bid_get_U(bid2);
3579 0 : U1 = l1==1? const_vec(lg(sprk1), cgetg(1,t_MAT))
3580 259 : : ZM_ZMV_mul(vecslice(U, 1, l1-1), U1);
3581 0 : U2 = l2==1? const_vec(lg(sprk2), cgetg(1,t_MAT))
3582 259 : : ZM_ZMV_mul(vecslice(U, l1, nbgen), U2);
3583 259 : U = shallowconcat(U1, U2);
3584 : }
3585 : else
3586 0 : U = const_vec(lg(sprk), cgetg(1,t_MAT));
3587 :
3588 259 : if (gen)
3589 : {
3590 259 : GEN uv = zkchinese1init2(nf, I2, I1, x);
3591 518 : gen = shallowconcat(zkVchinese1(gel(uv,1), abgrp_get_gen(G1)),
3592 259 : zkVchinese1(gel(uv,2), abgrp_get_gen(G2)));
3593 : }
3594 259 : sarch = bid_get_sarch(bid1); /* trivial */
3595 259 : y = bid_grp(nf, u1, cyc, gen, x, sarch);
3596 259 : x = mkvec2(x, bid_get_arch(bid1));
3597 259 : y = mkvec5(x, y, mkvec2(fa, fa2), mkvec2(sprk, sarch), U);
3598 259 : return gc_GEN(av,y);
3599 : }
3600 :
3601 : typedef struct _ideal_data {
3602 : GEN nf, emb, L, pr, prL, sgnU, archp;
3603 : } ideal_data;
3604 :
3605 : /* z <- ( z | f(v[i])_{i=1..#v} ) */
3606 : static void
3607 758573 : concat_join(GEN *pz, GEN v, GEN (*f)(ideal_data*,GEN), ideal_data *data)
3608 : {
3609 758573 : long i, nz, lv = lg(v);
3610 : GEN z, Z;
3611 758573 : if (lv == 1) return;
3612 223047 : z = *pz; nz = lg(z)-1;
3613 223047 : *pz = Z = cgetg(lv + nz, typ(z));
3614 371674 : for (i = 1; i <=nz; i++) gel(Z,i) = gel(z,i);
3615 223338 : Z += nz;
3616 492113 : for (i = 1; i < lv; i++) gel(Z,i) = f(data, gel(v,i));
3617 : }
3618 : static GEN
3619 469 : join_idealinit(ideal_data *D, GEN x)
3620 469 : { return join_bid(D->nf, x, D->prL); }
3621 : static GEN
3622 268509 : join_ideal(ideal_data *D, GEN x)
3623 268509 : { return idealmulpowprime(D->nf, x, D->pr, D->L); }
3624 : static GEN
3625 448 : join_unit(ideal_data *D, GEN x)
3626 : {
3627 448 : GEN bid = join_idealinit(D, gel(x,1)), u = gel(x,2), v = mkvec(D->emb);
3628 448 : if (lg(u) != 1) v = shallowconcat(u, v);
3629 448 : return mkvec2(bid, v);
3630 : }
3631 :
3632 : GEN
3633 49 : log_prk_units_init(GEN bnf)
3634 : {
3635 49 : GEN U = bnf_has_fu(bnf);
3636 49 : if (U) U = matalgtobasis(bnf_get_nf(bnf), U);
3637 21 : else if (!(U = bnf_compactfu_mat(bnf))) (void)bnf_build_units(bnf);
3638 49 : return mkvec2(bnf_get_tuU(bnf), U);
3639 : }
3640 : /* flag & nf_GEN : generators, otherwise no
3641 : * flag &2 : units, otherwise no
3642 : * flag &4 : ideals in HNF, otherwise bid
3643 : * flag &8 : omit ideals which cannot be conductors (pr^1 with Npr=2) */
3644 : static GEN
3645 11333 : Ideallist(GEN bnf, ulong bound, long flag)
3646 : {
3647 11333 : const long do_units = flag & 2, big_id = !(flag & 4), cond = flag & 8;
3648 11333 : const long istar_flag = (flag & nf_GEN) | nf_INIT;
3649 : pari_sp av;
3650 : long i, j;
3651 11333 : GEN nf, z, p, fa, id, BOUND, U, empty = cgetg(1,t_VEC);
3652 : forprime_t S;
3653 : ideal_data ID;
3654 : GEN (*join_z)(ideal_data*, GEN);
3655 :
3656 11333 : if (do_units)
3657 : {
3658 21 : bnf = checkbnf(bnf);
3659 21 : nf = bnf_get_nf(bnf);
3660 21 : join_z = &join_unit;
3661 : }
3662 : else
3663 : {
3664 11312 : nf = checknf(bnf);
3665 11312 : join_z = big_id? &join_idealinit: &join_ideal;
3666 : }
3667 11333 : if ((long)bound <= 0) return empty;
3668 11333 : id = matid(nf_get_degree(nf));
3669 11333 : if (big_id) id = Idealstar(nf,id, istar_flag);
3670 :
3671 : /* z[i] will contain all "objects" of norm i. Depending on flag, this means
3672 : * an ideal, a bid, or a couple [bid, log(units)]. Such objects are stored
3673 : * in vectors, computed one primary component at a time; join_z
3674 : * reconstructs the global object */
3675 11333 : BOUND = utoipos(bound);
3676 11333 : z = const_vec(bound, empty);
3677 11333 : U = do_units? log_prk_units_init(bnf): NULL;
3678 11333 : gel(z,1) = mkvec(U? mkvec2(id, empty): id);
3679 11333 : ID.nf = nf;
3680 :
3681 11333 : p = cgetipos(3);
3682 11333 : u_forprime_init(&S, 2, bound);
3683 11333 : av = avma;
3684 92940 : while ((p[2] = u_forprime_next(&S)))
3685 : {
3686 81616 : if (DEBUGLEVEL>1) err_printf("%ld ",p[2]);
3687 81616 : fa = idealprimedec_limit_norm(nf, p, BOUND);
3688 163121 : for (j = 1; j < lg(fa); j++)
3689 : {
3690 81514 : GEN pr = gel(fa,j), z2 = leafcopy(z);
3691 81515 : ulong Q, q = upr_norm(pr);
3692 : long l;
3693 81515 : ID.pr = ID.prL = pr;
3694 81515 : if (cond && q == 2) { l = 2; Q = 4; } else { l = 1; Q = q; }
3695 184549 : for (; Q <= bound; l++, Q *= q) /* add pr^l */
3696 : {
3697 : ulong iQ;
3698 103046 : ID.L = utoipos(l);
3699 103045 : if (big_id) {
3700 210 : ID.prL = Idealstarprk(nf, pr, l, istar_flag);
3701 210 : if (U)
3702 189 : ID.emb = Q == 2? empty
3703 189 : : log_prk_units(nf, U, gel(bid_get_sprk(ID.prL),1));
3704 : }
3705 861608 : for (iQ = Q,i = 1; iQ <= bound; iQ += Q,i++)
3706 758574 : concat_join(&gel(z,iQ), gel(z2,i), join_z, &ID);
3707 : }
3708 : }
3709 81607 : if (gc_needed(av,1))
3710 : {
3711 18 : if(DEBUGMEM>1) pari_warn(warnmem,"Ideallist");
3712 18 : z = gc_GEN(av, z);
3713 : }
3714 : }
3715 11333 : return z;
3716 : }
3717 : GEN
3718 63 : gideallist(GEN bnf, GEN B, long flag)
3719 : {
3720 63 : pari_sp av = avma;
3721 63 : if (typ(B) != t_INT)
3722 : {
3723 0 : B = gfloor(B);
3724 0 : if (typ(B) != t_INT) pari_err_TYPE("ideallist", B);
3725 0 : if (signe(B) < 0) B = gen_0;
3726 : }
3727 63 : if (signe(B) < 0)
3728 : {
3729 28 : if (flag != 4) pari_err_IMPL("ideallist with bid for single norm");
3730 28 : return gc_GEN(av, ideals_by_norm(checknf(bnf), absi(B)));
3731 : }
3732 35 : if (flag < 0 || flag > 15) pari_err_FLAG("ideallist");
3733 35 : return gc_GEN(av, Ideallist(bnf, itou(B), flag));
3734 : }
3735 : GEN
3736 11298 : ideallist0(GEN bnf, long bound, long flag)
3737 : {
3738 11298 : pari_sp av = avma;
3739 11298 : if (flag < 0 || flag > 15) pari_err_FLAG("ideallist");
3740 11298 : return gc_GEN(av, Ideallist(bnf, bound, flag));
3741 : }
3742 : GEN
3743 10563 : ideallist(GEN bnf,long bound) { return ideallist0(bnf,bound,4); }
3744 :
3745 : /* bid = for module m (without arch. part), arch = archimedean part.
3746 : * Output: bid for [m,arch] */
3747 : static GEN
3748 42 : join_bid_arch(GEN nf, GEN bid, GEN archp)
3749 : {
3750 42 : pari_sp av = avma;
3751 : GEN G, U;
3752 42 : GEN sprk, cyc, y, u1 = NULL, x, sarch, gen;
3753 :
3754 42 : checkbid(bid);
3755 42 : G = bid_get_grp(bid);
3756 42 : x = bid_get_ideal(bid);
3757 42 : sarch = nfarchstar(nf, bid_get_ideal(bid), archp);
3758 42 : sprk = bid_get_sprk(bid);
3759 :
3760 42 : gen = (lg(G)>3)? gel(G,3): NULL;
3761 42 : cyc = diagonal_shallow(shallowconcat(gel(G,2), sarch_get_cyc(sarch)));
3762 42 : cyc = ZM_snf_group(cyc, &U, gen? &u1: NULL);
3763 42 : y = bid_grp(nf, u1, cyc, gen, x, sarch);
3764 42 : U = split_U(U, sprk);
3765 42 : y = mkvec5(mkvec2(x, archp), y, gel(bid,3), mkvec2(sprk, sarch), U);
3766 42 : return gc_GEN(av,y);
3767 : }
3768 : static GEN
3769 42 : join_arch(ideal_data *D, GEN x) {
3770 42 : return join_bid_arch(D->nf, x, D->archp);
3771 : }
3772 : static GEN
3773 14 : join_archunit(ideal_data *D, GEN x) {
3774 14 : GEN bid = join_arch(D, gel(x,1)), u = gel(x,2), v = mkvec(D->emb);
3775 14 : if (lg(u) != 1) v = shallowconcat(u, v);
3776 14 : return mkvec2(bid, v);
3777 : }
3778 :
3779 : /* L from ideallist, add archimedean part */
3780 : GEN
3781 14 : ideallistarch(GEN bnf, GEN L, GEN arch)
3782 : {
3783 : pari_sp av;
3784 14 : long i, j, l = lg(L), lz;
3785 : GEN v, z, V, nf;
3786 : ideal_data ID;
3787 : GEN (*join_z)(ideal_data*, GEN);
3788 :
3789 14 : if (typ(L) != t_VEC) pari_err_TYPE("ideallistarch",L);
3790 14 : if (l == 1) return cgetg(1,t_VEC);
3791 14 : z = gel(L,1);
3792 14 : if (typ(z) != t_VEC) pari_err_TYPE("ideallistarch",z);
3793 14 : z = gel(z,1); /* either a bid or [bid,U] */
3794 14 : ID.archp = vec01_to_indices(arch);
3795 14 : if (lg(z) == 3)
3796 : { /* [bid,U]: do units */
3797 7 : bnf = checkbnf(bnf); nf = bnf_get_nf(bnf);
3798 7 : if (typ(z) != t_VEC) pari_err_TYPE("ideallistarch",z);
3799 7 : ID.emb = zm_to_ZM( rowpermute(nfsign_units(bnf,NULL,1), ID.archp) );
3800 7 : join_z = &join_archunit;
3801 : }
3802 : else
3803 : {
3804 7 : join_z = &join_arch;
3805 7 : nf = checknf(bnf);
3806 : }
3807 14 : ID.nf = nf;
3808 14 : av = avma; V = cgetg(l, t_VEC);
3809 63 : for (i = 1; i < l; i++)
3810 : {
3811 49 : z = gel(L,i); lz = lg(z);
3812 49 : gel(V,i) = v = cgetg(lz,t_VEC);
3813 91 : for (j=1; j<lz; j++) gel(v,j) = join_z(&ID, gel(z,j));
3814 : }
3815 14 : return gc_GEN(av,V);
3816 : }
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