Line data Source code
1 : /* Copyright (C) 2000-2003 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 : #include "pari.h"
16 : #include "paripriv.h"
17 :
18 : #define DEBUGLEVEL DEBUGLEVEL_galois
19 :
20 : /*************************************************************************/
21 : /** **/
22 : /** GALOIS CONJUGATES **/
23 : /** **/
24 : /*************************************************************************/
25 :
26 : static int
27 9426 : is2sparse(GEN x)
28 : {
29 9426 : long i, l = lg(x);
30 9426 : if (odd(l-3)) return 0;
31 28253 : for(i=3; i<l; i+=2)
32 18971 : if (signe(gel(x,i))) return 0;
33 9282 : return 1;
34 : }
35 :
36 : static GEN
37 30782 : galoisconj1(GEN nf)
38 : {
39 30782 : GEN x = get_nfpol(nf, &nf), f = nf? nf : x, y, z;
40 30782 : long i, lz, v = varn(x), nbmax;
41 30782 : pari_sp av = avma;
42 30782 : RgX_check_ZX(x, "nfgaloisconj");
43 30782 : nbmax = numberofconjugates(x, 2);
44 30782 : if (nbmax==1) retmkcol(pol_x(v));
45 9973 : if (nbmax==2 && is2sparse(x))
46 : {
47 9282 : GEN res = cgetg(3,t_COL);
48 9282 : gel(res,1) = deg1pol_shallow(gen_m1, gen_0, v);
49 9282 : gel(res,2) = pol_x(v);
50 9282 : return res;
51 : }
52 691 : x = leafcopy(x); setvarn(x, fetch_var_higher());
53 691 : z = nfroots(f, x); lz = lg(z);
54 691 : y = cgetg(lz, t_COL);
55 3338 : for (i = 1; i < lz; i++)
56 : {
57 2647 : GEN t = lift(gel(z,i));
58 2647 : if (typ(t) == t_POL) setvarn(t, v);
59 2647 : gel(y,i) = t;
60 : }
61 691 : (void)delete_var();
62 691 : return gc_upto(av, y);
63 : }
64 :
65 : /*************************************************************************/
66 : /** **/
67 : /** GALOISCONJ4 **/
68 : /** **/
69 : /*************************************************************************/
70 : /*DEBUGLEVEL:
71 : 1: timing
72 : 2: outline
73 : 4: complete outline
74 : 6: detail
75 : 7: memory
76 : 9: complete detail
77 : */
78 : struct galois_borne {
79 : GEN l;
80 : long valsol;
81 : long valabs;
82 : GEN bornesol;
83 : GEN ladicsol;
84 : GEN ladicabs;
85 : GEN dis;
86 : };
87 : struct galois_lift {
88 : GEN T;
89 : GEN den;
90 : GEN p;
91 : GEN L;
92 : GEN Lden;
93 : long e;
94 : GEN Q;
95 : GEN TQ;
96 : struct galois_borne *gb;
97 : };
98 : struct galois_testlift {
99 : long n;
100 : long f;
101 : long g;
102 : GEN bezoutcoeff;
103 : GEN pauto;
104 : GEN C;
105 : GEN Cd;
106 : };
107 : struct galois_test { /* data for permutation test */
108 : GEN order; /* order of tests pour galois_test_perm */
109 : GEN borne, lborne; /* coefficient bounds */
110 : GEN ladic;
111 : GEN PV; /* NULL or vector of test matrices (Vmatrix) */
112 : GEN TM; /* transpose of M */
113 : GEN L; /* p-adic roots, known mod ladic */
114 : GEN M; /* vandermonde inverse */
115 : };
116 : /* result of the study of Frobenius degrees */
117 : enum ga_code {ga_all_normal=1,ga_ext_2=2,ga_non_wss=4,
118 : ga_all_nilpotent=8,ga_easy=16};
119 : struct galois_analysis {
120 : long p; /* prime to be lifted */
121 : long deg; /* degree of the lift */
122 : long ord;
123 : long l; /* l: prime number such that T is totally split mod l */
124 : long p4;
125 : long group;
126 : };
127 : struct galois_frobenius {
128 : long p;
129 : long fp;
130 : long deg;
131 : GEN Tmod;
132 : GEN psi;
133 : };
134 :
135 : /* #r = r1 + r2 */
136 : GEN
137 10524 : embed_roots(GEN ro, long r1)
138 : {
139 10524 : long r2 = lg(ro)-1-r1;
140 : GEN L;
141 10524 : if (!r2) L = ro;
142 : else
143 : {
144 4960 : long i,j, N = r1+2*r2;
145 4960 : L = cgetg(N+1, t_VEC);
146 6256 : for (i = 1; i <= r1; i++) gel(L,i) = gel(ro,i);
147 14631 : for (j = i; j <= N; i++)
148 : {
149 9671 : GEN z = gel(ro,i);
150 9671 : gel(L,j++) = z;
151 9671 : gel(L,j++) = mkcomplex(gel(z,1), gneg(gel(z,2)));
152 : }
153 : }
154 10524 : return L;
155 : }
156 : GEN
157 112780 : embed_disc(GEN z, long r1, long prec)
158 : {
159 112780 : pari_sp av = avma;
160 112780 : GEN t = real_1(prec);
161 112780 : long i, j, n = lg(z)-1, r2 = n-r1;
162 184413 : for (i = 1; i < r1; i++)
163 : {
164 71633 : GEN zi = gel(z,i);
165 497226 : for (j = i+1; j <= r1; j++) t = gmul(t, gsub(zi, gel(z,j)));
166 : }
167 596986 : for (j = r1+1; j <= n; j++)
168 : {
169 484206 : GEN zj = gel(z,j), a = gel(zj,1), b = gel(zj,2), b2 = gsqr(b);
170 497772 : for (i = 1; i <= r1; i++)
171 : {
172 13566 : GEN zi = gel(z,i);
173 13566 : t = gmul(t, gadd(gsqr(gsub(zi, a)), b2));
174 : }
175 484206 : t = gmul(t, b);
176 : }
177 112780 : if (r2) t = gmul2n(t, r2);
178 112780 : if (r2 > 1)
179 : {
180 103780 : GEN T = real_1(prec);
181 483141 : for (i = r1+1; i < n; i++)
182 : {
183 379361 : GEN zi = gel(z,i), a = gel(zi,1), b = gel(zi,2);
184 1600951 : for (j = i+1; j <= n; j++)
185 : {
186 1221590 : GEN zj = gel(z,j), c = gel(zj,1), d = gel(zj,2);
187 1221590 : GEN f = gsqr(gsub(a,c)), g = gsqr(gsub(b,d)), h = gsqr(gadd(b,d));
188 1221590 : T = gmul(T, gmul(gadd(f,g), gadd(f,h)));
189 : }
190 : }
191 103780 : t = gmul(t, T);
192 : }
193 112780 : t = gsqr(t);
194 112780 : if (odd(r2)) t = gneg(t);
195 112780 : return gc_upto(av, t);
196 : }
197 :
198 : /* Compute bound for the coefficients of automorphisms.
199 : * T a ZX, den a t_INT denominator or NULL */
200 : GEN
201 72203 : initgaloisborne(GEN T, GEN den, long prec, GEN *pL, GEN *pprep, GEN *pD)
202 : {
203 : GEN L, prep, nf, r;
204 : pari_timer ti;
205 :
206 72203 : if (DEBUGLEVEL>=4) timer_start(&ti);
207 72203 : T = get_nfpol(T, &nf);
208 72203 : r = nf ? nf_get_roots(nf) : NULL;
209 72203 : if (nf && precision(gel(r, 1)) >= prec)
210 10524 : L = embed_roots(r, nf_get_r1(nf));
211 : else
212 61679 : L = QX_complex_roots(T, prec);
213 72203 : if (DEBUGLEVEL>=4) timer_printf(&ti,"roots");
214 72203 : prep = vandermondeinverseinit(L);
215 72203 : if (!den || pD)
216 : {
217 53033 : GEN res = RgV_prod(gabs(prep,prec));
218 53033 : GEN D = ZX_disc_all(T, 1 + gexpo(res)); /* +1 for safety */
219 53033 : if (pD) *pD = D;
220 53033 : if (!den) den = indexpartial(T,D);
221 : }
222 72203 : if (pprep) *pprep = prep;
223 72203 : *pL = L; return den;
224 : }
225 :
226 : /* ||| M ||| with respect to || x ||_oo, M t_MAT */
227 : GEN
228 413148 : matrixnorm(GEN M, long prec)
229 : {
230 413148 : long i,j,m, l = lg(M);
231 413148 : GEN B = real_0(prec);
232 :
233 413148 : if (l == 1) return B;
234 413148 : m = lgcols(M);
235 1462822 : for (i = 1; i < m; i++)
236 : {
237 1049674 : GEN z = gabs(gcoeff(M,i,1), prec);
238 4838362 : for (j = 2; j < l; j++) z = gadd(z, gabs(gcoeff(M,i,j), prec));
239 1049674 : if (gcmp(z, B) > 0) B = z;
240 : }
241 413148 : return B;
242 : }
243 :
244 : static GEN
245 26780 : galoisborne(GEN T, GEN dn, struct galois_borne *gb, long d)
246 : {
247 : pari_sp ltop, av2;
248 : GEN borne, borneroots, bornetrace, borneabs;
249 : long prec;
250 : GEN L, M, prep, den;
251 : pari_timer ti;
252 26780 : const long step=3;
253 :
254 26780 : prec = nbits2prec(bit_accuracy(ZX_max_lg(T)));
255 26780 : den = initgaloisborne(T,dn,prec, &L,&prep,&gb->dis);
256 26780 : if (!dn) dn = den;
257 26780 : ltop = avma;
258 26780 : if (DEBUGLEVEL>=4) timer_start(&ti);
259 26780 : M = vandermondeinverse(L, RgX_gtofp(T, prec), den, prep);
260 26780 : if (DEBUGLEVEL>=4) timer_printf(&ti,"vandermondeinverse");
261 26780 : borne = matrixnorm(M, prec);
262 26780 : borneroots = gsupnorm(L, prec); /*t_REAL*/
263 26780 : bornetrace = mulur((2*step)*degpol(T)/d,
264 26780 : powru(borneroots, minss(degpol(T), step)));
265 26780 : borneroots = ceil_safe(gmul(borne, borneroots));
266 26780 : borneabs = ceil_safe(gmax_shallow(gmul(borne, bornetrace),
267 : powru(bornetrace, d)));
268 26780 : av2 = avma;
269 : /*We use d-1 test, so we must overlift to 2^BITS_IN_LONG*/
270 26780 : gb->valsol = logint(shifti(borneroots,2+BITS_IN_LONG), gb->l) + 1;
271 26780 : gb->valabs = logint(shifti(borneabs,2), gb->l) + 1;
272 26780 : gb->valabs = maxss(gb->valsol, gb->valabs);
273 26780 : if (DEBUGLEVEL >= 4)
274 0 : err_printf("GaloisConj: val1=%ld val2=%ld\n", gb->valsol, gb->valabs);
275 26780 : set_avma(av2);
276 26780 : gb->bornesol = gc_INT(ltop, shifti(borneroots,1));
277 26780 : if (DEBUGLEVEL >= 9)
278 0 : err_printf("GaloisConj: Bound %Ps\n",borneroots);
279 26780 : gb->ladicsol = powiu(gb->l, gb->valsol);
280 26780 : gb->ladicabs = powiu(gb->l, gb->valabs);
281 26780 : return dn;
282 : }
283 :
284 : static GEN
285 25536 : makeLden(GEN L,GEN den, struct galois_borne *gb)
286 25536 : { return FpC_Fp_mul(L, den, gb->ladicsol); }
287 :
288 : /* Initialize the galois_lift structure */
289 : static void
290 25626 : initlift(GEN T, GEN den, ulong p, GEN L, GEN Lden, struct galois_borne *gb, struct galois_lift *gl)
291 : {
292 : pari_sp av;
293 : long e;
294 25626 : gl->gb = gb;
295 25626 : gl->T = T;
296 25626 : gl->den = is_pm1(den)? gen_1: den;
297 25626 : gl->p = utoipos(p);
298 25626 : gl->L = L;
299 25626 : gl->Lden = Lden;
300 25626 : av = avma;
301 25626 : e = logint(shifti(gb->bornesol, 2+BITS_IN_LONG), gl->p) + 1;
302 25626 : set_avma(av);
303 25626 : if (e < 2) e = 2;
304 25626 : gl->e = e;
305 25626 : gl->Q = powuu(p, e);
306 25626 : gl->TQ = FpX_red(T,gl->Q);
307 25626 : }
308 :
309 : /* Check whether f is (with high probability) a solution and compute its
310 : * permutation */
311 : static int
312 56173 : poltopermtest(GEN f, struct galois_lift *gl, GEN pf)
313 : {
314 : pari_sp av;
315 56173 : GEN fx, fp, B = gl->gb->bornesol;
316 : long i, j, ll;
317 242094 : for (i = 2; i < lg(f); i++)
318 192718 : if (abscmpii(gel(f,i),B) > 0)
319 : {
320 6797 : if (DEBUGLEVEL>=4) err_printf("GaloisConj: Solution too large.\n");
321 6797 : if (DEBUGLEVEL>=8) err_printf("f=%Ps\n borne=%Ps\n",f,B);
322 6797 : return 0;
323 : }
324 49376 : ll = lg(gl->L);
325 49376 : fp = const_vecsmall(ll-1, 1); /* left on stack */
326 49376 : av = avma;
327 243322 : for (i = 1; i < ll; i++, set_avma(av))
328 : {
329 194086 : fx = FpX_eval(f, gel(gl->L,i), gl->gb->ladicsol);
330 1058075 : for (j = 1; j < ll; j++)
331 1057935 : if (fp[j] && equalii(fx, gel(gl->Lden,j))) { pf[i]=j; fp[j]=0; break; }
332 194086 : if (j == ll) return 0;
333 : }
334 49236 : return 1;
335 : }
336 :
337 : static long
338 51738 : galoisfrobeniustest(GEN aut, struct galois_lift *gl, GEN frob)
339 : {
340 51738 : pari_sp av = avma;
341 51738 : GEN tlift = aut;
342 51738 : if (gl->den != gen_1) tlift = FpX_Fp_mul(tlift, gl->den, gl->Q);
343 51738 : tlift = FpX_center_i(tlift, gl->Q, shifti(gl->Q,-1));
344 51738 : return gc_long(av, poltopermtest(tlift, gl, frob));
345 : }
346 :
347 : static GEN
348 54175 : monoratlift(void *E, GEN S, GEN q)
349 : {
350 54175 : pari_sp ltop = avma;
351 54175 : struct galois_lift *gl = (struct galois_lift *) E;
352 54175 : GEN qm1 = sqrti(shifti(q,-2)), N = gl->Q;
353 54175 : GEN tlift = FpX_ratlift(S, q, qm1, qm1, gl->den);
354 54175 : if (tlift)
355 : {
356 23446 : pari_sp ltop = avma;
357 23446 : GEN frob = cgetg(lg(gl->L), t_VECSMALL);
358 23446 : if(DEBUGLEVEL>=4)
359 0 : err_printf("MonomorphismLift: trying early solution %Ps\n",tlift);
360 23446 : if (gl->den != gen_1)
361 20174 : tlift = FpX_Fp_mul(FpX_red(Q_muli_to_int(tlift, gl->den), N),
362 : Fp_inv(gl->den, N), N);
363 23446 : if (galoisfrobeniustest(tlift, gl, frob))
364 : {
365 23305 : if(DEBUGLEVEL>=4) err_printf("MonomorphismLift: true early solution.\n");
366 23305 : return gc_GEN(ltop, tlift);
367 : }
368 141 : if(DEBUGLEVEL>=4) err_printf("MonomorphismLift: false early solution.\n");
369 : }
370 30870 : set_avma(ltop);
371 30870 : return NULL;
372 : }
373 :
374 : static GEN
375 27225 : monomorphismratlift(GEN P, GEN S, struct galois_lift *gl)
376 : {
377 : pari_timer ti;
378 27225 : if (DEBUGLEVEL >= 1) timer_start(&ti);
379 27225 : S = ZpX_ZpXQ_liftroot_ea(P,S,gl->T,gl->p, gl->e, (void*)gl, monoratlift);
380 27225 : if (DEBUGLEVEL >= 1) timer_printf(&ti, "monomorphismlift()");
381 27225 : return S;
382 : }
383 :
384 : /* Let T be a polynomial in Z[X] , p a prime number, S in Fp[X]/(T) so
385 : * that T(S)=0 [p,T]. Lift S in S_0 so that T(S_0)=0 [T,p^e]
386 : * Unclean stack */
387 : static GEN
388 27225 : automorphismlift(GEN S, struct galois_lift *gl)
389 : {
390 27225 : return monomorphismratlift(gl->T, S, gl);
391 : }
392 :
393 : static GEN
394 25626 : galoisdolift(struct galois_lift *gl)
395 : {
396 25626 : pari_sp av = avma;
397 25626 : GEN Tp = FpX_red(gl->T, gl->p);
398 25626 : GEN S = FpX_Frobenius(Tp, gl->p);
399 25626 : return gc_upto(av, automorphismlift(S, gl));
400 : }
401 :
402 : static GEN
403 1113 : galoisdoliftn(struct galois_lift *gl, long e)
404 : {
405 1113 : pari_sp av = avma;
406 1113 : GEN Tp = FpX_red(gl->T, gl->p);
407 1113 : GEN S = FpXQ_autpow(FpX_Frobenius(Tp, gl->p), e, Tp, gl->p);
408 1113 : return gc_upto(av, automorphismlift(S, gl));
409 : }
410 :
411 : static ulong
412 59 : findpsi(GEN D, ulong pstart, GEN P, GEN S, long o, GEN *Tmod, GEN *Tpsi)
413 : {
414 : forprime_t iter;
415 : ulong p;
416 59 : long n = degpol(P), i, j, g = n/o;
417 59 : GEN psi = cgetg(g+1, t_VECSMALL);
418 59 : u_forprime_init(&iter, pstart, ULONG_MAX);
419 1866 : while ((p = u_forprime_next(&iter)))
420 : {
421 : GEN F, Sp;
422 1866 : long gp = 0;
423 1866 : if (smodis(D, p) == 0)
424 156 : continue;
425 1710 : F = gel(Flx_factor(ZX_to_Flx(P, p), p), 1);
426 1710 : if (lg(F)-1 != g) continue;
427 585 : Sp = RgX_to_Flx(S, p);
428 1515 : for (j = 1; j <= g; j++)
429 : {
430 1414 : GEN Fj = gel(F, j);
431 1414 : GEN Sj = Flx_rem(Sp, Fj, p);
432 1414 : GEN A = Flxq_autpowers(Flx_Frobenius(Fj, p), o, Fj, p);
433 4844 : for (i = 1; i <= o; i++)
434 4360 : if (gequal(Sj, gel(A,i+1)))
435 : {
436 930 : psi[j] = i; break;
437 : }
438 1414 : if (i > o) break;
439 930 : if (gp==0 && i==1) gp=j;
440 : }
441 585 : if (gp && j > g)
442 : {
443 : /* Normalize result so that psi[l]=1 */
444 59 : if (gp!=1)
445 : {
446 6 : swap(gel(F,1),gel(F,gp));
447 6 : lswap(uel(psi,1),uel(psi,gp));
448 : }
449 59 : *Tpsi = Flv_Fl_div(psi,psi[g],o);
450 59 : *Tmod = FlxV_to_ZXV(F);
451 59 : return p;
452 : }
453 : }
454 0 : return 0;
455 : }
456 :
457 : static void
458 1460 : inittestlift(GEN plift, GEN Tmod, struct galois_lift *gl,
459 : struct galois_testlift *gt)
460 : {
461 : pari_timer ti;
462 1460 : gt->n = lg(gl->L) - 1;
463 1460 : gt->g = lg(Tmod) - 1;
464 1460 : gt->f = gt->n / gt->g;
465 1460 : gt->bezoutcoeff = bezout_lift_fact(gl->T, Tmod, gl->p, gl->e);
466 1460 : if (DEBUGLEVEL >= 2) timer_start(&ti);
467 1460 : gt->pauto = FpXQ_autpowers(plift, gt->f-1, gl->TQ, gl->Q);
468 1460 : if (DEBUGLEVEL >= 2) timer_printf(&ti, "Frobenius power");
469 1460 : }
470 :
471 : /* Explanation of the intheadlong technique:
472 : * Let C be a bound, B = BITS_IN_LONG, M > C*2^B a modulus and 0 <= a_i < M for
473 : * i=1,...,n where n < 2^B. We want to test if there exist k,l, |k| < C < M/2^B,
474 : * such that sum a_i = k + l*M
475 : * We write a_i*2^B/M = b_i+c_i with b_i integer and 0<=c_i<1, so that
476 : * sum b_i - l*2^B = k*2^B/M - sum c_i
477 : * Since -1 < k*2^B/M < 1 and 0<=c_i<1, it follows that
478 : * -n-1 < sum b_i - l*2^B < 1 i.e. -n <= sum b_i -l*2^B <= 0
479 : * So we compute z = - sum b_i [mod 2^B] and check if 0 <= z <= n. */
480 :
481 : /* Assume 0 <= x < mod. */
482 : static ulong
483 1157489 : intheadlong(GEN x, GEN mod)
484 : {
485 1157489 : pari_sp av = avma;
486 1157489 : long res = (long) itou(divii(shifti(x,BITS_IN_LONG),mod));
487 1157489 : return gc_long(av,res);
488 : }
489 : static GEN
490 49730 : vecheadlong(GEN W, GEN mod)
491 : {
492 49730 : long i, l = lg(W);
493 49730 : GEN V = cgetg(l, t_VECSMALL);
494 1174104 : for(i=1; i<l; i++) V[i] = intheadlong(gel(W,i), mod);
495 49730 : return V;
496 : }
497 : static GEN
498 4928 : matheadlong(GEN W, GEN mod)
499 : {
500 4928 : long i, l = lg(W);
501 4928 : GEN V = cgetg(l,t_MAT);
502 54658 : for(i=1; i<l; i++) gel(V,i) = vecheadlong(gel(W,i), mod);
503 4928 : return V;
504 : }
505 : static ulong
506 33115 : polheadlong(GEN P, long n, GEN mod)
507 : {
508 33115 : return (lg(P)>n+2)? intheadlong(gel(P,n+2),mod): 0;
509 : }
510 :
511 : #define headlongisint(Z,N) (-(ulong)(Z)<=(ulong)(N))
512 :
513 : static long
514 1669 : frobeniusliftall(GEN sg, long el, GEN *psi, struct galois_lift *gl,
515 : struct galois_testlift *gt, GEN frob)
516 : {
517 1669 : pari_sp av, ltop2, ltop = avma;
518 1669 : long i,j,k, c = lg(sg)-1, n = lg(gl->L)-1, m = gt->g, d = m / c;
519 : GEN pf, u, v, C, Cd, SG, cache;
520 1669 : long N1, N2, R1, Ni, ord = gt->f, c_idx = gt->g-1;
521 : ulong headcache;
522 1669 : long hop = 0;
523 : GEN NN, NQ;
524 : pari_timer ti;
525 :
526 1669 : *psi = pf = cgetg(m, t_VECSMALL);
527 1669 : ltop2 = avma;
528 1669 : NN = diviiexact(mpfact(m), mului(c, powiu(mpfact(d), c)));
529 1669 : if (DEBUGLEVEL >= 4)
530 0 : err_printf("GaloisConj: I will try %Ps permutations\n", NN);
531 1669 : N1=10000000;
532 1669 : NQ=divis_rem(NN,N1,&R1);
533 1669 : if (abscmpiu(NQ,1000000000)>0)
534 : {
535 0 : pari_warn(warner,"Combinatorics too hard : would need %Ps tests!\n"
536 : "I will skip it, but it may induce an infinite loop",NN);
537 0 : *psi = NULL; return gc_long(ltop,0);
538 : }
539 1669 : N2=itos(NQ); if(!N2) N1=R1;
540 1669 : if (DEBUGLEVEL>=4) timer_start(&ti);
541 1669 : set_avma(ltop2);
542 1669 : C = gt->C;
543 1669 : Cd= gt->Cd;
544 1669 : v = FpXQ_mul(gel(gt->pauto, 1+el%ord), gel(gt->bezoutcoeff, m),gl->TQ,gl->Q);
545 1669 : if (gl->den != gen_1) v = FpX_Fp_mul(v, gl->den, gl->Q);
546 1669 : SG = cgetg(lg(sg),t_VECSMALL);
547 5475 : for(i=1; i<lg(SG); i++) SG[i] = (el*sg[i])%ord + 1;
548 1669 : cache = cgetg(m+1,t_VECSMALL); cache[m] = polheadlong(v,1,gl->Q);
549 1669 : headcache = polheadlong(v,2,gl->Q);
550 4448 : for (i = 1; i < m; i++) pf[i] = 1 + i/d;
551 1669 : av = avma;
552 1669 : for (Ni = 0, i = 0; ;i++)
553 : {
554 258283 : for (j = c_idx ; j > 0; j--)
555 : {
556 202958 : long h = SG[pf[j]];
557 202958 : if (!mael(C,h,j))
558 : {
559 4236 : pari_sp av3 = avma;
560 4236 : GEN r = FpXQ_mul(gel(gt->pauto,h), gel(gt->bezoutcoeff,j),gl->TQ,gl->Q);
561 4236 : if (gl->den != gen_1) r = FpX_Fp_mul(r, gl->den, gl->Q);
562 4236 : gmael(C,h,j) = gclone(r);
563 4236 : mael(Cd,h,j) = polheadlong(r,1,gl->Q);
564 4236 : set_avma(av3);
565 : }
566 202958 : uel(cache,j) = uel(cache,j+1)+umael(Cd,h,j);
567 : }
568 55325 : if (headlongisint(uel(cache,1),n))
569 : {
570 2835 : ulong head = headcache;
571 28376 : for (j = 1; j < m; j++) head += polheadlong(gmael(C,SG[pf[j]],j),2,gl->Q);
572 2835 : if (headlongisint(head,n))
573 : {
574 1431 : u = v;
575 3500 : for (j = 1; j < m; j++) u = ZX_add(u, gmael(C,SG[pf[j]],j));
576 1431 : u = FpX_center_i(FpX_red(u, gl->Q), gl->Q, shifti(gl->Q,-1));
577 1431 : if (poltopermtest(u, gl, frob))
578 : {
579 1425 : if (DEBUGLEVEL >= 4)
580 : {
581 0 : timer_printf(&ti, "");
582 0 : err_printf("GaloisConj: %d hops on %Ps tests\n",hop,addis(mulss(Ni,N1),i));
583 : }
584 1425 : return gc_long(ltop2,1);
585 : }
586 6 : if (DEBUGLEVEL >= 4) err_printf("M");
587 : }
588 1404 : else hop++;
589 : }
590 53900 : if (DEBUGLEVEL >= 4 && i % maxss(N1/20, 1) == 0)
591 0 : timer_printf(&ti, "GaloisConj:Testing %Ps", addis(mulss(Ni,N1),i));
592 53900 : set_avma(av);
593 53900 : if (i == N1-1)
594 : {
595 244 : if (Ni==N2-1) N1 = R1;
596 244 : if (Ni==N2) break;
597 0 : Ni++; i = 0;
598 0 : if (DEBUGLEVEL>=4) timer_start(&ti);
599 : }
600 146523 : for (j = 2; j < m && pf[j-1] >= pf[j]; j++)
601 : /*empty*/; /* to kill clang Warning */
602 87183 : for (k = 1; k < j-k && pf[k] != pf[j-k]; k++) { lswap(pf[k], pf[j-k]); }
603 94024 : for (k = j - 1; pf[k] >= pf[j]; k--)
604 : /*empty*/;
605 53656 : lswap(pf[j], pf[k]); c_idx = j;
606 : }
607 244 : if (DEBUGLEVEL>=4) err_printf("GaloisConj: not found, %d hops \n",hop);
608 244 : *psi = NULL; return gc_long(ltop,0);
609 : }
610 :
611 : /* Compute the test matrix for the i-th line of V. Clone. */
612 : static GEN
613 4928 : Vmatrix(long i, struct galois_test *td)
614 : {
615 4928 : pari_sp av = avma;
616 4928 : GEN m = gclone( matheadlong(FpC_FpV_mul(td->L, row(td->M,i), td->ladic), td->ladic));
617 4928 : set_avma(av); return m;
618 : }
619 :
620 : /* Initialize galois_test */
621 : static void
622 4725 : inittest(GEN L, GEN M, GEN borne, GEN ladic, struct galois_test *td)
623 : {
624 4725 : long i, n = lg(L)-1;
625 4725 : GEN p = cgetg(n+1, t_VECSMALL);
626 4725 : if (DEBUGLEVEL >= 8) err_printf("GaloisConj: Init Test\n");
627 4725 : td->order = p;
628 38137 : for (i = 1; i <= n-2; i++) p[i] = i+2;
629 4725 : p[n-1] = 1; p[n] = 2;
630 4725 : td->borne = borne;
631 4725 : td->lborne = subii(ladic, borne);
632 4725 : td->ladic = ladic;
633 4725 : td->L = L;
634 4725 : td->M = M;
635 4725 : td->TM = shallowtrans(M);
636 4725 : td->PV = zero_zv(n);
637 4725 : gel(td->PV, 2) = Vmatrix(2, td);
638 4725 : }
639 :
640 : /* Free clones stored inside galois_test */
641 : static void
642 4725 : freetest(struct galois_test *td)
643 : {
644 : long i;
645 47587 : for (i = 1; i < lg(td->PV); i++)
646 42862 : if (td->PV[i]) { gunclone(gel(td->PV,i)); td->PV[i] = 0; }
647 4725 : }
648 :
649 : /* Check if the integer P seen as a p-adic number is close to an integer less
650 : * than td->borne in absolute value */
651 : static long
652 80248 : padicisint(GEN P, struct galois_test *td)
653 : {
654 80248 : pari_sp ltop = avma;
655 80248 : GEN U = modii(P, td->ladic);
656 80248 : long r = cmpii(U, td->borne) <= 0 || cmpii(U, td->lborne) >= 0;
657 80248 : return gc_long(ltop, r);
658 : }
659 :
660 : /* Check if the permutation pf is valid according to td.
661 : * If not, update td to make subsequent test faster (hopefully) */
662 : static long
663 100667 : galois_test_perm(struct galois_test *td, GEN pf)
664 : {
665 100667 : pari_sp av = avma;
666 100667 : long i, j, n = lg(td->L)-1;
667 100667 : GEN V, P = NULL;
668 181941 : for (i = 1; i < n; i++)
669 : {
670 175959 : long ord = td->order[i];
671 175959 : GEN PW = gel(td->PV, ord);
672 175959 : if (PW)
673 : {
674 95711 : ulong head = umael(PW,1,pf[1]);
675 6103768 : for (j = 2; j <= n; j++) head += umael(PW,j,pf[j]);
676 95711 : if (!headlongisint(head,n)) break;
677 : } else
678 : {
679 80248 : if (!P) P = vecpermute(td->L, pf);
680 80248 : V = FpV_dotproduct(gel(td->TM,ord), P, td->ladic);
681 80248 : if (!padicisint(V, td)) {
682 203 : gel(td->PV, ord) = Vmatrix(ord, td);
683 203 : if (DEBUGLEVEL >= 4) err_printf("M");
684 203 : break;
685 : }
686 : }
687 : }
688 100667 : if (i == n) return gc_long(av,1);
689 94685 : if (DEBUGLEVEL >= 4) err_printf("%d.", i);
690 94685 : if (i > 1)
691 : {
692 636 : long z = td->order[i];
693 1308 : for (j = i; j > 1; j--) td->order[j] = td->order[j-1];
694 636 : td->order[1] = z;
695 636 : if (DEBUGLEVEL >= 8) err_printf("%Ps", td->order);
696 : }
697 94685 : return gc_long(av,0);
698 : }
699 : /*Compute a*b/c when a*b will overflow*/
700 : static long
701 0 : muldiv(long a,long b,long c)
702 : {
703 0 : return (long)((double)a*(double)b/c);
704 : }
705 :
706 : /* F = cycle decomposition of sigma,
707 : * B = cycle decomposition of cl(tau).
708 : * Check all permutations pf who can possibly correspond to tau, such that
709 : * tau*sigma*tau^-1 = sigma^s and tau^d = sigma^t, where d = ord cl(tau)
710 : * x: vector of choices,
711 : * G: vector allowing linear access to elts of F.
712 : * Choices multiple of e are not changed. */
713 : static GEN
714 7521 : testpermutation(GEN F, GEN B, GEN x, long s, long e, long cut,
715 : struct galois_test *td)
716 : {
717 7521 : pari_sp av, avm = avma;
718 : long a, b, c, d, n, p1, p2, p3, p4, p5, p6, l1, l2, N1, N2, R1;
719 7521 : long i, j, cx, hop = 0, start = 0;
720 : GEN pf, ar, G, W, NN, NQ;
721 : pari_timer ti;
722 7521 : if (DEBUGLEVEL >= 1) timer_start(&ti);
723 7521 : a = lg(F)-1; b = lg(gel(F,1))-1;
724 7521 : c = lg(B)-1; d = lg(gel(B,1))-1;
725 7521 : n = a*b;
726 7521 : s = (b+s) % b;
727 7521 : pf = cgetg(n+1, t_VECSMALL);
728 7521 : av = avma;
729 7521 : ar = cgetg(a+2, t_VECSMALL); ar[a+1]=0;
730 7521 : G = cgetg(a+1, t_VECSMALL);
731 7521 : W = gel(td->PV, td->order[n]);
732 49351 : for (cx=1, i=1, j=1; cx <= a; cx++, i++)
733 : {
734 41830 : gel(G,cx) = gel(F, coeff(B,i,j));
735 41830 : if (i == d) { i = 0; j++; }
736 : }
737 7521 : NN = divis(powuu(b, c * (d - d/e)), cut);
738 7521 : if (DEBUGLEVEL>=4) err_printf("GaloisConj: I will try %Ps permutations\n", NN);
739 7521 : N1 = 1000000;
740 7521 : NQ = divis_rem(NN,N1,&R1);
741 7521 : if (abscmpiu(NQ,100000000)>0)
742 : {
743 0 : set_avma(avm);
744 0 : pari_warn(warner,"Combinatorics too hard: would need %Ps tests!\n"
745 : "I'll skip it but you will get a partial result...",NN);
746 0 : return identity_perm(n);
747 : }
748 7521 : N2 = itos(NQ);
749 9303 : for (l2 = 0; l2 <= N2; l2++)
750 : {
751 7521 : long nbiter = (l2<N2) ? N1: R1;
752 7521 : if (DEBUGLEVEL >= 2 && N2) err_printf("%d%% ", muldiv(l2,100,N2));
753 8695577 : for (l1 = 0; l1 < nbiter; l1++)
754 : {
755 8693795 : if (start)
756 : {
757 15548668 : for (i=1, j=e; i < a;)
758 : {
759 15548668 : if ((++(x[i])) != b) break;
760 6862394 : x[i++] = 0;
761 6862394 : if (i == j) { i++; j += e; }
762 : }
763 : }
764 7521 : else { start=1; i = a-1; }
765 : /* intheadlong test: overflow in + is OK, we compute mod 2^BIL */
766 39494921 : for (p1 = i+1, p5 = p1%d - 1 ; p1 >= 1; p1--, p5--) /* p5 = (p1%d) - 1 */
767 : {
768 : GEN G1, G6;
769 30801126 : ulong V = 0;
770 30801126 : if (p5 == - 1) { p5 = d - 1; p6 = p1 + 1 - d; } else p6 = p1 + 1;
771 30801126 : G1 = gel(G,p1); G6 = gel(G,p6);
772 30801126 : p4 = p5 ? x[p1-1] : 0;
773 93754861 : for (p2 = 1+p4, p3 = 1 + x[p1]; p2 <= b; p2++)
774 : {
775 62953735 : V += umael(W,uel(G6,p3),uel(G1,p2));
776 62953735 : p3 += s; if (p3 > b) p3 -= b;
777 : }
778 30801126 : p3 = 1 + x[p1] - s; if (p3 <= 0) p3 += b;
779 43066263 : for (p2 = p4; p2 >= 1; p2--)
780 : {
781 12265137 : V += umael(W,uel(G6,p3),uel(G1,p2));
782 12265137 : p3 -= s; if (p3 <= 0) p3 += b;
783 : }
784 30801126 : uel(ar,p1) = uel(ar,p1+1) + V;
785 : }
786 8693795 : if (!headlongisint(uel(ar,1),n)) continue;
787 :
788 : /* intheadlong succeeds. Full computation */
789 3009240 : for (p1=1, p5=d; p1 <= a; p1++, p5++)
790 : {
791 2908816 : if (p5 == d) { p5 = 0; p4 = 0; } else p4 = x[p1-1];
792 2908816 : if (p5 == d-1) p6 = p1+1-d; else p6 = p1+1;
793 8156618 : for (p2 = 1+p4, p3 = 1 + x[p1]; p2 <= b; p2++)
794 : {
795 5247802 : pf[mael(G,p1,p2)] = mael(G,p6,p3);
796 5247802 : p3 += s; if (p3 > b) p3 -= b;
797 : }
798 2908816 : p3 = 1 + x[p1] - s; if (p3 <= 0) p3 += b;
799 3786206 : for (p2 = p4; p2 >= 1; p2--)
800 : {
801 877390 : pf[mael(G,p1,p2)] = mael(G,p6,p3);
802 877390 : p3 -= s; if (p3 <= 0) p3 += b;
803 : }
804 : }
805 100424 : if (galois_test_perm(td, pf))
806 : {
807 5739 : if (DEBUGLEVEL >= 1)
808 : {
809 0 : GEN nb = addis(mulss(l2,N1),l1);
810 0 : timer_printf(&ti, "testpermutation(%Ps)", nb);
811 0 : if (DEBUGLEVEL >= 2 && hop)
812 0 : err_printf("GaloisConj: %d hop over %Ps iterations\n", hop, nb);
813 : }
814 5739 : set_avma(av); return pf;
815 : }
816 94685 : hop++;
817 : }
818 : }
819 1782 : if (DEBUGLEVEL >= 1)
820 : {
821 0 : timer_printf(&ti, "testpermutation(%Ps)", NN);
822 0 : if (DEBUGLEVEL >= 2 && hop)
823 0 : err_printf("GaloisConj: %d hop over %Ps iterations\n", hop, NN);
824 : }
825 1782 : return gc_NULL(avm);
826 : }
827 :
828 : /* List of subgroups of (Z/mZ)^* whose order divide o, and return the list
829 : * of their elements, sorted by increasing order */
830 : static GEN
831 1465 : listznstarelts(long m, long o)
832 : {
833 1465 : pari_sp av = avma;
834 : GEN L, zn, zns;
835 : long i, phi, ind, l;
836 1465 : if (m == 2) retmkvec(mkvecsmall(1));
837 1453 : zn = znstar(stoi(m));
838 1453 : phi = itos(gel(zn,1));
839 1453 : o = ugcd(o, phi); /* do we impose this on input ? */
840 1453 : zns = znstar_small(zn);
841 1453 : L = cgetg(o+1, t_VEC);
842 4815 : for (i=1,ind = phi; ind; ind -= phi/o, i++) /* by *decreasing* exact index */
843 3362 : gel(L,i) = subgrouplist(gel(zn,2), mkvec(utoipos(ind)));
844 1453 : L = shallowconcat1(L); l = lg(L);
845 4575 : for (i = 1; i < l; i++) gel(L,i) = znstar_hnf_elts(zns, gel(L,i));
846 1453 : return gc_GEN(av, L);
847 : }
848 :
849 : /* A sympol is a symmetric polynomial
850 : *
851 : * Currently sympol are couple of t_VECSMALL [v,w]
852 : * v[1]...v[k], w[1]...w[k] represent the polynomial sum(i=1,k,v[i]*s_w[i])
853 : * where s_i(X_1,...,X_n) = sum(j=1,n,X_j^i) */
854 :
855 : static GEN
856 14982 : Flm_newtonsum(GEN M, ulong e, ulong p)
857 : {
858 14982 : long f = lg(M), g = lg(gel(M,1)), i, j;
859 14982 : GEN NS = cgetg(f, t_VECSMALL);
860 70098 : for(i=1; i<f; i++)
861 : {
862 55116 : ulong s = 0;
863 55116 : GEN Mi = gel(M,i);
864 246122 : for(j = 1; j < g; j++)
865 191006 : s = Fl_add(s, Fl_powu(uel(Mi,j), e, p), p);
866 55116 : uel(NS,i) = s;
867 : }
868 14982 : return NS;
869 : }
870 :
871 : static GEN
872 9482 : Flv_sympol_eval(GEN v, GEN NS, ulong p)
873 : {
874 9482 : pari_sp av = avma;
875 9482 : long i, l = lg(v);
876 9482 : GEN S = Flv_Fl_mul(gel(NS,1), uel(v,1), p);
877 10019 : for (i=2; i<l; i++)
878 537 : if (v[i]) S = Flv_add(S, Flv_Fl_mul(gel(NS,i), uel(v,i), p), p);
879 9482 : return gc_leaf(av, S);
880 : }
881 :
882 : static GEN
883 9482 : sympol_eval_newtonsum(long e, GEN O, GEN mod)
884 : {
885 9482 : long f = lg(O), g = lg(gel(O,1)), i, j;
886 9482 : GEN PL = cgetg(f, t_COL);
887 46077 : for(i=1; i<f; i++)
888 : {
889 36595 : pari_sp av = avma;
890 36595 : GEN s = gen_0;
891 150417 : for(j=1; j<g; j++) s = addii(s, Fp_powu(gmael(O,i,j), e, mod));
892 36595 : gel(PL,i) = gc_INT(av, remii(s,mod));
893 : }
894 9482 : return PL;
895 : }
896 :
897 : static GEN
898 9416 : sympol_eval(GEN sym, GEN O, GEN mod)
899 : {
900 9416 : pari_sp av = avma;
901 : long i;
902 9416 : GEN v = gel(sym,1), w = gel(sym,2);
903 9416 : GEN S = gen_0;
904 19303 : for (i=1; i<lg(v); i++)
905 9887 : if (v[i]) S = gadd(S, gmulsg(v[i], sympol_eval_newtonsum(w[i], O, mod)));
906 9416 : return gc_upto(av, S);
907 : }
908 :
909 : /* Let sigma be an automorphism of L (as a polynomial with rational coefs)
910 : * Let 'sym' be a symmetric polynomial defining alpha in L.
911 : * We have alpha = sym(x,sigma(x),,,sigma^(g-1)(x)). Compute alpha mod p */
912 : static GEN
913 4660 : sympol_aut_evalmod(GEN sym, long g, GEN sigma, GEN Tp, GEN p)
914 : {
915 4660 : pari_sp ltop=avma;
916 4660 : long i, j, npows = brent_kung_optpow(degpol(Tp)-1, g-1, 1);
917 4660 : GEN s, f, pows, v = zv_to_ZV(gel(sym,1)), w = zv_to_ZV(gel(sym,2));
918 4660 : sigma = RgX_to_FpX(sigma, p);
919 4660 : pows = FpXQ_powers(sigma,npows,Tp,p);
920 4660 : f = pol_x(varn(sigma));
921 4660 : s = pol_0(varn(sigma));
922 19038 : for(i=1; i<=g;i++)
923 : {
924 14378 : if (i > 1) f = FpX_FpXQV_eval(f,pows,Tp,p);
925 29074 : for(j=1; j<lg(v); j++)
926 14696 : s = FpX_add(s, FpX_Fp_mul(FpXQ_pow(f,gel(w,j),Tp,p),gel(v,j),p),p);
927 : }
928 4660 : return gc_upto(ltop, s);
929 : }
930 :
931 : /* Let Sp be as computed with sympol_aut_evalmod
932 : * Let Tmod be the factorisation of T mod p.
933 : * Return the factorisation of the minimal polynomial of S mod p w.r.t. Tmod */
934 : static GEN
935 4660 : fixedfieldfactmod(GEN Sp, GEN p, GEN Tmod)
936 : {
937 4660 : long i, l = lg(Tmod);
938 4660 : GEN F = cgetg(l,t_VEC);
939 15781 : for(i=1; i<l; i++)
940 : {
941 11121 : GEN Ti = gel(Tmod,i);
942 11121 : gel(F,i) = FpXQ_minpoly(FpX_rem(Sp,Ti,p), Ti,p);
943 : }
944 4660 : return F;
945 : }
946 :
947 : static GEN
948 9416 : fixedfieldsurmer(ulong l, GEN NS, GEN W)
949 : {
950 9416 : const long step=3;
951 9416 : long i, j, n = lg(W)-1, m = 1L<<((n-1)<<1);
952 9416 : GEN sym = cgetg(n+1,t_VECSMALL);
953 9887 : for (j=1;j<n;j++) sym[j] = step;
954 9416 : sym[n] = 0;
955 9416 : if (DEBUGLEVEL>=4) err_printf("FixedField: Weight: %Ps\n",W);
956 9482 : for (i=0; i<m; i++)
957 : {
958 9482 : pari_sp av = avma;
959 : GEN L;
960 9953 : for (j=1; sym[j]==step; j++) sym[j]=0;
961 9482 : sym[j]++;
962 9482 : if (DEBUGLEVEL>=6) err_printf("FixedField: Sym: %Ps\n",sym);
963 9482 : L = Flv_sympol_eval(sym, NS, l);
964 9482 : if (!vecsmall_is1to1(L)) { set_avma(av); continue; }
965 9416 : return mkvec2(sym,W);
966 : }
967 0 : return NULL;
968 : }
969 :
970 : /*Check whether the line of NS are pair-wise distinct.*/
971 : static long
972 9887 : sympol_is1to1_lg(GEN NS, long n)
973 : {
974 9887 : long i, j, k, l = lgcols(NS);
975 45322 : for (i=1; i<l; i++)
976 142014 : for(j=i+1; j<l; j++)
977 : {
978 109727 : for(k=1; k<n; k++)
979 109256 : if (mael(NS,k,j)!=mael(NS,k,i)) break;
980 106579 : if (k>=n) return 0;
981 : }
982 9416 : return 1;
983 : }
984 :
985 : /* Let O a set of orbits of roots (see fixedfieldorbits) modulo mod,
986 : * l | mod and p two prime numbers. Return a vector [sym,s,P] where:
987 : * sym is a sympol, s is the set of images of sym on O and
988 : * P is the polynomial with roots s. */
989 : static GEN
990 9416 : fixedfieldsympol(GEN O, ulong l)
991 : {
992 9416 : pari_sp ltop=avma;
993 9416 : const long n=(BITS_IN_LONG>>1)-1;
994 9416 : GEN NS = cgetg(n+1,t_MAT), sym = NULL, W = cgetg(n+1,t_VECSMALL);
995 9416 : long i, e=1;
996 9416 : if (DEBUGLEVEL>=4)
997 0 : err_printf("FixedField: Size: %ldx%ld\n",lg(O)-1,lg(gel(O,1))-1);
998 9416 : O = ZM_to_Flm(O,l);
999 19303 : for (i=1; !sym && i<=n; i++)
1000 : {
1001 9887 : GEN L = Flm_newtonsum(O, e++, l);
1002 9887 : if (lg(O)>2)
1003 14906 : while (vecsmall_isconst(L)) L = Flm_newtonsum(O, e++, l);
1004 9887 : W[i] = e-1; gel(NS,i) = L;
1005 9887 : if (sympol_is1to1_lg(NS,i+1))
1006 9416 : sym = fixedfieldsurmer(l,NS,vecsmall_shorten(W,i));
1007 : }
1008 9416 : if (!sym) pari_err_BUG("fixedfieldsympol [p too small]");
1009 9416 : if (DEBUGLEVEL>=2) err_printf("FixedField: Found: %Ps\n",gel(sym,1));
1010 9416 : return gc_GEN(ltop,sym);
1011 : }
1012 :
1013 : /* Let O a set of orbits as indices and L the corresponding roots.
1014 : * Return the set of orbits as roots. */
1015 : static GEN
1016 9416 : fixedfieldorbits(GEN O, GEN L)
1017 : {
1018 9416 : GEN S = cgetg(lg(O), t_MAT);
1019 : long i;
1020 44811 : for (i = 1; i < lg(O); i++) gel(S,i) = vecpermute(L, gel(O,i));
1021 9416 : return S;
1022 : }
1023 :
1024 : static GEN
1025 827 : fixedfieldinclusion(GEN O, GEN PL)
1026 : {
1027 827 : long i, j, f = lg(O)-1, g = lg(gel(O,1))-1;
1028 827 : GEN S = cgetg(f*g + 1, t_COL);
1029 5603 : for (i = 1; i <= f; i++)
1030 : {
1031 4776 : GEN Oi = gel(O,i);
1032 19824 : for (j = 1; j <= g; j++) gel(S, Oi[j]) = gel(PL, i);
1033 : }
1034 827 : return S;
1035 : }
1036 :
1037 : /* Polynomial attached to a vector of conjugates. Not stack clean */
1038 : static GEN
1039 44052 : vectopol(GEN v, GEN M, GEN den , GEN mod, GEN mod2, long x)
1040 : {
1041 44052 : long l = lg(v)+1, i;
1042 44052 : GEN z = cgetg(l,t_POL);
1043 44052 : z[1] = evalsigne(1)|evalvarn(x);
1044 336001 : for (i=2; i<l; i++)
1045 291949 : gel(z,i) = gdiv(centermodii(ZMrow_ZC_mul(M,v,i-1), mod, mod2), den);
1046 44052 : return normalizepol_lg(z, l);
1047 : }
1048 :
1049 : /* Polynomial associate to a permutation of the roots. Not stack clean */
1050 : static GEN
1051 42763 : permtopol(GEN p, GEN L, GEN M, GEN den, GEN mod, GEN mod2, long x)
1052 : {
1053 42763 : if (lg(p) != lg(L)) pari_err_TYPE("permtopol [permutation]", p);
1054 42763 : return vectopol(vecpermute(L,p), M, den, mod, mod2, x);
1055 : }
1056 :
1057 : static GEN
1058 7752 : galoisvecpermtopol(GEN gal, GEN vec, GEN mod, GEN mod2)
1059 : {
1060 7752 : long i, l = lg(vec);
1061 7752 : long v = varn(gal_get_pol(gal));
1062 7752 : GEN L = gal_get_roots(gal);
1063 7752 : GEN M = gal_get_invvdm(gal);
1064 7752 : GEN P = cgetg(l, t_MAT);
1065 40403 : for (i=1; i<l; i++)
1066 : {
1067 32657 : GEN p = gel(vec,i);
1068 32657 : if (typ(p) != t_VECSMALL) pari_err_TYPE("galoispermtopol", vec);
1069 32651 : gel(P, i) = vecpermute(L, p);
1070 : }
1071 7746 : P = RgM_to_RgXV(FpM_center(FpM_mul(M, P, mod), mod, mod2), v);
1072 7746 : return gdiv(P, gal_get_den(gal));
1073 : }
1074 :
1075 : static void
1076 57308 : notgalois(long p, struct galois_analysis *ga)
1077 : {
1078 57308 : if (DEBUGLEVEL >= 2) err_printf("GaloisAnalysis:non Galois for p=%ld\n", p);
1079 57308 : ga->p = p;
1080 57308 : ga->deg = 0;
1081 57308 : }
1082 :
1083 : /*Gather information about the group*/
1084 : static long
1085 82925 : init_group(long n, long np, GEN Fp, GEN Fe, long *porder)
1086 : {
1087 82925 : const long prim_nonwss_orders[] = { 48,56,60,72,75,80,196,200,216 };
1088 82925 : long i, phi_order = 1, order = 1, group = 0;
1089 : ulong p;
1090 :
1091 : /* non-WSS groups of this order? */
1092 829097 : for (i=0; i < (long)numberof(prim_nonwss_orders); i++)
1093 746189 : if (n % prim_nonwss_orders[i] == 0) { group |= ga_non_wss; break; }
1094 82925 : if (np == 2 && Fp[2] == 3 && Fe[2] == 1 && Fe[1] > 2) group |= ga_ext_2;
1095 :
1096 122826 : for (i = np; i > 0; i--)
1097 : {
1098 87875 : long p = Fp[i];
1099 87875 : if (phi_order % p == 0) { group |= ga_all_normal; break; }
1100 82965 : order *= p; phi_order *= p-1;
1101 82965 : if (Fe[i] > 1) break;
1102 : }
1103 82925 : if (uisprimepower(n, &p) || n == 135) group |= ga_all_nilpotent;
1104 82925 : if (n <= 104) group |= ga_easy; /* no need to use polynomial algo */
1105 82925 : *porder = order; return group;
1106 : }
1107 :
1108 : /*is a "better" than b ? (if so, update karma) */
1109 : static int
1110 140277 : improves(long a, long b, long plift, long p, long n, long *karma)
1111 : {
1112 140277 : if (!plift || a > b) { *karma = ugcd(p-1,n); return 1; }
1113 137220 : if (a == b) {
1114 134914 : long k = ugcd(p-1,n);
1115 134914 : if (k > *karma) { *karma = k; return 1; }
1116 : }
1117 121621 : return 0; /* worse */
1118 : }
1119 :
1120 : /* return 0 if not galois or not wss */
1121 : static int
1122 82925 : galoisanalysis(GEN T, struct galois_analysis *ga, long calcul_l, GEN bad)
1123 : {
1124 82925 : pari_sp ltop = avma, av;
1125 82925 : long group, linf, n, p, i, karma = 0;
1126 : GEN F, Fp, Fe, Fpe, O;
1127 : long np, order, plift, nbmax, nbtest, deg;
1128 : pari_timer ti;
1129 : forprime_t S;
1130 82925 : if (DEBUGLEVEL >= 1) timer_start(&ti);
1131 82925 : n = degpol(T);
1132 82925 : O = zero_zv(n);
1133 82925 : F = factoru_pow(n);
1134 82925 : Fp = gel(F,1); np = lg(Fp)-1;
1135 82925 : Fe = gel(F,2);
1136 82925 : Fpe= gel(F,3);
1137 82925 : group = init_group(n, np, Fp, Fe, &order);
1138 :
1139 : /*Now we study the orders of the Frobenius elements*/
1140 82925 : deg = Fp[np]; /* largest prime | n */
1141 82925 : plift = 0;
1142 82925 : nbtest = 0;
1143 82925 : nbmax = 8+(n>>1);
1144 82925 : u_forprime_init(&S, n*maxss(expu(n)-3, 2), ULONG_MAX);
1145 82925 : av = avma;
1146 426105 : while (!plift || (nbtest < nbmax && (nbtest <=8 || order < (n>>1)))
1147 25862 : || ((n == 24 || n==36) && O[6] == 0 && O[4] == 0)
1148 484639 : || ((group&ga_non_wss) && order == Fp[np]))
1149 : {
1150 432915 : long d, o, norm_o = 1;
1151 : GEN D, Tp;
1152 :
1153 432915 : if ((group&ga_non_wss) && nbtest >= 3*nbmax) break; /* in all cases */
1154 432915 : nbtest++; set_avma(av);
1155 432915 : p = u_forprime_next(&S);
1156 432915 : if (!p) pari_err_OVERFLOW("galoisanalysis [ran out of primes]");
1157 467881 : if (bad && dvdiu(bad, p)) continue;
1158 432915 : Tp = ZX_to_Flx(T,p);
1159 432915 : if (!Flx_is_squarefree(Tp,p)) { if (!--nbtest) nbtest = 1; continue; }
1160 :
1161 397949 : D = Flx_nbfact_by_degree(Tp, &d, p);
1162 397949 : o = n / d; /* d factors, all should have degree o */
1163 397949 : if (D[o] != d) { notgalois(p, ga); return gc_bool(ltop,0); }
1164 :
1165 340886 : if (!O[o]) O[o] = p;
1166 340886 : if (o % deg) goto ga_end; /* NB: deg > 1 */
1167 211381 : if ((group&ga_all_normal) && o < order) goto ga_end;
1168 :
1169 : /*Frob_p has order o > 1, find a power which generates a normal subgroup*/
1170 211260 : if (o * Fp[1] >= n)
1171 197973 : norm_o = o; /*subgroups of smallest index are normal*/
1172 : else
1173 : {
1174 16203 : for (i = np; i > 0; i--)
1175 : {
1176 16203 : if (o % Fpe[i]) break;
1177 2916 : norm_o *= Fpe[i];
1178 : }
1179 : }
1180 : /* Frob_p^(o/norm_o) generates a normal subgroup of order norm_o */
1181 211260 : if (norm_o != 1)
1182 : {
1183 200889 : if (!(group&ga_all_normal) || o > order)
1184 70983 : karma = ugcd(p-1,n);
1185 129906 : else if (!improves(norm_o, deg, plift,p,n, &karma)) goto ga_end;
1186 : /* karma0=0, deg0<=norm_o -> the first improves() returns 1 */
1187 87688 : deg = norm_o; group |= ga_all_normal; /* STORE */
1188 : }
1189 10371 : else if (group&ga_all_normal) goto ga_end;
1190 10371 : else if (!improves(o, order, plift,p,n, &karma)) goto ga_end;
1191 :
1192 89639 : order = o; plift = p; /* STORE */
1193 340886 : ga_end:
1194 340886 : if (DEBUGLEVEL >= 5)
1195 0 : err_printf("GaloisAnalysis:Nbtest=%ld,p=%ld,o=%ld,n_o=%d,best p=%ld,ord=%ld,k=%ld\n", nbtest, p, o, norm_o, plift, order,karma);
1196 : }
1197 : /* To avoid looping on non-WSS group.
1198 : * TODO: check for large groups. Would it be better to disable this check if
1199 : * we are in a good case (ga_all_normal && !(ga_ext_2) (e.g. 60)) ?*/
1200 25862 : ga->p = plift;
1201 25862 : if (!plift || ((group&ga_non_wss) && order == Fp[np]))
1202 : {
1203 0 : if (DEBUGLEVEL)
1204 0 : pari_warn(warner,"Galois group probably not weakly super solvable");
1205 0 : return 0;
1206 : }
1207 25862 : linf = 2*n*usqrt(n);
1208 25862 : if (calcul_l && O[1] <= linf)
1209 : {
1210 : pari_sp av2;
1211 : forprime_t S2;
1212 : ulong p;
1213 5850 : u_forprime_init(&S2, linf+1,ULONG_MAX);
1214 5850 : av2 = avma;
1215 80407 : while ((p = u_forprime_next(&S2)))
1216 : { /*find a totally split prime l > linf*/
1217 80407 : GEN Tp = ZX_to_Flx(T, p);
1218 80407 : long nb = Flx_nbroots(Tp, p);
1219 80407 : if (nb == n) { O[1] = p; break; }
1220 74802 : if (nb && Flx_is_squarefree(Tp,p)) { notgalois(p,ga); return gc_bool(ltop,0); }
1221 74557 : set_avma(av2);
1222 : }
1223 5605 : if (!p) pari_err_OVERFLOW("galoisanalysis [ran out of primes]");
1224 : }
1225 25617 : ga->group = group;
1226 25617 : ga->deg = deg;
1227 25617 : ga->ord = order;
1228 25617 : ga->l = O[1];
1229 25617 : ga->p4 = n >= 4 ? O[4] : 0;
1230 25617 : if (DEBUGLEVEL >= 4)
1231 0 : err_printf("GaloisAnalysis:p=%ld l=%ld group=%ld deg=%ld ord=%ld\n",
1232 0 : plift, O[1], group, deg, order);
1233 25617 : if (DEBUGLEVEL >= 1) timer_printf(&ti, "galoisanalysis()");
1234 25617 : return gc_bool(ltop,1);
1235 : }
1236 :
1237 : static GEN
1238 81 : a4galoisgen(struct galois_test *td)
1239 : {
1240 81 : const long n = 12;
1241 81 : pari_sp ltop = avma, av, av2;
1242 81 : long i, j, k, N, hop = 0;
1243 : GEN MT, O,O1,O2,O3, ar, mt, t, u, res, orb, pft, pfu, pfv;
1244 :
1245 81 : res = cgetg(3, t_VEC);
1246 81 : pft = cgetg(n+1, t_VECSMALL);
1247 81 : pfu = cgetg(n+1, t_VECSMALL);
1248 81 : pfv = cgetg(n+1, t_VECSMALL);
1249 81 : gel(res,1) = mkvec3(pft,pfu,pfv);
1250 81 : gel(res,2) = mkvecsmall3(2,2,3);
1251 81 : av = avma;
1252 81 : ar = cgetg(5, t_VECSMALL);
1253 81 : mt = gel(td->PV, td->order[n]);
1254 81 : t = identity_perm(n) + 1; /* Sorry for this hack */
1255 81 : u = cgetg(n+1, t_VECSMALL) + 1; /* too lazy to correct */
1256 81 : MT = cgetg(n+1, t_MAT);
1257 1053 : for (j = 1; j <= n; j++) gel(MT,j) = cgetg(n+1, t_VECSMALL);
1258 1053 : for (j = 1; j <= n; j++)
1259 6318 : for (i = 1; i < j; i++)
1260 5346 : ucoeff(MT,i,j) = ucoeff(MT,j,i) = ucoeff(mt,i,j)+ucoeff(mt,j,i);
1261 : /* MT(i,i) unused */
1262 :
1263 81 : av2 = avma;
1264 : /* N = itos(gdiv(mpfact(n), mpfact(n >> 1))) >> (n >> 1); */
1265 : /* n = 2k = 12; N = (2k)! / (k! * 2^k) = 10395 */
1266 81 : N = 10395;
1267 81 : if (DEBUGLEVEL>=4) err_printf("A4GaloisConj: will test %ld permutations\n", N);
1268 81 : uel(ar,4) = umael(MT,11,12);
1269 81 : uel(ar,3) = uel(ar,4) + umael(MT,9,10);
1270 81 : uel(ar,2) = uel(ar,3) + umael(MT,7,8);
1271 81 : uel(ar,1) = uel(ar,2) + umael(MT,5,6);
1272 194997 : for (i = 0; i < N; i++)
1273 : {
1274 : long g;
1275 194997 : if (i)
1276 : {
1277 194916 : long a, x = i, y = 1;
1278 274798 : do { y += 2; a = x%y; x = x/y; } while (!a);
1279 194916 : switch (y)
1280 : {
1281 129980 : case 3:
1282 129980 : lswap(t[2], t[2-a]);
1283 129980 : break;
1284 51971 : case 5:
1285 51971 : x = t[0]; t[0] = t[2]; t[2] = t[1]; t[1] = x;
1286 51971 : lswap(t[4], t[4-a]);
1287 51971 : uel(ar,1) = uel(ar,2) + umael(MT,t[4],t[5]);
1288 51971 : break;
1289 11149 : case 7:
1290 11149 : x = t[0]; t[0] = t[4]; t[4] = t[3]; t[3] = t[1]; t[1] = t[2]; t[2] = x;
1291 11149 : lswap(t[6], t[6-a]);
1292 11149 : uel(ar,2) = uel(ar,3) + umael(MT,t[6],t[7]);
1293 11149 : uel(ar,1) = uel(ar,2) + umael(MT,t[4],t[5]);
1294 11149 : break;
1295 1651 : case 9:
1296 1651 : x = t[0]; t[0] = t[6]; t[6] = t[5]; t[5] = t[3]; t[3] = x;
1297 1651 : lswap(t[1], t[4]);
1298 1651 : lswap(t[8], t[8-a]);
1299 1651 : uel(ar,3) = uel(ar,4) + umael(MT,t[8],t[9]);
1300 1651 : uel(ar,2) = uel(ar,3) + umael(MT,t[6],t[7]);
1301 1651 : uel(ar,1) = uel(ar,2) + umael(MT,t[4],t[5]);
1302 1651 : break;
1303 165 : case 11:
1304 165 : x = t[0]; t[0] = t[8]; t[8] = t[7]; t[7] = t[5]; t[5] = t[1];
1305 165 : t[1] = t[6]; t[6] = t[3]; t[3] = t[2]; t[2] = t[4]; t[4] = x;
1306 165 : lswap(t[10], t[10-a]);
1307 165 : uel(ar,4) = umael(MT,t[10],t[11]);
1308 165 : uel(ar,3) = uel(ar,4) + umael(MT,t[8],t[9]);
1309 165 : uel(ar,2) = uel(ar,3) + umael(MT,t[6],t[7]);
1310 165 : uel(ar,1) = uel(ar,2) + umael(MT,t[4],t[5]);
1311 : }
1312 : }
1313 194997 : g = uel(ar,1)+umael(MT,t[0],t[1])+umael(MT,t[2],t[3]);
1314 194997 : if (headlongisint(g,n))
1315 : {
1316 567 : for (k = 0; k < n; k += 2)
1317 : {
1318 486 : pft[t[k]] = t[k+1];
1319 486 : pft[t[k+1]] = t[k];
1320 : }
1321 81 : if (galois_test_perm(td, pft)) break;
1322 0 : hop++;
1323 : }
1324 194916 : set_avma(av2);
1325 : }
1326 81 : if (DEBUGLEVEL >= 1 && hop)
1327 0 : err_printf("A4GaloisConj: %ld hop over %ld iterations\n", hop, N);
1328 81 : if (i == N) return gc_NULL(ltop);
1329 : /* N = itos(gdiv(mpfact(n >> 1), mpfact(n >> 2))) >> 1; */
1330 81 : N = 60;
1331 81 : if (DEBUGLEVEL >= 4) err_printf("A4GaloisConj: sigma=%Ps \n", pft);
1332 324 : for (k = 0; k < n; k += 4)
1333 : {
1334 243 : u[k+3] = t[k+3];
1335 243 : u[k+2] = t[k+1];
1336 243 : u[k+1] = t[k+2];
1337 243 : u[k] = t[k];
1338 : }
1339 3893 : for (i = 0; i < N; i++)
1340 : {
1341 3893 : ulong g = 0;
1342 3893 : if (i)
1343 : {
1344 3812 : long a, x = i, y = -2;
1345 5998 : do { y += 4; a = x%y; x = x/y; } while (!a);
1346 3812 : lswap(u[0],u[2]);
1347 3812 : switch (y)
1348 : {
1349 1906 : case 2:
1350 1906 : break;
1351 1626 : case 6:
1352 1626 : lswap(u[4],u[6]);
1353 1626 : if (!(a & 1))
1354 : {
1355 667 : a = 4 - (a>>1);
1356 667 : lswap(u[6], u[a]);
1357 667 : lswap(u[4], u[a-2]);
1358 : }
1359 1626 : break;
1360 280 : case 10:
1361 280 : x = u[6];
1362 280 : u[6] = u[3];
1363 280 : u[3] = u[2];
1364 280 : u[2] = u[4];
1365 280 : u[4] = u[1];
1366 280 : u[1] = u[0];
1367 280 : u[0] = x;
1368 280 : if (a >= 3) a += 2;
1369 280 : a = 8 - a;
1370 280 : lswap(u[10],u[a]);
1371 280 : lswap(u[8], u[a-2]);
1372 280 : break;
1373 : }
1374 : }
1375 27251 : for (k = 0; k < n; k += 2) g += mael(MT,u[k],u[k+1]);
1376 3893 : if (headlongisint(g,n))
1377 : {
1378 567 : for (k = 0; k < n; k += 2)
1379 : {
1380 486 : pfu[u[k]] = u[k+1];
1381 486 : pfu[u[k+1]] = u[k];
1382 : }
1383 81 : if (galois_test_perm(td, pfu)) break;
1384 0 : hop++;
1385 : }
1386 3812 : set_avma(av2);
1387 : }
1388 81 : if (i == N) return gc_NULL(ltop);
1389 81 : if (DEBUGLEVEL >= 1 && hop)
1390 0 : err_printf("A4GaloisConj: %ld hop over %ld iterations\n", hop, N);
1391 81 : if (DEBUGLEVEL >= 4) err_printf("A4GaloisConj: tau=%Ps \n", pfu);
1392 81 : set_avma(av2);
1393 81 : orb = mkvec2(pft,pfu);
1394 81 : O = vecperm_orbits(orb, 12);
1395 81 : if (DEBUGLEVEL >= 4) {
1396 0 : err_printf("A4GaloisConj: orb=%Ps\n", orb);
1397 0 : err_printf("A4GaloisConj: O=%Ps \n", O);
1398 : }
1399 81 : av2 = avma;
1400 81 : O1 = gel(O,1); O2 = gel(O,2); O3 = gel(O,3);
1401 123 : for (j = 0; j < 2; j++)
1402 : {
1403 123 : pfv[O1[1]] = O2[1];
1404 123 : pfv[O1[2]] = O2[3+j];
1405 123 : pfv[O1[3]] = O2[4 - (j << 1)];
1406 123 : pfv[O1[4]] = O2[2+j];
1407 414 : for (i = 0; i < 4; i++)
1408 : {
1409 372 : ulong g = 0;
1410 372 : switch (i)
1411 : {
1412 123 : case 0: break;
1413 104 : case 1: lswap(O3[1], O3[2]); lswap(O3[3], O3[4]); break;
1414 86 : case 2: lswap(O3[1], O3[4]); lswap(O3[2], O3[3]); break;
1415 59 : case 3: lswap(O3[1], O3[2]); lswap(O3[3], O3[4]); break;
1416 : }
1417 372 : pfv[O2[1]] = O3[1];
1418 372 : pfv[O2[3+j]] = O3[4-j];
1419 372 : pfv[O2[4 - (j<<1)]] = O3[2 + (j<<1)];
1420 372 : pfv[O2[2+j]] = O3[3-j];
1421 372 : pfv[O3[1]] = O1[1];
1422 372 : pfv[O3[4-j]] = O1[2];
1423 372 : pfv[O3[2 + (j<<1)]] = O1[3];
1424 372 : pfv[O3[3-j]] = O1[4];
1425 4836 : for (k = 1; k <= n; k++) g += mael(mt,k,pfv[k]);
1426 372 : if (headlongisint(g,n) && galois_test_perm(td, pfv))
1427 : {
1428 81 : set_avma(av);
1429 81 : if (DEBUGLEVEL >= 1)
1430 0 : err_printf("A4GaloisConj: %ld hop over %d iterations max\n",
1431 : hop, 10395 + 68);
1432 81 : return res;
1433 : }
1434 291 : hop++; set_avma(av2);
1435 : }
1436 : }
1437 0 : return gc_NULL(ltop);
1438 : }
1439 :
1440 : /* S4 */
1441 : static GEN
1442 1223 : s4makelift(GEN u, struct galois_lift *gl)
1443 1223 : { return FpXQ_powers(u, degpol(gl->T)-1, gl->TQ, gl->Q); }
1444 :
1445 : static long
1446 204772 : s4test(GEN u, GEN liftpow, struct galois_lift *gl, GEN phi)
1447 : {
1448 204772 : pari_sp av = avma;
1449 : GEN res, Q, Q2;
1450 204772 : long bl, i, d = lg(u)-2;
1451 : pari_timer ti;
1452 204772 : if (DEBUGLEVEL >= 6) timer_start(&ti);
1453 204772 : if (!d) return 0;
1454 204772 : Q = gl->Q; Q2 = shifti(Q,-1);
1455 204772 : res = gel(u,2);
1456 7293690 : for (i = 2; i <= d; i++)
1457 7088918 : if (lg(gel(liftpow,i))>2)
1458 7088918 : res = addii(res, mulii(gmael(liftpow,i,2), gel(u,i+1)));
1459 204772 : res = remii(res,Q);
1460 204772 : if (gl->den != gen_1) res = mulii(res, gl->den);
1461 204772 : res = centermodii(res, Q,Q2);
1462 204772 : if (abscmpii(res, gl->gb->bornesol) > 0) return gc_long(av,0);
1463 3004 : res = scalar_ZX_shallow(gel(u,2),varn(u));
1464 100706 : for (i = 2; i <= d ; i++)
1465 97702 : if (lg(gel(liftpow,i))>2)
1466 97702 : res = ZX_add(res, ZX_Z_mul(gel(liftpow,i), gel(u,i+1)));
1467 3004 : res = FpX_red(res, Q);
1468 3004 : if (gl->den != gen_1) res = FpX_Fp_mul(res, gl->den, Q);
1469 3004 : res = FpX_center_i(res, Q, shifti(Q,-1));
1470 3004 : bl = poltopermtest(res, gl, phi);
1471 3004 : if (DEBUGLEVEL >= 6) timer_printf(&ti, "s4test()");
1472 3004 : return gc_long(av,bl);
1473 : }
1474 :
1475 : static GEN
1476 413 : s4releveauto(GEN M, GEN B, GEN T, GEN p,long a1,long a2,long a3,long a4,long a5,long a6)
1477 : {
1478 413 : GEN F = ZX_mul(gel(M,a1),gel(B,a2));
1479 413 : F = ZX_add(F, ZX_mul(gel(M,a2),gel(B,a1)));
1480 413 : F = ZX_add(F, ZX_mul(gel(M,a3),gel(B,a4)));
1481 413 : F = ZX_add(F, ZX_mul(gel(M,a4),gel(B,a3)));
1482 413 : F = ZX_add(F, ZX_mul(gel(M,a5),gel(B,a6)));
1483 413 : F = ZX_add(F, ZX_mul(gel(M,a6),gel(B,a5)));
1484 413 : return FpXQ_red(F, T, p);
1485 : }
1486 :
1487 : static GEN
1488 273186 : lincomb(GEN B, long a, long b, long j)
1489 : {
1490 273186 : long k = (-j) & 3;
1491 273186 : return ZX_add(gmael(B,a,j+1), gmael(B,b,k+1));
1492 : }
1493 :
1494 : static GEN
1495 75 : FpXV_ffisom(GEN V, GEN p)
1496 : {
1497 75 : pari_sp av = avma;
1498 75 : long i, j, l = lg(V);
1499 75 : GEN S = cgetg(l, t_VEC), Si = cgetg(l, t_VEC), M;
1500 561 : for (i = 1; i < l; i++)
1501 : {
1502 486 : gel(S,i) = FpX_ffisom(gel(V,1), gel(V,i), p);
1503 486 : gel(Si,i) = FpXQ_ffisom_inv(gel(S,i), gel(V,i), p);
1504 : }
1505 75 : M = cgetg(l, t_MAT);
1506 561 : for (j = 1; j < l; j++)
1507 486 : gel(M,j) = FpXC_FpXQ_eval(Si, gel(S,j), gel(V,j), p);
1508 75 : return gc_upto(av, M);
1509 : }
1510 :
1511 : static GEN
1512 75 : mkliftpow(GEN x, GEN T, GEN p, struct galois_lift *gl)
1513 561 : { pari_APPLY_same(automorphismlift(FpXV_chinese(gel(x,i), T, p, NULL), gl)) }
1514 :
1515 : #define rot3(x,y,z) {long _t=x; x=y; y=z; z=_t;}
1516 : #define rot4(x,y,z,t) {long _u=x; x=y; y=z; z=t; t=_u;}
1517 :
1518 : /* FIXME: could use the intheadlong technique */
1519 : static GEN
1520 63 : s4galoisgen(struct galois_lift *gl)
1521 : {
1522 63 : const long n = 24;
1523 : struct galois_testlift gt;
1524 63 : pari_sp av, ltop2, ltop = avma;
1525 : long i, j;
1526 63 : GEN sigma, tau, phi, res, r1,r2,r3,r4, pj, p = gl->p, Q = gl->Q, TQ = gl->TQ;
1527 : GEN sg, Tp, Tmod, misom, B, Bcoeff, liftpow, liftp, aut;
1528 :
1529 63 : res = cgetg(3, t_VEC);
1530 63 : r1 = cgetg(n+1, t_VECSMALL);
1531 63 : r2 = cgetg(n+1, t_VECSMALL);
1532 63 : r3 = cgetg(n+1, t_VECSMALL);
1533 63 : r4 = cgetg(n+1, t_VECSMALL);
1534 63 : gel(res,1)= mkvec4(r1,r2,r3,r4);
1535 63 : gel(res,2) = mkvecsmall4(2,2,3,2);
1536 63 : ltop2 = avma;
1537 63 : sg = identity_perm(6);
1538 63 : pj = zero_zv(6);
1539 63 : sigma = cgetg(n+1, t_VECSMALL);
1540 63 : tau = cgetg(n+1, t_VECSMALL);
1541 63 : phi = cgetg(n+1, t_VECSMALL);
1542 63 : Tp = FpX_red(gl->T,p);
1543 63 : Tmod = gel(FpX_factor(Tp,p), 1);
1544 63 : misom = FpXV_ffisom(Tmod, p);
1545 63 : aut = galoisdolift(gl);
1546 63 : inittestlift(aut, Tmod, gl, >);
1547 63 : B = FqC_FqV_mul(gt.pauto, gt.bezoutcoeff, gl->TQ, Q);
1548 63 : Bcoeff = gt.bezoutcoeff;
1549 63 : liftp = mkliftpow(shallowtrans(misom), Tmod, p, gl);
1550 63 : av = avma;
1551 104 : for (i = 0; i < 3; i++, set_avma(av))
1552 : {
1553 : pari_sp av1, av2, av3;
1554 : GEN u, u1, u2, u3;
1555 : long j1, j2, j3;
1556 104 : if (i)
1557 : {
1558 41 : if (i == 1) { lswap(sg[2],sg[3]); }
1559 1 : else { lswap(sg[1],sg[3]); }
1560 : }
1561 104 : u = s4releveauto(liftp,Bcoeff,TQ,Q,sg[1],sg[2],sg[3],sg[4],sg[5],sg[6]);
1562 104 : liftpow = s4makelift(u, gl);
1563 104 : av1 = avma;
1564 324 : for (j1 = 0; j1 < 4; j1++, set_avma(av1))
1565 : {
1566 283 : u1 = lincomb(B,sg[5],sg[6],j1);
1567 283 : av2 = avma;
1568 1261 : for (j2 = 0; j2 < 4; j2++, set_avma(av2))
1569 : {
1570 1041 : u2 = lincomb(B,sg[3],sg[4],j2);
1571 1041 : u2 = FpX_add(u1, u2, Q); av3 = avma;
1572 5081 : for (j3 = 0; j3 < 4; j3++, set_avma(av3))
1573 : {
1574 4103 : u3 = lincomb(B,sg[1],sg[2],j3);
1575 4103 : u3 = FpX_add(u2, u3, Q);
1576 4103 : if (DEBUGLEVEL >= 4)
1577 0 : err_printf("S4GaloisConj: Testing %d/3:%d/4:%d/4:%d/4:%Ps\n",
1578 : i,j1,j2,j3, sg);
1579 4103 : if (s4test(u3, liftpow, gl, sigma))
1580 : {
1581 63 : pj[1] = j3;
1582 63 : pj[2] = j2;
1583 63 : pj[3] = j1; goto suites4;
1584 : }
1585 : }
1586 : }
1587 : }
1588 : }
1589 0 : return gc_NULL(ltop);
1590 63 : suites4:
1591 63 : if (DEBUGLEVEL >= 4) err_printf("S4GaloisConj: sigma=%Ps\n", sigma);
1592 63 : if (DEBUGLEVEL >= 4) err_printf("S4GaloisConj: pj=%Ps\n", pj);
1593 63 : set_avma(av);
1594 135 : for (j = 1; j <= 3; j++)
1595 : {
1596 : pari_sp av2, av3;
1597 : GEN u;
1598 : long w, l;
1599 135 : rot3(sg[1], sg[3], sg[5])
1600 135 : rot3(sg[2], sg[4], sg[6])
1601 135 : rot3(pj[1], pj[2], pj[3])
1602 318 : for (l = 0; l < 2; l++, set_avma(av))
1603 : {
1604 246 : u = s4releveauto(liftp,Bcoeff,TQ,Q,sg[1],sg[3],sg[2],sg[4],sg[5],sg[6]);
1605 246 : liftpow = s4makelift(u, gl);
1606 246 : av2 = avma;
1607 675 : for (w = 0; w < 4; w += 2, set_avma(av2))
1608 : {
1609 : GEN uu;
1610 492 : pj[6] = (w + pj[3]) & 3;
1611 492 : uu = lincomb(B, sg[5], sg[6], pj[6]);
1612 492 : uu = FpX_red(uu, Q);
1613 492 : av3 = avma;
1614 2319 : for (i = 0; i < 4; i++, set_avma(av3))
1615 : {
1616 : GEN u;
1617 1890 : pj[4] = i;
1618 1890 : pj[5] = (i + pj[2] - pj[1]) & 3;
1619 1890 : if (DEBUGLEVEL >= 4)
1620 0 : err_printf("S4GaloisConj: Testing %d/3:%d/2:%d/2:%d/4:%Ps:%Ps\n",
1621 : j-1, w >> 1, l, i, sg, pj);
1622 1890 : u = ZX_add(lincomb(B,sg[1],sg[3],pj[4]),
1623 1890 : lincomb(B,sg[2],sg[4],pj[5]));
1624 1890 : u = FpX_add(uu,u,Q);
1625 1890 : if (s4test(u, liftpow, gl, tau)) goto suites4_2;
1626 : }
1627 : }
1628 183 : lswap(sg[3], sg[4]);
1629 183 : pj[2] = (-pj[2]) & 3;
1630 : }
1631 : }
1632 0 : return gc_NULL(ltop);
1633 63 : suites4_2:
1634 63 : set_avma(av);
1635 : {
1636 63 : long abc = (pj[1] + pj[2] + pj[3]) & 3;
1637 63 : long abcdef = ((abc + pj[4] + pj[5] - pj[6]) & 3) >> 1;
1638 : GEN u;
1639 : pari_sp av2;
1640 63 : u = s4releveauto(liftp,Bcoeff,TQ,Q,sg[1],sg[4],sg[2],sg[5],sg[3],sg[6]);
1641 63 : liftpow = s4makelift(u, gl);
1642 63 : av2 = avma;
1643 257 : for (j = 0; j < 8; j++)
1644 : {
1645 : long h, g, i;
1646 257 : h = j & 3;
1647 257 : g = (abcdef + ((j & 4) >> 1)) & 3;
1648 257 : i = (h + abc - g) & 3;
1649 257 : u = ZX_add(lincomb(B,sg[1],sg[4], g), lincomb(B,sg[2],sg[5], h));
1650 257 : u = FpX_add(u, lincomb(B,sg[3],sg[6], i),Q);
1651 257 : if (DEBUGLEVEL >= 4)
1652 0 : err_printf("S4GaloisConj: Testing %d/8 %d:%d:%d\n", j,g,h,i);
1653 257 : if (s4test(u, liftpow, gl, phi)) break;
1654 194 : set_avma(av2);
1655 : }
1656 : }
1657 63 : if (j == 8) return gc_NULL(ltop);
1658 1575 : for (i = 1; i <= n; i++)
1659 : {
1660 1512 : r1[i] = sigma[tau[i]];
1661 1512 : r2[i] = phi[sigma[tau[phi[i]]]];
1662 1512 : r3[i] = phi[sigma[i]];
1663 1512 : r4[i] = sigma[i];
1664 : }
1665 63 : set_avma(ltop2); return res;
1666 : }
1667 :
1668 : static GEN
1669 780 : f36releveauto2(GEN Bl, GEN T, GEN p,GEN a)
1670 : {
1671 780 : GEN F = gmael(Bl,a[1],a[1]);
1672 780 : F = ZX_add(F,gmael(Bl,a[2],a[3]));
1673 780 : F = ZX_add(F,gmael(Bl,a[3],a[2]));
1674 780 : F = ZX_add(F,gmael(Bl,a[4],a[5]));
1675 780 : F = ZX_add(F,gmael(Bl,a[5],a[4]));
1676 780 : F = ZX_add(F,gmael(Bl,a[6],a[7]));
1677 780 : F = ZX_add(F,gmael(Bl,a[7],a[6]));
1678 780 : F = ZX_add(F,gmael(Bl,a[8],a[9]));
1679 780 : F = ZX_add(F,gmael(Bl,a[9],a[8]));
1680 780 : return FpXQ_red(F, T, p);
1681 : }
1682 :
1683 : static GEN
1684 30 : f36releveauto4(GEN Bl, GEN T, GEN p,GEN a)
1685 : {
1686 30 : GEN F = gmael(Bl,a[1],a[1]);
1687 30 : F = ZX_add(F,gmael(Bl,a[2],a[3]));
1688 30 : F = ZX_add(F,gmael(Bl,a[3],a[4]));
1689 30 : F = ZX_add(F,gmael(Bl,a[4],a[5]));
1690 30 : F = ZX_add(F,gmael(Bl,a[5],a[2]));
1691 30 : F = ZX_add(F,gmael(Bl,a[6],a[7]));
1692 30 : F = ZX_add(F,gmael(Bl,a[7],a[8]));
1693 30 : F = ZX_add(F,gmael(Bl,a[8],a[9]));
1694 30 : F = ZX_add(F,gmael(Bl,a[9],a[6]));
1695 30 : return FpXQ_red(F, T, p);
1696 : }
1697 :
1698 : static GEN
1699 12 : f36galoisgen(struct galois_lift *gl)
1700 : {
1701 12 : const long n = 36;
1702 : struct galois_testlift gt;
1703 12 : pari_sp av, ltop2, ltop = avma;
1704 : long i;
1705 12 : GEN sigma, tau, rho, res, r1,r2,r3, pj, pk, p = gl->p, Q = gl->Q, TQ = gl->TQ;
1706 : GEN sg, s4, sp, Tp, Tmod, misom, Bcoeff, liftpow, aut, liftp, B, Bl, tam;
1707 12 : res = cgetg(3, t_VEC);
1708 12 : r1 = cgetg(n+1, t_VECSMALL);
1709 12 : r2 = cgetg(n+1, t_VECSMALL);
1710 12 : r3 = cgetg(n+1, t_VECSMALL);
1711 12 : gel(res,1)= mkvec3(r1,r2,r3);
1712 12 : gel(res,2) = mkvecsmall3(3,3,4);
1713 12 : ltop2 = avma;
1714 12 : sg = identity_perm(9);
1715 12 : s4 = identity_perm(9);
1716 12 : sp = identity_perm(9);
1717 12 : pj = zero_zv(4);
1718 12 : pk = zero_zv(2);
1719 12 : sigma = cgetg(n+1, t_VECSMALL);
1720 12 : tau = r3;
1721 12 : rho = cgetg(n+1, t_VECSMALL);
1722 12 : Tp = FpX_red(gl->T,p);
1723 12 : Tmod = gel(FpX_factor(Tp,p), 1);
1724 12 : misom = FpXV_ffisom(Tmod, p);
1725 12 : aut = galoisdolift(gl);
1726 12 : inittestlift(aut, Tmod, gl, >);
1727 12 : Bcoeff = gt.bezoutcoeff;
1728 12 : B = FqC_FqV_mul(gt.pauto, Bcoeff, gl->TQ, gl->Q);
1729 12 : liftp = mkliftpow(shallowtrans(misom), Tmod, p, gl);
1730 12 : Bl = FqC_FqV_mul(liftp,Bcoeff, gl->TQ, gl->Q);
1731 12 : av = avma;
1732 780 : for (i = 0; i < 105; i++, set_avma(av))
1733 : {
1734 : pari_sp av0, av1, av2, av3;
1735 : GEN u0, u1, u2, u3;
1736 : long j0, j1, j2, j3, s;
1737 780 : if (i)
1738 : {
1739 768 : rot3(sg[7],sg[8],sg[9])
1740 768 : if (i%3==0)
1741 : {
1742 252 : s=sg[5]; sg[5]=sg[6]; sg[6]=sg[7]; sg[7]=sg[8]; sg[8]=sg[9]; sg[9]=s;
1743 252 : if (i%15==0)
1744 : {
1745 42 : s=sg[3]; sg[3]=sg[4]; sg[4]=sg[5];
1746 42 : sg[5]=sg[6]; sg[6]=sg[7]; sg[7]=sg[8]; sg[8]=sg[9]; sg[9]=s;
1747 : }
1748 : }
1749 : }
1750 780 : liftpow = s4makelift(f36releveauto2(Bl, TQ, Q, sg), gl);
1751 780 : av0 = avma;
1752 3864 : for (j0 = 0; j0 < 4; j0++, set_avma(av0))
1753 : {
1754 3096 : u0 = lincomb(B,sg[8],sg[9],j0);
1755 3096 : u0 = FpX_add(u0, gmael(B,sg[1],3), Q); av1 = avma;
1756 15450 : for (j1 = 0; j1 < 4; j1++, set_avma(av1))
1757 : {
1758 12366 : u1 = lincomb(B,sg[6],sg[7],j1);
1759 12366 : u1 = FpX_add(u0, u1, Q); av2 = avma;
1760 61806 : for (j2 = 0; j2 < 4; j2++, set_avma(av2))
1761 : {
1762 49452 : u2 = lincomb(B,sg[4],sg[5],j2);
1763 49452 : u2 = FpX_add(u1, u2, Q); av3 = avma;
1764 247242 : for (j3 = 0; j3 < 4; j3++, set_avma(av3))
1765 : {
1766 197802 : u3 = lincomb(B,sg[2],sg[3],j3);
1767 197802 : u3 = FpX_add(u2, u3, Q);
1768 197802 : if (s4test(u3, liftpow, gl, sigma))
1769 : {
1770 12 : pj[1] = j3;
1771 12 : pj[2] = j2;
1772 12 : pj[3] = j1;
1773 12 : pj[4] = j0; goto suitef36;
1774 : }
1775 : }
1776 : }
1777 : }
1778 : }
1779 : }
1780 0 : return gc_NULL(ltop);
1781 12 : suitef36:
1782 12 : s4[1]=sg[1]; s4[2]=sg[2]; s4[4]=sg[3];
1783 12 : s4[3]=sg[4]; s4[5]=sg[5]; s4[6]=sg[6];
1784 12 : s4[8]=sg[7]; s4[7]=sg[8]; s4[9]=sg[9];
1785 12 : for (i = 0; i < 12; i++, set_avma(av))
1786 : {
1787 : pari_sp av0, av1;
1788 : GEN u0, u1;
1789 : long j0, j1;
1790 12 : if (i)
1791 : {
1792 0 : lswap(s4[3],s4[5]); pj[2] = (-pj[2])&3;
1793 0 : if (odd(i)) { lswap(s4[7],s4[9]); pj[4]=(-pj[4])&3; }
1794 0 : if (i%4==0)
1795 : {
1796 0 : rot3(s4[3],s4[6],s4[7]);
1797 0 : rot3(s4[5],s4[8],s4[9]);
1798 0 : rot3(pj[2],pj[3],pj[4]);
1799 : }
1800 : }
1801 12 : liftpow = s4makelift(f36releveauto4(Bl, TQ, Q, s4), gl);
1802 12 : av0 = avma;
1803 42 : for (j0 = 0; j0 < 4; j0++, set_avma(av0))
1804 : {
1805 42 : u0 = FpX_add(gmael(B,s4[1],2), gmael(B,s4[2],1+j0),Q);
1806 42 : u0 = FpX_add(u0, gmael(B,s4[3],1+smodss(pj[2]-j0,4)),Q);
1807 42 : u0 = FpX_add(u0, gmael(B,s4[4],1+smodss(j0-pj[1]-pj[2],4)),Q);
1808 42 : u0 = FpX_add(u0, gmael(B,s4[5],1+smodss(pj[1]-j0,4)),Q);
1809 42 : av1 = avma;
1810 174 : for (j1 = 0; j1 < 4; j1++, set_avma(av1))
1811 : {
1812 144 : u1 = FpX_add(u0, gmael(B,s4[6],1+j1),Q);
1813 144 : u1 = FpX_add(u1, gmael(B,s4[7],1+smodss(pj[4]-j1,4)),Q);
1814 144 : u1 = FpX_add(u1, gmael(B,s4[8],1+smodss(j1-pj[3]-pj[4],4)),Q);
1815 144 : u1 = FpX_add(u1, gmael(B,s4[9],1+smodss(pj[3]-j1,4)),Q);
1816 144 : if (s4test(u1, liftpow, gl, tau))
1817 : {
1818 12 : pk[1] = j0;
1819 12 : pk[2] = j1; goto suitef36_2;
1820 : }
1821 : }
1822 : }
1823 : }
1824 0 : return gc_NULL(ltop);
1825 12 : suitef36_2:
1826 12 : sp[1]=s4[9]; sp[2]=s4[1]; sp[3]=s4[2];
1827 12 : sp[4]=s4[7]; sp[5]=s4[3]; sp[6]=s4[8];
1828 12 : sp[8]=s4[4]; sp[7]=s4[5]; sp[9]=s4[6];
1829 18 : for (i = 0; i < 4; i++, set_avma(av))
1830 : {
1831 18 : const int w[4][6]={{0,0,1,3,0,2},{1,0,2,1,1,2},{3,3,2,0,3,1},{0,1,3,0,0,3}};
1832 : pari_sp av0, av1, av2;
1833 : GEN u0, u1, u2;
1834 : long j0, j1,j2,j3,j4,j5;
1835 18 : if (i)
1836 : {
1837 6 : rot4(sp[3],sp[5],sp[8],sp[7])
1838 6 : pk[1]=(-pk[1])&3;
1839 : }
1840 18 : liftpow = s4makelift(f36releveauto4(Bl,TQ,Q,sp), gl);
1841 18 : av0 = avma;
1842 48 : for (j0 = 0; j0 < 4; j0++, set_avma(av0))
1843 : {
1844 42 : u0 = FpX_add(gmael(B,sp[1],2), gmael(B,sp[2],1+j0),Q);
1845 42 : av1 = avma;
1846 174 : for (j1 = 0; j1 < 4; j1++, set_avma(av1))
1847 : {
1848 144 : u1 = FpX_add(u0, gmael(B,sp[3],1+j1),Q);
1849 144 : j3 = (-pk[1]-pj[3]+j0+j1-w[i][0]*pj[1]-w[i][3]*pj[2])&3;
1850 144 : u1 = FpX_add(u1, gmael(B,sp[6],1+j3),Q);
1851 144 : j5 = (-pk[1]+2*j0+2*j1-w[i][2]*pj[1]-w[i][5]*pj[2])&3;
1852 144 : u1 = FpX_add(u1, gmael(B,sp[8],1+j5),Q);
1853 144 : av2 = avma;
1854 708 : for (j2 = 0; j2 < 4; j2++, set_avma(av2))
1855 : {
1856 576 : u2 = FpX_add(u1, gmael(B,sp[4],1+j2),Q);
1857 576 : u2 = FpX_add(u2, gmael(B,sp[5],1+smodss(-j0-j1-j2,4)),Q);
1858 576 : j4 = (-pk[1]-pk[2]+pj[3]+pj[4]-j2-w[i][1]*pj[1]-w[i][4]*pj[2])&3;
1859 576 : u2 = FpX_add(u2, gmael(B,sp[7],1+j4),Q);
1860 576 : u2 = FpX_add(u2, gmael(B,sp[9],1+smodss(-j3-j4-j5,4)),Q);
1861 576 : if (s4test(u2, liftpow, gl, rho))
1862 12 : goto suitef36_3;
1863 : }
1864 : }
1865 : }
1866 : }
1867 0 : return gc_NULL(ltop);
1868 12 : suitef36_3:
1869 12 : tam = perm_inv(tau);
1870 444 : for (i = 1; i <= n; i++)
1871 : {
1872 432 : r1[tau[i]] = rho[i];
1873 432 : r2[i] = tam[rho[i]];
1874 : }
1875 12 : set_avma(ltop2); return res;
1876 : }
1877 :
1878 : /* return a vecvecsmall */
1879 : static GEN
1880 80 : galoisfindgroups(GEN lo, GEN sg, long f)
1881 : {
1882 80 : pari_sp ltop = avma;
1883 : long i, j, k;
1884 80 : GEN V = cgetg(lg(lo), t_VEC);
1885 234 : for(j=1,i=1; i<lg(lo); i++)
1886 : {
1887 154 : pari_sp av = avma;
1888 154 : GEN loi = gel(lo,i), W = cgetg(lg(loi),t_VECSMALL);
1889 388 : for (k=1; k<lg(loi); k++) W[k] = loi[k] % f;
1890 154 : W = vecsmall_uniq(W);
1891 154 : if (zv_equal(W, sg)) gel(V,j++) = loi;
1892 154 : set_avma(av);
1893 : }
1894 80 : setlg(V,j); return gc_GEN(ltop,V);
1895 : }
1896 :
1897 : static GEN
1898 1425 : galoismakepsi(long g, GEN sg, GEN pf)
1899 : {
1900 1425 : GEN psi=cgetg(g+1,t_VECSMALL);
1901 : long i;
1902 3482 : for (i = 1; i < g; i++) psi[i] = sg[pf[i]];
1903 1425 : psi[g] = sg[1]; return psi;
1904 : }
1905 :
1906 : static GEN
1907 23694 : galoisfrobeniuslift_nilp(GEN T, GEN den, GEN L, GEN Lden,
1908 : struct galois_frobenius *gf, struct galois_borne *gb)
1909 : {
1910 23694 : pari_sp ltop=avma, av2;
1911 : struct galois_lift gl;
1912 23694 : long i, k, deg = 1, g = lg(gf->Tmod)-1;
1913 23694 : GEN F,Fp,Fe, aut, frob, res = cgetg(lg(L), t_VECSMALL);
1914 23694 : gf->psi = const_vecsmall(g,1);
1915 23694 : av2 = avma;
1916 23694 : initlift(T, den, gf->p, L, Lden, gb, &gl);
1917 23694 : if (DEBUGLEVEL >= 4)
1918 0 : err_printf("GaloisConj: p=%ld e=%ld deg=%ld fp=%ld\n",
1919 : gf->p, gl.e, deg, gf->fp);
1920 23694 : aut = galoisdolift(&gl);
1921 23694 : if (galoisfrobeniustest(aut,&gl,res))
1922 : {
1923 22633 : set_avma(av2); gf->deg = gf->fp; return res;
1924 : }
1925 :
1926 1061 : F =factoru(gf->fp);
1927 1061 : Fp = gel(F,1);
1928 1061 : Fe = gel(F,2);
1929 1061 : frob = cgetg(lg(L), t_VECSMALL);
1930 2122 : for(k = lg(Fp)-1; k>=1; k--)
1931 : {
1932 1061 : pari_sp btop=avma;
1933 1061 : GEN fres=NULL;
1934 1061 : long el = gf->fp, dg = 1, dgf = 1, e, pr;
1935 2074 : for(e=1; e<=Fe[k]; e++)
1936 : {
1937 2074 : dg *= Fp[k]; el /= Fp[k];
1938 2074 : if (DEBUGLEVEL>=4) err_printf("Trying degre %d.\n",dg);
1939 2074 : if (el==1) break;
1940 1113 : aut = galoisdoliftn(&gl, el);
1941 1113 : if (!galoisfrobeniustest(aut,&gl,frob))
1942 100 : break;
1943 1013 : dgf = dg; fres = gcopy(frob);
1944 : }
1945 1061 : if (dgf == 1) { set_avma(btop); continue; }
1946 959 : pr = deg*dgf;
1947 959 : if (deg == 1)
1948 : {
1949 12987 : for(i=1;i<lg(res);i++) res[i]=fres[i];
1950 : }
1951 : else
1952 : {
1953 0 : GEN cp = perm_mul(res,fres);
1954 0 : for(i=1;i<lg(res);i++) res[i] = cp[i];
1955 : }
1956 959 : deg = pr; set_avma(btop);
1957 : }
1958 1061 : if (DEBUGLEVEL>=4 && res) err_printf("Best lift: %d\n",deg);
1959 1061 : if (deg==1) return gc_NULL(ltop);
1960 : else
1961 : {
1962 959 : set_avma(av2); gf->deg = deg; return res;
1963 : }
1964 : }
1965 :
1966 : static GEN
1967 1857 : galoisfrobeniuslift(GEN T, GEN den, GEN L, GEN Lden,
1968 : struct galois_frobenius *gf, struct galois_borne *gb)
1969 : {
1970 1857 : pari_sp ltop=avma, av2;
1971 : struct galois_testlift gt;
1972 : struct galois_lift gl;
1973 1857 : long i, j, k, n = lg(L)-1, deg = 1, g = lg(gf->Tmod)-1;
1974 1857 : GEN F,Fp,Fe, aut, frob, res = cgetg(lg(L), t_VECSMALL);
1975 1857 : gf->psi = const_vecsmall(g,1);
1976 1857 : av2 = avma;
1977 1857 : initlift(T, den, gf->p, L, Lden, gb, &gl);
1978 1857 : if (DEBUGLEVEL >= 4)
1979 0 : err_printf("GaloisConj: p=%ld e=%ld deg=%ld fp=%ld\n",
1980 : gf->p, gl.e, deg, gf->fp);
1981 1857 : aut = galoisdolift(&gl);
1982 1857 : if (galoisfrobeniustest(aut,&gl,res))
1983 : {
1984 472 : set_avma(av2); gf->deg = gf->fp; return res;
1985 : }
1986 1385 : inittestlift(aut,gf->Tmod, &gl, >);
1987 1385 : gt.C = cgetg(gf->fp+1,t_VEC);
1988 1385 : gt.Cd= cgetg(gf->fp+1,t_VEC);
1989 7747 : for (i = 1; i <= gf->fp; i++) {
1990 6362 : gel(gt.C,i) = zero_zv(gt.g);
1991 6362 : gel(gt.Cd,i) = zero_zv(gt.g);
1992 : }
1993 :
1994 1385 : F =factoru(gf->fp);
1995 1385 : Fp = gel(F,1);
1996 1385 : Fe = gel(F,2);
1997 1385 : frob = cgetg(lg(L), t_VECSMALL);
1998 2933 : for(k=lg(Fp)-1;k>=1;k--)
1999 : {
2000 1548 : pari_sp btop=avma;
2001 1548 : GEN psi=NULL, fres=NULL, sg = identity_perm(1);
2002 1548 : long el=gf->fp, dg=1, dgf=1, e, pr;
2003 3136 : for(e=1; e<=Fe[k]; e++)
2004 : {
2005 : GEN lo, pf;
2006 : long l;
2007 1628 : dg *= Fp[k]; el /= Fp[k];
2008 1628 : if (DEBUGLEVEL>=4) err_printf("Trying degre %d.\n",dg);
2009 1628 : if (galoisfrobeniustest(gel(gt.pauto,el+1),&gl,frob))
2010 : {
2011 163 : psi = const_vecsmall(g,1);
2012 163 : dgf = dg; fres = leafcopy(frob); continue;
2013 : }
2014 1465 : lo = listznstarelts(dg, n / gf->fp);
2015 1465 : if (e!=1) lo = galoisfindgroups(lo, sg, dgf);
2016 1465 : if (DEBUGLEVEL>=4) err_printf("Galoisconj:Subgroups list:%Ps\n", lo);
2017 3117 : for (l = 1; l < lg(lo); l++)
2018 3077 : if (lg(gel(lo,l))>2 && frobeniusliftall(gel(lo,l), el, &pf,&gl,>, frob))
2019 : {
2020 1425 : sg = leafcopy(gel(lo,l));
2021 1425 : psi = galoismakepsi(g,sg,pf);
2022 1425 : dgf = dg; fres = leafcopy(frob); break;
2023 : }
2024 1465 : if (l == lg(lo)) break;
2025 : }
2026 1548 : if (dgf == 1) { set_avma(btop); continue; }
2027 1520 : pr = deg*dgf;
2028 1520 : if (deg == 1)
2029 : {
2030 17085 : for(i=1;i<lg(res);i++) res[i]=fres[i];
2031 4769 : for(i=1;i<lg(psi);i++) gf->psi[i]=psi[i];
2032 : }
2033 : else
2034 : {
2035 135 : GEN cp = perm_mul(res,fres);
2036 2571 : for(i=1;i<lg(res);i++) res[i] = cp[i];
2037 439 : for(i=1;i<lg(psi);i++) gf->psi[i] = (dgf*gf->psi[i] + deg*psi[i]) % pr;
2038 : }
2039 1520 : deg = pr; set_avma(btop);
2040 : }
2041 7747 : for (i = 1; i <= gf->fp; i++)
2042 22062 : for (j = 1; j <= gt.g; j++) guncloneNULL(gmael(gt.C,i,j));
2043 1385 : if (DEBUGLEVEL>=4 && res) err_printf("Best lift: %d\n",deg);
2044 1385 : if (deg==1) return gc_NULL(ltop);
2045 : else
2046 : {
2047 : /* Normalize result so that psi[g]=1 */
2048 1385 : ulong im = Fl_inv(gf->psi[g], deg);
2049 1385 : GEN cp = perm_powu(res, im);
2050 17085 : for(i=1;i<lg(res);i++) res[i] = cp[i];
2051 4769 : for(i=1;i<lg(gf->psi);i++) gf->psi[i] = Fl_mul(im, gf->psi[i], deg);
2052 1385 : set_avma(av2); gf->deg = deg; return res;
2053 : }
2054 : }
2055 :
2056 : /* return NULL if not Galois */
2057 : static GEN
2058 25461 : galoisfindfrobenius(GEN T, GEN L, GEN den, GEN bad, struct galois_frobenius *gf,
2059 : struct galois_borne *gb, const struct galois_analysis *ga)
2060 : {
2061 25461 : pari_sp ltop = avma, av;
2062 25461 : long Try = 0, n = degpol(T), deg, gmask = (ga->group&ga_ext_2)? 3: 1;
2063 25461 : GEN frob, Lden = makeLden(L,den,gb);
2064 25461 : long is_nilpotent = ga->group&ga_all_nilpotent;
2065 : forprime_t S;
2066 :
2067 25461 : u_forprime_init(&S, ga->p, ULONG_MAX);
2068 25461 : av = avma;
2069 25461 : deg = gf->deg = ga->deg;
2070 25575 : while ((gf->p = u_forprime_next(&S)))
2071 : {
2072 : pari_sp lbot;
2073 : GEN Ti, Tp;
2074 : long nb, d;
2075 25575 : set_avma(av);
2076 25575 : Tp = ZX_to_Flx(T, gf->p);
2077 25575 : if (!Flx_is_squarefree(Tp, gf->p)) continue;
2078 25575 : if (bad && dvdiu(bad, gf->p)) continue;
2079 25575 : Ti = gel(Flx_factor(Tp, gf->p), 1);
2080 25575 : nb = lg(Ti)-1; d = degpol(gel(Ti,1));
2081 25575 : if (nb > 1 && degpol(gel(Ti,nb)) != d) return gc_NULL(ltop);
2082 25563 : if (((gmask&1)==0 || d % deg) && ((gmask&2)==0 || odd(d))) continue;
2083 25551 : if (DEBUGLEVEL >= 1) err_printf("GaloisConj: Trying p=%ld\n", gf->p);
2084 25551 : FlxV_to_ZXV_inplace(Ti);
2085 25551 : gf->fp = d;
2086 25551 : gf->Tmod = Ti; lbot = avma;
2087 25551 : if (is_nilpotent)
2088 23694 : frob = galoisfrobeniuslift_nilp(T, den, L, Lden, gf, gb);
2089 : else
2090 1857 : frob = galoisfrobeniuslift(T, den, L, Lden, gf, gb);
2091 25551 : if (frob)
2092 : {
2093 25449 : gf->Tmod = gcopy(Ti);
2094 25449 : return gc_all_unsafe(ltop, lbot, 3, &frob, &gf->Tmod, &gf->psi);
2095 : }
2096 102 : if (is_nilpotent) continue;
2097 0 : if ((ga->group&ga_all_normal) && d % deg == 0) gmask &= ~1;
2098 : /* The first prime degree is always divisible by deg, so we don't
2099 : * have to worry about ext_2 being used before regular supersolvable*/
2100 0 : if (!gmask) return gc_NULL(ltop);
2101 0 : if ((ga->group&ga_non_wss) && ++Try > ((3*n)>>1))
2102 : {
2103 0 : if (DEBUGLEVEL)
2104 0 : pari_warn(warner,"Galois group probably not weakly super solvable");
2105 0 : return NULL;
2106 : }
2107 : }
2108 0 : if (!gf->p) pari_err_OVERFLOW("galoisfindfrobenius [ran out of primes]");
2109 0 : return NULL;
2110 : }
2111 :
2112 : /* compute g such that tau(Pmod[#])= tau(Pmod[g]) */
2113 :
2114 : static long
2115 5661 : get_image(GEN tau, GEN P, GEN Pmod, GEN p)
2116 : {
2117 5661 : pari_sp av = avma;
2118 5661 : long g, gp = lg(Pmod)-1;
2119 5661 : tau = RgX_to_FpX(tau, p);
2120 5661 : tau = FpX_FpXQ_eval(gel(Pmod, gp), tau, P, p);
2121 5661 : tau = FpX_normalize(FpX_gcd(P, tau, p), p);
2122 9152 : for (g = 1; g <= gp; g++)
2123 9152 : if (ZX_equal(tau, gel(Pmod,g))) return gc_long(av,g);
2124 0 : return gc_long(av,0);
2125 : }
2126 :
2127 : static GEN
2128 29145 : gg_get_std(GEN G)
2129 : {
2130 29145 : return !G ? NULL: lg(G)==3 ? G: mkvec2(gel(G,1),gmael(G,5,1));
2131 : }
2132 :
2133 : static GEN galoisgen(GEN T, GEN L, GEN M, GEN den, GEN bad, struct galois_borne *gb,
2134 : const struct galois_analysis *ga);
2135 :
2136 : static GEN
2137 4660 : galoisgenfixedfield(GEN Tp, GEN Pmod, GEN PL, GEN P, GEN ip, GEN bad, struct galois_borne *gb)
2138 : {
2139 : GEN Pden, PM;
2140 : GEN tau, PG, Pg;
2141 : long g, lP;
2142 4660 : long x = varn(Tp);
2143 4660 : GEN Pp = FpX_red(P, ip);
2144 4660 : if (DEBUGLEVEL>=6)
2145 0 : err_printf("GaloisConj: Fixed field %Ps\n",P);
2146 4660 : if (degpol(P)==2 && !bad)
2147 : {
2148 3533 : PG=cgetg(3,t_VEC);
2149 3533 : gel(PG,1) = mkvec( mkvecsmall2(2,1) );
2150 3533 : gel(PG,2) = mkvecsmall(2);
2151 3533 : tau = deg1pol_shallow(gen_m1, negi(gel(P,3)), x);
2152 3533 : g = get_image(tau, Pp, Pmod, ip);
2153 3533 : if (!g) return NULL;
2154 3533 : Pg = mkvecsmall(g);
2155 : }
2156 : else
2157 : {
2158 : struct galois_analysis Pga;
2159 : struct galois_borne Pgb;
2160 : GEN mod, mod2;
2161 : long j;
2162 1132 : if (!galoisanalysis(P, &Pga, 0, NULL)) return NULL;
2163 1116 : if (bad) Pga.group &= ~ga_easy;
2164 1116 : Pgb.l = gb->l;
2165 1116 : Pden = galoisborne(P, NULL, &Pgb, degpol(P));
2166 :
2167 1116 : if (Pgb.valabs > gb->valabs)
2168 : {
2169 105 : if (DEBUGLEVEL>=4)
2170 0 : err_printf("GaloisConj: increase prec of p-adic roots of %ld.\n"
2171 0 : ,Pgb.valabs-gb->valabs);
2172 105 : PL = ZpX_liftroots(P,PL,gb->l,Pgb.valabs);
2173 : }
2174 1011 : else if (Pgb.valabs < gb->valabs)
2175 956 : PL = FpC_red(PL, Pgb.ladicabs);
2176 1116 : PM = FpV_invVandermonde(PL, Pden, Pgb.ladicabs);
2177 1116 : PG = galoisgen(P, PL, PM, Pden, bad ? lcmii(Pgb.dis, bad): NULL, &Pgb, &Pga);
2178 1116 : if (!PG) return NULL;
2179 1111 : lP = lg(gel(PG,1));
2180 1111 : mod = Pgb.ladicabs; mod2 = shifti(mod, -1);
2181 1111 : Pg = cgetg(lP, t_VECSMALL);
2182 3239 : for (j = 1; j < lP; j++)
2183 : {
2184 2128 : pari_sp btop=avma;
2185 2128 : tau = permtopol(gmael(PG,1,j), PL, PM, Pden, mod, mod2, x);
2186 2128 : g = get_image(tau, Pp, Pmod, ip);
2187 2128 : if (!g) return NULL;
2188 2128 : Pg[j] = g;
2189 2128 : set_avma(btop);
2190 : }
2191 : }
2192 4644 : return mkvec2(PG,Pg);
2193 : }
2194 :
2195 : static GEN
2196 4660 : galoisgenfixedfield0(GEN O, GEN L, GEN sigma, GEN T, GEN bad, GEN *pt_V,
2197 : struct galois_frobenius *gf, struct galois_borne *gb)
2198 : {
2199 4660 : pari_sp btop = avma;
2200 4660 : long vT = varn(T);
2201 4660 : GEN mod = gb->ladicabs, mod2 = shifti(gb->ladicabs,-1);
2202 : GEN OL, sym, P, PL, p, Tp, Sp, Pmod, PG;
2203 4660 : OL = fixedfieldorbits(O,L);
2204 4660 : sym = fixedfieldsympol(OL, itou(gb->l));
2205 4660 : PL = sympol_eval(sym, OL, mod);
2206 4660 : P = FpX_center_i(FpV_roots_to_pol(PL, mod, vT), mod, mod2);
2207 4660 : if (!FpX_is_squarefree(P,utoipos(gf->p)))
2208 : {
2209 59 : GEN badp = lcmii(bad? bad: gb->dis, ZX_disc(P));
2210 59 : gf->p = findpsi(badp, gf->p, T, sigma, gf->deg, &gf->Tmod, &gf->psi);
2211 : }
2212 4660 : p = utoipos(gf->p);
2213 4660 : Tp = FpX_red(T,p);
2214 4660 : Sp = sympol_aut_evalmod(sym, gf->deg, sigma, Tp, p);
2215 4660 : Pmod = fixedfieldfactmod(Sp, p, gf->Tmod);
2216 4660 : PG = galoisgenfixedfield(Tp, Pmod, PL, P, p, bad, gb);
2217 4660 : if (PG == NULL) return NULL;
2218 4644 : if (DEBUGLEVEL >= 4)
2219 0 : err_printf("GaloisConj: Back to Earth:%Ps\n", gg_get_std(gel(PG,1)));
2220 4644 : if (pt_V) *pt_V = mkvec3(sym, PL, P);
2221 4644 : return gc_all(btop, pt_V ? 4: 3, &PG, &gf->Tmod, &gf->psi, pt_V);
2222 : }
2223 :
2224 : /* Let sigma^m=1, tau*sigma*tau^-1=sigma^s. Return n = sum_{0<=k<e,0} s^k mod m
2225 : * so that (sigma*tau)^e = sigma^n*tau^e. N.B. n*(1-s) = 1-s^e mod m,
2226 : * unfortunately (1-s) may not invertible mod m */
2227 : static long
2228 12576 : stpow(long s, long e, long m)
2229 : {
2230 12576 : long i, n = 1;
2231 19657 : for (i = 1; i < e; i++) n = (1 + n * s) % m;
2232 12576 : return n;
2233 : }
2234 :
2235 : static GEN
2236 5661 : wpow(long s, long m, long e, long n)
2237 : {
2238 5661 : GEN w = cgetg(n+1,t_VECSMALL);
2239 5661 : long si = s;
2240 : long i;
2241 5661 : w[1] = 1;
2242 6288 : for(i=2; i<=n; i++) w[i] = w[i-1]*e;
2243 11949 : for(i=n; i>=1; i--)
2244 : {
2245 6288 : si = Fl_powu(si,e,m);
2246 6288 : w[i] = Fl_mul(s-1, stpow(si, w[i], m), m);
2247 : }
2248 5661 : return w;
2249 : }
2250 :
2251 : static GEN
2252 5661 : galoisgenliftauto(GEN O, GEN gj, long s, long n, struct galois_test *td)
2253 : {
2254 5661 : pari_sp av = avma;
2255 : long sr, k;
2256 5661 : long deg = lg(gel(O,1))-1;
2257 5661 : GEN X = cgetg(lg(O), t_VECSMALL);
2258 5661 : GEN oX = cgetg(lg(O), t_VECSMALL);
2259 5661 : GEN B = perm_cycles(gj);
2260 5661 : long oj = lg(gel(B,1)) - 1;
2261 5661 : GEN F = factoru(oj);
2262 5661 : GEN Fp = gel(F,1);
2263 5661 : GEN Fe = gel(F,2);
2264 5661 : GEN pf = identity_perm(n);
2265 5661 : if (DEBUGLEVEL >= 6)
2266 0 : err_printf("GaloisConj: %Ps of relative order %d\n", gj, oj);
2267 10773 : for (k=lg(Fp)-1; k>=1; k--)
2268 : {
2269 5661 : long f, dg = 1, el = oj, osel = 1, a = 0;
2270 5661 : long p = Fp[k], e = Fe[k], op = oj / upowuu(p,e);
2271 : long i;
2272 5661 : GEN pf1 = NULL, w, wg, Be = cgetg(e+1,t_VEC);
2273 5661 : gel(Be,e) = cyc_pow(B, op);
2274 6288 : for(i=e-1; i>=1; i--) gel(Be,i) = cyc_pow(gel(Be,i+1), p);
2275 5661 : w = wpow(Fl_powu(s,op,deg),deg,p,e);
2276 5661 : wg = cgetg(e+2,t_VECSMALL);
2277 5661 : wg[e+1] = deg;
2278 11949 : for (i=e; i>=1; i--) wg[i] = ugcd(wg[i+1], w[i]);
2279 31109 : for (i=1; i<lg(O); i++) oX[i] = 0;
2280 11400 : for (f=1; f<=e; f++)
2281 : {
2282 : long sel, t;
2283 6288 : GEN Bel = gel(Be,f);
2284 6288 : dg *= p; el /= p;
2285 6288 : sel = Fl_powu(s,el,deg);
2286 6288 : if (DEBUGLEVEL >= 6) err_printf("GaloisConj: B=%Ps\n", Bel);
2287 6288 : sr = ugcd(stpow(sel,p,deg),deg);
2288 6288 : if (DEBUGLEVEL >= 6)
2289 0 : err_printf("GaloisConj: exp %d: s=%ld [%ld] a=%ld w=%ld wg=%ld sr=%ld\n",
2290 0 : dg, sel, deg, a, w[f], wg[f+1], sr);
2291 8130 : for (t = 0; t < sr; t++)
2292 7581 : if ((a+t*w[f])%wg[f+1]==0)
2293 : {
2294 : long i, j, k, st;
2295 49351 : for (i = 1; i < lg(X); i++) X[i] = 0;
2296 25937 : for (i = 0; i < lg(X)-1; i+=dg)
2297 39226 : for (j = 1, k = p, st = t; k <= dg; j++, k += p)
2298 : {
2299 20810 : X[k+i] = (oX[j+i] + st)%deg;
2300 20810 : st = (t + st*osel)%deg;
2301 : }
2302 7521 : pf1 = testpermutation(O, Bel, X, sel, p, sr, td);
2303 7521 : if (pf1) break;
2304 : }
2305 6288 : if (!pf1) return NULL;
2306 36657 : for (i=1; i<lg(O); i++) oX[i] = X[i];
2307 5739 : osel = sel; a = (a+t*w[f])%deg;
2308 : }
2309 5112 : pf = perm_mul(pf, perm_powu(pf1, el));
2310 : }
2311 5112 : return gc_leaf(av, pf);
2312 : }
2313 :
2314 : static GEN
2315 0 : FlxV_Flx_gcd(GEN x, GEN T, ulong p)
2316 0 : { pari_APPLY_same(Flx_normalize(Flx_gcd(gel(x,i),T,p),p)) }
2317 :
2318 : static GEN
2319 0 : Flx_FlxV_minpolymod(GEN y, GEN x, ulong p)
2320 0 : { pari_APPLY_same(Flxq_minpoly(Flx_rem(y, gel(x,i), p), gel(x,i), p)) }
2321 :
2322 : static GEN
2323 0 : FlxV_minpolymod(GEN x, GEN y, ulong p)
2324 0 : { pari_APPLY_same(Flx_FlxV_minpolymod(gel(x,i), y, p)) }
2325 :
2326 : static GEN
2327 0 : factperm(GEN x)
2328 : {
2329 0 : pari_APPLY_same(gen_indexsort(gel(x,i), (void*)cmp_Flx, cmp_nodata))
2330 : }
2331 :
2332 : /* compute (prod p_i^e_i)(1) */
2333 :
2334 : static long
2335 0 : permprodeval(GEN p, GEN e, long s)
2336 : {
2337 0 : long i, j, l = lg(p);
2338 0 : for (i=l-1; i>=1; i--)
2339 : {
2340 0 : GEN pi = gel(p,i);
2341 0 : long ei = uel(e,i);
2342 0 : for(j = 1; j <= ei; j++)
2343 0 : s = uel(pi, s);
2344 : }
2345 0 : return s;
2346 : }
2347 :
2348 : static GEN
2349 0 : pc_to_perm(GEN pc, GEN gen, long n)
2350 : {
2351 0 : long i, l = lg(pc);
2352 0 : GEN s = identity_perm(n);
2353 0 : for (i=1; i<l; i++)
2354 0 : s = perm_mul(gel(gen,pc[i]),s);
2355 0 : return s;
2356 : }
2357 :
2358 : static GEN
2359 0 : genorbit(GEN ordH, GEN permfact_Hp, long fr, long n, long k, long j)
2360 : {
2361 0 : pari_sp av = avma;
2362 0 : long l = lg(gel(permfact_Hp,1))-1, no = 1, b, i;
2363 0 : GEN W = zero_zv(l);
2364 0 : GEN orb = cgetg(l+1, t_VECSMALL);
2365 0 : GEN gen = cgetg(l+1, t_VEC);
2366 0 : GEN E = cgetg(k+1, t_VECSMALL);
2367 0 : for(b = 0; b < n; b++)
2368 : {
2369 0 : long bb = b, s;
2370 0 : for(i = 1; i <= k; i++)
2371 : {
2372 0 : uel(E,i) = bb % uel(ordH,i);
2373 0 : bb /= uel(ordH,i);
2374 : }
2375 0 : if (E[j]) continue;
2376 0 : s = permprodeval(permfact_Hp, E, fr);
2377 0 : if (s>lg(W)-1) pari_err_BUG("W1");
2378 0 : if (W[s]) continue;
2379 0 : W[s] = 1;
2380 0 : if (no > l) pari_err_BUG("genorbit");
2381 0 : uel(orb,no) = s;
2382 0 : gel(gen,no) = zv_copy(E);
2383 0 : no++;
2384 : }
2385 0 : if(no<l) pari_err_BUG("genorbit");
2386 0 : return gc_GEN(av, mkvec2(orb,gen));
2387 : }
2388 :
2389 0 : INLINE GEN br_get(GEN br, long i, long j) { return gmael(br,j,i-j); }
2390 0 : static GEN pcgrp_get_ord(GEN G) { return gel(G,1); }
2391 0 : static GEN pcgrp_get_pow(GEN G) { return gel(G,2); }
2392 0 : static GEN pcgrp_get_br(GEN G) { return gel(G,3); }
2393 :
2394 : static GEN
2395 20789 : cyclic_pc(long n)
2396 : {
2397 20789 : return mkvec3(mkvecsmall(n),mkvec(cgetg(1,t_VECSMALL)), mkvec(cgetg(1,t_VEC)));
2398 : }
2399 :
2400 : static GEN
2401 0 : pc_normalize(GEN g, GEN G)
2402 : {
2403 0 : long i, l = lg(g)-1, o = 1;
2404 0 : GEN ord = pcgrp_get_ord(G), pw = pcgrp_get_pow(G), br = pcgrp_get_br(G);
2405 0 : for (i = 1; i < l; i++)
2406 : {
2407 0 : if (g[i] == g[i+1])
2408 : {
2409 0 : if (++o == ord[g[i]])
2410 : {
2411 0 : GEN v = vecsmall_concat(vecslice(g,1,i-o+1),gel(pw,g[i]));
2412 0 : GEN w = vecsmall_concat(v,vecslice(g,i+2,l));
2413 0 : return pc_normalize(w, G);
2414 : }
2415 : }
2416 0 : else if (g[i] > g[i+1])
2417 : {
2418 0 : GEN v = vecsmall_concat(vecslice(g,1,i-1), br_get(br,g[i],g[i+1]));
2419 0 : GEN w = vecsmall_concat(mkvecsmall2(g[i+1],g[i]),vecslice(g,i+2,l));
2420 0 : v = vecsmall_concat(v, w);
2421 0 : return pc_normalize(v, G);
2422 : }
2423 0 : else o = 1;
2424 : }
2425 0 : return g;
2426 : }
2427 :
2428 : static GEN
2429 0 : pc_inv(GEN g, GEN G)
2430 : {
2431 0 : long i, l = lg(g);
2432 0 : GEN ord = pcgrp_get_ord(G), pw = pcgrp_get_pow(G);
2433 0 : GEN v = cgetg(l, t_VEC);
2434 0 : if (l==1) return v;
2435 0 : for(i = 1; i < l; i++)
2436 : {
2437 0 : ulong gi = uel(g,i);
2438 0 : gel(v,l-i) = vecsmall_concat(pc_inv(gel(pw, gi), G),
2439 0 : const_vecsmall(uel(ord,gi)-1,gi));
2440 : }
2441 0 : return pc_normalize(shallowconcat1(v), G);
2442 : }
2443 :
2444 : static GEN
2445 0 : pc_mul(GEN g, GEN h, GEN G)
2446 : {
2447 0 : return pc_normalize(vecsmall_concat(g,h), G);
2448 : }
2449 :
2450 : static GEN
2451 0 : pc_bracket(GEN g, GEN h, GEN G)
2452 : {
2453 0 : GEN gh = pc_mul(g, h, G);
2454 0 : GEN hg = pc_mul(h, g, G);
2455 0 : long i, l1 = lg(gh), l2 = lg(hg), lm = minss(l1,l2);
2456 0 : for (i = 1; i < lm; i++)
2457 0 : if (gh[l1-i] != hg[l2-i]) break;
2458 0 : return pc_mul(vecsmall_shorten(gh,l1-i), pc_inv(vecsmall_shorten(hg,l2-i), G), G);
2459 : }
2460 :
2461 : static GEN
2462 0 : pc_exp(GEN v)
2463 : {
2464 0 : long i, l = lg(v);
2465 0 : GEN w = cgetg(l, t_VEC);
2466 0 : if (l==1) return w;
2467 0 : for (i = 1; i < l; i++)
2468 0 : gel(w,i) = const_vecsmall(v[i], i+1);
2469 0 : return shallowconcat1(w);
2470 : }
2471 : static GEN
2472 0 : vecsmall_increase(GEN x)
2473 0 : { pari_APPLY_ulong(x[i]+1) }
2474 :
2475 : static GEN
2476 0 : vecvecsmall_increase(GEN x)
2477 0 : { pari_APPLY_same(vecsmall_increase(gel(x,i))) }
2478 :
2479 : static GEN
2480 0 : pcgrp_lift(GEN G, long deg)
2481 : {
2482 0 : GEN ord = pcgrp_get_ord(G), pw = pcgrp_get_pow(G), br = pcgrp_get_br(G);
2483 0 : long i, l = lg(br);
2484 0 : GEN Ord = vecsmall_prepend(ord, deg);
2485 0 : GEN Pw = vec_prepend(vecvecsmall_increase(pw), cgetg(1,t_VECSMALL));
2486 0 : GEN Br = cgetg(l+1, t_VEC);
2487 0 : gel(Br,1) = const_vec(l-1, cgetg(1, t_VECSMALL));
2488 0 : for (i = 1; i < l; i++)
2489 0 : gel(Br,i+1) = vecvecsmall_increase(gel(br, i));
2490 0 : return mkvec3(Ord, Pw, Br);
2491 : }
2492 :
2493 : static GEN
2494 0 : brl_add(GEN x, GEN a)
2495 : {
2496 0 : pari_APPLY_same(vecsmall_concat(const_vecsmall(uel(a,i),1),gel(x,i)))
2497 : }
2498 :
2499 : static void
2500 0 : pcgrp_insert(GEN G, long j, GEN a)
2501 : {
2502 0 : GEN pw = pcgrp_get_pow(G), br = pcgrp_get_br(G);
2503 0 : gel(pw,j) = vecsmall_concat(gel(a,1),gel(pw, j));
2504 0 : gel(br,j) = brl_add(gel(br, j), gel(a,2));
2505 0 : }
2506 :
2507 : static long
2508 0 : getfr(GEN f, GEN h)
2509 : {
2510 0 : long i, l = lg(f);
2511 0 : for (i = 1; i < l; i++)
2512 0 : if (zv_equal(gel(f,i), h)) return i;
2513 0 : pari_err_BUG("galoisinit");
2514 0 : return 0;
2515 : }
2516 :
2517 : static long
2518 0 : get_pow(GEN pf, ulong o, GEN pw, GEN gen)
2519 : {
2520 0 : long i, n = lg(pf)-1;
2521 0 : GEN p1 = perm_powu(pf, o);
2522 0 : GEN p2 = pc_to_perm(pw, gen, n);
2523 0 : for(i = 0; ; i++)
2524 : {
2525 0 : if (zv_equal(p1, p2)) break;
2526 0 : p2 = perm_mul(gel(gen,1), p2);
2527 : }
2528 0 : return i;
2529 : }
2530 :
2531 : struct galois_perm
2532 : {
2533 : GEN L;
2534 : GEN M;
2535 : GEN den;
2536 : GEN mod, mod2;
2537 : long x;
2538 : GEN cache;
2539 : };
2540 :
2541 : static void
2542 0 : galoisperm_init(struct galois_perm *gp, GEN L, GEN M, GEN den, GEN mod, GEN mod2, long x)
2543 : {
2544 0 : gp->L = L;
2545 0 : gp->M = M;
2546 0 : gp->den = den;
2547 0 : gp->mod = mod;
2548 0 : gp->mod2 = mod2;
2549 0 : gp->x = x;
2550 0 : gp->cache = zerovec(lg(L)-1);
2551 0 : }
2552 :
2553 : static void
2554 0 : galoisperm_free(struct galois_perm *gp)
2555 : {
2556 0 : long i, l = lg(gp->cache);
2557 0 : for (i=1; i<l; i++)
2558 0 : if (!isintzero(gel(gp->cache,i)))
2559 0 : gunclone(gel(gp->cache,i));
2560 0 : }
2561 :
2562 : static GEN
2563 0 : permtoaut(GEN p, struct galois_perm *gp)
2564 : {
2565 0 : pari_sp av = avma;
2566 0 : if (isintzero(gel(gp->cache,p[1])))
2567 : {
2568 0 : GEN pol = permtopol(p, gp->L, gp->M, gp->den, gp->mod, gp->mod2, gp->x);
2569 0 : gel(gp->cache,p[1]) = gclone(pol);
2570 : }
2571 0 : set_avma(av);
2572 0 : return gel(gp->cache,p[1]);
2573 : }
2574 :
2575 : static GEN
2576 0 : pc_evalcache(GEN W, GEN u, GEN sp, GEN T, GEN p, struct galois_perm *gp)
2577 : {
2578 : GEN v;
2579 0 : long ns = sp[1];
2580 0 : if (!isintzero(gel(W,ns))) return gel(W,ns);
2581 0 : v = RgX_to_FpX(permtoaut(sp, gp), p);
2582 0 : gel(W,ns) = FpX_FpXQV_eval(v, u, T, p);
2583 0 : return gel(W,ns);
2584 : }
2585 :
2586 : static ulong
2587 0 : findp(GEN D, GEN P, GEN S, long o, GEN *Tmod)
2588 : {
2589 : forprime_t iter;
2590 : ulong p;
2591 0 : long n = degpol(P);
2592 0 : u_forprime_init(&iter, n*maxss(expu(n)-3, 2), ULONG_MAX);
2593 0 : while ((p = u_forprime_next(&iter)))
2594 : {
2595 : GEN F, F1, Sp;
2596 0 : if (smodis(D, p) == 0)
2597 0 : continue;
2598 0 : F = gel(Flx_factor(ZX_to_Flx(P, p), p), 1);
2599 0 : F1 = gel(F,1);
2600 0 : if (degpol(F1) != o)
2601 0 : continue;
2602 0 : Sp = RgX_to_Flx(S, p);
2603 0 : if (gequal(Flx_rem(Sp, F1, p), Flx_Frobenius(F1, p)))
2604 : {
2605 0 : *Tmod = FlxV_to_ZXV(F);
2606 0 : return p;
2607 : }
2608 : }
2609 0 : return 0;
2610 : }
2611 :
2612 : static GEN
2613 0 : nilp_froblift(GEN genG, GEN autH, long j, GEN pcgrp,
2614 : GEN idp, GEN incl, GEN H, struct galois_lift *gl, struct galois_perm *gp)
2615 : {
2616 0 : pari_sp av = avma;
2617 0 : GEN T = gl->T, p = gl->p, pe = gl->Q;
2618 0 : ulong pp = itou(p);
2619 0 : long e = gl->e;
2620 0 : GEN pf = cgetg(lg(gl->L), t_VECSMALL);
2621 0 : GEN Tp = ZX_to_Flx(T, pp);
2622 0 : GEN Hp = ZX_to_Flx(H, pp);
2623 0 : GEN ord = pcgrp_get_ord(pcgrp);
2624 0 : GEN pcp = gel(pcgrp_get_pow(pcgrp),j+1);
2625 0 : long o = uel(ord,1);
2626 0 : GEN ordH = vecslice(ord,2,lg(ord)-1);
2627 0 : long n = zv_prod(ordH), k = lg(ordH)-1, l = k-j, m = upowuu(o, l), v = varn(T);
2628 0 : GEN factTp = gel(Flx_factor(Tp, pp), 1);
2629 0 : long fp = degpol(gel(factTp, 1));
2630 0 : GEN frobp = Flxq_autpow(Flx_Frobenius(Tp, pp), fp-1, Tp, pp);
2631 0 : GEN frob = ZpX_ZpXQ_liftroot(T, Flx_to_ZX(frobp), T, p, e);
2632 0 : if (galoisfrobeniustest(frob, gl, pf))
2633 : {
2634 0 : GEN pfi = perm_inv(pf);
2635 0 : long d = get_pow(pfi, uel(ord,j+1), pcp, genG);
2636 0 : return mkvec3(pfi, mkvec2(const_vecsmall(d,1),zero_zv(l+1)), gel(factTp, 1));
2637 : }
2638 : else
2639 : {
2640 0 : GEN frobG = FpXQ_powers(frob, usqrt(degpol(T)), T, pe);
2641 0 : GEN autHp = RgXV_to_FlxV(autH,pp);
2642 0 : GEN inclp = RgX_to_Flx(incl,pp);
2643 0 : GEN factHp = gel(Flx_factor(Hp, pp),1);
2644 0 : long fr = getfr(factHp, idp);
2645 0 : GEN minHp = FlxV_minpolymod(autHp, factHp, pp);
2646 0 : GEN permfact_Hp = factperm(minHp);
2647 0 : GEN permfact_Gp = FlxV_Flx_gcd(FlxC_Flxq_eval(factHp, inclp, Tp, pp), Tp, pp);
2648 0 : GEN bezout_Gpe = bezout_lift_fact(T, FlxV_to_ZXV(permfact_Gp), p, e);
2649 0 : GEN id = gmael(Flx_factor(gel(permfact_Gp, fr),pp),1,1);
2650 0 : GEN orbgen = genorbit(ordH, permfact_Hp, fr, n, k, j);
2651 0 : GEN orb = gel(orbgen,1), gen = gel(orbgen,2);
2652 0 : long nborb = lg(orb)-1;
2653 0 : GEN A = cgetg(l+1, t_VECSMALL);
2654 0 : GEN W = zerovec(lg(gl->L)-1);
2655 0 : GEN U = zeromatcopy(nborb,degpol(T));
2656 0 : GEN br = pcgrp_get_br(pcgrp), brj = gcopy(gel(br, j+1));
2657 0 : GEN Ui = cgetg(nborb+1, t_VEC);
2658 : long a, b, i;
2659 0 : for(a = 0; a < m; a++)
2660 : {
2661 : pari_timer ti;
2662 : pari_sp av2;
2663 0 : GEN B = pol_0(v);
2664 0 : long aa = a;
2665 0 : if (DEBUGLEVEL>=4) timer_start(&ti);
2666 0 : for(i = 1; i <= l; i++)
2667 : {
2668 0 : uel(A,i) = aa % o;
2669 0 : aa /= o;
2670 : }
2671 0 : gel(br,j+1) = brl_add(brj, A);
2672 0 : for(b = 1; b <= nborb; b++)
2673 : {
2674 0 : GEN br = pc_bracket(pc_exp(gel(gen,b)), mkvecsmall(j+1), pcgrp);
2675 0 : GEN sp = pc_to_perm(br, genG, degpol(T));
2676 0 : long u = sp[1];
2677 0 : long s = permprodeval(permfact_Hp, gel(gen,b), fr);
2678 0 : if (isintzero(gmael(U,u,s)))
2679 : {
2680 0 : GEN Ub = pc_evalcache(W, frobG, sp, T, pe, gp);
2681 0 : gmael(U,u,s) = FpXQ_mul(Ub, gel(bezout_Gpe,s), T, pe);
2682 : }
2683 0 : gel(Ui, b) = gmael(U,u,s);
2684 : }
2685 0 : av2 = avma;
2686 0 : for(b = 1; b <= nborb; b++)
2687 0 : B = FpX_add(B, gel(Ui,b), pe);
2688 0 : if (DEBUGLEVEL >= 4) timer_printf(&ti,"Testing candidate %ld",a);
2689 0 : if (galoisfrobeniustest(B, gl, pf))
2690 : {
2691 0 : GEN pfi = perm_inv(pf);
2692 0 : long d = get_pow(pfi, uel(ord,j+1), pcp, genG);
2693 0 : gel(br,j+1) = brj;
2694 0 : return gc_GEN(av,mkvec3(pfi,mkvec2(const_vecsmall(d,1),A),id));
2695 : }
2696 0 : set_avma(av2);
2697 : }
2698 0 : return gc_NULL(av);
2699 : }
2700 : }
2701 :
2702 : static GEN
2703 0 : galoisgenlift_nilp(GEN PG, GEN O, GEN V, GEN T, GEN frob, GEN sigma,
2704 : struct galois_borne *gb, struct galois_frobenius *gf, struct galois_perm *gp)
2705 : {
2706 0 : long j, n = degpol(T), deg = gf->deg;
2707 0 : ulong p = gf->p;
2708 0 : GEN L = gp->L, M = gp->M, den = gp->den;
2709 0 : GEN S = fixedfieldinclusion(O, gel(V,2));
2710 0 : GEN incl = vectopol(S, M, den, gb->ladicabs, shifti(gb->ladicabs,-1), varn(T));
2711 0 : GEN H = gel(V,3);
2712 0 : GEN PG1 = gmael(PG, 1, 1);
2713 0 : GEN PG2 = gmael(PG, 1, 2);
2714 0 : GEN PG3 = gmael(PG, 1, 3);
2715 0 : GEN PG4 = gmael(PG, 1, 4);
2716 0 : long lP = lg(PG1);
2717 0 : GEN PG5 = pcgrp_lift(gmael(PG, 1, 5), deg);
2718 0 : GEN res = cgetg(6, t_VEC), res1, res2, res3;
2719 0 : gel(res,1) = res1 = cgetg(lP + 1, t_VEC);
2720 0 : gel(res,2) = res2 = cgetg(lP + 1, t_VEC);
2721 0 : gel(res,3) = res3 = cgetg(lP + 1, t_VEC);
2722 0 : gel(res,4) = vecsmall_prepend(PG4, p);
2723 0 : gel(res,5) = PG5;
2724 0 : gel(res1, 1) = frob;
2725 0 : gel(res2, 1) = ZX_to_Flx(gel(gf->Tmod,1), p);
2726 0 : gel(res3, 1) = sigma;
2727 0 : for (j = 1; j < lP; j++)
2728 : {
2729 : struct galois_lift gl;
2730 0 : GEN Lden = makeLden(L,den,gb);
2731 : GEN pf;
2732 0 : initlift(T, den, uel(PG4,j), L, Lden, gb, &gl);
2733 0 : pf = nilp_froblift(vecslice(res1,1,j), PG3, j, PG5, gel(PG2,j), incl, H, &gl, gp);
2734 0 : if (!pf) return NULL;
2735 0 : if (DEBUGLEVEL>=2)
2736 0 : err_printf("found: %ld/%ld: %Ps: %Ps\n", n, j+1, gel(pf,2),gel(pf,1));
2737 0 : pcgrp_insert(PG5, j+1, gel(pf,2));
2738 0 : gel(res1, j+1) = gel(pf,1);
2739 0 : gel(res2, j+1) = gel(pf,3);
2740 0 : gel(res3, j+1) = gcopy(permtoaut(gel(pf,1), gp));
2741 : }
2742 0 : if (DEBUGLEVEL >= 4) err_printf("GaloisConj: Fini!\n");
2743 0 : return res;
2744 : }
2745 :
2746 : static GEN
2747 4644 : galoisgenlift(GEN PG, GEN Pg, GEN O, GEN L, GEN M, GEN frob,
2748 : struct galois_borne *gb, struct galois_frobenius *gf)
2749 : {
2750 : struct galois_test td;
2751 : GEN res, res1;
2752 4644 : GEN PG1 = gel(PG, 1), PG2 = gel(PG, 2);
2753 4644 : long lP = lg(PG1), j, n = lg(L)-1;
2754 4644 : inittest(L, M, gb->bornesol, gb->ladicsol, &td);
2755 4644 : res = cgetg(3, t_VEC);
2756 4644 : gel(res,1) = res1 = cgetg(lP + 1, t_VEC);
2757 4644 : gel(res,2) = vecsmall_prepend(PG2, gf->deg);
2758 4644 : gel(res1, 1) = vecsmall_copy(frob);
2759 9756 : for (j = 1; j < lP; j++)
2760 : {
2761 5661 : GEN pf = galoisgenliftauto(O, gel(PG1, j), gf->psi[Pg[j]], n, &td);
2762 5661 : if (!pf) { freetest(&td); return NULL; }
2763 5112 : gel(res1, j+1) = pf;
2764 : }
2765 4095 : if (DEBUGLEVEL >= 4) err_printf("GaloisConj: Fini!\n");
2766 4095 : freetest(&td);
2767 4095 : return res;
2768 : }
2769 :
2770 : static ulong
2771 25449 : psi_order(GEN psi, ulong d)
2772 : {
2773 25449 : long i, l = lg(psi);
2774 25449 : ulong s = 1;
2775 57253 : for (i=1; i<l; i++)
2776 31804 : s = clcm(s, d/cgcd(uel(psi, i)-1, d));
2777 25449 : return s;
2778 : }
2779 :
2780 : static GEN
2781 25617 : galoisgen(GEN T, GEN L, GEN M, GEN den, GEN bad, struct galois_borne *gb,
2782 : const struct galois_analysis *ga)
2783 : {
2784 : struct galois_test td;
2785 : struct galois_frobenius gf, ogf;
2786 25617 : pari_sp ltop = avma;
2787 25617 : long x, n = degpol(T), is_central;
2788 : ulong po;
2789 25617 : GEN sigma, res, frob, O, PG, V, ofrob = NULL;
2790 :
2791 25617 : if (!ga->deg) return NULL;
2792 25617 : x = varn(T);
2793 25617 : if (DEBUGLEVEL >= 9) err_printf("GaloisConj: denominator:%Ps\n", den);
2794 25617 : if (n == 12 && ga->ord==3 && !ga->p4)
2795 : { /* A4 is very probable: test it first */
2796 81 : pari_sp av = avma;
2797 81 : if (DEBUGLEVEL >= 4) err_printf("GaloisConj: Testing A4 first\n");
2798 81 : inittest(L, M, gb->bornesol, gb->ladicsol, &td);
2799 81 : PG = a4galoisgen(&td);
2800 81 : freetest(&td);
2801 81 : if (PG) return gc_upto(ltop, PG);
2802 0 : set_avma(av);
2803 : }
2804 25536 : if (n == 24 && ga->ord==3 && ga->p4)
2805 : { /* S4 is very probable: test it first */
2806 63 : pari_sp av = avma;
2807 : struct galois_lift gl;
2808 63 : if (DEBUGLEVEL >= 4) err_printf("GaloisConj: Testing S4 first\n");
2809 63 : initlift(T, den, ga->p4, L, makeLden(L,den,gb), gb, &gl);
2810 63 : PG = s4galoisgen(&gl);
2811 63 : if (PG) return gc_upto(ltop, PG);
2812 0 : set_avma(av);
2813 : }
2814 25473 : if (n == 36 && ga->ord==3 && ga->p4)
2815 : { /* F36 is very probable: test it first */
2816 12 : pari_sp av = avma;
2817 : struct galois_lift gl;
2818 12 : if (DEBUGLEVEL >= 4) err_printf("GaloisConj: Testing 3x3:4 first (p=%ld)\n",ga->p4);
2819 12 : initlift(T, den, ga->p4, L, makeLden(L,den,gb), gb, &gl);
2820 12 : PG = f36galoisgen(&gl);
2821 12 : if (PG) return gc_upto(ltop, PG);
2822 0 : set_avma(av);
2823 : }
2824 25461 : frob = galoisfindfrobenius(T, L, den, bad, &gf, gb, ga);
2825 25461 : if (!frob) return gc_NULL(ltop);
2826 25449 : po = psi_order(gf.psi, gf.deg);
2827 25449 : if (!(ga->group&ga_easy) && po < (ulong) gf.deg && gf.deg/radicalu(gf.deg)%po == 0)
2828 : {
2829 0 : is_central = 1;
2830 0 : if (!bad) bad = gb->dis;
2831 0 : if (po > 1)
2832 : {
2833 0 : ofrob = frob; ogf = gf;
2834 0 : frob = perm_powu(frob, po);
2835 0 : gf.deg /= po;
2836 : }
2837 25449 : } else is_central = 0;
2838 25449 : sigma = permtopol(frob, L, M, den, gb->ladicabs, shifti(gb->ladicabs,-1), x);
2839 25449 : if (is_central && gf.fp != gf.deg)
2840 0 : { gf.p = findp(bad, T, sigma, gf.deg, &gf.Tmod); gf.fp = gf.deg;
2841 0 : gf.psi = const_vecsmall(lg(gf.Tmod)-1, 1);
2842 : }
2843 25449 : if (gf.deg == n) /* cyclic */
2844 : {
2845 20789 : GEN Tp = ZX_to_Flx(gel(gf.Tmod,1), gf.p);
2846 20789 : res = mkvec5(mkvec(frob), mkvec(Tp), mkvec(sigma), mkvecsmall(gf.p), cyclic_pc(n));
2847 20789 : return gc_GEN(ltop, res);
2848 : }
2849 4660 : O = perm_cycles(frob);
2850 4660 : if (DEBUGLEVEL >= 9) err_printf("GaloisConj: Frobenius:%Ps\n", sigma);
2851 4660 : PG = galoisgenfixedfield0(O, L, sigma, T, is_central ? bad: NULL,
2852 : is_central ? &V: NULL, &gf, gb);
2853 4660 : if (PG == NULL) return gc_NULL(ltop);
2854 4644 : if (is_central && lg(gel(PG,1))!=3)
2855 0 : {
2856 : struct galois_perm gp;
2857 0 : galoisperm_init(&gp, L, M, den, gb->ladicabs, shifti(gb->ladicabs,-1), varn(T));
2858 0 : res = galoisgenlift_nilp(PG, O, V, T, frob, sigma, gb, &gf, &gp);
2859 0 : galoisperm_free(&gp);
2860 : }
2861 : else
2862 : {
2863 4644 : if (is_central && po > 1)
2864 : { /* backtrack powering of frob */
2865 0 : frob = ofrob; gf = ogf;
2866 0 : O = perm_cycles(ofrob);
2867 0 : sigma = permtopol(ofrob, L, M, den, gb->ladicabs, shifti(gb->ladicabs,-1), x);
2868 0 : PG = galoisgenfixedfield0(O, L, sigma, T, NULL, NULL, &gf, gb);
2869 0 : if (PG == NULL) return gc_NULL(ltop);
2870 : }
2871 4644 : res = galoisgenlift(gg_get_std(gel(PG,1)), gel(PG,2), O, L, M, frob, gb, &gf);
2872 : }
2873 4644 : if (!res) return gc_NULL(ltop);
2874 4095 : return gc_GEN(ltop, res);
2875 : }
2876 :
2877 : /* T = polcyclo(N) */
2878 : static GEN
2879 831 : conjcyclo(GEN T, long N)
2880 : {
2881 831 : pari_sp av = avma;
2882 831 : long i, k = 1, d = eulerphiu(N), v = varn(T);
2883 831 : GEN L = cgetg(d+1,t_COL);
2884 12515 : for (i=1; i<=N; i++)
2885 11684 : if (ugcd(i, N)==1)
2886 : {
2887 5468 : GEN s = pol_xn(i, v);
2888 5468 : if (i >= d) s = ZX_rem(s, T);
2889 5468 : gel(L,k++) = s;
2890 : }
2891 831 : return gc_upto(av, gen_sort(L, (void*)&gcmp, &gen_cmp_RgX));
2892 : }
2893 :
2894 : static GEN
2895 1067 : aut_to_groupelts(GEN aut, GEN L, ulong p)
2896 : {
2897 1067 : pari_sp av = avma;
2898 1067 : long i, d = lg(aut)-1;
2899 1067 : GEN P = ZV_to_Flv(L, p);
2900 1067 : GEN N = FlxV_Flv_multieval(aut, P, p);
2901 1067 : GEN q = perm_inv(vecsmall_indexsort(P));
2902 1067 : GEN G = cgetg(d+1, t_VEC);
2903 30808 : for (i=1; i<=d; i++)
2904 29741 : gel(G,i) = perm_mul(vecsmall_indexsort(gel(N,i)), q);
2905 1067 : return gc_GEN(av, vecvecsmall_sort_shallow(G));
2906 : }
2907 :
2908 : static ulong
2909 6 : galois_find_totally_split(GEN P, GEN Q)
2910 : {
2911 6 : pari_sp av = avma;
2912 : forprime_t iter;
2913 : ulong p;
2914 6 : long n = degpol(P);
2915 6 : u_forprime_init(&iter, n*maxss(expu(n)-3, 2), ULONG_MAX);
2916 612 : while ((p = u_forprime_next(&iter)))
2917 : {
2918 612 : if (Flx_is_totally_split(ZX_to_Flx(P, p), p)
2919 6 : && (!Q || Flx_is_squarefree(ZX_to_Flx(Q, p), p)))
2920 6 : return gc_ulong(av, p);
2921 606 : set_avma(av);
2922 : }
2923 0 : return 0;
2924 : }
2925 :
2926 : GEN
2927 1073 : galoisinitfromaut(GEN T, GEN aut, ulong l)
2928 : {
2929 1073 : pari_sp ltop = avma;
2930 1073 : GEN nf, A, G, L, M, grp, den=NULL;
2931 : struct galois_analysis ga;
2932 : struct galois_borne gb;
2933 : long n;
2934 : pari_timer ti;
2935 :
2936 1073 : T = get_nfpol(T, &nf);
2937 1073 : n = degpol(T);
2938 1073 : if (nf)
2939 0 : { if (!den) den = nf_get_zkden(nf); }
2940 : else
2941 : {
2942 1073 : if (n <= 0) pari_err_IRREDPOL("galoisinit",T);
2943 1073 : RgX_check_ZX(T, "galoisinit");
2944 1073 : if (!ZX_is_squarefree(T))
2945 0 : pari_err_DOMAIN("galoisinit","issquarefree(pol)","=",gen_0,T);
2946 1073 : if (!gequal1(gel(T,n+2))) pari_err_IMPL("galoisinit(nonmonic)");
2947 : }
2948 1073 : if (lg(aut)-1 != n)
2949 6 : return gen_0;
2950 1067 : ga.l = l? l: galois_find_totally_split(T, NULL);
2951 1067 : if (!l) aut = RgXV_to_FlxV(aut, ga.l);
2952 1067 : gb.l = utoipos(ga.l);
2953 1067 : if (DEBUGLEVEL >= 1) timer_start(&ti);
2954 1067 : den = galoisborne(T, den, &gb, degpol(T));
2955 1067 : if (DEBUGLEVEL >= 1) timer_printf(&ti, "galoisborne()");
2956 1067 : L = ZpX_roots(T, gb.l, gb.valabs);
2957 1067 : if (DEBUGLEVEL >= 1) timer_printf(&ti, "ZpX_roots");
2958 1067 : M = FpV_invVandermonde(L, den, gb.ladicabs);
2959 1067 : if (DEBUGLEVEL >= 1) timer_printf(&ti, "FpV_invVandermonde()");
2960 1067 : A = aut_to_groupelts(aut, L, ga.l);
2961 1067 : G = groupelts_to_group(A);
2962 1067 : if (!G) G = trivialgroup();
2963 1061 : else A = group_elts(G,n);
2964 1067 : grp = cgetg(9, t_VEC);
2965 1067 : gel(grp,1) = T;
2966 1067 : gel(grp,2) = mkvec3(utoipos(ga.l), utoipos(gb.valabs), gb.ladicabs);
2967 1067 : gel(grp,3) = L;
2968 1067 : gel(grp,4) = M;
2969 1067 : gel(grp,5) = den;
2970 1067 : gel(grp,6) = A;
2971 1067 : gel(grp,7) = gel(G,1);
2972 1067 : gel(grp,8) = gel(G,2);
2973 1067 : return gc_GEN(ltop, grp);
2974 : }
2975 :
2976 : GEN
2977 1060 : galoissplittinginit(GEN T, GEN D)
2978 : {
2979 1060 : pari_sp av = avma;
2980 1060 : GEN R = nfsplitting0(T, D, 3), P = gel(R,1), aut = gel(R,2);
2981 1055 : ulong p = itou(gel(R,3));
2982 1055 : return gc_upto(av, galoisinitfromaut(P, aut, p));
2983 : }
2984 :
2985 : /* T: polynomial or nf, den multiple of common denominator of solutions or
2986 : * NULL (unknown). If T is nf, and den unknown, use den = denom(nf.zk) */
2987 : static GEN
2988 83055 : galoisconj4_main(GEN T, GEN den, long flag)
2989 : {
2990 83055 : pari_sp ltop = avma;
2991 : GEN nf, G, L, M, aut, grp;
2992 : struct galois_analysis ga;
2993 : struct galois_borne gb;
2994 : long n;
2995 : pari_timer ti;
2996 :
2997 83055 : T = get_nfpol(T, &nf);
2998 83055 : n = poliscyclo(T);
2999 83055 : if (n) return flag? galoiscyclo(n, varn(T)): conjcyclo(T, n);
3000 81828 : n = degpol(T);
3001 81828 : if (nf)
3002 46500 : { if (!den) den = nf_get_zkden(nf); }
3003 : else
3004 : {
3005 35328 : if (n <= 0) pari_err_IRREDPOL("galoisinit",T);
3006 35328 : RgX_check_ZX(T, "galoisinit");
3007 35328 : if (!ZX_is_squarefree(T))
3008 6 : pari_err_DOMAIN("galoisinit","issquarefree(pol)","=",gen_0,T);
3009 35322 : if (!gequal1(gel(T,n+2))) pari_err_IMPL("galoisinit(nonmonic)");
3010 : }
3011 81816 : if (n == 1)
3012 : {
3013 18 : if (!flag) { G = cgetg(2, t_COL); gel(G,1) = pol_x(varn(T)); return G;}
3014 18 : ga.l = 3;
3015 18 : ga.deg = 1;
3016 18 : den = gen_1;
3017 : }
3018 81798 : else if (!galoisanalysis(T, &ga, 1, NULL)) return gc_NULL(ltop);
3019 :
3020 24519 : if (den)
3021 : {
3022 15514 : if (typ(den) != t_INT) pari_err_TYPE("galoisconj4 [2nd arg integer]", den);
3023 15514 : den = absi_shallow(den);
3024 : }
3025 24519 : gb.l = utoipos(ga.l);
3026 24519 : if (DEBUGLEVEL >= 1) timer_start(&ti);
3027 24519 : den = galoisborne(T, den, &gb, degpol(T));
3028 24519 : if (DEBUGLEVEL >= 1) timer_printf(&ti, "galoisborne()");
3029 24519 : L = ZpX_roots(T, gb.l, gb.valabs);
3030 24519 : if (DEBUGLEVEL >= 1) timer_printf(&ti, "ZpX_roots");
3031 24519 : M = FpV_invVandermonde(L, den, gb.ladicabs);
3032 24519 : if (DEBUGLEVEL >= 1) timer_printf(&ti, "FpV_invVandermonde()");
3033 24519 : if (n == 1)
3034 : {
3035 18 : G = cgetg(3, t_VEC);
3036 18 : gel(G,1) = cgetg(1, t_VEC);
3037 18 : gel(G,2) = cgetg(1, t_VECSMALL);
3038 : }
3039 : else
3040 24501 : G = gg_get_std(galoisgen(T, L, M, den, NULL, &gb, &ga));
3041 24519 : if (DEBUGLEVEL >= 6) err_printf("GaloisConj: %Ps\n", G);
3042 24519 : if (!G) return gc_NULL(ltop);
3043 23947 : if (DEBUGLEVEL >= 1) timer_start(&ti);
3044 23947 : grp = cgetg(9, t_VEC);
3045 23947 : gel(grp,1) = T;
3046 23947 : gel(grp,2) = mkvec3(utoipos(ga.l), utoipos(gb.valabs), gb.ladicabs);
3047 23947 : gel(grp,3) = L;
3048 23947 : gel(grp,4) = M;
3049 23947 : gel(grp,5) = den;
3050 23947 : gel(grp,6) = group_elts(G,n);
3051 23947 : gel(grp,7) = gel(G,1);
3052 23947 : gel(grp,8) = gel(G,2);
3053 23947 : if (flag) return gc_GEN(ltop, grp);
3054 7542 : aut = galoisvecpermtopol(grp, gal_get_group(grp), gb.ladicabs, shifti(gb.ladicabs,-1));
3055 7542 : settyp(aut, t_COL);
3056 7542 : if (DEBUGLEVEL >= 1) timer_printf(&ti, "Computation of polynomials");
3057 7542 : return gc_upto(ltop, gen_sort(aut, (void*)&gcmp, &gen_cmp_RgX));
3058 : }
3059 :
3060 : /* Heuristic computation of #Aut(T), pinit = first prime to be tested */
3061 : long
3062 30782 : numberofconjugates(GEN T, long pinit)
3063 : {
3064 30782 : pari_sp av = avma;
3065 30782 : long c, nbtest, nbmax, n = degpol(T);
3066 : ulong p;
3067 : forprime_t S;
3068 :
3069 30782 : if (n == 1) return 1;
3070 30782 : nbmax = (n < 10)? 20: (n<<1) + 1;
3071 30782 : nbtest = 0;
3072 : #if 0
3073 : c = ZX_sturm(T); c = ugcd(c, n-c); /* too costly: finite primes are cheaper */
3074 : #else
3075 30782 : c = n;
3076 : #endif
3077 30782 : u_forprime_init(&S, pinit, ULONG_MAX);
3078 291135 : while((p = u_forprime_next(&S)))
3079 : {
3080 291135 : GEN L, Tp = ZX_to_Flx(T,p);
3081 : long i, nb;
3082 291135 : if (!Flx_is_squarefree(Tp, p)) continue;
3083 : /* unramified */
3084 241057 : nbtest++;
3085 241057 : L = Flx_nbfact_by_degree(Tp, &nb, p); /* L[i] = #factors of degree i */
3086 241057 : if (L[n/nb] == nb) {
3087 201085 : if (c == n && nbtest > 10) break; /* probably Galois */
3088 : }
3089 : else
3090 : {
3091 70712 : c = ugcd(c, L[1]);
3092 247725 : for (i = 2; i <= n; i++)
3093 197822 : if (L[i]) { c = ugcd(c, L[i]*i); if (c == 1) break; }
3094 70712 : if (c == 1) break;
3095 : }
3096 220206 : if (nbtest == nbmax) break;
3097 210275 : if (DEBUGLEVEL >= 6)
3098 0 : err_printf("NumberOfConjugates [%ld]:c=%ld,p=%ld\n", nbtest,c,p);
3099 210275 : set_avma(av);
3100 : }
3101 30782 : if (DEBUGLEVEL >= 2) err_printf("NumberOfConjugates:c=%ld,p=%ld\n", c, p);
3102 30782 : return gc_long(av,c);
3103 : }
3104 : static GEN
3105 0 : galoisconj4(GEN nf, GEN d)
3106 : {
3107 0 : pari_sp av = avma;
3108 : GEN G, T;
3109 0 : G = galoisconj4_main(nf, d, 0);
3110 0 : if (G) return G; /* Success */
3111 0 : set_avma(av); T = get_nfpol(nf, &nf);
3112 0 : G = cgetg(2, t_COL); gel(G,1) = pol_x(varn(T)); return G; /* Fail */
3113 :
3114 : }
3115 :
3116 : /* d multiplicative bound for the automorphism's denominators */
3117 : static GEN
3118 60207 : galoisconj_monic(GEN nf, GEN d)
3119 : {
3120 60207 : pari_sp av = avma;
3121 60207 : GEN G, NF, T = get_nfpol(nf,&NF);
3122 60207 : if (degpol(T) == 2)
3123 : { /* fast shortcut */
3124 21058 : GEN b = gel(T,3);
3125 21058 : long v = varn(T);
3126 21058 : G = cgetg(3, t_COL);
3127 21058 : gel(G,1) = deg1pol_shallow(gen_m1, negi(b), v);
3128 21058 : gel(G,2) = pol_x(v);
3129 21058 : return G;
3130 : }
3131 39149 : G = galoisconj4_main(nf, d, 0);
3132 39149 : if (G) return G; /* Success */
3133 30776 : set_avma(av); return galoisconj1(nf);
3134 : }
3135 :
3136 : GEN
3137 60207 : galoisconj(GEN nf, GEN d)
3138 : {
3139 : pari_sp av;
3140 60207 : GEN NF, S, L, T = get_nfpol(nf,&NF);
3141 60207 : if (NF) return galoisconj_monic(NF, d);
3142 60 : RgX_check_QX(T, "galoisconj");
3143 60 : av = avma;
3144 60 : T = Q_primpart(T);
3145 60 : if (ZX_is_monic(T)) return galoisconj_monic(T, d);
3146 0 : S = galoisconj_monic(poltomonic(T,&L), NULL);
3147 0 : return gc_upto(av, gdiv(RgXV_unscale(S, L),L));
3148 : }
3149 :
3150 : /* FIXME: obsolete, use galoisconj(nf, d) directly */
3151 : GEN
3152 54 : galoisconj0(GEN nf, long flag, GEN d, long prec)
3153 : {
3154 : (void)prec;
3155 54 : switch(flag) {
3156 48 : case 2:
3157 48 : case 0: return galoisconj(nf, d);
3158 6 : case 1: return galoisconj1(nf);
3159 0 : case 4: return galoisconj4(nf, d);
3160 : }
3161 0 : pari_err_FLAG("nfgaloisconj");
3162 : return NULL; /*LCOV_EXCL_LINE*/
3163 : }
3164 :
3165 : /******************************************************************************/
3166 : /* Galois theory related algorithms */
3167 : /******************************************************************************/
3168 : GEN
3169 26412 : checkgal(GEN gal)
3170 : {
3171 26412 : if (typ(gal) == t_POL) pari_err_TYPE("checkgal [apply galoisinit first]",gal);
3172 26412 : if (typ(gal) != t_VEC || lg(gal) != 9) pari_err_TYPE("checkgal",gal);
3173 26406 : return gal;
3174 : }
3175 :
3176 : GEN
3177 43918 : galoisinit(GEN nf, GEN den)
3178 : {
3179 : GEN G;
3180 43918 : if (is_vec_t(typ(nf)) && lg(nf)==3 && is_vec_t(typ(gel(nf,2))))
3181 12 : return galoisinitfromaut(gel(nf,1), gel(nf,2), 0);
3182 43906 : G = galoisconj4_main(nf, den, 1);
3183 43894 : return G? G: gen_0;
3184 : }
3185 :
3186 : static GEN
3187 15396 : galoispermtopol_i(GEN gal, GEN perm, GEN mod, GEN mod2)
3188 : {
3189 15396 : switch (typ(perm))
3190 : {
3191 15186 : case t_VECSMALL:
3192 15186 : return permtopol(perm, gal_get_roots(gal), gal_get_invvdm(gal),
3193 : gal_get_den(gal), mod, mod2,
3194 15186 : varn(gal_get_pol(gal)));
3195 210 : case t_VEC: case t_COL: case t_MAT:
3196 210 : return galoisvecpermtopol(gal, perm, mod, mod2);
3197 : }
3198 0 : pari_err_TYPE("galoispermtopol", perm);
3199 : return NULL; /* LCOV_EXCL_LINE */
3200 : }
3201 :
3202 : GEN
3203 15396 : galoispermtopol(GEN gal, GEN perm)
3204 : {
3205 15396 : pari_sp av = avma;
3206 : GEN mod, mod2;
3207 15396 : gal = checkgal(gal);
3208 15396 : mod = gal_get_mod(gal);
3209 15396 : mod2 = shifti(mod,-1);
3210 15396 : return gc_GEN(av, galoispermtopol_i(gal, perm, mod, mod2));
3211 : }
3212 :
3213 : GEN
3214 78 : galoiscosets(GEN O, GEN perm)
3215 : {
3216 78 : long i, j, k, u, f, l = lg(O);
3217 78 : GEN RC, C = cgetg(l,t_VECSMALL), o = gel(O,1);
3218 78 : pari_sp av = avma;
3219 78 : f = lg(o); u = o[1]; RC = zero_zv(lg(perm)-1);
3220 318 : for(i=1,j=1; j<l; i++)
3221 : {
3222 240 : GEN p = gel(perm,i);
3223 240 : if (RC[ p[u] ]) continue;
3224 654 : for(k=1; k<f; k++) RC[ p[ o[k] ] ] = 1;
3225 192 : C[j++] = i;
3226 : }
3227 78 : set_avma(av); return C;
3228 : }
3229 :
3230 : static GEN
3231 78 : fixedfieldfactor(GEN L, GEN O, GEN perm, GEN M, GEN den, GEN mod, GEN mod2,
3232 : long x,long y)
3233 : {
3234 78 : pari_sp ltop = avma;
3235 78 : long i, j, k, l = lg(O), lo = lg(gel(O,1));
3236 78 : GEN V, res, cosets = galoiscosets(O,perm), F = cgetg(lo+1,t_COL);
3237 :
3238 78 : gel(F, lo) = gen_1;
3239 78 : if (DEBUGLEVEL>=4) err_printf("GaloisFixedField:cosets=%Ps \n",cosets);
3240 78 : if (DEBUGLEVEL>=6) err_printf("GaloisFixedField:den=%Ps mod=%Ps \n",den,mod);
3241 78 : V = cgetg(l,t_COL); res = cgetg(l,t_VEC);
3242 270 : for (i = 1; i < l; i++)
3243 : {
3244 192 : pari_sp av = avma;
3245 192 : GEN G = cgetg(l,t_VEC), Lp = vecpermute(L, gel(perm, cosets[i]));
3246 804 : for (k = 1; k < l; k++)
3247 612 : gel(G,k) = FpV_roots_to_pol(vecpermute(Lp, gel(O,k)), mod, x);
3248 654 : for (j = 1; j < lo; j++)
3249 : {
3250 1572 : for(k = 1; k < l; k++) gel(V,k) = gmael(G,k,j+1);
3251 462 : gel(F,j) = vectopol(V, M, den, mod, mod2, y);
3252 : }
3253 192 : gel(res,i) = gc_upto(av,gtopolyrev(F,x));
3254 : }
3255 78 : return gc_upto(ltop,res);
3256 : }
3257 :
3258 : static void
3259 6085 : chk_perm(GEN perm, long n)
3260 : {
3261 6085 : if (typ(perm) != t_VECSMALL || lg(perm)!=n+1)
3262 0 : pari_err_TYPE("galoisfixedfield", perm);
3263 6085 : }
3264 :
3265 : static int
3266 11201 : is_group(GEN g)
3267 : {
3268 11201 : if (typ(g) == t_VEC && lg(g) == 3)
3269 : {
3270 2191 : GEN a = gel(g,1), o = gel(g,2);
3271 2191 : return typ(a)==t_VEC && typ(o)==t_VECSMALL && lg(a) == lg(o);
3272 : }
3273 9010 : return 0;
3274 : }
3275 :
3276 : GEN
3277 4756 : galoisfixedfield(GEN gal, GEN perm, long flag, long y)
3278 : {
3279 4756 : pari_sp ltop = avma;
3280 : GEN T, L, P, S, PL, O, res, mod, mod2, OL, sym;
3281 : long vT, n, i;
3282 4756 : if (flag<0 || flag>2) pari_err_FLAG("galoisfixedfield");
3283 4756 : gal = checkgal(gal); T = gal_get_pol(gal);
3284 4756 : vT = varn(T);
3285 4756 : L = gal_get_roots(gal); n = lg(L)-1;
3286 4756 : mod = gal_get_mod(gal);
3287 4756 : if (typ(perm) == t_VEC)
3288 : {
3289 3861 : if (is_group(perm)) perm = gel(perm, 1);
3290 9051 : for (i = 1; i < lg(perm); i++) chk_perm(gel(perm,i), n);
3291 3861 : O = vecperm_orbits(perm, n);
3292 : }
3293 : else
3294 : {
3295 895 : chk_perm(perm, n);
3296 895 : O = perm_cycles(perm);
3297 : }
3298 4756 : mod2 = shifti(mod,-1);
3299 4756 : OL = fixedfieldorbits(O, L);
3300 4756 : sym = fixedfieldsympol(OL, itou(gal_get_p(gal)));
3301 4756 : PL = sympol_eval(sym, OL, mod);
3302 4756 : P = FpX_center_i(FpV_roots_to_pol(PL, mod, vT), mod, mod2);
3303 4756 : if (flag==1) return gc_GEN(ltop,P);
3304 827 : S = fixedfieldinclusion(O, PL);
3305 827 : S = vectopol(S, gal_get_invvdm(gal), gal_get_den(gal), mod, mod2, vT);
3306 827 : if (flag==0)
3307 749 : res = cgetg(3, t_VEC);
3308 : else
3309 : {
3310 : GEN PM, Pden;
3311 : struct galois_borne Pgb;
3312 78 : long val = itos(gal_get_e(gal));
3313 78 : Pgb.l = gal_get_p(gal);
3314 78 : Pden = galoisborne(P, NULL, &Pgb, degpol(T)/degpol(P));
3315 78 : if (Pgb.valabs > val)
3316 : {
3317 6 : if (DEBUGLEVEL>=4)
3318 0 : err_printf("GaloisConj: increase p-adic prec by %ld.\n", Pgb.valabs-val);
3319 6 : PL = ZpX_liftroots(P, PL, Pgb.l, Pgb.valabs);
3320 6 : L = ZpX_liftroots(T, L, Pgb.l, Pgb.valabs);
3321 6 : mod = Pgb.ladicabs; mod2 = shifti(mod,-1);
3322 : }
3323 78 : PM = FpV_invVandermonde(PL, Pden, mod);
3324 78 : if (y < 0) y = 1;
3325 78 : if (varncmp(y, vT) <= 0)
3326 0 : pari_err_PRIORITY("galoisfixedfield", T, "<=", y);
3327 78 : setvarn(P, y);
3328 78 : res = cgetg(4, t_VEC);
3329 78 : gel(res,3) = fixedfieldfactor(L,O,gal_get_group(gal), PM,Pden,mod,mod2,vT,y);
3330 : }
3331 827 : gel(res,1) = gcopy(P);
3332 827 : gel(res,2) = gmodulo(S, T);
3333 827 : return gc_upto(ltop, res);
3334 : }
3335 :
3336 : /* gal a galois group output the underlying wss group */
3337 : GEN
3338 2852 : galois_group(GEN gal) { return mkvec2(gal_get_gen(gal), gal_get_orders(gal)); }
3339 :
3340 : GEN
3341 3632 : checkgroup(GEN g, GEN *S)
3342 : {
3343 3632 : if (is_group(g)) { *S = NULL; return g; }
3344 2726 : g = checkgal(g);
3345 2720 : *S = gal_get_group(g); return galois_group(g);
3346 : }
3347 :
3348 : static GEN
3349 7144 : group_is_elt(GEN G)
3350 : {
3351 7144 : long i, n = lg(G)-1;
3352 7144 : if (n==0) pari_err_DIM("checkgroupelts");
3353 7144 : if (lg(G)==9 && typ(gel(G,1))==t_POL)
3354 5800 : if (lg(gal_get_gen(G))==1 && lg(gal_get_group(G))>2)
3355 12 : return gal_get_group(G);
3356 7132 : if (typ(G)==t_VEC && typ(gel(G,1))==t_VECSMALL)
3357 : {
3358 5352 : for (i = 1; i <= n; i++)
3359 : {
3360 5094 : if (typ(gel(G,i)) != t_VECSMALL)
3361 0 : pari_err_TYPE("checkgroupelts (element)", gel(G,i));
3362 5094 : if (lg(gel(G,i)) != lg(gel(G,1)))
3363 12 : pari_err_DIM("checkgroupelts [length of permutations]");
3364 : }
3365 258 : return G;
3366 : }
3367 6862 : return NULL;
3368 : }
3369 :
3370 : GEN
3371 3972 : checkgroupelts(GEN G)
3372 : {
3373 3972 : GEN S = group_is_elt(G);
3374 3960 : if (S) return S;
3375 3708 : if (is_group(G))
3376 : { /* subgroup of S_n */
3377 318 : if (lg(gel(G,1))==1) return mkvec(mkvecsmall(1));
3378 318 : return group_elts(G, group_domain(G));
3379 : }
3380 3390 : if (lg(G)==9 && typ(gel(G,1))==t_POL)
3381 3354 : return gal_get_group(G); /* galoisinit */
3382 36 : pari_err_TYPE("checkgroupelts",G);
3383 : return NULL; /* LCOV_EXCL_LINE */
3384 : }
3385 :
3386 : GEN
3387 192 : galoisisabelian(GEN gal, long flag)
3388 : {
3389 192 : pari_sp av = avma;
3390 192 : GEN S, G = checkgroup(gal,&S);
3391 192 : if (!group_isabelian(G)) { set_avma(av); return gen_0; }
3392 174 : switch(flag)
3393 : {
3394 42 : case 0: return gc_upto(av, group_abelianHNF(G,S));
3395 42 : case 1: set_avma(av); return gen_1;
3396 90 : case 2: return gc_upto(av, group_abelianSNF(G,S));
3397 0 : default: pari_err_FLAG("galoisisabelian");
3398 : }
3399 : return NULL; /* LCOV_EXCL_LINE */
3400 : }
3401 :
3402 : long
3403 48 : galoisisnormal(GEN gal, GEN sub)
3404 : {
3405 48 : pari_sp av = avma;
3406 48 : GEN S, G = checkgroup(gal, &S), H = checkgroup(sub, &S);
3407 48 : long res = group_subgroup_isnormal(G, H);
3408 48 : set_avma(av);
3409 48 : return res;
3410 : }
3411 :
3412 : static GEN
3413 264 : conjclasses_count(GEN conj, long nb)
3414 : {
3415 264 : long i, l = lg(conj);
3416 264 : GEN c = zero_zv(nb);
3417 3462 : for (i = 1; i < l; i++) c[conj[i]]++;
3418 264 : return c;
3419 : }
3420 : GEN
3421 264 : galoisconjclasses(GEN G)
3422 : {
3423 264 : pari_sp av = avma;
3424 264 : GEN c, e, cc = group_to_cc(G);
3425 264 : GEN elts = gel(cc,1), conj = gel(cc,2), repr = gel(cc,3);
3426 264 : long i, l = lg(conj), lc = lg(repr);
3427 264 : c = conjclasses_count(conj, lc-1);
3428 264 : e = cgetg(lc, t_VEC);
3429 2694 : for (i = 1; i < lc; i++) gel(e,i) = cgetg(c[i]+1, t_VEC);
3430 3462 : for (i = 1; i < l; i++)
3431 : {
3432 3198 : long ci = conj[i];
3433 3198 : gmael(e, ci, c[ci]) = gel(elts, i);
3434 3198 : c[ci]--;
3435 : }
3436 264 : return gc_GEN(av, e);
3437 : }
3438 :
3439 : static GEN
3440 708 : groupelts_to_group_or_elts(GEN elts)
3441 : {
3442 708 : GEN G = groupelts_to_group(elts);
3443 708 : return G ? G: gcopy(elts);
3444 : }
3445 :
3446 : static GEN
3447 12 : vec_groupelts_to_group_or_elts(GEN x)
3448 720 : { pari_APPLY_same(groupelts_to_group_or_elts(gel(x,i))) }
3449 :
3450 : GEN
3451 2740 : galoissubgroups(GEN gal)
3452 : {
3453 2740 : pari_sp av = avma;
3454 2740 : GEN S = group_is_elt(gal), G;
3455 2740 : if (S) return gc_upto(av,
3456 : vec_groupelts_to_group_or_elts(groupelts_solvablesubgroups(S)));
3457 2728 : G = checkgroup(gal, &S);
3458 2728 : return gc_upto(av, group_subgroups(G));
3459 : }
3460 :
3461 : GEN
3462 64 : galoissubfields(GEN G, long flag, long v)
3463 : {
3464 64 : pari_sp av = avma;
3465 64 : GEN L = galoissubgroups(G);
3466 64 : long i, l = lg(L);
3467 64 : GEN S = cgetg(l, t_VEC);
3468 990 : for (i = 1; i < l; ++i) gel(S,i) = galoisfixedfield(G, gmael(L,i,1), flag, v);
3469 64 : return gc_upto(av, S);
3470 : }
3471 :
3472 : GEN
3473 24 : galoisexport(GEN gal, long format)
3474 : {
3475 24 : pari_sp av = avma;
3476 24 : GEN S, G = checkgroup(gal,&S);
3477 24 : return gc_upto(av, group_export(G,format));
3478 : }
3479 :
3480 : GEN
3481 432 : galoisidentify(GEN gal)
3482 : {
3483 432 : pari_sp av = avma;
3484 : long idx, card;
3485 432 : GEN S = group_is_elt(gal), G;
3486 432 : G = S ? S: checkgroup(gal,&S);
3487 426 : idx = group_ident(G,S);
3488 426 : card = S ? lg(S)-1: group_order(G);
3489 426 : set_avma(av); return mkvec2s(card, idx);
3490 : }
3491 :
3492 : /* index of conjugacy class containing g */
3493 : static long
3494 31662 : cc_id(GEN cc, GEN g)
3495 : {
3496 31662 : GEN conj = gel(cc,2);
3497 31662 : long k = signe(gel(cc,4))? g[1]: vecvecsmall_search(gel(cc,1), g);
3498 31662 : return conj[k];
3499 : }
3500 :
3501 : static GEN
3502 3588 : Qevproj_RgX(GEN c, long d, GEN pro)
3503 3588 : { return RgV_to_RgX(Qevproj_down(RgX_to_RgC(c,d), pro), varn(c)); }
3504 : /* c in Z[X] / (X^o-1), To = polcyclo(o), T = polcyclo(expo), e = expo/o
3505 : * return c(X^e) mod T as an element of Z[X] / (To) */
3506 : static GEN
3507 3360 : chival(GEN c, GEN T, GEN To, long e, GEN pro, long phie)
3508 : {
3509 3360 : c = ZX_rem(c, To);
3510 3360 : if (e != 1) c = ZX_rem(RgX_inflate(c,e), T);
3511 3360 : if (pro) c = Qevproj_RgX(c, phie, pro);
3512 3360 : return c;
3513 : }
3514 : /* chi(g^l) = sum_{k=0}^{o-1} a_k zeta_o^{l*k} for all l;
3515 : * => a_k = 1/o sum_{l=0}^{o-1} chi(g^l) zeta_o^{-k*l}. Assume o > 1 */
3516 : static GEN
3517 738 : chiFT(GEN cp, GEN jg, GEN vze, long e, long o, ulong p, ulong pov2)
3518 : {
3519 738 : const long var = 1;
3520 738 : ulong oinv = Fl_inv(o,p);
3521 : long k, l;
3522 738 : GEN c = cgetg(o+2, t_POL);
3523 4836 : for (k = 0; k < o; k++)
3524 : {
3525 4098 : ulong a = 0;
3526 44124 : for (l=0; l<o; l++)
3527 : {
3528 40026 : ulong z = vze[Fl_mul(k,l,o)*e + 1];/* zeta_o^{-k*l} */
3529 40026 : a = Fl_add(a, Fl_mul(uel(cp,jg[l+1]), z, p), p);
3530 : }
3531 4098 : gel(c,k+2) = stoi(Fl_center(Fl_mul(a,oinv,p), p, pov2)); /* a_k */
3532 : }
3533 738 : c[1] = evalvarn(var) | evalsigne(1); return ZX_renormalize(c,o+2);
3534 : }
3535 : static GEN
3536 468 : cc_chartable(GEN cc)
3537 : {
3538 : GEN al, elts, rep, ctp, ct, dec, id, vjg, H, vord, operm;
3539 : long i, j, k, f, l, expo, lcl, n;
3540 : ulong p, pov2;
3541 :
3542 468 : elts = gel(cc,1); n = lg(elts)-1;
3543 468 : if (n == 1) return mkvec2(mkmat(mkcol(gen_1)), gen_1);
3544 456 : rep = gel(cc,3);
3545 456 : lcl = lg(rep);
3546 456 : vjg = cgetg(lcl, t_VEC);
3547 456 : vord = cgetg(lcl,t_VECSMALL);
3548 456 : id = identity_perm(lg(gel(elts,1))-1);
3549 456 : expo = 1;
3550 4182 : for(j=1;j<lcl;j++)
3551 : {
3552 3726 : GEN jg, h = id, g = gel(elts,rep[j]);
3553 : long o;
3554 3726 : vord[j] = o = perm_orderu(g);
3555 3726 : expo = ulcm(expo, o);
3556 3726 : gel(vjg,j) = jg = cgetg(o+1,t_VECSMALL);
3557 23718 : for (l=1; l<=o; l++)
3558 : {
3559 19992 : jg[l] = cc_id(cc, h); /* index of conjugacy class of g^(l-1) */
3560 19992 : if (l < o) h = perm_mul(h, g);
3561 : }
3562 : }
3563 : /* would sort conjugacy classes by inc. order */
3564 456 : operm = vecsmall_indexsort(vord);
3565 :
3566 : /* expo > 1, exponent of G */
3567 456 : p = unextprime(2*n+1);
3568 894 : while (p%expo != 1) p = unextprime(p+1);
3569 : /* compute character table modulo p: idempotents of Z(KG) */
3570 456 : al = conjclasses_algcenter(cc, utoipos(p));
3571 456 : dec = algsimpledec_ss(al,1);
3572 456 : ctp = cgetg(lcl,t_VEC);
3573 4182 : for(i=1; i<lcl; i++)
3574 : {
3575 3726 : GEN e = ZV_to_Flv(gmael3(dec,i,3,1), p); /*(1/n)[(dim chi)chi(g): g in G]*/
3576 3726 : ulong d = usqrt(Fl_mul(e[1], n, p)); /* = chi(1) <= sqrt(n) < sqrt(p) */
3577 3726 : gel(ctp,i) = Flv_Fl_mul(e,Fl_div(n,d,p), p); /*[chi(g): g in G]*/
3578 : }
3579 : /* Find minimal f such that table is defined over Q(zeta(f)): the conductor
3580 : * of the class field Q(\zeta_e)^H defined by subgroup
3581 : * H = { k in (Z/e)^*: g^k ~ g, for all g } */
3582 456 : H = coprimes_zv(expo);
3583 2964 : for (k = 2; k < expo; k++)
3584 : {
3585 2508 : if (!H[k]) continue;
3586 2184 : for (j = 2; j < lcl; j++) /* skip g ~ 1 */
3587 2028 : if (umael(vjg,j,(k % vord[j])+1) != umael(vjg,j,2)) { H[k] = 0; break; }
3588 : }
3589 456 : f = znstar_conductor_bits(Flv_to_F2v(H));
3590 : /* lift character table to Z[zeta_f] */
3591 456 : pov2 = p>>1;
3592 456 : ct = cgetg(lcl, t_MAT);
3593 456 : if (f == 1)
3594 : { /* rational representation */
3595 804 : for (j=1; j<lcl; j++) gel(ct,j) = cgetg(lcl,t_COL);
3596 804 : for(j=1; j<lcl; j++)
3597 : {
3598 678 : GEN jg = gel(vjg,j); /* jg[l+1] = class of g^l */
3599 678 : long t = lg(jg) > 2? jg[2]: jg[1];
3600 5748 : for(i=1; i<lcl; i++)
3601 : {
3602 5070 : GEN cp = gel(ctp,i); /* cp[i] = chi(g_i) mod \P */
3603 5070 : gcoeff(ct,j,i) = stoi(Fl_center(cp[t], p, pov2));
3604 : }
3605 : }
3606 : }
3607 : else
3608 : {
3609 330 : const long var = 1;
3610 330 : ulong ze = Fl_powu(pgener_Fl(p),(p-1)/expo, p); /* seen as zeta_e^(-1) */
3611 330 : GEN vze = Fl_powers(ze, expo-1, p); /* vze[i] = ze^(i-1) */
3612 330 : GEN vzeZX = const_vec(p, gen_0);
3613 330 : GEN T = polcyclo(expo, var), vT = const_vec(expo,NULL), pro = NULL;
3614 330 : long phie = degpol(T), id1 = gel(vjg,1)[1]; /* index of 1_G, always 1 ? */
3615 330 : gel(vT, expo) = T;
3616 330 : if (f != expo)
3617 : {
3618 126 : long phif = eulerphiu(f);
3619 126 : GEN zf = ZX_rem(pol_xn(expo/f,var), T), zfj = zf;
3620 126 : GEN M = cgetg(phif+1, t_MAT);
3621 126 : gel(M,1) = col_ei(phie,1);
3622 444 : for (j = 2; j <= phif; j++)
3623 : {
3624 318 : gel(M,j) = RgX_to_RgC(zfj, phie);
3625 318 : if (j < phif) zfj = ZX_rem(ZX_mul(zfj, zf), T);
3626 : }
3627 126 : pro = Qevproj_init(M);
3628 : }
3629 330 : gel(vzeZX,1) = pol_1(var);
3630 2928 : for (i = 2; i <= expo; i++)
3631 : {
3632 2598 : GEN t = ZX_rem(pol_xn(expo-(i-1), var), T);
3633 2598 : if (pro) t = Qevproj_RgX(t, phie, pro);
3634 2598 : gel(vzeZX, vze[i]) = t;
3635 : }
3636 3378 : for(i=1; i<lcl; i++)
3637 : { /* loop over characters */
3638 3048 : GEN cp = gel(ctp,i), C, cj; /* cp[j] = chi(g_j) mod \P */
3639 3048 : long dim = cp[id1];
3640 3048 : gel(ct, i) = C = const_col(lcl-1, NULL);
3641 3048 : gel(C,operm[1]) = utoi(dim); /* chi(1_G) */
3642 35124 : for (j=lcl-1; j > 1; j--)
3643 : { /* loop over conjugacy classes, decreasing order: skip 1_G */
3644 32076 : long e, jperm = operm[j], o = vord[jperm];
3645 32076 : GEN To, jg = gel(vjg,jperm); /* jg[l+1] = class of g^l */
3646 :
3647 32076 : if (gel(C, jperm)) continue; /* done already */
3648 30774 : if (dim == 1) { gel(C, jperm) = gel(vzeZX, cp[jg[2]]); continue; }
3649 738 : e = expo / o;
3650 738 : cj = chiFT(cp, jg, vze, e, o, p, pov2);
3651 738 : To = gel(vT, o); if (!To) To = gel(vT,o) = polcyclo(o, var);
3652 738 : gel(C, jperm) = chival(cj, T, To, e, pro, phie);
3653 3360 : for (k = 2; k < o; k++)
3654 : {
3655 2622 : GEN ck = RgX_inflate(cj, k); /* chi(g^k) */
3656 2622 : gel(C, jg[k+1]) = chival(ck, T, To, e, pro, phie);
3657 : }
3658 : }
3659 : }
3660 : }
3661 456 : ct = gen_sort_shallow(ct,(void*)cmp_universal,cmp_nodata);
3662 1488 : i = 1; while (!vec_isconst(gel(ct,i))) i++;
3663 456 : if (i > 1) swap(gel(ct,1), gel(ct,i));
3664 456 : return mkvec2(ct, utoipos(f));
3665 : }
3666 : GEN
3667 474 : galoischartable(GEN gal)
3668 : {
3669 474 : pari_sp av = avma;
3670 474 : GEN cc = group_to_cc(gal);
3671 468 : return gc_GEN(av, cc_chartable(cc));
3672 : }
3673 :
3674 : static void
3675 1278 : checkgaloischar(GEN ch, GEN repr)
3676 : {
3677 1278 : if (gvar(ch) == 0) pari_err_PRIORITY("galoischarpoly",ch,"=",0);
3678 1278 : if (!is_vec_t(typ(ch))) pari_err_TYPE("galoischarpoly", ch);
3679 1278 : if (lg(repr) != lg(ch)) pari_err_DIM("galoischarpoly");
3680 1278 : }
3681 :
3682 : static long
3683 1326 : galoischar_dim(GEN ch)
3684 : {
3685 1326 : pari_sp av = avma;
3686 1326 : long d = gtos(simplify_shallow(lift_shallow(gel(ch,1))));
3687 1326 : return gc_long(av,d);
3688 : }
3689 :
3690 : static GEN
3691 10590 : galoischar_aut_charpoly(GEN cc, GEN ch, GEN p, long d)
3692 : {
3693 10590 : GEN q = p, V = cgetg(d+2, t_POL);
3694 : long i;
3695 10590 : V[1] = evalsigne(1)|evalvarn(0);
3696 22260 : for (i = 1; i <= d; i++)
3697 : {
3698 11670 : gel(V,i+1) = gel(ch, cc_id(cc,q));
3699 11670 : if (i < d) q = perm_mul(q, p);
3700 : }
3701 10590 : return liftpol_shallow(RgXn_expint(RgX_neg(V),d+1));
3702 : }
3703 :
3704 : static GEN
3705 1278 : galoischar_charpoly(GEN cc, GEN ch, long o)
3706 : {
3707 1278 : GEN chm, V, elts = gel(cc,1), repr = gel(cc,3);
3708 1278 : long i, d, l = lg(ch), v = gvar(ch);
3709 1278 : checkgaloischar(ch, repr);
3710 1278 : chm = v < 0 ? ch: gmodulo(ch, polcyclo(o, v));
3711 1278 : V = cgetg(l, t_COL); d = galoischar_dim(ch);
3712 11868 : for (i = 1; i < l; i++)
3713 10590 : gel(V,i) = galoischar_aut_charpoly(cc, chm, gel(elts,repr[i]), d);
3714 1278 : return V;
3715 : }
3716 :
3717 : GEN
3718 1230 : galoischarpoly(GEN gal, GEN ch, long o)
3719 : {
3720 1230 : pari_sp av = avma;
3721 1230 : GEN cc = group_to_cc(gal);
3722 1230 : return gc_GEN(av, galoischar_charpoly(cc, ch, o));
3723 : }
3724 :
3725 : static GEN
3726 48 : cc_char_det(GEN cc, GEN ch, long o)
3727 : {
3728 48 : long i, l = lg(ch), d = galoischar_dim(ch);
3729 48 : GEN V = galoischar_charpoly(cc, ch, o);
3730 240 : for (i = 1; i < l; i++) gel(V,i) = leading_coeff(gel(V,i));
3731 48 : return odd(d)? gneg(V): V;
3732 : }
3733 :
3734 : GEN
3735 48 : galoischardet(GEN gal, GEN ch, long o)
3736 : {
3737 48 : pari_sp av = avma;
3738 48 : GEN cc = group_to_cc(gal);
3739 48 : return gc_GEN(av, cc_char_det(cc, ch, o));
3740 : }
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