Line data Source code
1 : /* Copyright (C) 2000 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 : #include "pari.h"
15 : #include "paripriv.h"
16 :
17 : #define DEBUGLEVEL DEBUGLEVEL_bnf
18 :
19 : /*******************************************************************/
20 : /* */
21 : /* CLASS GROUP AND REGULATOR (McCURLEY, BUCHMANN) */
22 : /* GENERAL NUMBER FIELDS */
23 : /* */
24 : /*******************************************************************/
25 : /* get_random_ideal */
26 : static const long RANDOM_BITS = 4;
27 : /* Buchall */
28 : static const long RELSUP = 5;
29 : static const long FAIL_DIVISOR = 32;
30 : static const long MINFAIL = 10;
31 : /* small_norm */
32 : static const long BNF_RELPID = 4;
33 : static const long BMULT = 8;
34 : static const long maxtry_ELEMENT = 1000*1000;
35 : static const long maxtry_FACT = 500;
36 : /* rnd_rel */
37 : static const long RND_REL_RELPID = 1;
38 : /* random relations */
39 : static const long MINSFB = 3;
40 : static const long SFB_MAX = 3;
41 : static const long DEPSIZESFBMULT = 16;
42 : static const long DEPSFBDIV = 10;
43 : /* add_rel_i */
44 : static const ulong mod_p = 27449UL;
45 : /* be_honest */
46 : static const long maxtry_HONEST = 50;
47 :
48 : typedef struct FACT {
49 : long pr, ex;
50 : } FACT;
51 :
52 : typedef struct subFB_t {
53 : GEN subFB;
54 : struct subFB_t *old;
55 : } subFB_t;
56 :
57 : /* a factor base contains only noninert primes
58 : * KC = # of P in factor base (p <= n, NP <= n2)
59 : * KC2= # of P assumed to generate class group (NP <= n2)
60 : *
61 : * KCZ = # of rational primes under ideals counted by KC
62 : * KCZ2= same for KC2 */
63 :
64 : typedef struct FB_t {
65 : GEN FB; /* FB[i] = i-th rational prime used in factor base */
66 : GEN LP; /* vector of all prime ideals in FB */
67 : GEN LV; /* LV[p] = vector of P|p, NP <= n2
68 : * isclone() is set for LV[p] iff all P|p are in FB
69 : * LV[i], i not prime or i > n2, is undefined! */
70 : GEN iLP; /* iLP[p] = i such that LV[p] = [LP[i],...] */
71 : GEN L_jid; /* indexes of "useful" prime ideals for rnd_rel */
72 : long KC, KCZ, KCZ2;
73 : GEN prodZ; /* product of the primes in KCZ*/
74 : GEN subFB; /* LP o subFB = part of FB used to build random relations */
75 : int sfb_chg; /* need to change subFB ? */
76 : GEN perm; /* permutation of LP used to represent relations [updated by
77 : hnfspec/hnfadd: dense rows come first] */
78 : GEN idealperm; /* permutation of ideals under field automorphisms */
79 : GEN minidx; /* minidx[i] min ideal in orbit of LP[i] under field autom */
80 : subFB_t *allsubFB; /* all subFB's used */
81 : GEN embperm; /* permutations of the complex embeddings */
82 : long MAXDEPSIZESFB; /* # trials before increasing subFB */
83 : long MAXDEPSFB; /* MAXDEPSIZESFB / DEPSFBDIV, # trials befor rotating subFB */
84 : } FB_t;
85 :
86 : enum { sfb_CHANGE = 1, sfb_INCREASE = 2 };
87 :
88 : typedef struct REL_t {
89 : GEN R; /* relation vector as t_VECSMALL; clone */
90 : long nz; /* index of first nonzero elt in R (hash) */
91 : GEN m; /* pseudo-minimum yielding the relation; clone */
92 : long relorig; /* relation this one is an image of */
93 : long relaut; /* automorphim used to compute this relation from the original */
94 : GEN emb; /* archimedean embeddings */
95 : GEN junk[2]; /*make sure sizeof(struct) is a power of two.*/
96 : } REL_t;
97 :
98 : typedef struct RELCACHE_t {
99 : REL_t *chk; /* last checkpoint */
100 : REL_t *base; /* first rel found */
101 : REL_t *last; /* last rel found so far */
102 : REL_t *end; /* target for last relation. base <= last <= end */
103 : size_t len; /* number of rels pre-allocated in base */
104 : long relsup; /* how many linearly dependent relations we allow */
105 : GEN basis; /* mod p basis (generating family actually) */
106 : ulong missing; /* missing vectors in generating family above */
107 : } RELCACHE_t;
108 :
109 : typedef struct FP_t {
110 : double **q;
111 : GEN x;
112 : double *y;
113 : double *z;
114 : double *v;
115 : } FP_t;
116 :
117 : typedef struct RNDREL_t {
118 : long jid;
119 : GEN ex;
120 : } RNDREL_t;
121 :
122 : static void
123 0 : wr_rel(GEN e)
124 : {
125 0 : long i, l = lg(e);
126 0 : for (i = 1; i < l; i++)
127 0 : if (e[i]) err_printf("%ld^%ld ",i,e[i]);
128 0 : }
129 : static void
130 0 : dbg_newrel(RELCACHE_t *cache)
131 : {
132 0 : if (DEBUGLEVEL > 1)
133 : {
134 0 : err_printf("\n++++ cglob = %ld\nrel = ", cache->last - cache->base);
135 0 : wr_rel(cache->last->R);
136 0 : err_printf("\n");
137 : }
138 : else
139 0 : err_printf("%ld ", cache->last - cache->base);
140 0 : }
141 :
142 : static void
143 63603 : delete_cache(RELCACHE_t *M)
144 : {
145 : REL_t *rel;
146 1028259 : for (rel = M->base+1; rel <= M->last; rel++)
147 : {
148 964656 : gunclone(rel->R);
149 964659 : if (rel->m) gunclone(rel->m);
150 : }
151 63603 : pari_free((void*)M->base); M->base = NULL;
152 63603 : }
153 :
154 : static void
155 65779 : delete_FB(FB_t *F)
156 : {
157 : subFB_t *s, *sold;
158 132071 : for (s = F->allsubFB; s; s = sold) { sold = s->old; pari_free(s); }
159 65780 : gunclone(F->minidx);
160 65780 : gunclone(F->idealperm);
161 65780 : }
162 :
163 : static void
164 63688 : reallocate(RELCACHE_t *M, long len)
165 : {
166 63688 : M->len = len;
167 63688 : if (!M->base)
168 63602 : M->base = (REL_t*)pari_malloc((len+1) * sizeof(REL_t));
169 : else
170 : {
171 86 : size_t l = M->last - M->base, c = M->chk - M->base, e = M->end - M->base;
172 86 : pari_realloc_ip((void**)&M->base, (len+1) * sizeof(REL_t));
173 86 : M->last = M->base + l;
174 86 : M->chk = M->base + c;
175 86 : M->end = M->base + e;
176 : }
177 63689 : }
178 :
179 : #define pr_get_smallp(pr) gel(pr,1)[2]
180 :
181 : /* don't take P|p all other Q|p are already there */
182 : static int
183 271337 : bad_subFB(FB_t *F, long t)
184 : {
185 271337 : GEN LP, P = gel(F->LP,t);
186 271337 : long p = pr_get_smallp(P);
187 271337 : LP = gel(F->LV,p);
188 271337 : return (isclone(LP) && t == F->iLP[p] + lg(LP)-1);
189 : }
190 :
191 : static void
192 66292 : assign_subFB(FB_t *F, GEN yes, long iyes)
193 : {
194 66292 : long i, lv = sizeof(subFB_t) + iyes*sizeof(long); /* for struct + GEN */
195 66292 : subFB_t *s = (subFB_t *)pari_malloc(lv);
196 66293 : s->subFB = (GEN)&s[1];
197 66293 : s->old = F->allsubFB; F->allsubFB = s;
198 286012 : for (i = 0; i < iyes; i++) s->subFB[i] = yes[i];
199 66293 : F->subFB = s->subFB;
200 66293 : F->MAXDEPSIZESFB = (iyes-1) * DEPSIZESFBMULT;
201 66293 : F->MAXDEPSFB = F->MAXDEPSIZESFB / DEPSFBDIV;
202 66293 : }
203 :
204 : /* Determine the permutation of the ideals made by each field automorphism */
205 : static GEN
206 65779 : FB_aut_perm(FB_t *F, GEN auts, GEN cyclic)
207 : {
208 65779 : long i, j, m, KC = F->KC, nauts = lg(auts)-1;
209 65779 : GEN minidx, perm = zero_Flm_copy(KC, nauts);
210 :
211 65779 : if (!nauts) { F->minidx = gclone(identity_zv(KC)); return cgetg(1,t_MAT); }
212 41517 : minidx = zero_Flv(KC);
213 90292 : for (m = 1; m < lg(cyclic); m++)
214 : {
215 48776 : GEN thiscyc = gel(cyclic, m);
216 48776 : long k0 = thiscyc[1];
217 48776 : GEN aut = gel(auts, k0), permk0 = gel(perm, k0), ppermk;
218 48776 : i = 1;
219 209422 : while (i <= KC)
220 : {
221 160647 : pari_sp av2 = avma;
222 160647 : GEN seen = zero_Flv(KC), P = gel(F->LP, i);
223 160649 : long imin = i, p, f, l;
224 160649 : p = pr_get_smallp(P);
225 160649 : f = pr_get_f(P);
226 : do
227 : {
228 474036 : if (++i > KC) break;
229 425259 : P = gel(F->LP, i);
230 : }
231 425259 : while (p == pr_get_smallp(P) && f == pr_get_f(P));
232 634667 : for (j = imin; j < i; j++)
233 : {
234 474030 : GEN img = ZM_ZC_mul(aut, pr_get_gen(gel(F->LP, j)));
235 1655505 : for (l = imin; l < i; l++)
236 1655505 : if (!seen[l] && ZC_prdvd(img, gel(F->LP, l)))
237 : {
238 474018 : seen[l] = 1; permk0[j] = l; break;
239 : }
240 : }
241 160637 : set_avma(av2);
242 : }
243 67796 : for (ppermk = permk0, i = 2; i < lg(thiscyc); i++)
244 : {
245 19021 : GEN permk = gel(perm, thiscyc[i]);
246 382700 : for (j = 1; j <= KC; j++) permk[j] = permk0[ppermk[j]];
247 19021 : ppermk = permk;
248 : }
249 : }
250 306200 : for (j = 1; j <= KC; j++)
251 : {
252 264684 : if (minidx[j]) continue;
253 127256 : minidx[j] = j;
254 355291 : for (i = 1; i <= nauts; i++) minidx[coeff(perm, j, i)] = j;
255 : }
256 41516 : F->minidx = gclone(minidx); return perm;
257 : }
258 :
259 : /* set subFB.
260 : * Fill F->perm (if != NULL): primes ideals sorted by increasing norm (except
261 : * the ones in subFB come first [dense rows for hnfspec]) */
262 : static void
263 65776 : subFBgen(FB_t *F, GEN auts, GEN cyclic, double PROD, long minsFB)
264 : {
265 : GEN y, perm, yes, no;
266 65776 : long i, j, k, iyes, ino, lv = F->KC + 1;
267 : double prod;
268 : pari_sp av;
269 :
270 65776 : F->LP = cgetg(lv, t_VEC);
271 65779 : F->L_jid = F->perm = cgetg(lv, t_VECSMALL);
272 65779 : av = avma;
273 65779 : y = cgetg(lv,t_COL); /* Norm P */
274 309756 : for (k=0, i=1; i <= F->KCZ; i++)
275 : {
276 243977 : GEN LP = gel(F->LV,F->FB[i]);
277 243977 : long l = lg(LP);
278 705434 : for (j = 1; j < l; j++)
279 : {
280 461468 : GEN P = gel(LP,j);
281 461468 : k++;
282 461468 : gel(y,k) = pr_norm(P);
283 461457 : gel(F->LP,k) = P;
284 : }
285 : }
286 : /* perm sorts LP by increasing norm */
287 65779 : perm = indexsort(y);
288 65779 : no = cgetg(lv, t_VECSMALL); ino = 1;
289 65779 : yes = cgetg(lv, t_VECSMALL); iyes = 1;
290 65779 : prod = 1.0;
291 301199 : for (i = 1; i < lv; i++)
292 : {
293 271338 : long t = perm[i];
294 271338 : if (bad_subFB(F, t)) { no[ino++] = t; continue; }
295 :
296 151796 : yes[iyes++] = t;
297 151796 : prod *= (double)itos(gel(y,t));
298 151797 : if (iyes > minsFB && prod > PROD) break;
299 : }
300 65779 : setlg(yes, iyes);
301 217576 : for (j=1; j<iyes; j++) F->perm[j] = yes[j];
302 185321 : for (i=1; i<ino; i++, j++) F->perm[j] = no[i];
303 255938 : for ( ; j<lv; j++) F->perm[j] = perm[j];
304 65779 : F->allsubFB = NULL;
305 65779 : F->idealperm = gclone(FB_aut_perm(F, auts, cyclic));
306 65779 : if (iyes) assign_subFB(F, yes, iyes);
307 65780 : set_avma(av);
308 65780 : }
309 : static int
310 2400 : subFB_change(FB_t *F)
311 : {
312 2400 : long i, iyes, minsFB, lv = F->KC + 1, l = lg(F->subFB)-1;
313 2400 : pari_sp av = avma;
314 2400 : GEN yes, L_jid = F->L_jid, present = zero_zv(lv-1);
315 :
316 2400 : switch (F->sfb_chg)
317 : {
318 128 : case sfb_INCREASE: minsFB = l + 1; break;
319 2272 : default: minsFB = l; break;
320 : }
321 :
322 2400 : yes = cgetg(minsFB+1, t_VECSMALL); iyes = 1;
323 2400 : if (L_jid)
324 : {
325 7452 : for (i = 1; i < lg(L_jid); i++)
326 : {
327 6786 : long l = L_jid[i];
328 6786 : yes[iyes++] = l;
329 6786 : present[l] = 1;
330 6786 : if (iyes > minsFB) break;
331 : }
332 : }
333 0 : else i = 1;
334 2400 : if (iyes <= minsFB)
335 : {
336 968 : for ( ; i < lv; i++)
337 : {
338 954 : long l = F->perm[i];
339 954 : if (present[l]) continue;
340 755 : yes[iyes++] = l;
341 755 : if (iyes > minsFB) break;
342 : }
343 666 : if (i == lv) return 0;
344 : }
345 2386 : if (zv_equal(F->subFB, yes))
346 : {
347 1873 : if (DEBUGLEVEL) err_printf("\n*** NOT Changing sub factor base\n");
348 : }
349 : else
350 : {
351 513 : if (DEBUGLEVEL) err_printf("\n*** Changing sub factor base\n");
352 513 : assign_subFB(F, yes, iyes);
353 : }
354 2386 : F->sfb_chg = 0; return gc_bool(av, 1);
355 : }
356 :
357 : /* make sure enough room to store n more relations */
358 : static void
359 128545 : pre_allocate(RELCACHE_t *cache, size_t n)
360 : {
361 128545 : size_t len = (cache->last - cache->base) + n;
362 128545 : if (len >= cache->len) reallocate(cache, len << 1);
363 128545 : }
364 :
365 : void
366 133793 : init_GRHcheck(GRHcheck_t *S, long N, long R1, double LOGD)
367 : {
368 133793 : const double c1 = M_PI*M_PI/2;
369 133793 : const double c2 = 3.663862376709;
370 133793 : const double c3 = 3.801387092431; /* Euler + log(8*Pi)*/
371 133793 : S->clone = 0;
372 133793 : S->cN = R1*c2 + N*c1;
373 133793 : S->cD = LOGD - N*c3 - R1*M_PI/2;
374 133793 : S->maxprimes = 16000; /* sufficient for LIMC=176081*/
375 133793 : S->primes = (GRHprime_t*)pari_malloc(S->maxprimes*sizeof(*S->primes));
376 133796 : S->nprimes = 0;
377 133796 : S->limp = 0;
378 133796 : u_forprime_init(&S->P, 2, ULONG_MAX);
379 133794 : }
380 :
381 : void
382 133797 : free_GRHcheck(GRHcheck_t *S)
383 : {
384 133797 : if (S->clone)
385 : {
386 63560 : long i = S->nprimes;
387 : GRHprime_t *pr;
388 7519806 : for (pr = S->primes, i = S->nprimes; i > 0; pr++, i--) gunclone(pr->dec);
389 : }
390 133801 : pari_free(S->primes);
391 133797 : }
392 :
393 : int
394 1525495 : GRHok(GRHcheck_t *S, double L, double SA, double SB)
395 : {
396 1525495 : return (S->cD + (S->cN + 2*SB) / L - 2*SA < -1e-8);
397 : }
398 :
399 : /* Return factorization pattern of p: [f,n], where n[i] primes of
400 : * residue degree f[i] */
401 : static GEN
402 7451103 : get_fs(GEN nf, GEN P, GEN index, ulong p)
403 : {
404 : long j, k, f, n, l;
405 : GEN fs, ns;
406 :
407 7451103 : if (umodiu(index, p))
408 : { /* easy case: p does not divide index */
409 7413330 : GEN F = Flx_degfact(ZX_to_Flx(P,p), p);
410 7414245 : fs = gel(F,1); l = lg(fs);
411 : }
412 : else
413 : {
414 37654 : GEN F = idealprimedec(nf, utoipos(p));
415 37701 : l = lg(F);
416 37701 : fs = cgetg(l, t_VECSMALL);
417 118264 : for (j = 1; j < l; j++) fs[j] = pr_get_f(gel(F,j));
418 : }
419 7451946 : ns = cgetg(l, t_VECSMALL);
420 7448785 : f = fs[1]; n = 1;
421 13782504 : for (j = 2, k = 1; j < l; j++)
422 6333719 : if (fs[j] == f)
423 4612926 : n++;
424 : else
425 : {
426 1720793 : ns[k] = n; fs[k] = f; k++;
427 1720793 : f = fs[j]; n = 1;
428 : }
429 7448785 : ns[k] = n; fs[k] = f; k++;
430 7448785 : setlg(fs, k);
431 7448691 : setlg(ns, k); return mkvec2(fs,ns);
432 : }
433 :
434 : /* cache data for all rational primes up to the LIM */
435 : static void
436 914322 : cache_prime_dec(GRHcheck_t *S, ulong LIM, GEN nf)
437 : {
438 914322 : pari_sp av = avma;
439 : GRHprime_t *pr;
440 : GEN index, P;
441 : double nb;
442 :
443 914322 : if (S->limp >= LIM) return;
444 327213 : S->clone = 1;
445 327213 : nb = primepi_upper_bound((double)LIM); /* #{p <= LIM} <= nb */
446 327229 : GRH_ensure(S, nb+1); /* room for one extra prime */
447 327228 : P = nf_get_pol(nf);
448 327224 : index = nf_get_index(nf);
449 327222 : for (pr = S->primes + S->nprimes;;)
450 7124587 : {
451 7451809 : ulong p = u_forprime_next(&(S->P));
452 7450450 : pr->p = p;
453 7450450 : pr->logp = log((double)p);
454 7450450 : pr->dec = gclone(get_fs(nf, P, index, p));
455 7451912 : S->nprimes++;
456 7451912 : pr++;
457 7451912 : set_avma(av);
458 : /* store up to nextprime(LIM) included */
459 7451810 : if (p >= LIM) { S->limp = p; break; }
460 : }
461 : }
462 :
463 : static double
464 2243074 : tailresback(long R1, long R2, double rK, long C, double C2, double C3, double r1K, double r2K, double logC, double logC2, double logC3)
465 : {
466 2243074 : const double rQ = 1.83787706641;
467 2243074 : const double r1Q = 1.98505372441;
468 2243074 : const double r2Q = 1.07991541347;
469 4486148 : return fabs((R1+R2-1)*(12*logC3+4*logC2-9*logC-6)/(2*C*logC3)
470 2243074 : + (rK-rQ)*(6*logC2 + 5*logC + 2)/(C*logC3)
471 2243074 : - R2*(6*logC2+11*logC+6)/(C2*logC2)
472 2243074 : - 2*(r1K-r1Q)*(3*logC2 + 4*logC + 2)/(C2*logC3)
473 2243074 : + (R1+R2-1)*(12*logC3+40*logC2+45*logC+18)/(6*C3*logC3)
474 2243074 : + (r2K-r2Q)*(2*logC2 + 3*logC + 2)/(C3*logC3));
475 : }
476 :
477 : static double
478 1121541 : tailres(long R1, long R2, double al2K, double rKm, double rKM, double r1Km,
479 : double r1KM, double r2Km, double r2KM, double C, long i)
480 : {
481 : /* C >= 3*2^i, lower bound for eint1(log(C)/2) */
482 : /* for(i=0,30,print(eint1(log(3*2^i)/2))) */
483 : static double tab[] = {
484 : 0.50409264803,
485 : 0.26205336997,
486 : 0.14815491171,
487 : 0.08770540561,
488 : 0.05347651832,
489 : 0.03328934284,
490 : 0.02104510690,
491 : 0.01346475900,
492 : 0.00869778586,
493 : 0.00566279855,
494 : 0.00371111950,
495 : 0.00244567837,
496 : 0.00161948049,
497 : 0.00107686891,
498 : 0.00071868750,
499 : 0.00048119961,
500 : 0.00032312188,
501 : 0.00021753772,
502 : 0.00014679818,
503 : 9.9272855581E-5,
504 : 6.7263969995E-5,
505 : 4.5656812967E-5,
506 : 3.1041124593E-5,
507 : 2.1136011590E-5,
508 : 1.4411645381E-5,
509 : 9.8393304088E-6,
510 : 6.7257395409E-6,
511 : 4.6025878272E-6,
512 : 3.1529719271E-6,
513 : 2.1620490021E-6,
514 : 1.4839266071E-6
515 : };
516 1121541 : const double logC = log(C), logC2 = logC*logC, logC3 = logC*logC2;
517 1121541 : const double C2 = C*C, C3 = C*C2;
518 1121541 : double E1 = i >30? 0: tab[i];
519 1121541 : return al2K*((33*logC2+22*logC+8)/(8*logC3*sqrt(C))+15*E1/16)
520 1121541 : + maxdd(tailresback(rKm,r1KM,r2Km, C,C2,C3,R1,R2,logC,logC2,logC3),
521 1121548 : tailresback(rKM,r1Km,r2KM, C,C2,C3,R1,R2,logC,logC2,logC3))/2
522 1121548 : + ((R1+R2-1)*4*C+R2)*(C2+6*logC)/(4*C2*C2*logC2);
523 : }
524 :
525 : static long
526 63558 : primeneeded(long N, long R1, long R2, double LOGD)
527 : {
528 63558 : const double lim = 0.25; /* should be log(2)/2 == 0.34657... */
529 63558 : const double al2K = 0.3526*LOGD - 0.8212*N + 4.5007;
530 63558 : const double rKm = -1.0155*LOGD + 2.1042*N - 8.3419;
531 63558 : const double rKM = -0.5 *LOGD + 1.2076*N + 1;
532 63558 : const double r1Km = - LOGD + 1.4150*N;
533 63558 : const double r1KM = - LOGD + 1.9851*N;
534 63558 : const double r2Km = - LOGD + 0.9151*N;
535 63558 : const double r2KM = - LOGD + 1.0800*N;
536 63558 : long Cmin = 3, Cmax = 3, i = 0;
537 570199 : while (tailres(R1, R2, al2K, rKm, rKM, r1Km, r1KM, r2Km, r2KM, Cmax, i) > lim)
538 : {
539 506641 : Cmin = Cmax;
540 506641 : Cmax *= 2;
541 506641 : i++;
542 : }
543 63557 : i--;
544 614929 : while (Cmax - Cmin > 1)
545 : {
546 551370 : long t = (Cmin + Cmax)/2;
547 551370 : if (tailres(R1, R2, al2K, rKm, rKM, r1Km, r1KM, r2Km, r2KM, t, i) > lim)
548 341635 : Cmin = t;
549 : else
550 209737 : Cmax = t;
551 : }
552 63559 : return Cmax;
553 : }
554 :
555 : /* ~ 1 / Res(s = 1, zeta_K) */
556 : static GEN
557 63560 : compute_invres(GRHcheck_t *S, long LIMC)
558 : {
559 63560 : pari_sp av = avma;
560 63560 : double loginvres = 0.;
561 : GRHprime_t *pr;
562 : long i;
563 63560 : double logLIMC = log((double)LIMC);
564 63560 : double logLIMC2 = logLIMC*logLIMC, denc;
565 : double c0, c1, c2;
566 63560 : denc = 1/(pow((double)LIMC, 3.) * logLIMC * logLIMC2);
567 63560 : c2 = ( logLIMC2 + 3 * logLIMC / 2 + 1) * denc;
568 63560 : denc *= LIMC;
569 63560 : c1 = (3 * logLIMC2 + 4 * logLIMC + 2) * denc;
570 63560 : denc *= LIMC;
571 63560 : c0 = (3 * logLIMC2 + 5 * logLIMC / 2 + 1) * denc;
572 7463547 : for (pr = S->primes, i = S->nprimes; i > 0; pr++, i--)
573 : {
574 : GEN dec, fs, ns;
575 : long addpsi;
576 : double addpsi1, addpsi2;
577 7455679 : double logp = pr->logp, NPk;
578 7455679 : long j, k, limp = logLIMC/logp;
579 7455679 : ulong p = pr->p, p2 = p*p;
580 7455679 : if (limp < 1) break;
581 7399987 : dec = pr->dec;
582 7399987 : fs = gel(dec, 1); ns = gel(dec, 2);
583 7399987 : loginvres += 1./p;
584 : /* NB: limp = 1 nearly always and limp > 2 for very few primes */
585 8756678 : for (k=2, NPk = p; k <= limp; k++) { NPk *= p; loginvres += 1/(k * NPk); }
586 7399987 : addpsi = limp;
587 7399987 : addpsi1 = p *(pow((double)p , (double)limp)-1)/(p -1);
588 7399987 : addpsi2 = p2*(pow((double)p2, (double)limp)-1)/(p2-1);
589 7399987 : j = lg(fs);
590 16510195 : while (--j > 0)
591 : {
592 : long f, nb, kmax;
593 : double NP, NP2, addinvres;
594 9110208 : f = fs[j]; if (f > limp) continue;
595 3961148 : nb = ns[j];
596 3961148 : NP = pow((double)p, (double)f);
597 3961148 : addinvres = 1/NP;
598 3961148 : kmax = limp / f;
599 4833868 : for (k=2, NPk = NP; k <= kmax; k++) { NPk *= NP; addinvres += 1/(k*NPk); }
600 3961148 : NP2 = NP*NP;
601 3961148 : loginvres -= nb * addinvres;
602 3961148 : addpsi -= nb * f * kmax;
603 3961148 : addpsi1 -= nb*(f*NP *(pow(NP ,(double)kmax)-1)/(NP -1));
604 3961148 : addpsi2 -= nb*(f*NP2*(pow(NP2,(double)kmax)-1)/(NP2-1));
605 : }
606 7399987 : loginvres -= (addpsi*c0 - addpsi1*c1 + addpsi2*c2)*logp;
607 : }
608 63560 : return gerepileuptoleaf(av, mpexp(dbltor(loginvres)));
609 : }
610 :
611 : static long
612 63559 : nthideal(GRHcheck_t *S, GEN nf, long n)
613 : {
614 63559 : pari_sp av = avma;
615 63559 : GEN P = nf_get_pol(nf);
616 63558 : ulong p = 0, *vecN = (ulong*)const_vecsmall(n, LONG_MAX);
617 63558 : long i, N = poldegree(P, -1);
618 63557 : for (i = 0; ; i++)
619 228628 : {
620 : GRHprime_t *pr;
621 : GEN fs;
622 292185 : cache_prime_dec(S, p+1, nf);
623 292187 : pr = S->primes + i;
624 292187 : fs = gel(pr->dec, 1);
625 292187 : p = pr->p;
626 292187 : if (fs[1] != N)
627 : {
628 196047 : GEN ns = gel(pr->dec, 2);
629 196047 : long k, l, j = lg(fs);
630 439721 : while (--j > 0)
631 : {
632 243673 : ulong NP = upowuu(p, fs[j]);
633 : long nf;
634 243674 : if (!NP) continue;
635 747839 : for (k = 1; k <= n; k++) if (vecN[k] > NP) break;
636 243282 : if (k > n) continue;
637 : /* vecN[k] <= NP */
638 157643 : nf = ns[j]; /*#{primes of norme NP} = nf, insert them here*/
639 352799 : for (l = k+nf; l <= n; l++) vecN[l] = vecN[l-nf];
640 398174 : for (l = 0; l < nf && k+l <= n; l++) vecN[k+l] = NP;
641 362562 : while (l <= k) vecN[l++] = NP;
642 : }
643 : }
644 292188 : if (p > vecN[n]) break;
645 : }
646 63560 : return gc_long(av, vecN[n]);
647 : }
648 :
649 : /* Compute FB, LV, iLP + KC*. Reset perm
650 : * C2: bound for norm of tested prime ideals (includes be_honest())
651 : * C1: bound for p, such that P|p (NP <= C2) used to build relations */
652 : static void
653 65780 : FBgen(FB_t *F, GEN nf, long N, ulong C1, ulong C2, GRHcheck_t *S)
654 : {
655 : GRHprime_t *pr;
656 : long i, ip;
657 : GEN prim;
658 65780 : const double L = log((double)C2 + 0.5);
659 :
660 65780 : cache_prime_dec(S, C2, nf);
661 65780 : pr = S->primes;
662 65780 : F->sfb_chg = 0;
663 65780 : F->FB = cgetg(C2+1, t_VECSMALL);
664 65780 : F->iLP = cgetg(C2+1, t_VECSMALL);
665 65780 : F->LV = zerovec(C2);
666 :
667 65780 : prim = icopy(gen_1);
668 65781 : i = ip = 0;
669 65781 : F->KC = F->KCZ = 0;
670 432417 : for (;; pr++) /* p <= C2 */
671 432417 : {
672 498198 : ulong p = pr->p;
673 : long k, l, m;
674 : GEN LP, nb, f;
675 :
676 498198 : if (!F->KC && p > C1) { F->KCZ = i; F->KC = ip; }
677 498198 : if (p > C2) break;
678 :
679 461083 : if (DEBUGLEVEL>1) err_printf(" %ld",p);
680 :
681 461085 : f = gel(pr->dec, 1); nb = gel(pr->dec, 2);
682 461085 : if (f[1] == N)
683 : {
684 144961 : if (p == C2) break;
685 136449 : continue; /* p inert */
686 : }
687 316124 : l = (long)(L/pr->logp); /* p^f <= C2 <=> f <= l */
688 576795 : for (k=0, m=1; m < lg(f) && f[m]<=l; m++) k += nb[m];
689 316124 : if (!k)
690 : { /* too inert to appear in FB */
691 72125 : if (p == C2) break;
692 71495 : continue;
693 : }
694 243999 : prim[2] = p; LP = idealprimedec_limit_f(nf,prim, l);
695 : /* keep noninert ideals with Norm <= C2 */
696 243996 : if (m == lg(f)) setisclone(LP); /* flag it: all prime divisors in FB */
697 243996 : F->FB[++i]= p;
698 243996 : gel(F->LV,p) = LP;
699 243996 : F->iLP[p] = ip; ip += k;
700 243996 : if (p == C2) break;
701 : }
702 65780 : if (!F->KC) { F->KCZ = i; F->KC = ip; }
703 : /* Note F->KC > 0 otherwise GRHchk is false */
704 65780 : setlg(F->FB, F->KCZ+1); F->KCZ2 = i;
705 65780 : F->prodZ = zv_prod_Z(F->FB);
706 65775 : if (DEBUGLEVEL>1)
707 : {
708 0 : err_printf("\n");
709 0 : if (DEBUGLEVEL>6)
710 : {
711 0 : err_printf("########## FACTORBASE ##########\n\n");
712 0 : err_printf("KC2=%ld, KC=%ld, KCZ=%ld, KCZ2=%ld\n",
713 : ip, F->KC, F->KCZ, F->KCZ2);
714 0 : for (i=1; i<=F->KCZ; i++) err_printf("++ LV[%ld] = %Ps",i,gel(F->LV,F->FB[i]));
715 : }
716 : }
717 65775 : F->perm = NULL; F->L_jid = NULL;
718 65775 : }
719 :
720 : static int
721 492814 : GRHchk(GEN nf, GRHcheck_t *S, ulong LIMC)
722 : {
723 492814 : double logC = log((double)LIMC), SA = 0, SB = 0;
724 492814 : GRHprime_t *pr = S->primes;
725 :
726 492814 : cache_prime_dec(S, LIMC, nf);
727 492809 : for (pr = S->primes;; pr++)
728 3030831 : {
729 3523640 : ulong p = pr->p;
730 : GEN dec, fs, ns;
731 : double logCslogp;
732 : long j;
733 :
734 3523640 : if (p > LIMC) break;
735 3136355 : dec = pr->dec; fs = gel(dec, 1); ns = gel(dec,2);
736 3136355 : logCslogp = logC/pr->logp;
737 4937017 : for (j = 1; j < lg(fs); j++)
738 : {
739 3864471 : long f = fs[j], M, nb;
740 : double logNP, q, A, B;
741 3864471 : if (f > logCslogp) break;
742 1800658 : logNP = f * pr->logp;
743 1800658 : q = 1/sqrt((double)upowuu(p, f));
744 1800662 : A = logNP * q; B = logNP * A; M = (long)(logCslogp/f);
745 1800662 : if (M > 1)
746 : {
747 374065 : double inv1_q = 1 / (1-q);
748 374065 : A *= (1 - pow(q, (double)M)) * inv1_q;
749 374065 : B *= (1 - pow(q, (double)M)*(M+1 - M*q)) * inv1_q * inv1_q;
750 : }
751 1800662 : nb = ns[j];
752 1800662 : SA += nb * A;
753 1800662 : SB += nb * B;
754 : }
755 3136359 : if (p == LIMC) break;
756 : }
757 492813 : return GRHok(S, logC, SA, SB);
758 : }
759 :
760 : /* SMOOTH IDEALS */
761 : static void
762 12433454 : store(long i, long e, FACT *fact)
763 : {
764 12433454 : ++fact[0].pr;
765 12433454 : fact[fact[0].pr].pr = i; /* index */
766 12433454 : fact[fact[0].pr].ex = e; /* exponent */
767 12433454 : }
768 :
769 : /* divide out x by all P|p, where x as in can_factor(). k = v_p(Nx) */
770 : static int
771 7263089 : divide_p_elt(GEN LP, long ip, long k, GEN m, FACT *fact)
772 : {
773 7263089 : long j, l = lg(LP);
774 30535617 : for (j=1; j<l; j++)
775 : {
776 30503543 : GEN P = gel(LP,j);
777 30503543 : long v = ZC_nfval(m, P);
778 30501416 : if (!v) continue;
779 11687936 : store(ip + j, v, fact); /* v = v_P(m) > 0 */
780 11689352 : k -= v * pr_get_f(P);
781 11689617 : if (!k) return 1;
782 : }
783 32074 : return 0;
784 : }
785 : static int
786 163547 : divide_p_id(GEN LP, long ip, long k, GEN nf, GEN I, FACT *fact)
787 : {
788 163547 : long j, l = lg(LP);
789 246548 : for (j=1; j<l; j++)
790 : {
791 238708 : GEN P = gel(LP,j);
792 238708 : long v = idealval(nf,I, P);
793 238707 : if (!v) continue;
794 158946 : store(ip + j, v, fact); /* v = v_P(I) > 0 */
795 158947 : k -= v * pr_get_f(P);
796 158947 : if (!k) return 1;
797 : }
798 7840 : return 0;
799 : }
800 : static int
801 542471 : divide_p_quo(GEN LP, long ip, long k, GEN nf, GEN I, GEN m, FACT *fact)
802 : {
803 542471 : long j, l = lg(LP);
804 816866 : for (j=1; j<l; j++)
805 : {
806 816525 : GEN P = gel(LP,j);
807 816525 : long v = ZC_nfval(m, P);
808 816525 : if (!v) continue;
809 573527 : v -= idealval(nf,I, P);
810 573527 : if (!v) continue;
811 566861 : store(ip + j, v, fact); /* v = v_P(m / I) > 0 */
812 566861 : k -= v * pr_get_f(P);
813 566861 : if (!k) return 1;
814 : }
815 341 : return 0;
816 : }
817 :
818 : /* |*N| != 0 is the norm of a primitive ideal, in particular not divisible by
819 : * any inert prime. Is |*N| a smooth rational integer wrt F ?
820 : */
821 : static int
822 32717361 : Z_issmooth_prod(GEN N, GEN P)
823 : {
824 32717361 : P = gcdii(P,N);
825 101519000 : while (!is_pm1(P))
826 : {
827 68802374 : N = diviiexact(N, P);
828 68802246 : P = gcdii(N, P);
829 : }
830 32714144 : return is_pm1(N);
831 : }
832 :
833 : static int
834 7969033 : divide_p(FB_t *F, long p, long k, GEN nf, GEN I, GEN m, FACT *fact)
835 : {
836 7969033 : GEN LP = gel(F->LV,p);
837 7969033 : long ip = F->iLP[p];
838 7969033 : if (!m) return divide_p_id (LP,ip,k,nf,I,fact);
839 7805486 : if (!I) return divide_p_elt(LP,ip,k,m,fact);
840 542441 : return divide_p_quo(LP,ip,k,nf,I,m,fact);
841 : }
842 :
843 : /* Let x = m if I == NULL,
844 : * I if m == NULL,
845 : * m/I otherwise.
846 : * Can we factor the integral primitive ideal x ? |N| = Norm x > 0 */
847 : static long
848 33483654 : can_factor(FB_t *F, GEN nf, GEN I, GEN m, GEN N, FACT *fact)
849 : {
850 : GEN f, p, e;
851 : long i, l;
852 33483654 : fact[0].pr = 0;
853 33483654 : if (is_pm1(N)) return 1;
854 32717357 : if (!Z_issmooth_prod(N, F->prodZ)) return 0;
855 3927572 : f = absZ_factor(N); p = gel(f,1); e = gel(f,2); l = lg(p);
856 11857100 : for (i = 1; i < l; i++)
857 7969012 : if (!divide_p(F, itou(gel(p,i)), itou(gel(e,i)), nf, I, m, fact))
858 : {
859 39671 : if (DEBUGLEVEL > 1) err_printf(".");
860 39671 : return 0;
861 : }
862 3888088 : return 1;
863 : }
864 :
865 : /* can we factor m/I ? [m in I from idealpseudomin_nonscalar], NI = norm I */
866 : static long
867 1398349 : factorgen(FB_t *F, GEN nf, GEN I, GEN NI, GEN m, FACT *fact)
868 : {
869 1398349 : long e, r1 = nf_get_r1(nf);
870 1398350 : GEN M = nf_get_M(nf);
871 1398350 : GEN N = divri(embed_norm(RgM_RgC_mul(M,m), r1), NI); /* ~ N(m/I) */
872 1398352 : N = grndtoi(N, &e);
873 1398351 : if (e > -32)
874 : {
875 0 : if (DEBUGLEVEL > 1) err_printf("+");
876 0 : return 0;
877 : }
878 1398351 : return can_factor(F, nf, I, m, N, fact);
879 : }
880 :
881 : /* FUNDAMENTAL UNITS */
882 :
883 : /* a, y real. Return (Re(x) + a) + I * (Im(x) % y) */
884 : static GEN
885 6566689 : addRe_modIm(GEN x, GEN a, GEN y, GEN iy)
886 : {
887 : GEN z;
888 6566689 : if (typ(x) == t_COMPLEX)
889 : {
890 4556271 : GEN re, im = modRr_i(gel(x,2), y, iy);
891 4556193 : if (!im) return NULL;
892 4556192 : re = gadd(gel(x,1), a);
893 4556182 : z = gequal0(im)? re: mkcomplex(re, im);
894 : }
895 : else
896 2010418 : z = gadd(x, a);
897 6566631 : return z;
898 : }
899 : static GEN
900 200770 : modIm(GEN x, GEN y, GEN iy)
901 : {
902 200770 : if (typ(x) == t_COMPLEX)
903 : {
904 183847 : GEN im = modRr_i(gel(x,2), y, iy);
905 183834 : if (!im) return NULL;
906 183834 : x = gequal0(im)? gel(x,1): mkcomplex(gel(x,1), im);
907 : }
908 200761 : return x;
909 : }
910 :
911 : /* clean archimedean components. ipi = 2^n / pi (n arbitrary); its
912 : * exponent may be modified */
913 : static GEN
914 2922785 : cleanarch(GEN x, long N, GEN ipi, long prec)
915 : {
916 : long i, l, R1, RU;
917 2922785 : GEN s, y = cgetg_copy(x, &l);
918 :
919 2922799 : if (!ipi) ipi = invr(mppi(prec));
920 2922796 : if (typ(x) == t_MAT)
921 : {
922 523028 : for (i = 1; i < l; i++)
923 459322 : if (!(gel(y,i) = cleanarch(gel(x,i), N, ipi, prec))) return NULL;
924 63706 : return y;
925 : }
926 2859083 : RU = l-1; R1 = (RU<<1) - N;
927 2859083 : s = gdivgs(RgV_sum(real_i(x)), -N); /* -log |norm(x)| / N */
928 2859063 : i = 1;
929 2859063 : if (R1)
930 : {
931 2380316 : GEN pi2 = Pi2n(1, prec);
932 2380326 : setexpo(ipi, -3); /* 1/(2pi) */
933 7311902 : for (; i <= R1; i++)
934 4931605 : if (!(gel(y,i) = addRe_modIm(gel(x,i), s, pi2, ipi))) return NULL;
935 : }
936 2859044 : if (i <= RU)
937 : {
938 1075703 : GEN pi4 = Pi2n(2, prec), s2 = gmul2n(s, 1);
939 1075714 : setexpo(ipi, -4); /* 1/(4pi) */
940 2710808 : for (; i <= RU; i++)
941 1635085 : if (!(gel(y,i) = addRe_modIm(gel(x,i), s2, pi4, ipi))) return NULL;
942 : }
943 2859064 : return y;
944 : }
945 : GEN
946 195096 : nf_cxlog_normalize(GEN nf, GEN x, long prec)
947 : {
948 195096 : long N = nf_get_degree(nf);
949 195096 : return cleanarch(x, N, NULL, prec);
950 : }
951 :
952 : /* clean unit archimedean components. ipi = 2^n / pi (n arbitrary); its
953 : * exponent may be modified */
954 : static GEN
955 132404 : cleanarchunit(GEN x, long N, GEN ipi, long prec)
956 : {
957 : long i, l, R1, RU;
958 132404 : GEN y = cgetg_copy(x, &l);
959 :
960 132406 : if (!ipi) ipi = invr(mppi(prec));
961 132406 : if (typ(x) == t_MAT)
962 : {
963 132400 : for (i = 1; i < l; i++)
964 68841 : if (!(gel(y,i) = cleanarchunit(gel(x,i), N, ipi, prec))) return NULL;
965 63559 : return y;
966 : }
967 68842 : if (gexpo(RgV_sum(real_i(x))) > -10) return NULL;
968 68832 : RU = l-1; R1 = (RU<<1) - N;
969 68832 : i = 1;
970 68832 : if (R1)
971 : {
972 54430 : GEN pi2 = Pi2n(1, prec);
973 54432 : setexpo(ipi, -3); /* 1/(2pi) */
974 184805 : for (; i <= R1; i++)
975 130382 : if (!(gel(y,i) = modIm(gel(x,i), pi2, ipi))) return NULL;
976 : }
977 68825 : if (i <= RU)
978 : {
979 34354 : GEN pi4 = Pi2n(2, prec);
980 34354 : setexpo(ipi, -4); /* 1/(4pi) */
981 104753 : for (; i <= RU; i++)
982 70388 : if (!(gel(y,i) = modIm(gel(x,i), pi4, ipi))) return NULL;
983 : }
984 68836 : return y;
985 : }
986 :
987 : static GEN
988 388 : not_given(long reason)
989 : {
990 388 : if (DEBUGLEVEL)
991 0 : switch(reason)
992 : {
993 0 : case fupb_LARGE:
994 0 : pari_warn(warner,"fundamental units too large, not given");
995 0 : break;
996 0 : case fupb_PRECI:
997 0 : pari_warn(warner,"insufficient precision for fundamental units, not given");
998 0 : break;
999 : }
1000 388 : return NULL;
1001 : }
1002 :
1003 : /* check whether exp(x) will 1) get too big (real(x) large), 2) require
1004 : * large accuracy for argument reduction (imag(x) large) */
1005 : static long
1006 2674728 : expbitprec(GEN x, long *e)
1007 : {
1008 : GEN re, im;
1009 2674728 : if (typ(x) != t_COMPLEX) re = x;
1010 : else
1011 : {
1012 1685268 : im = gel(x,2); *e = maxss(*e, expo(im) + 5 - bit_prec(im));
1013 1685269 : re = gel(x,1);
1014 : }
1015 2674729 : return (expo(re) <= 20);
1016 :
1017 : }
1018 : static long
1019 1165828 : RgC_expbitprec(GEN x)
1020 : {
1021 1165828 : long l = lg(x), i, e = - (long)HIGHEXPOBIT;
1022 3639454 : for (i = 1; i < l; i++)
1023 2474073 : if (!expbitprec(gel(x,i), &e)) return LONG_MAX;
1024 1165381 : return e;
1025 : }
1026 : static long
1027 48377 : RgM_expbitprec(GEN x)
1028 : {
1029 48377 : long i, j, I, J, e = - (long)HIGHEXPOBIT;
1030 48377 : RgM_dimensions(x, &I,&J);
1031 117145 : for (j = 1; j <= J; j++)
1032 269423 : for (i = 1; i <= I; i++)
1033 200655 : if (!expbitprec(gcoeff(x,i,j), &e)) return LONG_MAX;
1034 48321 : return e;
1035 : }
1036 :
1037 : static GEN
1038 1345 : FlxqX_chinese_unit(GEN X, GEN U, GEN invzk, GEN D, GEN T, ulong p)
1039 : {
1040 1345 : long i, lU = lg(U), lX = lg(X), d = lg(invzk)-1;
1041 1345 : GEN M = cgetg(lU, t_MAT);
1042 1345 : if (D)
1043 : {
1044 1228 : D = Flv_inv(D, p);
1045 69240 : for (i = 1; i < lX; i++)
1046 68012 : if (uel(D, i) != 1)
1047 57030 : gel(X,i) = Flx_Fl_mul(gel(X,i), uel(D,i), p);
1048 : }
1049 3808 : for (i = 1; i < lU; i++)
1050 : {
1051 2463 : GEN H = FlxqV_factorback(X, gel(U, i), T, p);
1052 2463 : gel(M, i) = Flm_Flc_mul(invzk, Flx_to_Flv(H, d), p);
1053 : }
1054 1345 : return M;
1055 : }
1056 :
1057 : static GEN
1058 271 : chinese_unit_slice(GEN A, GEN U, GEN B, GEN D, GEN C, GEN P, GEN *mod)
1059 : {
1060 271 : pari_sp av = avma;
1061 271 : long i, n = lg(P)-1, v = varn(C);
1062 : GEN H, T;
1063 271 : if (n == 1)
1064 : {
1065 0 : ulong p = uel(P,1);
1066 0 : GEN a = ZXV_to_FlxV(A, p), b = ZM_to_Flm(B, p), c = ZX_to_Flx(C, p);
1067 0 : GEN d = D ? ZV_to_Flv(D, p): NULL;
1068 0 : GEN Hp = FlxqX_chinese_unit(a, U, b, d, c, p);
1069 0 : H = gerepileupto(av, Flm_to_ZM(Hp));
1070 0 : *mod = utoi(p);
1071 0 : return H;
1072 : }
1073 271 : T = ZV_producttree(P);
1074 271 : A = ZXC_nv_mod_tree(A, P, T, v);
1075 271 : B = ZM_nv_mod_tree(B, P, T);
1076 271 : D = D ? ZV_nv_mod_tree(D, P, T): NULL;
1077 271 : C = ZX_nv_mod_tree(C, P, T);
1078 :
1079 271 : H = cgetg(n+1, t_VEC);
1080 1616 : for(i=1; i <= n; i++)
1081 : {
1082 1345 : ulong p = P[i];
1083 1345 : GEN a = gel(A,i), b = gel(B,i), c = gel(C,i), d = D ? gel(D,i): NULL;
1084 1345 : gel(H,i) = FlxqX_chinese_unit(a, U, b, d, c, p);
1085 : }
1086 271 : H = nmV_chinese_center_tree_seq(H, P, T, ZV_chinesetree(P, T));
1087 271 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
1088 : }
1089 :
1090 : GEN
1091 271 : chinese_unit_worker(GEN P, GEN A, GEN U, GEN B, GEN D, GEN C)
1092 : {
1093 271 : GEN V = cgetg(3, t_VEC);
1094 271 : gel(V,1) = chinese_unit_slice(A, U, B, isintzero(D) ? NULL: D, C, P, &gel(V,2));
1095 271 : return V;
1096 : }
1097 :
1098 : /* Let x = \prod X[i]^E[i] = u, return u.
1099 : * If dX != NULL, X[i] = nX[i] / dX[i] where nX[i] is a ZX, dX[i] in Z */
1100 : static GEN
1101 100 : chinese_unit(GEN nf, GEN nX, GEN dX, GEN U, ulong bnd)
1102 : {
1103 100 : pari_sp av = avma;
1104 100 : GEN f = nf_get_index(nf), T = nf_get_pol(nf), invzk = nf_get_invzk(nf);
1105 : GEN H, mod;
1106 : forprime_t S;
1107 100 : GEN worker = snm_closure(is_entry("_chinese_unit_worker"),
1108 : mkcol5(nX, U, invzk, dX? dX: gen_0, T));
1109 100 : init_modular_big(&S);
1110 100 : H = gen_crt("chinese_units", worker, &S, f, bnd, 0, &mod, nmV_chinese_center, FpM_center);
1111 100 : settyp(H, t_VEC); return gerepilecopy(av, H);
1112 : }
1113 :
1114 : /* *pE a ZM */
1115 : static void
1116 170 : ZM_remove_unused(GEN *pE, GEN *pX)
1117 : {
1118 170 : long j, k, l = lg(*pX);
1119 170 : GEN E = *pE, v = cgetg(l, t_VECSMALL);
1120 19295 : for (j = k = 1; j < l; j++)
1121 19125 : if (!ZMrow_equal0(E, j)) v[k++] = j;
1122 170 : if (k < l)
1123 : {
1124 170 : setlg(v, k);
1125 170 : *pX = vecpermute(*pX,v);
1126 170 : *pE = rowpermute(E,v);
1127 : }
1128 170 : }
1129 :
1130 : /* s = -log|norm(x)|/N */
1131 : static GEN
1132 1234647 : fixarch(GEN x, GEN s, long R1)
1133 : {
1134 : long i, l;
1135 1234647 : GEN y = cgetg_copy(x, &l);
1136 3408387 : for (i = 1; i <= R1; i++) gel(y,i) = gadd(s, gel(x,i));
1137 1736180 : for ( ; i < l; i++) gel(y,i) = gadd(s, gmul2n(gel(x,i),-1));
1138 1234655 : return y;
1139 : }
1140 :
1141 : static GEN
1142 63559 : getfu(GEN nf, GEN *ptA, GEN *ptU, long prec)
1143 : {
1144 63559 : GEN U, y, matep, A, T = nf_get_pol(nf), M = nf_get_M(nf);
1145 63559 : long e, j, R1, RU, N = degpol(T);
1146 :
1147 63559 : R1 = nf_get_r1(nf); RU = (N+R1) >> 1;
1148 63560 : if (RU == 1) return cgetg(1,t_VEC);
1149 :
1150 48376 : A = *ptA;
1151 48376 : matep = cgetg(RU,t_MAT);
1152 117205 : for (j = 1; j < RU; j++)
1153 : {
1154 68831 : GEN Aj = gel(A,j), s = gdivgs(RgV_sum(real_i(Aj)), -N);
1155 68830 : gel(matep,j) = fixarch(Aj, s, R1);
1156 : }
1157 48374 : U = lll(real_i(matep));
1158 48376 : if (lg(U) < RU) return not_given(fupb_PRECI);
1159 48376 : if (ptU) { *ptU = U; *ptA = A = RgM_ZM_mul(A,U); }
1160 48377 : y = RgM_ZM_mul(matep,U);
1161 48377 : e = RgM_expbitprec(y);
1162 48377 : if (e >= 0) return not_given(e == LONG_MAX? fupb_LARGE: fupb_PRECI);
1163 48321 : if (prec <= 0) prec = gprecision(A);
1164 48321 : y = RgM_solve_realimag(M, gexp(y,prec));
1165 48321 : if (!y) return not_given(fupb_PRECI);
1166 48321 : y = grndtoi(y, &e); if (e >= 0) return not_given(fupb_PRECI);
1167 48010 : settyp(y, t_VEC);
1168 :
1169 48010 : if (!ptU) *ptA = A = RgM_ZM_mul(A, U);
1170 116108 : for (j = 1; j < RU; j++)
1171 : { /* y[i] are hopefully unit generators. Normalize: smallest T2 norm */
1172 68119 : GEN u = gel(y,j), v = zk_inv(nf, u);
1173 68119 : if (!v || !is_pm1(Q_denom(v)) || ZV_isscalar(u))
1174 21 : return not_given(fupb_PRECI);
1175 68098 : if (gcmp(RgC_fpnorml2(v,DEFAULTPREC), RgC_fpnorml2(u,DEFAULTPREC)) < 0)
1176 : {
1177 27823 : gel(A,j) = RgC_neg(gel(A,j));
1178 27823 : if (ptU) gel(U,j) = ZC_neg(gel(U,j));
1179 27823 : u = v;
1180 : }
1181 68098 : gel(y,j) = nf_to_scalar_or_alg(nf, u);
1182 : }
1183 47989 : return y;
1184 : }
1185 :
1186 : static void
1187 0 : err_units() { pari_err_PREC("makeunits [cannot get units, use bnfinit(,1)]"); }
1188 :
1189 : /* bound for log2 |sigma(u)|, sigma complex embedding, u fundamental unit
1190 : * attached to bnf_get_logfu */
1191 : static double
1192 100 : log2fubound(GEN bnf)
1193 : {
1194 100 : GEN LU = bnf_get_logfu(bnf);
1195 100 : long i, j, l = lg(LU), r1 = nf_get_r1(bnf_get_nf(bnf));
1196 100 : double e = 0.0;
1197 355 : for (j = 1; j < l; j++)
1198 : {
1199 255 : GEN u = gel(LU,j);
1200 637 : for (i = 1; i <= r1; i++)
1201 : {
1202 382 : GEN E = real_i(gel(u,i));
1203 382 : e = maxdd(e, gtodouble(E));
1204 : }
1205 945 : for ( ; i <= l; i++)
1206 : {
1207 690 : GEN E = real_i(gel(u,i));
1208 690 : e = maxdd(e, gtodouble(E) / 2);
1209 : }
1210 : }
1211 100 : return e / M_LN2;
1212 : }
1213 : /* bound for log2(|RgM_solve_realimag(M, y)|_oo / |y|_oo)*/
1214 : static double
1215 100 : log2Mbound(GEN nf)
1216 : {
1217 100 : GEN G = nf_get_G(nf), D = nf_get_disc(nf);
1218 100 : long r2 = nf_get_r2(nf), l = lg(G), i;
1219 100 : double e, d = dbllog2(D)/2 - r2 * M_LN2; /* log2 |det(split_realimag(M))| */
1220 100 : e = log2(nf_get_degree(nf));
1221 588 : for (i = 2; i < l; i++) e += dbllog2(gnorml2(gel(G,i))); /* Hadamard bound */
1222 100 : return e / 2 - d;
1223 : }
1224 :
1225 : static GEN
1226 100 : vec_chinese_units(GEN bnf)
1227 : {
1228 100 : GEN nf = bnf_get_nf(bnf), SUnits = bnf_get_sunits(bnf);
1229 100 : double bnd = ceil(log2Mbound(nf) + log2fubound(bnf));
1230 100 : GEN X, dX, Y, U, f = nf_get_index(nf);
1231 100 : long j, l, v = nf_get_varn(nf);
1232 100 : if (!SUnits) err_units(); /* no compact units */
1233 100 : Y = gel(SUnits,1);
1234 100 : U = gel(SUnits,2);
1235 100 : ZM_remove_unused(&U, &Y); l = lg(Y); X = cgetg(l, t_VEC);
1236 100 : if (is_pm1(f)) f = dX = NULL; else dX = cgetg(l, t_VEC);
1237 6280 : for (j = 1; j < l; j++)
1238 : {
1239 6180 : GEN t = nf_to_scalar_or_alg(nf, gel(Y,j));
1240 6180 : if (f)
1241 : {
1242 : GEN den;
1243 5260 : t = Q_remove_denom(t, &den);
1244 5260 : gel(dX,j) = den ? den: gen_1;
1245 : }
1246 6180 : gel(X,j) = typ(t) == t_INT? scalarpol_shallow(t,v): t;
1247 : }
1248 100 : if (dblexpo(bnd) >= BITS_IN_LONG)
1249 0 : pari_err_OVERFLOW("vec_chinese_units [units too large]");
1250 100 : return chinese_unit(nf, X, dX, U, (ulong)bnd);
1251 : }
1252 :
1253 : static GEN
1254 24901 : makeunits(GEN bnf)
1255 : {
1256 24901 : GEN nf = bnf_get_nf(bnf), fu = bnf_get_fu_nocheck(bnf);
1257 24901 : GEN tu = nf_to_scalar_or_basis(nf, bnf_get_tuU(bnf));
1258 24901 : fu = (typ(fu) == t_MAT)? vec_chinese_units(bnf): matalgtobasis(nf, fu);
1259 24901 : return vec_prepend(fu, tu);
1260 : }
1261 :
1262 : /*******************************************************************/
1263 : /* */
1264 : /* PRINCIPAL IDEAL ALGORITHM (DISCRETE LOG) */
1265 : /* */
1266 : /*******************************************************************/
1267 :
1268 : /* G: prime ideals, E: vector of nonnegative exponents.
1269 : * C = possible extra prime (^1) or NULL
1270 : * Return Norm (product) */
1271 : static GEN
1272 77 : get_norm_fact_primes(GEN G, GEN E, GEN C)
1273 : {
1274 77 : pari_sp av=avma;
1275 77 : GEN N = gen_1, P, p;
1276 77 : long i, c = lg(E);
1277 77 : for (i=1; i<c; i++)
1278 : {
1279 0 : GEN ex = gel(E,i);
1280 0 : long s = signe(ex);
1281 0 : if (!s) continue;
1282 :
1283 0 : P = gel(G,i); p = pr_get_p(P);
1284 0 : N = mulii(N, powii(p, mului(pr_get_f(P), ex)));
1285 : }
1286 77 : if (C) N = mulii(N, pr_norm(C));
1287 77 : return gerepileuptoint(av, N);
1288 : }
1289 :
1290 : /* gen: HNF ideals */
1291 : static GEN
1292 1160193 : get_norm_fact(GEN gen, GEN ex, GEN *pd)
1293 : {
1294 1160193 : long i, c = lg(ex);
1295 : GEN d,N,I,e,n,ne,de;
1296 1160193 : d = N = gen_1;
1297 1462406 : for (i=1; i<c; i++)
1298 302214 : if (signe(gel(ex,i)))
1299 : {
1300 181683 : I = gel(gen,i); e = gel(ex,i); n = ZM_det_triangular(I);
1301 181683 : ne = powii(n,e);
1302 181683 : de = equalii(n, gcoeff(I,1,1))? ne: powii(gcoeff(I,1,1), e);
1303 181683 : N = mulii(N, ne);
1304 181683 : d = mulii(d, de);
1305 : }
1306 1160192 : *pd = d; return N;
1307 : }
1308 :
1309 : static GEN
1310 1321111 : get_pr_lists(GEN FB, long N, int list_pr)
1311 : {
1312 : GEN pr, L;
1313 1321111 : long i, l = lg(FB), p, pmax;
1314 :
1315 1321111 : pmax = 0;
1316 9227828 : for (i=1; i<l; i++)
1317 : {
1318 7906717 : pr = gel(FB,i); p = pr_get_smallp(pr);
1319 7906717 : if (p > pmax) pmax = p;
1320 : }
1321 1321111 : L = const_vec(pmax, NULL);
1322 1321116 : if (list_pr)
1323 : {
1324 0 : for (i=1; i<l; i++)
1325 : {
1326 0 : pr = gel(FB,i); p = pr_get_smallp(pr);
1327 0 : if (!L[p]) gel(L,p) = vectrunc_init(N+1);
1328 0 : vectrunc_append(gel(L,p), pr);
1329 : }
1330 0 : for (p=1; p<=pmax; p++)
1331 0 : if (L[p]) gen_sort_inplace(gel(L,p), (void*)&cmp_prime_over_p,
1332 : &cmp_nodata, NULL);
1333 : }
1334 : else
1335 : {
1336 9227842 : for (i=1; i<l; i++)
1337 : {
1338 7906726 : pr = gel(FB,i); p = pr_get_smallp(pr);
1339 7906726 : if (!L[p]) gel(L,p) = vecsmalltrunc_init(N+1);
1340 7906725 : vecsmalltrunc_append(gel(L,p), i);
1341 : }
1342 : }
1343 1321116 : return L;
1344 : }
1345 :
1346 : /* recover FB, LV, iLP, KCZ from Vbase */
1347 : static GEN
1348 1321111 : recover_partFB(FB_t *F, GEN Vbase, long N)
1349 : {
1350 1321111 : GEN FB, LV, iLP, L = get_pr_lists(Vbase, N, 0);
1351 1321117 : long l = lg(L), p, ip, i;
1352 :
1353 1321117 : i = ip = 0;
1354 1321117 : FB = cgetg(l, t_VECSMALL);
1355 1321116 : iLP= cgetg(l, t_VECSMALL);
1356 1321116 : LV = cgetg(l, t_VEC);
1357 20140425 : for (p = 2; p < l; p++)
1358 : {
1359 18819308 : if (!L[p]) continue;
1360 4318541 : FB[++i] = p;
1361 4318541 : gel(LV,p) = vecpermute(Vbase, gel(L,p));
1362 4318541 : iLP[p]= ip; ip += lg(gel(L,p))-1;
1363 : }
1364 1321117 : F->KCZ = i;
1365 1321117 : F->KC = ip;
1366 1321117 : F->FB = FB; setlg(FB, i+1);
1367 1321117 : F->prodZ = zv_prod_Z(F->FB);
1368 1321113 : F->LV = LV;
1369 1321113 : F->iLP= iLP; return L;
1370 : }
1371 :
1372 : /* add v^e to factorization */
1373 : static void
1374 19919 : add_to_fact(long v, long e, FACT *fact)
1375 : {
1376 19919 : long i, l = fact[0].pr;
1377 34685 : for (i=1; i<=l && fact[i].pr < v; i++)/*empty*/;
1378 19919 : if (i <= l && fact[i].pr == v) fact[i].ex += e; else store(v, e, fact);
1379 19919 : }
1380 : static void
1381 0 : inv_fact(FACT *fact)
1382 : {
1383 0 : long i, l = fact[0].pr;
1384 0 : for (i=1; i<=l; i++) fact[i].ex = -fact[i].ex;
1385 0 : }
1386 :
1387 : /* L (small) list of primes above the same p including pr. Return pr index */
1388 : static int
1389 3250 : pr_index(GEN L, GEN pr)
1390 : {
1391 3250 : long j, l = lg(L);
1392 3250 : GEN al = pr_get_gen(pr);
1393 3251 : for (j=1; j<l; j++)
1394 3251 : if (ZV_equal(al, pr_get_gen(gel(L,j)))) return j;
1395 0 : pari_err_BUG("codeprime");
1396 : return 0; /* LCOV_EXCL_LINE */
1397 : }
1398 :
1399 : static long
1400 3250 : Vbase_to_FB(FB_t *F, GEN pr)
1401 : {
1402 3250 : long p = pr_get_smallp(pr);
1403 3250 : return F->iLP[p] + pr_index(gel(F->LV,p), pr);
1404 : }
1405 :
1406 : /* x, y 2 extended ideals whose first component is an integral HNF and second
1407 : * a famat */
1408 : static GEN
1409 3476 : idealHNF_mulred(GEN nf, GEN x, GEN y)
1410 : {
1411 3476 : GEN A = idealHNF_mul(nf, gel(x,1), gel(y,1));
1412 3476 : GEN F = famat_mul_shallow(gel(x,2), gel(y,2));
1413 3476 : return idealred(nf, mkvec2(A, F));
1414 : }
1415 : /* idealred(x * pr^n), n > 0 is small, x extended ideal. Reduction in order to
1416 : * avoid prec pb: don't let id become too large as lgsub increases */
1417 : static GEN
1418 4510 : idealmulpowprime2(GEN nf, GEN x, GEN pr, ulong n)
1419 : {
1420 4510 : GEN A = idealmulpowprime(nf, gel(x,1), pr, utoipos(n));
1421 4510 : return mkvec2(A, gel(x,2));
1422 : }
1423 : static GEN
1424 65258 : init_famat(GEN x) { return mkvec2(x, trivial_fact()); }
1425 : /* optimized idealfactorback + reduction; z = init_famat() */
1426 : static GEN
1427 28711 : genback(GEN z, GEN nf, GEN P, GEN E)
1428 : {
1429 28711 : long i, l = lg(E);
1430 28711 : GEN I = NULL;
1431 75993 : for (i = 1; i < l; i++)
1432 47279 : if (signe(gel(E,i)))
1433 : {
1434 : GEN J;
1435 32187 : gel(z,1) = gel(P,i);
1436 32187 : J = idealpowred(nf, z, gel(E,i));
1437 32190 : I = I? idealHNF_mulred(nf, I, J): J;
1438 : }
1439 28714 : return I; /* != NULL since a generator */
1440 : }
1441 :
1442 : /* return famat y (principal ideal) such that y / x is smooth [wrt Vbase] */
1443 : static GEN
1444 1337459 : SPLIT(FB_t *F, GEN nf, GEN x, GEN Vbase, FACT *fact)
1445 : {
1446 1337459 : GEN vecG, ex, Ly, y, x0, Nx = ZM_det_triangular(x);
1447 : long nbtest_lim, nbtest, i, j, k, ru, lgsub;
1448 : pari_sp av;
1449 :
1450 : /* try without reduction if x is small */
1451 2674721 : if (gexpo(gcoeff(x,1,1)) < 100 &&
1452 1487420 : can_factor(F, nf, x, NULL, Nx, fact)) return NULL;
1453 :
1454 1187303 : av = avma;
1455 1187303 : Ly = idealpseudominvec(x, nf_get_roundG(nf));
1456 1236350 : for(k=1; k<lg(Ly); k++)
1457 : {
1458 1227320 : y = gel(Ly,k);
1459 1227320 : if (factorgen(F, nf, x, Nx, y, fact)) return y;
1460 : }
1461 9030 : set_avma(av);
1462 :
1463 : /* reduce in various directions */
1464 9030 : ru = lg(nf_get_roots(nf));
1465 9030 : vecG = cgetg(ru, t_VEC);
1466 14469 : for (j=1; j<ru; j++)
1467 : {
1468 12770 : gel(vecG,j) = nf_get_Gtwist1(nf, j);
1469 12770 : av = avma;
1470 12770 : Ly = idealpseudominvec(x, gel(vecG,j));
1471 42867 : for(k=1; k<lg(Ly); k++)
1472 : {
1473 37428 : y = gel(Ly,k);
1474 37428 : if (factorgen(F, nf, x, Nx, y, fact)) return y;
1475 : }
1476 5439 : set_avma(av);
1477 : }
1478 :
1479 : /* tough case, multiply by random products */
1480 1699 : lgsub = 3;
1481 1699 : ex = cgetg(lgsub, t_VECSMALL);
1482 1699 : x0 = init_famat(x);
1483 1699 : nbtest = 1; nbtest_lim = 4;
1484 : for(;;)
1485 630 : {
1486 2329 : GEN Ired, I, NI, id = x0;
1487 2329 : av = avma;
1488 2329 : if (DEBUGLEVEL>2) err_printf("# ideals tried = %ld\n",nbtest);
1489 7125 : for (i=1; i<lgsub; i++)
1490 : {
1491 4796 : ex[i] = random_bits(RANDOM_BITS);
1492 4796 : if (ex[i]) id = idealmulpowprime2(nf, id, gel(Vbase,i), ex[i]);
1493 : }
1494 2329 : if (id == x0) continue;
1495 : /* I^(-1) * \prod Vbase[i]^ex[i] = (id[2]) / x */
1496 :
1497 2329 : I = gel(id,1); NI = ZM_det_triangular(I);
1498 2329 : if (can_factor(F, nf, I, NULL, NI, fact))
1499 : {
1500 0 : inv_fact(fact); /* I^(-1) */
1501 0 : for (i=1; i<lgsub; i++)
1502 0 : if (ex[i]) add_to_fact(Vbase_to_FB(F,gel(Vbase,i)), ex[i], fact);
1503 0 : return gel(id,2);
1504 : }
1505 2329 : Ired = ru == 2? I: ZM_lll(I, 0.99, LLL_INPLACE);
1506 3997 : for (j=1; j<ru; j++)
1507 : {
1508 3367 : pari_sp av2 = avma;
1509 3367 : Ly = idealpseudominvec(Ired, gel(vecG,j));
1510 13674 : for (k=1; k < lg(Ly); k++)
1511 : {
1512 12006 : y = gel(Ly,k);
1513 12006 : if (factorgen(F, nf, I, NI, y, fact))
1514 : {
1515 5130 : for (i=1; i<lgsub; i++)
1516 3431 : if (ex[i]) add_to_fact(Vbase_to_FB(F,gel(Vbase,i)), ex[i], fact);
1517 1699 : return famat_mul_shallow(gel(id,2), y);
1518 : }
1519 : }
1520 1668 : set_avma(av2);
1521 : }
1522 630 : set_avma(av);
1523 630 : if (++nbtest > nbtest_lim)
1524 : {
1525 33 : nbtest = 0;
1526 33 : if (++lgsub < minss(8, lg(Vbase)-1))
1527 : {
1528 33 : nbtest_lim <<= 1;
1529 33 : ex = cgetg(lgsub, t_VECSMALL);
1530 : }
1531 0 : else nbtest_lim = LONG_MAX; /* don't increase further */
1532 33 : if (DEBUGLEVEL>2) err_printf("SPLIT: increasing factor base [%ld]\n",lgsub);
1533 : }
1534 : }
1535 : }
1536 :
1537 : INLINE GEN
1538 1326038 : bnf_get_W(GEN bnf) { return gel(bnf,1); }
1539 : INLINE GEN
1540 2642115 : bnf_get_B(GEN bnf) { return gel(bnf,2); }
1541 : INLINE GEN
1542 2671292 : bnf_get_C(GEN bnf) { return gel(bnf,4); }
1543 : INLINE GEN
1544 1321132 : bnf_get_vbase(GEN bnf) { return gel(bnf,5); }
1545 : INLINE GEN
1546 1321050 : bnf_get_Ur(GEN bnf) { return gmael(bnf,9,1); }
1547 : INLINE GEN
1548 277614 : bnf_get_ga(GEN bnf) { return gmael(bnf,9,2); }
1549 : INLINE GEN
1550 282583 : bnf_get_GD(GEN bnf) { return gmael(bnf,9,3); }
1551 :
1552 : /* Return y (as an elt of K or a t_MAT representing an elt in Z[K])
1553 : * such that x / (y) is smooth and store the exponents of its factorization
1554 : * on g_W and g_B in Wex / Bex; return NULL for y = 1 */
1555 : static GEN
1556 1321048 : split_ideal(GEN bnf, GEN x, GEN *pWex, GEN *pBex)
1557 : {
1558 1321048 : GEN L, y, Vbase = bnf_get_vbase(bnf);
1559 1321048 : GEN Wex, W = bnf_get_W(bnf);
1560 1321048 : GEN Bex, B = bnf_get_B(bnf);
1561 : long p, j, i, l, nW, nB;
1562 : FACT *fact;
1563 : FB_t F;
1564 :
1565 1321048 : L = recover_partFB(&F, Vbase, lg(x)-1);
1566 1321050 : fact = (FACT*)stack_malloc((F.KC+1)*sizeof(FACT));
1567 1321051 : y = SPLIT(&F, bnf_get_nf(bnf), x, Vbase, fact);
1568 1321051 : nW = lg(W)-1; *pWex = Wex = zero_zv(nW);
1569 1321051 : nB = lg(B)-1; *pBex = Bex = zero_zv(nB); l = lg(F.FB);
1570 1321051 : p = j = 0; /* -Wall */
1571 1980376 : for (i = 1; i <= fact[0].pr; i++)
1572 : { /* decode index C = ip+j --> (p,j) */
1573 659325 : long a, b, t, C = fact[i].pr;
1574 1946190 : for (t = 1; t < l; t++)
1575 : {
1576 1871476 : long q = F.FB[t], k = C - F.iLP[q];
1577 1871476 : if (k <= 0) break;
1578 1286865 : p = q;
1579 1286865 : j = k;
1580 : }
1581 659325 : a = gel(L, p)[j];
1582 659325 : b = a - nW;
1583 659325 : if (b <= 0) Wex[a] = y? -fact[i].ex: fact[i].ex;
1584 506452 : else Bex[b] = y? -fact[i].ex: fact[i].ex;
1585 : }
1586 1321051 : return y;
1587 : }
1588 :
1589 : GEN
1590 1038758 : init_red_mod_units(GEN bnf, long prec)
1591 : {
1592 1038758 : GEN s = gen_0, p1,s1,mat, logfu = bnf_get_logfu(bnf);
1593 1038758 : long i,j, RU = lg(logfu);
1594 :
1595 1038758 : if (RU == 1) return NULL;
1596 1038758 : mat = cgetg(RU,t_MAT);
1597 2347466 : for (j=1; j<RU; j++)
1598 : {
1599 1308708 : p1 = cgetg(RU+1,t_COL); gel(mat,j) = p1;
1600 1308708 : s1 = gen_0;
1601 3228045 : for (i=1; i<RU; i++)
1602 : {
1603 1919338 : gel(p1,i) = real_i(gcoeff(logfu,i,j));
1604 1919338 : s1 = mpadd(s1, mpsqr(gel(p1,i)));
1605 : }
1606 1308707 : gel(p1,RU) = gen_0; if (mpcmp(s1,s) > 0) s = s1;
1607 : }
1608 1038758 : s = gsqrt(gmul2n(s,RU),prec);
1609 1038758 : if (expo(s) < 27) s = utoipos(1UL << 27);
1610 1038758 : return mkvec2(mat, s);
1611 : }
1612 :
1613 : /* z computed above. Return unit exponents that would reduce col (arch) */
1614 : GEN
1615 1038758 : red_mod_units(GEN col, GEN z)
1616 : {
1617 : long i,RU;
1618 : GEN x,mat,N2;
1619 :
1620 1038758 : if (!z) return NULL;
1621 1038758 : mat= gel(z,1);
1622 1038758 : N2 = gel(z,2);
1623 1038758 : RU = lg(mat); x = cgetg(RU+1,t_COL);
1624 2347466 : for (i=1; i<RU; i++) gel(x,i) = real_i(gel(col,i));
1625 1038758 : gel(x,RU) = N2;
1626 1038758 : x = lll(shallowconcat(mat,x));
1627 1038758 : if (typ(x) != t_MAT || lg(x) <= RU) return NULL;
1628 1038758 : x = gel(x,RU);
1629 1038758 : if (signe(gel(x,RU)) < 0) x = gneg_i(x);
1630 1038758 : if (!gequal1(gel(x,RU))) pari_err_BUG("red_mod_units");
1631 1038758 : setlg(x,RU); return x;
1632 : }
1633 :
1634 : static GEN
1635 2158915 : add(GEN a, GEN t) { return a = a? RgC_add(a,t): t; }
1636 :
1637 : /* [x] archimedian components, A column vector. return [x] A */
1638 : static GEN
1639 1996037 : act_arch(GEN A, GEN x)
1640 : {
1641 : GEN a;
1642 1996037 : long i,l = lg(A), tA = typ(A);
1643 1996037 : if (tA == t_MAT)
1644 : { /* assume lg(x) >= l */
1645 190968 : a = cgetg(l, t_MAT);
1646 280620 : for (i=1; i<l; i++) gel(a,i) = act_arch(gel(A,i), x);
1647 190970 : return a;
1648 : }
1649 1805069 : if (l==1) return cgetg(1, t_COL);
1650 1805069 : a = NULL;
1651 1805069 : if (tA == t_VECSMALL)
1652 : {
1653 6871712 : for (i=1; i<l; i++)
1654 : {
1655 5711519 : long c = A[i];
1656 5711519 : if (c) a = add(a, gmulsg(c, gel(x,i)));
1657 : }
1658 : }
1659 : else
1660 : { /* A a t_COL of t_INT. Assume lg(A)==lg(x) */
1661 1403012 : for (i=1; i<l; i++)
1662 : {
1663 758138 : GEN c = gel(A,i);
1664 758138 : if (signe(c)) a = add(a, gmul(c, gel(x,i)));
1665 : }
1666 : }
1667 1805067 : return a? a: zerocol(lgcols(x)-1);
1668 : }
1669 : /* act_arch(matdiagonal(v), x) */
1670 : static GEN
1671 63657 : diagact_arch(GEN v, GEN x)
1672 : {
1673 63657 : long i, l = lg(v);
1674 63657 : GEN a = cgetg(l, t_MAT);
1675 92441 : for (i = 1; i < l; i++) gel(a,i) = gmul(gel(x,i), gel(v,i));
1676 63657 : return a;
1677 : }
1678 :
1679 : static long
1680 1336435 : prec_arch(GEN bnf)
1681 : {
1682 1336435 : GEN a = bnf_get_C(bnf);
1683 1336435 : long i, l = lg(a), prec;
1684 :
1685 1336435 : for (i=1; i<l; i++)
1686 1336351 : if ( (prec = gprecision(gel(a,i))) ) return prec;
1687 84 : return DEFAULTPREC;
1688 : }
1689 :
1690 : static long
1691 3925 : needed_bitprec(GEN x)
1692 : {
1693 3925 : long i, e = 0, l = lg(x);
1694 22629 : for (i = 1; i < l; i++)
1695 : {
1696 18704 : GEN c = gel(x,i);
1697 18704 : long f = gexpo(c) - prec2nbits(gprecision(c));
1698 18704 : if (f > e) e = f;
1699 : }
1700 3925 : return e;
1701 : }
1702 :
1703 : /* col = archimedian components of x, Nx its norm, dx a multiple of its
1704 : * denominator. Return x or NULL (fail) */
1705 : GEN
1706 1165825 : isprincipalarch(GEN bnf, GEN col, GEN kNx, GEN e, GEN dx, long *pe)
1707 : {
1708 : GEN nf, x, y, logfu, s, M;
1709 1165825 : long N, prec = gprecision(col);
1710 1165825 : bnf = checkbnf(bnf); nf = bnf_get_nf(bnf); M = nf_get_M(nf);
1711 1165826 : if (!prec) prec = prec_arch(bnf);
1712 1165826 : *pe = 128;
1713 1165826 : logfu = bnf_get_logfu(bnf);
1714 1165826 : N = nf_get_degree(nf);
1715 1165826 : if (!(col = cleanarch(col,N,NULL,prec))) return NULL;
1716 1165824 : if (lg(col) > 2)
1717 : { /* reduce mod units */
1718 1038758 : GEN u, z = init_red_mod_units(bnf,prec);
1719 1038758 : if (!(u = red_mod_units(col,z))) return NULL;
1720 1038758 : col = RgC_add(col, RgM_RgC_mul(logfu, u));
1721 1038757 : if (!(col = cleanarch(col,N,NULL,prec))) return NULL;
1722 : }
1723 1165824 : s = divru(mulir(e, glog(kNx,prec)), N);
1724 1165815 : col = fixarch(col, s, nf_get_r1(nf));
1725 1165827 : if (RgC_expbitprec(col) >= 0) return NULL;
1726 1165380 : col = gexp(col, prec);
1727 : /* d.alpha such that x = alpha \prod gj^ej */
1728 1165381 : x = RgM_solve_realimag(M,col); if (!x) return NULL;
1729 1165379 : x = RgC_Rg_mul(x, dx);
1730 1165375 : y = grndtoi(x, pe);
1731 1165379 : if (*pe > -5) { *pe = needed_bitprec(x); return NULL; }
1732 1161454 : return RgC_Rg_div(y, dx);
1733 : }
1734 :
1735 : /* y = C \prod g[i]^e[i] ? */
1736 : static int
1737 1155433 : fact_ok(GEN nf, GEN y, GEN C, GEN g, GEN e)
1738 : {
1739 1155433 : pari_sp av = avma;
1740 1155433 : long i, c = lg(e);
1741 1155433 : GEN z = C? C: gen_1;
1742 1435974 : for (i=1; i<c; i++)
1743 280541 : if (signe(gel(e,i))) z = idealmul(nf, z, idealpow(nf, gel(g,i), gel(e,i)));
1744 1155433 : if (typ(z) != t_MAT) z = idealhnf_shallow(nf,z);
1745 1155434 : if (typ(y) != t_MAT) y = idealhnf_shallow(nf,y);
1746 1155434 : return gc_bool(av, ZM_equal(y,z));
1747 : }
1748 : static GEN
1749 1321051 : ZV_divrem(GEN A, GEN B, GEN *pR)
1750 : {
1751 1321051 : long i, l = lg(A);
1752 1321051 : GEN Q = cgetg(l, t_COL), R = cgetg(l, t_COL);
1753 1832975 : for (i = 1; i < l; i++) gel(Q,i) = truedvmdii(gel(A,i), gel(B,i), &gel(R,i));
1754 1321052 : *pR = R; return Q;
1755 : }
1756 :
1757 : static GEN
1758 1321049 : Ur_ZC_mul(GEN bnf, GEN v)
1759 : {
1760 1321049 : GEN w, U = bnf_get_Ur(bnf);
1761 1321050 : long i, l = lg(bnf_get_cyc(bnf)); /* may be < lgcols(U) */
1762 :
1763 1321050 : w = cgetg(l, t_COL);
1764 1832975 : for (i = 1; i < l; i++) gel(w,i) = ZMrow_ZC_mul(U, v, i);
1765 1321052 : return w;
1766 : }
1767 :
1768 : static GEN
1769 9184 : ZV_mul(GEN x, GEN y)
1770 : {
1771 9184 : long i, l = lg(x);
1772 9184 : GEN z = cgetg(l, t_COL);
1773 35631 : for (i = 1; i < l; i++) gel(z,i) = mulii(gel(x,i), gel(y,i));
1774 9184 : return z;
1775 : }
1776 : static int
1777 1156786 : dump_gen(GEN SUnits, GEN x, long flag)
1778 : {
1779 : GEN d;
1780 : long e;
1781 1156786 : if (!(flag & nf_GENMAT) || !SUnits) return 0;
1782 274588 : e = gexpo(gel(SUnits,2)); if (e > 64) return 0; /* U large */
1783 274395 : x = Q_remove_denom(x, &d);
1784 274394 : return (d && expi(d) > 32) || gexpo(x) > 32;
1785 : }
1786 :
1787 : /* assume x in HNF; cf class_group_gen for notations. Return NULL iff
1788 : * flag & nf_FORCE and computation of principal ideal generator fails */
1789 : static GEN
1790 1334837 : isprincipalall(GEN bnf, GEN x, long *pprec, long flag)
1791 : {
1792 : GEN xar, Wex, Bex, gen, xc, col, A, Q, R, UA, SUnits;
1793 1334837 : GEN C = bnf_get_C(bnf), nf = bnf_get_nf(bnf), cyc = bnf_get_cyc(bnf);
1794 : long nB, nW, e;
1795 :
1796 1334837 : if (lg(cyc) == 1 && !(flag & (nf_GEN|nf_GENMAT|nf_GEN_IF_PRINCIPAL)))
1797 4725 : return cgetg(1,t_COL);
1798 1330112 : if (lg(x) == 2)
1799 : { /* nf = Q */
1800 84 : col = gel(x,1);
1801 84 : if (flag & nf_GENMAT) col = to_famat_shallow(col, gen_1);
1802 84 : return (flag & nf_GEN_IF_PRINCIPAL)? col: mkvec2(cgetg(1,t_COL), col);
1803 : }
1804 :
1805 1330028 : x = Q_primitive_part(x, &xc);
1806 1330023 : if (equali1(gcoeff(x,1,1))) /* trivial ideal */
1807 : {
1808 8974 : R = zerocol(lg(cyc)-1);
1809 8974 : if (!(flag & (nf_GEN|nf_GENMAT|nf_GEN_IF_PRINCIPAL))) return R;
1810 8960 : if (flag & nf_GEN_IF_PRINCIPAL)
1811 6454 : return scalarcol_shallow(xc? xc: gen_1, nf_get_degree(nf));
1812 2506 : if (flag & nf_GENMAT)
1813 1869 : col = xc? to_famat_shallow(xc, gen_1): trivial_fact();
1814 : else
1815 637 : col = scalarcol_shallow(xc? xc: gen_1, nf_get_degree(nf));
1816 2506 : return mkvec2(R, col);
1817 : }
1818 1321048 : xar = split_ideal(bnf, x, &Wex, &Bex);
1819 : /* x = g_W Wex + g_B Bex + [xar] = g_W (Wex - B*Bex) + [xar] + [C_B]Bex */
1820 1321051 : A = zc_to_ZC(Wex); nB = lg(Bex)-1;
1821 1321050 : if (nB) A = ZC_sub(A, ZM_zc_mul(bnf_get_B(bnf), Bex));
1822 1321049 : UA = Ur_ZC_mul(bnf, A);
1823 1321052 : Q = ZV_divrem(UA, cyc, &R);
1824 : /* g_W (Wex - B*Bex) = G Ur A - [ga]A = G R + [GD]Q - [ga]A
1825 : * Finally: x = G R + [xar] + [C_B]Bex + [GD]Q - [ga]A */
1826 1321052 : if (!(flag & (nf_GEN|nf_GENMAT|nf_GEN_IF_PRINCIPAL))) return R;
1827 1160765 : if ((flag & nf_GEN_IF_PRINCIPAL) && !ZV_equal0(R)) return gen_0;
1828 :
1829 1160758 : nW = lg(Wex)-1;
1830 1160758 : gen = bnf_get_gen(bnf);
1831 1160758 : col = NULL;
1832 1160758 : SUnits = bnf_get_sunits(bnf);
1833 1160758 : if (lg(R) == 1
1834 278179 : || abscmpiu(gel(R,vecindexmax(R)), 4 * bit_accuracy(*pprec)) < 0)
1835 : { /* q = N (x / prod gj^ej) = N(alpha), denom(alpha) | d */
1836 1160193 : GEN d, q = gdiv(ZM_det_triangular(x), get_norm_fact(gen, R, &d));
1837 1160192 : col = xar? nf_cxlog(nf, xar, *pprec): NULL;
1838 1160194 : if (nB) col = add(col, act_arch(Bex, nW? vecslice(C,nW+1,lg(C)-1): C));
1839 1160191 : if (nW) col = add(col, RgC_sub(act_arch(Q, bnf_get_GD(bnf)),
1840 : act_arch(A, bnf_get_ga(bnf))));
1841 1160190 : col = isprincipalarch(bnf, col, q, gen_1, d, &e);
1842 1160195 : if (col && (dump_gen(SUnits, col, flag)
1843 1156785 : || !fact_ok(nf,x, col,gen,R))) col = NULL;
1844 : }
1845 1160759 : if (!col && (flag & nf_GENMAT))
1846 : {
1847 9910 : if (SUnits)
1848 : {
1849 9420 : GEN X = gel(SUnits,1), U = gel(SUnits,2), C = gel(SUnits,3);
1850 9420 : GEN v = gel(bnf,9), Ge = gel(v,4), M1 = gel(v,5), M2 = gel(v,6);
1851 9420 : GEN z = NULL, F = NULL;
1852 9420 : if (nB)
1853 : {
1854 9420 : GEN C2 = nW? vecslice(C, nW+1, lg(C)-1): C;
1855 9420 : z = ZM_zc_mul(C2, Bex);
1856 : }
1857 9420 : if (nW)
1858 : { /* [GD]Q - [ga]A = ([X]M1 - [Ge]D) Q - ([X]M2 - [Ge]Ur) A */
1859 9184 : GEN C1 = vecslice(C, 1, nW);
1860 9184 : GEN v = ZC_sub(ZM_ZC_mul(M1,Q), ZM_ZC_mul(M2,A));
1861 9184 : z = add(z, ZM_ZC_mul(C1, v));
1862 9184 : F = famat_reduce(famatV_factorback(Ge, ZC_sub(UA, ZV_mul(cyc,Q))));
1863 9184 : if (lgcols(F) == 1) F = NULL;
1864 : }
1865 : /* reduce modulo units and Q^* */
1866 9420 : if (lg(U) != 1) z = ZC_sub(z, ZM_ZC_mul(U, RgM_Babai(U,z)));
1867 9420 : col = mkmat2(X, z);
1868 9420 : if (F) col = famat_mul_shallow(col, F);
1869 9420 : col = famat_remove_trivial(col);
1870 9420 : if (xar) col = famat_mul_shallow(col, xar);
1871 : }
1872 490 : else if (!ZV_equal0(R))
1873 : { /* in case isprincipalfact calls bnfinit() due to prec trouble...*/
1874 483 : GEN y = isprincipalfact(bnf, x, gen, ZC_neg(R), flag);
1875 483 : if (typ(y) != t_VEC) return y;
1876 483 : col = gel(y,2);
1877 : }
1878 : }
1879 1160759 : if (col)
1880 : { /* add back missing content */
1881 1161214 : if (xc) col = (typ(col)==t_MAT)? famat_mul_shallow(col,xc)
1882 546 : : RgC_Rg_mul(col,xc);
1883 1160668 : if (typ(col) != t_MAT && lg(col) != 1 && (flag & nf_GENMAT))
1884 1137093 : col = to_famat_shallow(col, gen_1);
1885 : }
1886 : else
1887 : {
1888 91 : if (e < 0) e = 0;
1889 91 : *pprec += nbits2extraprec(e + 128);
1890 91 : if (flag & nf_FORCE)
1891 : {
1892 77 : if (DEBUGLEVEL)
1893 0 : pari_warn(warner,"precision too low for generators, e = %ld",e);
1894 77 : return NULL;
1895 : }
1896 14 : pari_warn(warner,"precision too low for generators, not given");
1897 14 : col = cgetg(1, t_COL);
1898 : }
1899 1160682 : return (flag & nf_GEN_IF_PRINCIPAL)? col: mkvec2(R, col);
1900 : }
1901 :
1902 : static GEN
1903 460942 : triv_gen(GEN bnf, GEN x, long flag)
1904 : {
1905 460942 : pari_sp av = avma;
1906 460942 : GEN nf = bnf_get_nf(bnf);
1907 : long c;
1908 460942 : if (flag & nf_GEN_IF_PRINCIPAL)
1909 : {
1910 7 : if (!(flag & nf_GENMAT)) return algtobasis(nf,x);
1911 7 : x = nf_to_scalar_or_basis(nf,x);
1912 7 : if (typ(x) == t_INT && is_pm1(x)) return trivial_fact();
1913 0 : return gerepilecopy(av, to_famat_shallow(x, gen_1));
1914 : }
1915 460935 : c = lg(bnf_get_cyc(bnf)) - 1;
1916 460935 : if (flag & nf_GENMAT)
1917 451338 : retmkvec2(zerocol(c), to_famat_shallow(algtobasis(nf,x), gen_1));
1918 9597 : if (flag & nf_GEN)
1919 21 : retmkvec2(zerocol(c), algtobasis(nf,x));
1920 9576 : return zerocol(c);
1921 : }
1922 :
1923 : GEN
1924 1766069 : bnfisprincipal0(GEN bnf,GEN x,long flag)
1925 : {
1926 1766069 : pari_sp av = avma;
1927 : GEN c, nf;
1928 : long pr;
1929 :
1930 1766069 : bnf = checkbnf(bnf);
1931 1766069 : nf = bnf_get_nf(bnf);
1932 1766069 : switch( idealtyp(&x, NULL) )
1933 : {
1934 56070 : case id_PRINCIPAL:
1935 56070 : if (gequal0(x)) pari_err_DOMAIN("bnfisprincipal","ideal","=",gen_0,x);
1936 56070 : return triv_gen(bnf, x, flag);
1937 1686319 : case id_PRIME:
1938 1686319 : if (pr_is_inert(x)) return triv_gen(bnf, pr_get_p(x), flag);
1939 1281454 : x = pr_hnf(nf, x);
1940 1281453 : break;
1941 23681 : case id_MAT:
1942 23681 : if (lg(x)==1) pari_err_DOMAIN("bnfisprincipal","ideal","=",gen_0,x);
1943 23681 : if (nf_get_degree(nf) != lg(x)-1)
1944 0 : pari_err_TYPE("idealtyp [dimension != degree]", x);
1945 : }
1946 1305133 : pr = prec_arch(bnf); /* precision of unit matrix */
1947 1305134 : c = getrand();
1948 : for (;;)
1949 7 : {
1950 1305144 : pari_sp av1 = avma;
1951 1305144 : GEN y = isprincipalall(bnf,x,&pr,flag);
1952 1305142 : if (y) return gerepilecopy(av, y);
1953 :
1954 7 : if (DEBUGLEVEL) pari_warn(warnprec,"isprincipal",pr);
1955 7 : set_avma(av1); bnf = bnfnewprec_shallow(bnf,pr); setrand(c);
1956 : }
1957 : }
1958 : GEN
1959 174553 : isprincipal(GEN bnf,GEN x) { return bnfisprincipal0(bnf,x,0); }
1960 :
1961 : /* FIXME: OBSOLETE */
1962 : GEN
1963 0 : isprincipalgen(GEN bnf,GEN x)
1964 0 : { return bnfisprincipal0(bnf,x,nf_GEN); }
1965 : GEN
1966 0 : isprincipalforce(GEN bnf,GEN x)
1967 0 : { return bnfisprincipal0(bnf,x,nf_FORCE); }
1968 : GEN
1969 0 : isprincipalgenforce(GEN bnf,GEN x)
1970 0 : { return bnfisprincipal0(bnf,x,nf_GEN | nf_FORCE); }
1971 :
1972 : /* lg(u) > 1 */
1973 : static int
1974 91 : RgV_is1(GEN u) { return isint1(gel(u,1)) && RgV_isscalar(u); }
1975 : static GEN
1976 29623 : add_principal_part(GEN nf, GEN u, GEN v, long flag)
1977 : {
1978 29623 : if (flag & nf_GENMAT)
1979 14237 : return (typ(u) == t_COL && RgV_is1(u))? v: famat_mul_shallow(v,u);
1980 : else
1981 15386 : return nfmul(nf, v, u);
1982 : }
1983 :
1984 : #if 0
1985 : /* compute C prod P[i]^e[i], e[i] >=0 for all i. C may be NULL (omitted)
1986 : * e destroyed ! */
1987 : static GEN
1988 : expand(GEN nf, GEN C, GEN P, GEN e)
1989 : {
1990 : long i, l = lg(e), done = 1;
1991 : GEN id = C;
1992 : for (i=1; i<l; i++)
1993 : {
1994 : GEN ei = gel(e,i);
1995 : if (signe(ei))
1996 : {
1997 : if (mod2(ei)) id = id? idealmul(nf, id, gel(P,i)): gel(P,i);
1998 : ei = shifti(ei,-1);
1999 : if (signe(ei)) done = 0;
2000 : gel(e,i) = ei;
2001 : }
2002 : }
2003 : if (id != C) id = idealred(nf, id);
2004 : if (done) return id;
2005 : return idealmulred(nf, id, idealsqr(nf, expand(nf,id,P,e)));
2006 : }
2007 : /* C is an extended ideal, possibly with C[1] = NULL */
2008 : static GEN
2009 : expandext(GEN nf, GEN C, GEN P, GEN e)
2010 : {
2011 : long i, l = lg(e), done = 1;
2012 : GEN A = gel(C,1);
2013 : for (i=1; i<l; i++)
2014 : {
2015 : GEN ei = gel(e,i);
2016 : if (signe(ei))
2017 : {
2018 : if (mod2(ei)) A = A? idealmul(nf, A, gel(P,i)): gel(P,i);
2019 : ei = shifti(ei,-1);
2020 : if (signe(ei)) done = 0;
2021 : gel(e,i) = ei;
2022 : }
2023 : }
2024 : if (A == gel(C,1))
2025 : A = C;
2026 : else
2027 : A = idealred(nf, mkvec2(A, gel(C,2)));
2028 : if (done) return A;
2029 : return idealmulred(nf, A, idealsqr(nf, expand(nf,A,P,e)));
2030 : }
2031 : #endif
2032 :
2033 : static GEN
2034 0 : expand(GEN nf, GEN C, GEN P, GEN e)
2035 : {
2036 0 : long i, l = lg(e);
2037 0 : GEN B, A = C;
2038 0 : for (i=1; i<l; i++) /* compute prod P[i]^e[i] */
2039 0 : if (signe(gel(e,i)))
2040 : {
2041 0 : B = idealpowred(nf, gel(P,i), gel(e,i));
2042 0 : A = A? idealmulred(nf,A,B): B;
2043 : }
2044 0 : return A;
2045 : }
2046 : static GEN
2047 29636 : expandext(GEN nf, GEN C, GEN P, GEN e)
2048 : {
2049 29636 : long i, l = lg(e);
2050 29636 : GEN B, A = gel(C,1), C1 = A;
2051 89542 : for (i=1; i<l; i++) /* compute prod P[i]^e[i] */
2052 59905 : if (signe(gel(e,i)))
2053 : {
2054 32363 : gel(C,1) = gel(P,i);
2055 32363 : B = idealpowred(nf, C, gel(e,i));
2056 32364 : A = A? idealmulred(nf,A,B): B;
2057 : }
2058 29637 : return A == C1? C: A;
2059 : }
2060 :
2061 : /* isprincipal for C * \prod P[i]^e[i] (C omitted if NULL) */
2062 : GEN
2063 29637 : isprincipalfact(GEN bnf, GEN C, GEN P, GEN e, long flag)
2064 : {
2065 29637 : const long gen = flag & (nf_GEN|nf_GENMAT|nf_GEN_IF_PRINCIPAL);
2066 : long prec;
2067 29637 : pari_sp av = avma;
2068 29637 : GEN C0, Cext, c, id, nf = bnf_get_nf(bnf);
2069 :
2070 29637 : if (gen)
2071 : {
2072 14244 : Cext = (flag & nf_GENMAT)? trivial_fact()
2073 29637 : : mkpolmod(gen_1,nf_get_pol(nf));
2074 29637 : C0 = mkvec2(C, Cext);
2075 29636 : id = expandext(nf, C0, P, e);
2076 : } else {
2077 0 : Cext = NULL;
2078 0 : C0 = C;
2079 0 : id = expand(nf, C, P, e);
2080 : }
2081 29637 : if (id == C0) /* e = 0 */
2082 : {
2083 12415 : if (!C) return bnfisprincipal0(bnf, gen_1, flag);
2084 12408 : switch(typ(C))
2085 : {
2086 7 : case t_INT: case t_FRAC: case t_POL: case t_POLMOD: case t_COL:
2087 7 : return triv_gen(bnf, C, flag);
2088 : }
2089 12401 : C = idealhnf_shallow(nf,C);
2090 : }
2091 : else
2092 : {
2093 17222 : if (gen) { C = gel(id,1); Cext = gel(id,2); } else C = id;
2094 : }
2095 29623 : prec = prec_arch(bnf);
2096 29623 : c = getrand();
2097 : for (;;)
2098 70 : {
2099 29693 : pari_sp av1 = avma;
2100 29693 : GEN y = isprincipalall(bnf, C, &prec, flag);
2101 29693 : if (y)
2102 : {
2103 29623 : if (flag & nf_GEN_IF_PRINCIPAL)
2104 : {
2105 20545 : if (typ(y) == t_INT) return gc_NULL(av);
2106 20545 : y = add_principal_part(nf, y, Cext, flag);
2107 : }
2108 : else
2109 : {
2110 9078 : GEN u = gel(y,2);
2111 9078 : if (!gen || typ(y) != t_VEC) return gerepileupto(av,y);
2112 9078 : if (lg(u) != 1) gel(y,2) = add_principal_part(nf, u, Cext, flag);
2113 : }
2114 29623 : return gerepilecopy(av, y);
2115 : }
2116 70 : if (DEBUGLEVEL) pari_warn(warnprec,"isprincipal",prec);
2117 70 : set_avma(av1); bnf = bnfnewprec_shallow(bnf,prec); setrand(c);
2118 : }
2119 : }
2120 : GEN
2121 0 : isprincipalfact_or_fail(GEN bnf, GEN C, GEN P, GEN e)
2122 : {
2123 0 : const long flag = nf_GENMAT|nf_FORCE;
2124 : long prec;
2125 0 : pari_sp av = avma;
2126 0 : GEN u, y, id, C0, Cext, nf = bnf_get_nf(bnf);
2127 :
2128 0 : Cext = trivial_fact();
2129 0 : C0 = mkvec2(C, Cext);
2130 0 : id = expandext(nf, C0, P, e);
2131 0 : if (id == C0) /* e = 0 */
2132 0 : C = idealhnf_shallow(nf,C);
2133 : else {
2134 0 : C = gel(id,1); Cext = gel(id,2);
2135 : }
2136 0 : prec = prec_arch(bnf);
2137 0 : y = isprincipalall(bnf, C, &prec, flag);
2138 0 : if (!y) return gc_utoipos(av, prec);
2139 0 : u = gel(y,2);
2140 0 : if (lg(u) != 1) gel(y,2) = add_principal_part(nf, u, Cext, flag);
2141 0 : return gerepilecopy(av, y);
2142 : }
2143 :
2144 : GEN
2145 148627 : nfsign_from_logarch(GEN LA, GEN invpi, GEN archp)
2146 : {
2147 148627 : long l = lg(archp), i;
2148 148627 : GEN y = cgetg(l, t_VECSMALL);
2149 148628 : pari_sp av = avma;
2150 :
2151 279246 : for (i=1; i<l; i++)
2152 : {
2153 130618 : GEN c = ground( gmul(imag_i(gel(LA,archp[i])), invpi) );
2154 130617 : y[i] = mpodd(c)? 1: 0;
2155 : }
2156 148628 : set_avma(av); return y;
2157 : }
2158 :
2159 : GEN
2160 226809 : nfsign_tu(GEN bnf, GEN archp)
2161 : {
2162 : long n;
2163 226809 : if (bnf_get_tuN(bnf) != 2) return cgetg(1, t_VECSMALL);
2164 159806 : n = archp? lg(archp) - 1: nf_get_r1(bnf_get_nf(bnf));
2165 159806 : return const_vecsmall(n, 1);
2166 : }
2167 : GEN
2168 228011 : nfsign_fu(GEN bnf, GEN archp)
2169 : {
2170 228011 : GEN invpi, y, A = bnf_get_logfu(bnf), nf = bnf_get_nf(bnf);
2171 228032 : long j = 1, RU = lg(A);
2172 :
2173 228032 : if (!archp) archp = identity_perm( nf_get_r1(nf) );
2174 228032 : invpi = invr( mppi(nf_get_prec(nf)) );
2175 228031 : y = cgetg(RU,t_MAT);
2176 376583 : for (j = 1; j < RU; j++)
2177 148529 : gel(y,j) = nfsign_from_logarch(gel(A,j), invpi, archp);
2178 228054 : return y;
2179 : }
2180 : GEN
2181 35 : nfsign_units(GEN bnf, GEN archp, int add_zu)
2182 : {
2183 35 : GEN sfu = nfsign_fu(bnf, archp);
2184 35 : return add_zu? vec_prepend(sfu, nfsign_tu(bnf, archp)): sfu;
2185 : }
2186 :
2187 : /* obsolete */
2188 : GEN
2189 7 : signunits(GEN bnf)
2190 : {
2191 : pari_sp av;
2192 : GEN S, y, nf;
2193 : long i, j, r1, r2;
2194 :
2195 7 : bnf = checkbnf(bnf); nf = bnf_get_nf(bnf);
2196 7 : nf_get_sign(nf, &r1,&r2);
2197 7 : S = zeromatcopy(r1, r1+r2-1); av = avma;
2198 7 : y = nfsign_fu(bnf, NULL);
2199 14 : for (j = 1; j < lg(y); j++)
2200 : {
2201 7 : GEN Sj = gel(S,j), yj = gel(y,j);
2202 21 : for (i = 1; i <= r1; i++) gel(Sj,i) = yj[i]? gen_m1: gen_1;
2203 : }
2204 7 : set_avma(av); return S;
2205 : }
2206 :
2207 : static GEN
2208 728477 : get_log_embed(REL_t *rel, GEN M, long RU, long R1, long prec)
2209 : {
2210 728477 : GEN arch, C, z = rel->m;
2211 : long i;
2212 728477 : arch = typ(z) == t_COL? RgM_RgC_mul(M, z): const_col(nbrows(M), z);
2213 728479 : C = cgetg(RU+1, t_COL); arch = glog(arch, prec);
2214 1676152 : for (i=1; i<=R1; i++) gel(C,i) = gel(arch,i);
2215 1592669 : for ( ; i<=RU; i++) gel(C,i) = gmul2n(gel(arch,i), 1);
2216 728467 : return C;
2217 : }
2218 : static GEN
2219 1008082 : rel_embed(REL_t *rel, FB_t *F, GEN embs, long ind, GEN M, long RU, long R1,
2220 : long prec)
2221 : {
2222 : GEN C, D, perm;
2223 : long i, n;
2224 1008082 : if (!rel->relaut) return get_log_embed(rel, M, RU, R1, prec);
2225 : /* image of another relation by automorphism */
2226 279611 : C = gel(embs, ind - rel->relorig);
2227 279611 : perm = gel(F->embperm, rel->relaut);
2228 279611 : D = cgetg_copy(C, &n);
2229 1237001 : for (i = 1; i < n; i++)
2230 : {
2231 957377 : long v = perm[i];
2232 957377 : gel(D,i) = (v > 0)? gel(C,v): conj_i(gel(C,-v));
2233 : }
2234 279624 : return D;
2235 : }
2236 : static GEN
2237 114042 : get_embs(FB_t *F, RELCACHE_t *cache, GEN nf, GEN embs, long PREC)
2238 : {
2239 114042 : long ru, j, k, l = cache->last - cache->chk + 1, r1 = nf_get_r1(nf);
2240 114042 : GEN M = nf_get_M(nf), nembs = cgetg(cache->last - cache->base+1, t_MAT);
2241 : REL_t *rel;
2242 :
2243 5181041 : for (k = 1; k <= cache->chk - cache->base; k++) gel(nembs,k) = gel(embs,k);
2244 114042 : embs = nembs; ru = nbrows(M);
2245 1089684 : for (j=1,rel = cache->chk + 1; j < l; rel++,j++,k++)
2246 975648 : gel(embs,k) = rel_embed(rel, F, embs, k, M, ru, r1, PREC);
2247 114036 : return embs;
2248 : }
2249 : static void
2250 902111 : set_rel_alpha(REL_t *rel, GEN auts, GEN vA, long ind)
2251 : {
2252 : GEN u;
2253 902111 : if (!rel->relaut)
2254 653258 : u = rel->m;
2255 : else
2256 248853 : u = ZM_ZC_mul(gel(auts, rel->relaut), gel(vA, ind - rel->relorig));
2257 902110 : gel(vA, ind) = u;
2258 902110 : }
2259 : static GEN
2260 3317250 : set_fact(FB_t *F, FACT *fact, GEN e, long *pnz)
2261 : {
2262 3317250 : long n = fact[0].pr;
2263 3317250 : GEN c = zero_Flv(F->KC);
2264 3317380 : if (!n) /* trivial factorization */
2265 0 : *pnz = F->KC+1;
2266 : else
2267 : {
2268 3317380 : long i, nz = minss(fact[1].pr, fact[n].pr);
2269 14960033 : for (i = 1; i <= n; i++) c[fact[i].pr] = fact[i].ex;
2270 3317384 : if (e)
2271 : {
2272 16669 : long l = lg(e);
2273 70855 : for (i = 1; i < l; i++)
2274 54186 : if (e[i]) { long v = F->subFB[i]; c[v] += e[i]; if (v < nz) nz = v; }
2275 : }
2276 3317384 : *pnz = nz;
2277 : }
2278 3317384 : return c;
2279 : }
2280 :
2281 : /* Is cols already in the cache ? bs = index of first non zero coeff in cols
2282 : * General check for colinearity useless since exceedingly rare */
2283 : static int
2284 3972464 : already_known(RELCACHE_t *cache, long bs, GEN cols)
2285 : {
2286 : REL_t *r;
2287 3972464 : long l = lg(cols);
2288 260831036 : for (r = cache->last; r > cache->base; r--)
2289 257806208 : if (bs == r->nz)
2290 : {
2291 30212457 : GEN coll = r->R;
2292 30212457 : long b = bs;
2293 220291869 : while (b < l && cols[b] == coll[b]) b++;
2294 30212457 : if (b == l) return 1;
2295 : }
2296 3024828 : return 0;
2297 : }
2298 :
2299 : /* Add relation R to cache, nz = index of first non zero coeff in R.
2300 : * If relation is a linear combination of the previous ones, return 0.
2301 : * Otherwise, update basis and return > 0. Compute mod p (much faster)
2302 : * so some kernel vector might not be genuine. */
2303 : static int
2304 3976436 : add_rel_i(RELCACHE_t *cache, GEN R, long nz, GEN m, long orig, long aut, REL_t **relp, long in_rnd_rel)
2305 : {
2306 3976436 : long i, k, n = lg(R)-1;
2307 :
2308 3976436 : if (nz == n+1) { k = 0; goto ADD_REL; }
2309 3972432 : if (already_known(cache, nz, R)) return -1;
2310 3024862 : if (cache->last >= cache->base + cache->len) return 0;
2311 3024862 : if (DEBUGLEVEL>6)
2312 : {
2313 0 : err_printf("adding vector = %Ps\n",R);
2314 0 : err_printf("generators =\n%Ps\n", cache->basis);
2315 : }
2316 3024885 : if (cache->missing)
2317 : {
2318 2650900 : GEN a = leafcopy(R), basis = cache->basis;
2319 2650897 : k = lg(a);
2320 134859627 : do --k; while (!a[k]);
2321 8524763 : while (k)
2322 : {
2323 6337927 : GEN c = gel(basis, k);
2324 6337927 : if (c[k])
2325 : {
2326 5873866 : long ak = a[k];
2327 401558913 : for (i=1; i < k; i++) if (c[i]) a[i] = (a[i] + ak*(mod_p-c[i])) % mod_p;
2328 5873866 : a[k] = 0;
2329 219955994 : do --k; while (!a[k]); /* k cannot go below 0: codeword is a sentinel */
2330 : }
2331 : else
2332 : {
2333 464061 : ulong invak = Fl_inv(uel(a,k), mod_p);
2334 : /* Cleanup a */
2335 13958171 : for (i = k; i-- > 1; )
2336 : {
2337 13494114 : long j, ai = a[i];
2338 13494114 : c = gel(basis, i);
2339 13494114 : if (!ai || !c[i]) continue;
2340 251031 : ai = mod_p-ai;
2341 4639967 : for (j = 1; j < i; j++) if (c[j]) a[j] = (a[j] + ai*c[j]) % mod_p;
2342 251031 : a[i] = 0;
2343 : }
2344 : /* Insert a/a[k] as k-th column */
2345 464057 : c = gel(basis, k);
2346 13958172 : for (i = 1; i<k; i++) if (a[i]) c[i] = (a[i] * invak) % mod_p;
2347 464057 : c[k] = 1; a = c;
2348 : /* Cleanup above k */
2349 13559137 : for (i = k+1; i<n; i++)
2350 : {
2351 : long j, ck;
2352 13095080 : c = gel(basis, i);
2353 13095080 : ck = c[k];
2354 13095080 : if (!ck) continue;
2355 2465490 : ck = mod_p-ck;
2356 84871638 : for (j = 1; j < k; j++) if (a[j]) c[j] = (c[j] + ck*a[j]) % mod_p;
2357 2465490 : c[k] = 0;
2358 : }
2359 464057 : cache->missing--;
2360 464057 : break;
2361 : }
2362 : }
2363 : }
2364 : else
2365 373985 : k = (cache->last - cache->base) + 1;
2366 3024878 : if (k || cache->relsup > 0 || (m && in_rnd_rel))
2367 : {
2368 : REL_t *rel;
2369 :
2370 960649 : ADD_REL:
2371 964653 : rel = ++cache->last;
2372 964653 : if (!k && cache->relsup && nz < n+1)
2373 : {
2374 122478 : cache->relsup--;
2375 122478 : k = (rel - cache->base) + cache->missing;
2376 : }
2377 964653 : rel->R = gclone(R);
2378 964645 : rel->m = m ? gclone(m) : NULL;
2379 964644 : rel->nz = nz;
2380 964644 : if (aut)
2381 : {
2382 275492 : rel->relorig = (rel - cache->base) - orig;
2383 275492 : rel->relaut = aut;
2384 : }
2385 : else
2386 689152 : rel->relaut = 0;
2387 964644 : if (relp) *relp = rel;
2388 964644 : if (DEBUGLEVEL) dbg_newrel(cache);
2389 : }
2390 3028873 : return k;
2391 : }
2392 :
2393 : static int
2394 3488006 : add_rel(RELCACHE_t *cache, FB_t *F, GEN R, long nz, GEN m, long in_rnd_rel)
2395 : {
2396 : REL_t *rel;
2397 : long k, l, reln;
2398 3488006 : const long lauts = lg(F->idealperm), KC = F->KC;
2399 :
2400 3488006 : k = add_rel_i(cache, R, nz, m, 0, 0, &rel, in_rnd_rel);
2401 3488051 : if (k > 0 && typ(m) != t_INT)
2402 : {
2403 518407 : GEN Rl = cgetg(KC+1, t_VECSMALL);
2404 518401 : reln = rel - cache->base;
2405 1006858 : for (l = 1; l < lauts; l++)
2406 : {
2407 488447 : GEN perml = gel(F->idealperm, l);
2408 488447 : long i, nzl = perml[nz];
2409 :
2410 22619390 : for (i = 1; i <= KC; i++) Rl[i] = 0;
2411 20070193 : for (i = nz; i <= KC; i++)
2412 19581746 : if (R[i])
2413 : {
2414 1362992 : long v = perml[i];
2415 :
2416 1362992 : if (v < nzl) nzl = v;
2417 1362992 : Rl[v] = R[i];
2418 : }
2419 488447 : (void)add_rel_i(cache, Rl, nzl, NULL, reln, l, NULL, in_rnd_rel);
2420 : }
2421 : }
2422 3488055 : return k;
2423 : }
2424 :
2425 : INLINE void
2426 58309434 : step(GEN x, double *y, GEN inc, long k)
2427 : {
2428 58309434 : if (!y[k])
2429 5848369 : x[k]++; /* leading coeff > 0 */
2430 : else
2431 : {
2432 52461065 : long i = inc[k];
2433 52461065 : x[k] += i;
2434 52461065 : inc[k] = (i > 0)? -1-i: 1-i;
2435 : }
2436 58309434 : }
2437 :
2438 : INLINE long
2439 534568 : Fincke_Pohst_ideal(RELCACHE_t *cache, FB_t *F, GEN nf, GEN M, GEN I,
2440 : GEN NI, FACT *fact, long Nrelid, FP_t *fp, RNDREL_t *rr, long prec,
2441 : long *Nsmall, long *Nfact)
2442 : {
2443 : pari_sp av;
2444 534568 : const long N = nf_get_degree(nf), R1 = nf_get_r1(nf);
2445 534570 : GEN G = nf_get_G(nf), G0 = nf_get_roundG(nf), r, u, gx, inc, ideal;
2446 : double BOUND, B1, B2;
2447 534569 : long j, k, skipfirst, relid=0, try_elt=0, try_factor=0;
2448 :
2449 534569 : inc = const_vecsmall(N, 1);
2450 534567 : u = ZM_lll(ZM_mul(G0, I), 0.99, LLL_IM);
2451 534565 : ideal = ZM_mul(I,u); /* approximate T2-LLL reduction */
2452 534553 : r = gaussred_from_QR(RgM_mul(G, ideal), prec); /* Cholesky for T2 | ideal */
2453 534569 : if (!r) pari_err_BUG("small_norm (precision too low)");
2454 :
2455 3223966 : for (k=1; k<=N; k++)
2456 : {
2457 2689399 : if (!gisdouble(gcoeff(r,k,k),&(fp->v[k]))) return 0;
2458 10273621 : for (j=1; j<k; j++) if (!gisdouble(gcoeff(r,j,k),&(fp->q[j][k]))) return 0;
2459 2689398 : if (DEBUGLEVEL>3) err_printf("v[%ld]=%.4g ",k,fp->v[k]);
2460 : }
2461 534567 : B1 = fp->v[1]; /* T2(ideal[1]) */
2462 534567 : B2 = fp->v[2] + B1 * fp->q[1][2] * fp->q[1][2]; /* T2(ideal[2]) */
2463 534567 : if (ZV_isscalar(gel(ideal,1))) /* probable */
2464 : {
2465 148167 : skipfirst = 1;
2466 148167 : BOUND = maxdd(BMULT * B1, 2 * B2);
2467 : }
2468 : else
2469 : {
2470 386399 : skipfirst = 0;
2471 386399 : BOUND = mindd(BMULT * B1, 2 * B2);
2472 : }
2473 : /* BOUND at most BMULT fp->x smallest known vector */
2474 534570 : if (DEBUGLEVEL>1)
2475 : {
2476 0 : if (DEBUGLEVEL>3) err_printf("\n");
2477 0 : err_printf("BOUND = %.4g\n",BOUND);
2478 : }
2479 534567 : BOUND *= 1 + 1e-6;
2480 534567 : k = N; fp->y[N] = fp->z[N] = 0; fp->x[N] = 0;
2481 31770660 : for (av = avma;; set_avma(av), step(fp->x,fp->y,inc,k))
2482 31236063 : {
2483 : GEN R;
2484 : long nz;
2485 : do
2486 : { /* look for primitive element of small norm, cf minim00 */
2487 43931035 : int fl = 0;
2488 : double p;
2489 43931035 : if (k > 1)
2490 : {
2491 12694993 : long l = k-1;
2492 12694993 : fp->z[l] = 0;
2493 101055465 : for (j=k; j<=N; j++) fp->z[l] += fp->q[l][j]*fp->x[j];
2494 12694993 : p = (double)fp->x[k] + fp->z[k];
2495 12694993 : fp->y[l] = fp->y[k] + p*p*fp->v[k];
2496 12694993 : if (l <= skipfirst && !fp->y[1]) fl = 1;
2497 12694993 : fp->x[l] = (long)floor(-fp->z[l] + 0.5);
2498 12694993 : k = l;
2499 : }
2500 12236407 : for(;; step(fp->x,fp->y,inc,k))
2501 : {
2502 56167394 : if (!fl)
2503 : {
2504 56509928 : if (++try_elt > maxtry_ELEMENT) return 0;
2505 56019211 : p = (double)fp->x[k] + fp->z[k];
2506 56019211 : if (fp->y[k] + p*p*fp->v[k] <= BOUND) break;
2507 :
2508 14836928 : step(fp->x,fp->y,inc,k);
2509 :
2510 14837019 : p = (double)fp->x[k] + fp->z[k];
2511 14837019 : if (fp->y[k] + p*p*fp->v[k] <= BOUND) break;
2512 : }
2513 12585216 : fl = 0; inc[k] = 1;
2514 12585216 : if (++k > N) return 0;
2515 : }
2516 43582269 : } while (k > 1);
2517 :
2518 : /* element complete */
2519 61800939 : if (zv_content(fp->x) !=1) continue; /* not primitive */
2520 30942834 : gx = ZM_zc_mul(ideal,fp->x);
2521 30942741 : if (ZV_isscalar(gx)) continue;
2522 30904718 : if (++try_factor > maxtry_FACT) return 0;
2523 :
2524 30859105 : if (!Nrelid)
2525 : {
2526 63 : if (!factorgen(F,nf,I,NI,gx,fact)) continue;
2527 14 : return 1;
2528 : }
2529 30859042 : else if (rr)
2530 : {
2531 121532 : if (!factorgen(F,nf,I,NI,gx,fact)) continue;
2532 16669 : add_to_fact(rr->jid, 1, fact);
2533 : }
2534 : else
2535 : {
2536 30737510 : GEN Nx, xembed = RgM_RgC_mul(M, gx);
2537 : long e;
2538 30737542 : if (Nsmall) (*Nsmall)++;
2539 30737542 : Nx = grndtoi(embed_norm(xembed, R1), &e);
2540 30737516 : if (e >= 0) {
2541 0 : if (DEBUGLEVEL > 1) err_printf("+");
2542 27445032 : continue;
2543 : }
2544 30737516 : if (!can_factor(F, nf, NULL, gx, Nx, fact)) continue;
2545 : }
2546 :
2547 : /* smooth element */
2548 3309006 : R = set_fact(F, fact, rr ? rr->ex : NULL, &nz);
2549 : /* make sure we get maximal rank first, then allow all relations */
2550 3309185 : if (add_rel(cache, F, R, nz, gx, rr ? 1 : 0) <= 0)
2551 : { /* probably Q-dependent from previous ones: forget it */
2552 2791102 : if (DEBUGLEVEL>1) err_printf("*");
2553 2791099 : if (DEBUGLEVEL && Nfact && rr) (*Nfact)++;
2554 2791099 : continue;
2555 : }
2556 518138 : if (DEBUGLEVEL && Nfact) (*Nfact)++;
2557 518138 : if (cache->last >= cache->end) return 1; /* we have enough */
2558 421857 : if (++relid == Nrelid) break;
2559 : }
2560 43856 : return 0;
2561 : }
2562 :
2563 : static void
2564 114669 : small_norm(RELCACHE_t *cache, FB_t *F, GEN nf, long Nrelid, GEN M,
2565 : FACT *fact, GEN p0)
2566 : {
2567 114669 : const long prec = nf_get_prec(nf);
2568 : FP_t fp;
2569 : pari_sp av;
2570 114669 : GEN L_jid = F->L_jid, Np0;
2571 114669 : long Nsmall, Nfact, n = lg(L_jid);
2572 : pari_timer T;
2573 :
2574 114669 : if (DEBUGLEVEL)
2575 : {
2576 0 : timer_start(&T);
2577 0 : err_printf("#### Look for %ld relations in %ld ideals (small_norm)\n",
2578 0 : cache->end - cache->last, lg(L_jid)-1);
2579 0 : if (p0) err_printf("Look in p0 = %Ps\n", vecslice(p0,1,4));
2580 : }
2581 114669 : Nsmall = Nfact = 0;
2582 114669 : minim_alloc(lg(M), &fp.q, &fp.x, &fp.y, &fp.z, &fp.v);
2583 114669 : Np0 = p0? pr_norm(p0): NULL;
2584 550161 : for (av = avma; --n; set_avma(av))
2585 : {
2586 518316 : long j = L_jid[n];
2587 518316 : GEN id = gel(F->LP, j), Nid;
2588 518316 : if (DEBUGLEVEL>1)
2589 0 : err_printf("\n*** Ideal no %ld: %Ps\n", j, vecslice(id,1,4));
2590 518317 : if (p0)
2591 278129 : { Nid = mulii(Np0, pr_norm(id)); id = idealmul(nf, p0, id); }
2592 : else
2593 240188 : { Nid = pr_norm(id); id = pr_hnf(nf, id);}
2594 518316 : if (Fincke_Pohst_ideal(cache, F, nf, M, id, Nid, fact, Nrelid, &fp,
2595 82825 : NULL, prec, &Nsmall, &Nfact)) break;
2596 : }
2597 114671 : if (DEBUGLEVEL && Nsmall)
2598 : {
2599 0 : if (DEBUGLEVEL == 1)
2600 0 : { if (Nfact) err_printf("\n"); }
2601 : else
2602 0 : err_printf(" \nnb. fact./nb. small norm = %ld/%ld = %.3f\n",
2603 0 : Nfact,Nsmall,((double)Nfact)/Nsmall);
2604 0 : if (timer_get(&T)>1) timer_printf(&T,"small_norm");
2605 : }
2606 114671 : }
2607 :
2608 : static GEN
2609 13665 : get_random_ideal(FB_t *F, GEN nf, GEN ex)
2610 : {
2611 13665 : long i, l = lg(ex);
2612 : for (;;)
2613 11 : {
2614 13676 : GEN I = NULL;
2615 60011 : for (i = 1; i < l; i++)
2616 46335 : if ((ex[i] = random_bits(RANDOM_BITS)))
2617 : {
2618 43457 : GEN pr = gel(F->LP, F->subFB[i]), e = utoipos(ex[i]);
2619 43457 : I = I? idealmulpowprime(nf, I, pr, e): idealpow(nf, pr, e);
2620 : }
2621 13676 : if (I && !ZM_isscalar(I,NULL)) return I; /* != (n)Z_K */
2622 : }
2623 : }
2624 :
2625 : static void
2626 13665 : rnd_rel(RELCACHE_t *cache, FB_t *F, GEN nf, FACT *fact)
2627 : {
2628 : pari_timer T;
2629 13665 : GEN L_jid = F->L_jid, M = nf_get_M(nf), R, NR;
2630 13665 : long i, l = lg(L_jid), prec = nf_get_prec(nf), Nfact = 0;
2631 : RNDREL_t rr;
2632 : FP_t fp;
2633 : pari_sp av;
2634 :
2635 13665 : if (DEBUGLEVEL) {
2636 0 : timer_start(&T);
2637 0 : err_printf("#### Look for %ld relations in %ld ideals (rnd_rel)\n",
2638 0 : cache->end - cache->last, l-1);
2639 : }
2640 13665 : rr.ex = cgetg(lg(F->subFB), t_VECSMALL);
2641 13665 : R = get_random_ideal(F, nf, rr.ex); /* random product from subFB */
2642 13665 : NR = ZM_det_triangular(R);
2643 13665 : minim_alloc(lg(M), &fp.q, &fp.x, &fp.y, &fp.z, &fp.v);
2644 16449 : for (av = avma, i = 1; i < l; i++, set_avma(av))
2645 : { /* try P[j] * base */
2646 16240 : long j = L_jid[i];
2647 16240 : GEN P = gel(F->LP, j), Nid = mulii(NR, pr_norm(P));
2648 16240 : if (DEBUGLEVEL>1) err_printf("\n*** Ideal %ld: %Ps\n", j, vecslice(P,1,4));
2649 16240 : rr.jid = j;
2650 16240 : if (Fincke_Pohst_ideal(cache, F, nf, M, idealHNF_mul(nf, R, P), Nid, fact,
2651 13456 : RND_REL_RELPID, &fp, &rr, prec, NULL, &Nfact)) break;
2652 : }
2653 13665 : if (DEBUGLEVEL)
2654 : {
2655 0 : if (Nfact) err_printf("\n");
2656 0 : if (timer_get(&T)>=0) timer_printf(&T,"rnd_rel");
2657 : }
2658 13665 : }
2659 :
2660 : static GEN
2661 63558 : automorphism_perms(GEN M, GEN auts, GEN cyclic, long r1, long r2, long N)
2662 : {
2663 63558 : long L = lgcols(M), lauts = lg(auts), lcyc = lg(cyclic), i, j, l, m;
2664 63558 : GEN Mt, perms = cgetg(lauts, t_VEC);
2665 : pari_sp av;
2666 :
2667 127278 : for (l = 1; l < lauts; l++) gel(perms, l) = cgetg(L, t_VECSMALL);
2668 63559 : av = avma;
2669 63559 : Mt = shallowtrans(gprec_w(M, LOWDEFAULTPREC));
2670 63558 : Mt = shallowconcat(Mt, conj_i(vecslice(Mt, r1+1, r1+r2)));
2671 110040 : for (l = 1; l < lcyc; l++)
2672 : {
2673 46484 : GEN thiscyc = gel(cyclic, l), thisperm, perm, prev, Nt;
2674 46484 : long k = thiscyc[1];
2675 :
2676 46484 : Nt = RgM_mul(shallowtrans(gel(auts, k)), Mt);
2677 46485 : perm = gel(perms, k);
2678 152629 : for (i = 1; i < L; i++)
2679 : {
2680 106145 : GEN v = gel(Nt, i), minD;
2681 106145 : minD = gnorml2(gsub(v, gel(Mt, 1)));
2682 106148 : perm[i] = 1;
2683 561488 : for (j = 2; j <= N; j++)
2684 : {
2685 455344 : GEN D = gnorml2(gsub(v, gel(Mt, j)));
2686 455337 : if (gcmp(D, minD) < 0) { minD = D; perm[i] = j >= L ? r2-j : j; }
2687 : }
2688 : }
2689 64929 : for (prev = perm, m = 2; m < lg(thiscyc); m++, prev = thisperm)
2690 : {
2691 18445 : thisperm = gel(perms, thiscyc[m]);
2692 93604 : for (i = 1; i < L; i++)
2693 : {
2694 75159 : long pp = labs(prev[i]);
2695 75159 : thisperm[i] = prev[i] < 0 ? -perm[pp] : perm[pp];
2696 : }
2697 : }
2698 : }
2699 63556 : set_avma(av); return perms;
2700 : }
2701 :
2702 : /* Determine the field automorphisms as matrices on the integral basis */
2703 : static GEN
2704 63620 : automorphism_matrices(GEN nf, GEN *cycp)
2705 : {
2706 63620 : pari_sp av = avma;
2707 63620 : GEN auts = galoisconj(nf, NULL), mats, cyclic, cyclicidx;
2708 63620 : long nauts = lg(auts)-1, i, j, k, l;
2709 :
2710 63620 : cyclic = cgetg(nauts+1, t_VEC);
2711 63619 : cyclicidx = zero_Flv(nauts);
2712 97680 : for (l = 1; l <= nauts; l++)
2713 : {
2714 97680 : GEN aut = gel(auts, l);
2715 97680 : if (gequalX(aut)) { swap(gel(auts, l), gel(auts, nauts)); break; }
2716 : }
2717 : /* trivial automorphism is last */
2718 190988 : for (l = 1; l <= nauts; l++) gel(auts, l) = algtobasis(nf, gel(auts, l));
2719 : /* Compute maximal cyclic subgroups */
2720 127369 : for (l = nauts; --l > 0; ) if (!cyclicidx[l])
2721 : {
2722 47984 : GEN elt = gel(auts, l), aut = elt, cyc = cgetg(nauts+1, t_VECSMALL);
2723 47984 : cyc[1] = cyclicidx[l] = l; j = 1;
2724 : do
2725 : {
2726 66987 : elt = galoisapply(nf, elt, aut);
2727 217203 : for (k = 1; k <= nauts; k++) if (gequal(elt, gel(auts, k))) break;
2728 66983 : cyclicidx[k] = l; cyc[++j] = k;
2729 : }
2730 66983 : while (k != nauts);
2731 47980 : setlg(cyc, j);
2732 47982 : gel(cyclic, l) = cyc;
2733 : }
2734 127368 : for (i = j = 1; i < nauts; i++)
2735 63748 : if (cyclicidx[i] == i) cyclic[j++] = cyclic[i];
2736 63620 : setlg(cyclic, j);
2737 63621 : mats = cgetg(nauts, t_VEC);
2738 110135 : while (--j > 0)
2739 : {
2740 46514 : GEN cyc = gel(cyclic, j);
2741 46514 : long id = cyc[1];
2742 46514 : GEN M, Mi, aut = gel(auts, id);
2743 :
2744 46514 : gel(mats, id) = Mi = M = nfgaloismatrix(nf, aut);
2745 64960 : for (i = 2; i < lg(cyc); i++) gel(mats, cyc[i]) = Mi = ZM_mul(Mi, M);
2746 : }
2747 63621 : gerepileall(av, 2, &mats, &cyclic);
2748 63620 : if (cycp) *cycp = cyclic;
2749 63620 : return mats;
2750 : }
2751 :
2752 : /* vP a list of maximal ideals above the same p from idealprimedec: f(P/p) is
2753 : * increasing; 1 <= j <= #vP; orbit a zc of length <= #vP; auts a vector of
2754 : * automorphisms in ZM form.
2755 : * Set orbit[i] = 1 for all vP[i], i >= j, in the orbit of pr = vP[j] wrt auts.
2756 : * N.B.1 orbit need not be initialized to 0: useful to incrementally run
2757 : * through successive orbits
2758 : * N.B.2 i >= j, so primes with index < j will be missed; run incrementally
2759 : * starting from j = 1 ! */
2760 : static void
2761 11865 : pr_orbit_fill(GEN orbit, GEN auts, GEN vP, long j)
2762 : {
2763 11865 : GEN pr = gel(vP,j), gen = pr_get_gen(pr);
2764 11865 : long i, l = lg(auts), J = lg(orbit), f = pr_get_f(pr);
2765 11865 : orbit[j] = 1;
2766 23730 : for (i = 1; i < l; i++)
2767 : {
2768 11865 : GEN g = ZM_ZC_mul(gel(auts,i), gen);
2769 : long k;
2770 11886 : for (k = j+1; k < J; k++)
2771 : {
2772 35 : GEN prk = gel(vP,k);
2773 35 : if (pr_get_f(prk) > f) break; /* f(P[k]) increases with k */
2774 : /* don't check that e matches: (almost) always 1 ! */
2775 35 : if (!orbit[k] && ZC_prdvd(g, prk)) { orbit[k] = 1; break; }
2776 : }
2777 : }
2778 11865 : }
2779 : /* remark: F->KCZ changes if be_honest() fails */
2780 : static int
2781 7 : be_honest(FB_t *F, GEN nf, GEN auts, FACT *fact)
2782 : {
2783 : long i, iz, nbtest;
2784 7 : long lgsub = lg(F->subFB), KCZ0 = F->KCZ;
2785 7 : long N = nf_get_degree(nf), prec = nf_get_prec(nf);
2786 7 : GEN M = nf_get_M(nf);
2787 : FP_t fp;
2788 : pari_sp av;
2789 :
2790 7 : if (DEBUGLEVEL) {
2791 0 : err_printf("Be honest for %ld primes from %ld to %ld\n", F->KCZ2 - F->KCZ,
2792 0 : F->FB[ F->KCZ+1 ], F->FB[ F->KCZ2 ]);
2793 : }
2794 7 : minim_alloc(N+1, &fp.q, &fp.x, &fp.y, &fp.z, &fp.v);
2795 7 : if (lg(auts) == 1) auts = NULL;
2796 7 : av = avma;
2797 14 : for (iz=F->KCZ+1; iz<=F->KCZ2; iz++, set_avma(av))
2798 : {
2799 7 : long p = F->FB[iz];
2800 7 : GEN pr_orbit, P = gel(F->LV,p);
2801 7 : long j, J = lg(P); /* > 1 */
2802 : /* the P|p, NP > C2 are assumed in subgroup generated by FB + last P
2803 : * with NP <= C2 is unramified --> check all but last */
2804 7 : if (pr_get_e(gel(P,J-1)) == 1) J--;
2805 7 : if (J == 1) continue;
2806 7 : if (DEBUGLEVEL>1) err_printf("%ld ", p);
2807 7 : pr_orbit = auts? zero_zv(J-1): NULL;
2808 28 : for (j = 1; j < J; j++)
2809 : {
2810 : GEN Nid, id, id0;
2811 21 : if (pr_orbit)
2812 : {
2813 21 : if (pr_orbit[j]) continue;
2814 : /* discard all primes in automorphism orbit simultaneously */
2815 14 : pr_orbit_fill(pr_orbit, auts, P, j);
2816 : }
2817 14 : id = id0 = pr_hnf(nf,gel(P,j));
2818 14 : Nid = pr_norm(gel(P,j));
2819 14 : for (nbtest=0;;)
2820 : {
2821 14 : if (Fincke_Pohst_ideal(NULL, F, nf, M, id, Nid, fact, 0, &fp,
2822 14 : NULL, prec, NULL, NULL)) break;
2823 0 : if (++nbtest > maxtry_HONEST)
2824 : {
2825 0 : if (DEBUGLEVEL)
2826 0 : pari_warn(warner,"be_honest() failure on prime %Ps\n", gel(P,j));
2827 0 : return 0;
2828 : }
2829 : /* occurs at most once in the whole function */
2830 0 : for (i = 1, id = id0; i < lgsub; i++)
2831 : {
2832 0 : long ex = random_bits(RANDOM_BITS);
2833 0 : if (ex)
2834 : {
2835 0 : GEN pr = gel(F->LP, F->subFB[i]);
2836 0 : id = idealmulpowprime(nf, id, pr, utoipos(ex));
2837 : }
2838 : }
2839 0 : if (!equali1(gcoeff(id,N,N))) id = Q_primpart(id);
2840 0 : if (expi(gcoeff(id,1,1)) > 100) id = idealred(nf, id);
2841 0 : Nid = ZM_det_triangular(id);
2842 : }
2843 : }
2844 7 : F->KCZ++; /* SUCCESS, "enlarge" factorbase */
2845 : }
2846 7 : F->KCZ = KCZ0; return gc_bool(av,1);
2847 : }
2848 :
2849 : /* all primes with N(P) <= BOUND factor on factorbase ? */
2850 : void
2851 63 : bnftestprimes(GEN bnf, GEN BOUND)
2852 : {
2853 63 : pari_sp av0 = avma, av;
2854 63 : ulong count = 0;
2855 63 : GEN auts, p, nf = bnf_get_nf(bnf), Vbase = bnf_get_vbase(bnf);
2856 63 : GEN fb = gen_sort_shallow(Vbase, (void*)&cmp_prime_ideal, cmp_nodata);
2857 63 : ulong pmax = pr_get_smallp(gel(fb, lg(fb)-1)); /*largest p in factorbase*/
2858 : forprime_t S;
2859 : FACT *fact;
2860 : FB_t F;
2861 :
2862 63 : (void)recover_partFB(&F, Vbase, nf_get_degree(nf));
2863 63 : fact = (FACT*)stack_malloc((F.KC+1)*sizeof(FACT));
2864 63 : forprime_init(&S, gen_2, BOUND);
2865 63 : auts = automorphism_matrices(nf, NULL);
2866 63 : if (lg(auts) == 1) auts = NULL;
2867 63 : av = avma;
2868 37604 : while (( p = forprime_next(&S) ))
2869 : {
2870 : GEN pr_orbit, vP;
2871 : long j, J;
2872 :
2873 37541 : if (DEBUGLEVEL == 1 && ++count > 1000)
2874 : {
2875 0 : err_printf("passing p = %Ps / %Ps\n", p, BOUND);
2876 0 : count = 0;
2877 : }
2878 37541 : set_avma(av);
2879 37541 : vP = idealprimedec_limit_norm(nf, p, BOUND);
2880 37541 : J = lg(vP);
2881 : /* if last is unramified, all P|p in subgroup generated by FB: skip last */
2882 37541 : if (J > 1 && pr_get_e(gel(vP,J-1)) == 1) J--;
2883 37541 : if (J == 1) continue;
2884 14525 : if (DEBUGLEVEL>1) err_printf("*** p = %Ps\n",p);
2885 14525 : pr_orbit = auts? zero_zv(J-1): NULL;
2886 31549 : for (j = 1; j < J; j++)
2887 : {
2888 17024 : GEN P = gel(vP,j);
2889 17024 : long k = 0;
2890 17024 : if (pr_orbit)
2891 : {
2892 11858 : if (pr_orbit[j]) continue;
2893 : /* discard all primes in automorphism orbit simultaneously */
2894 11851 : pr_orbit_fill(pr_orbit, auts, vP, j);
2895 : }
2896 17017 : if (abscmpiu(p, pmax) > 0 || !(k = tablesearch(fb, P, &cmp_prime_ideal)))
2897 16408 : (void)SPLIT(&F, nf, pr_hnf(nf,P), Vbase, fact);
2898 17017 : if (DEBUGLEVEL>1)
2899 : {
2900 0 : err_printf(" Testing P = %Ps\n",P);
2901 0 : if (k) err_printf(" #%ld in factor base\n",k);
2902 0 : else err_printf(" is %Ps\n", isprincipal(bnf,P));
2903 : }
2904 : }
2905 : }
2906 63 : set_avma(av0);
2907 63 : }
2908 :
2909 : /* A t_MAT of complex floats, in fact reals. Extract a submatrix B
2910 : * whose columns are definitely nonzero, i.e. gexpo(A[j]) >= -2
2911 : *
2912 : * If possible precision problem (t_REAL 0 with large exponent), set
2913 : * *precpb to 1 */
2914 : static GEN
2915 95015 : clean_cols(GEN A, int *precpb)
2916 : {
2917 95015 : long l = lg(A), h, i, j, k;
2918 : GEN B;
2919 95015 : *precpb = 0;
2920 95015 : if (l == 1) return A;
2921 95015 : h = lgcols(A);;
2922 95015 : B = cgetg(l, t_MAT);
2923 3674574 : for (i = k = 1; i < l; i++)
2924 : {
2925 3579559 : GEN Ai = gel(A,i);
2926 3579559 : int non0 = 0;
2927 17907574 : for (j = 1; j < h; j++)
2928 : {
2929 14328016 : GEN c = gel(Ai,j);
2930 14328016 : if (gexpo(c) >= -2)
2931 : {
2932 12365801 : if (gequal0(c)) *precpb = 1; else non0 = 1;
2933 : }
2934 : }
2935 3579558 : if (non0) gel(B, k++) = Ai;
2936 : }
2937 95015 : setlg(B, k); return B;
2938 : }
2939 :
2940 : static long
2941 3035017 : compute_multiple_of_R_pivot(GEN X, GEN x0/*unused*/, long ix, GEN c)
2942 : {
2943 3035017 : GEN x = gel(X,ix);
2944 3035017 : long i, k = 0, ex = - (long)HIGHEXPOBIT, lx = lg(x);
2945 : (void)x0;
2946 15800800 : for (i=1; i<lx; i++)
2947 12765787 : if (!c[i] && !gequal0(gel(x,i)))
2948 : {
2949 3036971 : long e = gexpo(gel(x,i));
2950 3036968 : if (e > ex) { ex = e; k = i; }
2951 : }
2952 3035013 : return (k && ex > -32)? k: lx;
2953 : }
2954 :
2955 : /* Ar = (log |sigma_i(u_j)|) for units (u_j) found so far;
2956 : * RU = R1+R2 = target rank for unit matrix, after adding [1 x r1, 2 x r2];
2957 : * N = field degree, need = unit rank defect;
2958 : * L = NULL (prec problem) or B^(-1) * A with approximate rational entries
2959 : * (as t_REAL), B a submatrix of A, with (probably) maximal rank RU */
2960 : static GEN
2961 110234 : compute_multiple_of_R(GEN Ar, long RU, long N, long *pneed, long *bit, GEN *ptL)
2962 : {
2963 : GEN T, d, mdet, Im_mdet, kR, L;
2964 110234 : long i, j, r, R1 = 2*RU - N;
2965 : int precpb;
2966 110234 : pari_sp av = avma;
2967 :
2968 110234 : if (RU == 1) { *ptL = zeromat(0, lg(Ar)-1); return gen_1; }
2969 :
2970 95015 : if (DEBUGLEVEL) err_printf("\n#### Computing regulator multiple\n");
2971 95015 : mdet = clean_cols(Ar, &precpb);
2972 : /* will cause precision to increase on later failure, but we may succeed! */
2973 95015 : *ptL = precpb? NULL: gen_1;
2974 95015 : T = cgetg(RU+1,t_COL);
2975 255259 : for (i=1; i<=R1; i++) gel(T,i) = gen_1;
2976 205161 : for ( ; i<=RU; i++) gel(T,i) = gen_2;
2977 95015 : mdet = shallowconcat(T, mdet); /* det(Span(mdet)) = N * R */
2978 :
2979 : /* could be using indexrank(), but need custom "get_pivot" function */
2980 95015 : d = RgM_pivots(mdet, NULL, &r, &compute_multiple_of_R_pivot);
2981 : /* # of independent columns = target rank ? */
2982 95014 : if (lg(mdet)-1 - r != RU)
2983 : {
2984 37129 : if (DEBUGLEVEL)
2985 0 : err_printf("Units matrix target rank = %ld < %ld\n",lg(mdet)-1 - r, RU);
2986 37129 : *pneed = RU - (lg(mdet)-1-r); return gc_NULL(av);
2987 : }
2988 :
2989 57885 : Im_mdet = cgetg(RU+1, t_MAT); /* extract independent columns */
2990 : /* N.B: d[1] = 1, corresponding to T above */
2991 57885 : gel(Im_mdet, 1) = T;
2992 251747 : for (i = j = 2; i <= RU; j++)
2993 193862 : if (d[j]) gel(Im_mdet, i++) = gel(mdet,j);
2994 :
2995 : /* integral multiple of R: the cols we picked form a Q-basis, they have an
2996 : * index in the full lattice. First column is T */
2997 57885 : kR = divru(det2(Im_mdet), N);
2998 : /* R > 0.2 uniformly */
2999 57885 : if (!signe(kR) || expo(kR) < -3)
3000 : {
3001 1 : if (DEBUGLEVEL) err_printf("Regulator is zero.\n");
3002 1 : *pneed = 0; return gc_NULL(av);
3003 : }
3004 57884 : d = det2(rowslice(vecslice(Im_mdet, 2, RU), 2, RU));
3005 57884 : setabssign(d); setabssign(kR);
3006 57884 : if (gexpo(gsub(d,kR)) - gexpo(d) > -20) { *ptL = NULL; return gc_NULL(av); }
3007 57850 : L = RgM_inv(Im_mdet);
3008 : /* estimate # of correct bits in result */
3009 57851 : if (!L || (*bit = -gexpo(RgM_Rg_sub_shallow(RgM_mul(L,Im_mdet), gen_1))) < 16)
3010 18 : { *ptL = NULL; return gc_NULL(av); }
3011 :
3012 57833 : *ptL = RgM_mul(rowslice(L,2,RU), Ar); /* approximate rational entries */
3013 57833 : return gc_all(av,2, &kR, ptL);
3014 : }
3015 :
3016 : /* leave small integer n as is, convert huge n to t_REAL (for readability) */
3017 : static GEN
3018 0 : i2print(GEN n)
3019 0 : { return lgefint(n) <= DEFAULTPREC? n: itor(n,LOWDEFAULTPREC); }
3020 :
3021 : static long
3022 72989 : bad_check(GEN c)
3023 : {
3024 72989 : long ec = gexpo(c);
3025 72989 : if (DEBUGLEVEL) err_printf("\n ***** check = %.28Pg\n",c);
3026 : /* safe check for c < 0.75 : avoid underflow in gtodouble() */
3027 72989 : if (ec < -1 || (ec == -1 && gtodouble(c) < 0.75)) return fupb_PRECI;
3028 : /* safe check for c > 1.3 : avoid overflow */
3029 72989 : if (ec > 0 || (ec == 0 && gtodouble(c) > 1.3)) return fupb_RELAT;
3030 63567 : return fupb_NONE;
3031 : }
3032 : /* Input:
3033 : * lambda = approximate rational entries: coords of units found so far on a
3034 : * sublattice of maximal rank (sublambda)
3035 : * *ptkR = regulator of sublambda = multiple of regulator of lambda
3036 : * Compute R = true regulator of lambda.
3037 : *
3038 : * If c := Rz ~ 1, by Dirichlet's formula, then lambda is the full group of
3039 : * units AND the full set of relations for the class group has been computed.
3040 : * In fact z is a very rough approximation and we only expect 0.75 < Rz < 1.3
3041 : *
3042 : * Output: *ptkR = R, *ptL = numerator(units) (in terms of lambda) */
3043 : static long
3044 73049 : compute_R(GEN lambda, GEN z, GEN *ptL, GEN *ptkR)
3045 : {
3046 73049 : pari_sp av = avma;
3047 73049 : long bit, r, reason, RU = lg(lambda) == 1? 1: lgcols(lambda);
3048 : GEN L, H, D, den, R, c;
3049 :
3050 73050 : *ptL = NULL;
3051 73050 : if (DEBUGLEVEL) err_printf("\n#### Computing check\n");
3052 73050 : if (RU == 1) { *ptkR = gen_1; *ptL = lambda; return bad_check(z); }
3053 57831 : D = gmul2n(mpmul(*ptkR,z), 1); /* bound for denom(lambda) */
3054 57832 : if (expo(D) < 0 && rtodbl(D) < 0.95) return fupb_PRECI;
3055 57832 : L = bestappr(lambda,D);
3056 57833 : if (lg(L) == 1)
3057 : {
3058 1 : if (DEBUGLEVEL) err_printf("truncation error in bestappr\n");
3059 1 : return fupb_PRECI;
3060 : }
3061 57832 : den = Q_denom(L);
3062 57832 : if (mpcmp(den,D) > 0)
3063 : {
3064 14 : if (DEBUGLEVEL) err_printf("D = %Ps\nden = %Ps\n",D, i2print(den));
3065 14 : return fupb_PRECI;
3066 : }
3067 57817 : bit = -gexpo(gsub(L, lambda)); /* input accuracy */
3068 57818 : L = Q_muli_to_int(L, den);
3069 57815 : if (gexpo(L) + expi(den) > bit - 32)
3070 : {
3071 48 : if (DEBUGLEVEL) err_printf("dubious bestappr; den = %Ps\n", i2print(den));
3072 48 : return fupb_PRECI;
3073 : }
3074 57770 : H = ZM_hnf(L); r = lg(H)-1;
3075 57769 : if (!r || r != nbrows(H))
3076 0 : R = gen_0; /* wrong rank */
3077 : else
3078 57769 : R = gmul(*ptkR, gdiv(ZM_det_triangular(H), powiu(den, r)));
3079 : /* R = tentative regulator; regulator > 0.2 uniformly */
3080 57767 : if (gexpo(R) < -3) {
3081 0 : if (DEBUGLEVEL) err_printf("\n#### Tentative regulator: %.28Pg\n", R);
3082 0 : return gc_long(av, fupb_PRECI);
3083 : }
3084 57770 : c = gmul(R,z); /* should be n (= 1 if we are done) */
3085 57770 : if (DEBUGLEVEL) err_printf("\n#### Tentative regulator: %.28Pg\n", R);
3086 57770 : if ((reason = bad_check(c))) return gc_long(av, reason);
3087 48383 : *ptkR = R; *ptL = L; return fupb_NONE;
3088 : }
3089 : static GEN
3090 63657 : get_clg2(GEN cyc, GEN Ga, GEN C, GEN Ur, GEN Ge, GEN M1, GEN M2)
3091 : {
3092 63657 : GEN GD = gsub(act_arch(M1, C), diagact_arch(cyc, Ga));
3093 63656 : GEN ga = gsub(act_arch(M2, C), act_arch(Ur, Ga));
3094 63657 : return mkvecn(6, Ur, ga, GD, Ge, M1, M2);
3095 : }
3096 : /* compute class group (clg1) + data for isprincipal (clg2) */
3097 : static GEN
3098 63560 : class_group_gen(GEN nf,GEN W,GEN C,GEN Vbase,long prec, GEN *pclg2)
3099 : {
3100 : GEN M1, M2, z, G, Ga, Ge, cyc, X, Y, D, U, V, Ur, Ui, Uir;
3101 : long j, l;
3102 :
3103 63560 : D = ZM_snfall(W,&U,&V); /* UWV=D, D diagonal, G = g Ui (G=new gens, g=old) */
3104 63560 : Ui = ZM_inv(U, NULL);
3105 63560 : l = lg(D); cyc = cgetg(l, t_VEC); /* elementary divisors */
3106 92274 : for (j = 1; j < l; j++)
3107 : {
3108 30319 : gel(cyc,j) = gcoeff(D,j,j); /* strip useless components */
3109 30319 : if (is_pm1(gel(cyc,j))) break;
3110 : }
3111 63560 : l = j;
3112 63560 : Ur = ZM_hnfdivrem(U, D, &Y);
3113 63559 : Uir = ZM_hnfdivrem(Ui,W, &X);
3114 : /* {x} = logarithmic embedding of x (arch. component)
3115 : * NB: [J,z] = idealred(I) --> I = y J, with {y} = - z
3116 : * G = g Uir - {Ga}, Uir = Ui + WX
3117 : * g = G Ur - {ga}, Ur = U + DY */
3118 63559 : G = cgetg(l,t_VEC);
3119 63558 : Ga= cgetg(l,t_MAT);
3120 63558 : Ge= cgetg(l,t_COL);
3121 63558 : z = init_famat(NULL);
3122 92270 : for (j = 1; j < l; j++)
3123 : {
3124 28711 : GEN I = genback(z, nf, Vbase, gel(Uir,j));
3125 28714 : gel(G,j) = gel(I,1); /* generator, order cyc[j] */
3126 28714 : gel(Ge,j)= gel(I,2);
3127 28714 : gel(Ga,j)= nf_cxlog(nf, gel(I,2), prec);
3128 28714 : if (!gel(Ga,j)) pari_err_PREC("class_group_gen");
3129 : }
3130 : /* {ga} = {GD}Y + G U - g = {GD}Y - {Ga} U + gW X U
3131 : = gW (X Ur + V Y) - {Ga}Ur */
3132 63559 : M2 = ZM_add(ZM_mul(X,Ur), ZM_mul(V,Y));
3133 63559 : setlg(cyc,l); setlg(V,l); setlg(D,l);
3134 : /* G D =: {GD} = g (Ui + W X) D - {Ga}D = g W (V + X D) - {Ga}D
3135 : * NB: Ui D = W V. gW is given by (first l-1 cols of) C */
3136 63559 : M1 = ZM_add(V, ZM_mul(X,D));
3137 63558 : *pclg2 = get_clg2(cyc, Ga, C, Ur, Ge, M1, M2);
3138 63560 : return mkvec3(ZV_prod(cyc), cyc, G);
3139 : }
3140 :
3141 : /* compute principal ideals corresponding to (gen[i]^cyc[i]) */
3142 : static GEN
3143 4969 : makecycgen(GEN bnf)
3144 : {
3145 4969 : GEN cyc = bnf_get_cyc(bnf), gen = bnf_get_gen(bnf), nf = bnf_get_nf(bnf);
3146 4969 : GEN h, y, GD = bnf_get_GD(bnf), W = bnf_get_W(bnf); /* HNF */
3147 4969 : GEN Sunits = bnf_get_sunits(bnf);
3148 4969 : GEN X = Sunits? gel(Sunits,1): NULL, C = Sunits? gel(Sunits,3): NULL;
3149 : long e, i, l;
3150 :
3151 4969 : if (DEBUGLEVEL) pari_warn(warner,"completing bnf (building cycgen)");
3152 4969 : h = cgetg_copy(gen, &l);
3153 11639 : for (i = 1; i < l; i++)
3154 : {
3155 6670 : GEN gi = gel(gen,i), ci = gel(cyc,i);
3156 6670 : if (X && equalii(ci, gcoeff(W,i,i)))
3157 : {
3158 : long j;
3159 8557 : for (j = i+1; j < l; j++)
3160 3186 : if (signe(gcoeff(W,i,j))) break;
3161 5579 : if (j == i) { gel(h,i) = mkmat2(X, gel(C,i)); continue; }
3162 : }
3163 6670 : if (abscmpiu(ci, 5) < 0)
3164 : {
3165 5557 : GEN N = ZM_det_triangular(gi);
3166 5557 : y = isprincipalarch(bnf,gel(GD,i), N, ci, gen_1, &e);
3167 5557 : if (y && fact_ok(nf,y,NULL,mkvec(gi),mkvec(ci)))
3168 : {
3169 4592 : gel(h,i) = to_famat_shallow(y,gen_1);
3170 4592 : continue;
3171 : }
3172 : }
3173 2078 : y = isprincipalfact(bnf, NULL, mkvec(gi), mkvec(ci), nf_GENMAT|nf_FORCE);
3174 2078 : gel(h,i) = gel(y,2);
3175 : }
3176 4969 : return h;
3177 : }
3178 :
3179 : static GEN
3180 77 : get_y(GEN bnf, GEN W, GEN B, GEN C, GEN pFB, long j)
3181 : {
3182 77 : GEN y, nf = bnf_get_nf(bnf);
3183 77 : long e, lW = lg(W)-1;
3184 77 : GEN ex = (j<=lW)? gel(W,j): gel(B,j-lW);
3185 77 : GEN P = (j<=lW)? NULL: gel(pFB,j);
3186 77 : if (C)
3187 : { /* archimedean embeddings known: cheap trial */
3188 77 : GEN Nx = get_norm_fact_primes(pFB, ex, P);
3189 77 : y = isprincipalarch(bnf,gel(C,j), Nx,gen_1, gen_1, &e);
3190 77 : if (y && fact_ok(nf,y,P,pFB,ex)) return y;
3191 : }
3192 0 : y = isprincipalfact_or_fail(bnf, P, pFB, ex);
3193 0 : return typ(y) == t_INT? y: gel(y,2);
3194 : }
3195 : /* compute principal ideals corresponding to bnf relations */
3196 : static GEN
3197 21 : makematal(GEN bnf)
3198 : {
3199 21 : GEN W = bnf_get_W(bnf), B = bnf_get_B(bnf), C = bnf_get_C(bnf);
3200 : GEN pFB, ma, retry;
3201 21 : long lma, j, prec = 0;
3202 :
3203 21 : if (DEBUGLEVEL) pari_warn(warner,"completing bnf (building matal)");
3204 21 : lma=lg(W)+lg(B)-1;
3205 21 : pFB = bnf_get_vbase(bnf);
3206 21 : ma = cgetg(lma,t_VEC);
3207 21 : retry = vecsmalltrunc_init(lma);
3208 98 : for (j=lma-1; j>0; j--)
3209 : {
3210 77 : pari_sp av = avma;
3211 77 : GEN y = get_y(bnf, W, B, C, pFB, j);
3212 77 : if (typ(y) == t_INT)
3213 : {
3214 0 : long E = itos(y);
3215 0 : if (DEBUGLEVEL>1) err_printf("\n%ld done later at prec %ld\n",j,E);
3216 0 : set_avma(av);
3217 0 : vecsmalltrunc_append(retry, j);
3218 0 : if (E > prec) prec = E;
3219 : }
3220 : else
3221 : {
3222 77 : if (DEBUGLEVEL>1) err_printf("%ld ",j);
3223 77 : gel(ma,j) = gerepileupto(av,y);
3224 : }
3225 : }
3226 21 : if (prec)
3227 : {
3228 0 : long k, l = lg(retry);
3229 0 : GEN y, nf = bnf_get_nf(bnf);
3230 0 : if (DEBUGLEVEL) pari_warn(warnprec,"makematal",prec);
3231 0 : nf = nfnewprec_shallow(nf,prec);
3232 0 : bnf = Buchall(nf, nf_FORCE, prec);
3233 0 : if (DEBUGLEVEL) err_printf("makematal, adding missing entries:");
3234 0 : for (k=1; k<l; k++)
3235 : {
3236 0 : pari_sp av = avma;
3237 0 : long j = retry[k];
3238 0 : y = get_y(bnf,W,B,NULL, pFB, j);
3239 0 : if (typ(y) == t_INT) pari_err_PREC("makematal");
3240 0 : if (DEBUGLEVEL>1) err_printf("%ld ",j);
3241 0 : gel(ma,j) = gerepileupto(av,y);
3242 : }
3243 : }
3244 21 : if (DEBUGLEVEL>1) err_printf("\n");
3245 21 : return ma;
3246 : }
3247 :
3248 : enum { MATAL = 1, CYCGEN, UNITS };
3249 : GEN
3250 26739 : bnf_build_cycgen(GEN bnf)
3251 26739 : { return obj_checkbuild(bnf, CYCGEN, &makecycgen); }
3252 : GEN
3253 21 : bnf_build_matalpha(GEN bnf)
3254 21 : { return obj_checkbuild(bnf, MATAL, &makematal); }
3255 : GEN
3256 31898 : bnf_build_units(GEN bnf)
3257 31898 : { return obj_checkbuild(bnf, UNITS, &makeunits); }
3258 :
3259 : /* return fu in compact form if available; in terms of a fixed basis
3260 : * of S-units */
3261 : GEN
3262 70 : bnf_compactfu_mat(GEN bnf)
3263 : {
3264 70 : GEN X, U, SUnits = bnf_get_sunits(bnf);
3265 70 : if (!SUnits) return NULL;
3266 70 : X = gel(SUnits,1);
3267 70 : U = gel(SUnits,2); ZM_remove_unused(&U, &X);
3268 70 : return mkvec2(X, U);
3269 : }
3270 : /* return fu in compact form if available; individually as famat */
3271 : GEN
3272 37127 : bnf_compactfu(GEN bnf)
3273 : {
3274 37127 : GEN fu, X, U, SUnits = bnf_get_sunits(bnf);
3275 : long i, l;
3276 37127 : if (!SUnits) return NULL;
3277 36896 : X = gel(SUnits,1);
3278 36896 : U = gel(SUnits,2); l = lg(U); fu = cgetg(l, t_VEC);
3279 60177 : for (i = 1; i < l; i++)
3280 23282 : gel(fu,i) = famat_remove_trivial(mkmat2(X, gel(U,i)));
3281 36895 : return fu;
3282 : }
3283 : /* return expanded fu if available */
3284 : GEN
3285 263576 : bnf_has_fu(GEN bnf)
3286 : {
3287 263576 : GEN fu = obj_check(bnf, UNITS);
3288 263571 : if (fu) return vecsplice(fu, 1);
3289 262769 : fu = bnf_get_fu_nocheck(bnf);
3290 262769 : return (typ(fu) == t_MAT)? NULL: fu;
3291 : }
3292 : /* return expanded fu if available; build if cheap */
3293 : GEN
3294 263297 : bnf_build_cheapfu(GEN bnf)
3295 : {
3296 : GEN fu, SUnits;
3297 263297 : if ((fu = bnf_has_fu(bnf))) return fu;
3298 149 : if ((SUnits = bnf_get_sunits(bnf)))
3299 : {
3300 149 : pari_sp av = avma;
3301 149 : long e = gexpo(real_i(bnf_get_logfu(bnf)));
3302 149 : set_avma(av); if (e < 13) return vecsplice(bnf_build_units(bnf), 1);
3303 : }
3304 77 : return NULL;
3305 : }
3306 :
3307 : static GEN
3308 63659 : get_regulator(GEN A)
3309 : {
3310 63659 : pari_sp av = avma;
3311 : GEN R;
3312 :
3313 63659 : if (lg(A) == 1) return gen_1;
3314 48468 : R = det( rowslice(real_i(A), 1, lgcols(A)-2) );
3315 48467 : setabssign(R); return gerepileuptoleaf(av, R);
3316 : }
3317 :
3318 : /* return corrected archimedian components for elts of x (vector)
3319 : * (= log(sigma_i(x)) - log(|Nx|) / [K:Q]) */
3320 : static GEN
3321 42 : get_archclean(GEN nf, GEN x, long prec, int units)
3322 : {
3323 42 : long k, N, l = lg(x);
3324 42 : GEN M = cgetg(l, t_MAT);
3325 :
3326 42 : if (l == 1) return M;
3327 28 : N = nf_get_degree(nf);
3328 126 : for (k = 1; k < l; k++)
3329 : {
3330 98 : pari_sp av = avma;
3331 98 : GEN c = nf_cxlog(nf, gel(x,k), prec);
3332 98 : if (!c || (!units && !(c = cleanarch(c, N, NULL,prec)))) return NULL;
3333 98 : gel(M,k) = gerepilecopy(av, c);
3334 : }
3335 28 : return M;
3336 : }
3337 : static void
3338 77 : Sunits_archclean(GEN nf, GEN Sunits, GEN *pmun, GEN *pC, long prec)
3339 : {
3340 77 : GEN ipi, M, X = gel(Sunits,1), U = gel(Sunits,2), G = gel(Sunits,3);
3341 77 : long k, N = nf_get_degree(nf), l = lg(X);
3342 :
3343 77 : M = cgetg(l, t_MAT);
3344 3640 : for (k = 1; k < l; k++)
3345 3563 : if (!(gel(M,k) = nf_cxlog(nf, gel(X,k), prec))) return;
3346 77 : ipi = invr(mppi(prec));
3347 77 : *pmun = cleanarch(RgM_ZM_mul(M, U), N, ipi, prec); /* not cleanarchunit ! */
3348 77 : if (*pmun) *pC = cleanarch(RgM_ZM_mul(M, G), N, ipi, prec);
3349 : }
3350 :
3351 : GEN
3352 98 : bnfnewprec_shallow(GEN bnf, long prec)
3353 : {
3354 98 : GEN nf0 = bnf_get_nf(bnf), nf, v, fu, matal, y, A, C;
3355 98 : GEN Sunits = bnf_get_sunits(bnf), Ur, Ga, Ge, M1, M2;
3356 98 : long r1, r2, prec0 = prec;
3357 :
3358 98 : nf_get_sign(nf0, &r1, &r2);
3359 98 : if (Sunits)
3360 : {
3361 77 : fu = matal = NULL;
3362 77 : prec += nbits2extraprec(gexpo(Sunits));
3363 : }
3364 : else
3365 : {
3366 21 : fu = bnf_build_units(bnf);
3367 21 : fu = vecslice(fu, 2, lg(fu)-1);
3368 21 : if (r1 + r2 > 1) {
3369 14 : long e = gexpo(bnf_get_logfu(bnf)) + 1 - TWOPOTBITS_IN_LONG;
3370 14 : if (e >= 0) prec += nbits2extraprec(e);
3371 : }
3372 21 : matal = bnf_build_matalpha(bnf);
3373 : }
3374 :
3375 98 : if (DEBUGLEVEL && prec0 != prec) pari_warn(warnprec,"bnfnewprec",prec);
3376 98 : for(C = NULL;;)
3377 0 : {
3378 98 : pari_sp av = avma;
3379 98 : nf = nfnewprec_shallow(nf0,prec);
3380 98 : if (Sunits)
3381 77 : Sunits_archclean(nf, Sunits, &A, &C, prec);
3382 : else
3383 : {
3384 21 : A = get_archclean(nf, fu, prec, 1);
3385 21 : if (A) C = get_archclean(nf, matal, prec, 0);
3386 : }
3387 98 : if (C) break;
3388 0 : set_avma(av); prec = precdbl(prec);
3389 0 : if (DEBUGLEVEL) pari_warn(warnprec,"bnfnewprec(extra)",prec);
3390 : }
3391 98 : y = leafcopy(bnf);
3392 98 : gel(y,3) = A;
3393 98 : gel(y,4) = C;
3394 98 : gel(y,7) = nf;
3395 98 : gel(y,8) = v = leafcopy(gel(bnf,8));
3396 98 : gel(v,2) = get_regulator(A);
3397 98 : v = gel(bnf,9);
3398 98 : if (lg(v) < 7) pari_err_TYPE("bnfnewprec [obsolete bnf format]", bnf);
3399 98 : Ur = gel(v,1);
3400 98 : Ge = gel(v,4);
3401 98 : Ga = nfV_cxlog(nf, Ge, prec);
3402 98 : M1 = gel(v,5);
3403 98 : M2 = gel(v,6);
3404 98 : gel(y,9) = get_clg2(bnf_get_cyc(bnf), Ga, C, Ur, Ge, M1, M2);
3405 98 : return y;
3406 : }
3407 : GEN
3408 21 : bnfnewprec(GEN bnf, long prec)
3409 : {
3410 21 : pari_sp av = avma;
3411 21 : return gerepilecopy(av, bnfnewprec_shallow(checkbnf(bnf), prec));
3412 : }
3413 :
3414 : GEN
3415 0 : bnrnewprec_shallow(GEN bnr, long prec)
3416 : {
3417 0 : GEN y = cgetg(7,t_VEC);
3418 : long i;
3419 0 : gel(y,1) = bnfnewprec_shallow(bnr_get_bnf(bnr), prec);
3420 0 : for (i=2; i<7; i++) gel(y,i) = gel(bnr,i);
3421 0 : return y;
3422 : }
3423 : GEN
3424 7 : bnrnewprec(GEN bnr, long prec)
3425 : {
3426 7 : GEN y = cgetg(7,t_VEC);
3427 : long i;
3428 7 : checkbnr(bnr);
3429 7 : gel(y,1) = bnfnewprec(bnr_get_bnf(bnr), prec);
3430 42 : for (i=2; i<7; i++) gel(y,i) = gcopy(gel(bnr,i));
3431 7 : return y;
3432 : }
3433 :
3434 : static GEN
3435 64687 : buchall_end(GEN nf,GEN res, GEN clg2, GEN W, GEN B, GEN A, GEN C,GEN Vbase)
3436 : {
3437 64687 : GEN z = obj_init(9, 3);
3438 64686 : gel(z,1) = W;
3439 64686 : gel(z,2) = B;
3440 64686 : gel(z,3) = A;
3441 64686 : gel(z,4) = C;
3442 64686 : gel(z,5) = Vbase;
3443 64686 : gel(z,6) = gen_0;
3444 64686 : gel(z,7) = nf;
3445 64686 : gel(z,8) = res;
3446 64686 : gel(z,9) = clg2;
3447 64686 : return z;
3448 : }
3449 :
3450 : GEN
3451 2499 : bnfinit0(GEN P, long flag, GEN data, long prec)
3452 : {
3453 2499 : double c1 = 0., c2 = 0.;
3454 2499 : long fl, relpid = BNF_RELPID;
3455 :
3456 2499 : if (data)
3457 : {
3458 21 : long lx = lg(data);
3459 21 : if (typ(data) != t_VEC || lx > 5) pari_err_TYPE("bnfinit",data);
3460 21 : switch(lx)
3461 : {
3462 0 : case 4: relpid = itos(gel(data,3));
3463 14 : case 3: c2 = gtodouble(gel(data,2));
3464 21 : case 2: c1 = gtodouble(gel(data,1));
3465 : }
3466 : }
3467 2499 : switch(flag)
3468 : {
3469 1722 : case 2:
3470 1722 : case 0: fl = 0; break;
3471 777 : case 1: fl = nf_FORCE; break;
3472 0 : default: pari_err_FLAG("bnfinit");
3473 : return NULL; /* LCOV_EXCL_LINE */
3474 : }
3475 2499 : return Buchall_param(P, c1, c2, relpid, fl, prec);
3476 : }
3477 : GEN
3478 62191 : Buchall(GEN P, long flag, long prec)
3479 62191 : { return Buchall_param(P, 0., 0., BNF_RELPID, flag & nf_FORCE, prec); }
3480 :
3481 : static GEN
3482 1127 : Buchall_deg1(GEN nf)
3483 : {
3484 1127 : GEN v = cgetg(1,t_VEC), m = cgetg(1,t_MAT);
3485 1127 : GEN res, W, A, B, C, Vbase = cgetg(1,t_COL);
3486 1127 : GEN fu = v, R = gen_1, zu = mkvec2(gen_2, gen_m1);
3487 1127 : GEN clg1 = mkvec3(gen_1,v,v), clg2 = mkvecn(6, m,m,m,v,m,m);
3488 :
3489 1127 : W = A = B = C = m; res = mkvec5(clg1, R, gen_1, zu, fu);
3490 1127 : return buchall_end(nf,res,clg2,W,B,A,C,Vbase);
3491 : }
3492 :
3493 : /* return (small set of) indices of columns generating the same lattice as x.
3494 : * Assume HNF(x) is inexpensive (few rows, many columns).
3495 : * Dichotomy approach since interesting columns may be at the very end */
3496 : GEN
3497 63567 : extract_full_lattice(GEN x)
3498 : {
3499 63567 : long dj, j, k, l = lg(x);
3500 : GEN h, h2, H, v;
3501 :
3502 63567 : if (l < 200) return NULL; /* not worth it */
3503 :
3504 9 : v = vecsmalltrunc_init(l);
3505 9 : H = ZM_hnf(x);
3506 9 : h = cgetg(1, t_MAT);
3507 9 : dj = 1;
3508 431 : for (j = 1; j < l; )
3509 : {
3510 431 : pari_sp av = avma;
3511 431 : long lv = lg(v);
3512 :
3513 3351 : for (k = 0; k < dj; k++) v[lv+k] = j+k;
3514 431 : setlg(v, lv + dj);
3515 431 : h2 = ZM_hnf(vecpermute(x, v));
3516 431 : if (ZM_equal(h, h2))
3517 : { /* these dj columns can be eliminated */
3518 206 : set_avma(av); setlg(v, lv);
3519 206 : j += dj;
3520 206 : if (j >= l) break;
3521 206 : dj <<= 1;
3522 206 : if (j + dj >= l) { dj = (l - j) >> 1; if (!dj) dj = 1; }
3523 : }
3524 225 : else if (dj > 1)
3525 : { /* at least one interesting column, try with first half of this set */
3526 164 : set_avma(av); setlg(v, lv);
3527 164 : dj >>= 1; /* > 0 */
3528 : }
3529 : else
3530 : { /* this column should be kept */
3531 61 : if (ZM_equal(h2, H)) break;
3532 52 : h = h2; j++;
3533 : }
3534 : }
3535 9 : return v;
3536 : }
3537 :
3538 : static void
3539 63602 : init_rel(RELCACHE_t *cache, FB_t *F, long add_need)
3540 : {
3541 63602 : const long n = F->KC + add_need; /* expected # of needed relations */
3542 : long i, j, k, p;
3543 : GEN c, P;
3544 : GEN R;
3545 :
3546 63602 : if (DEBUGLEVEL) err_printf("KCZ = %ld, KC = %ld, n = %ld\n", F->KCZ,F->KC,n);
3547 63602 : reallocate(cache, 10*n + 50); /* make room for lots of relations */
3548 63603 : cache->chk = cache->base;
3549 63603 : cache->end = cache->base + n;
3550 63603 : cache->relsup = add_need;
3551 63603 : cache->last = cache->base;
3552 63603 : cache->missing = lg(cache->basis) - 1;
3553 302625 : for (i = 1; i <= F->KCZ; i++)
3554 : { /* trivial relations (p) = prod P^e */
3555 239021 : p = F->FB[i]; P = gel(F->LV,p);
3556 239021 : if (!isclone(P)) continue;
3557 :
3558 : /* all prime divisors in FB */
3559 166613 : c = zero_Flv(F->KC); k = F->iLP[p];
3560 166613 : R = c; c += k;
3561 532056 : for (j = lg(P)-1; j; j--) c[j] = pr_get_e(gel(P,j));
3562 166612 : add_rel(cache, F, R, k+1, pr_get_p(gel(P,1)), 0);
3563 : }
3564 63604 : }
3565 :
3566 : /* Let z = \zeta_n in nf. List of not-obviously-dependent generators for
3567 : * cyclotomic units modulo torsion in Q(z) [independent when n a prime power]:
3568 : * - z^a - 1, n/(a,n) not a prime power, a \nmid n unless a=1, 1 <= a < n/2
3569 : * - (Z^a - 1)/(Z - 1), p^k || n, Z = z^{n/p^k}, (p,a) = 1, 1 < a <= (p^k-1)/2
3570 : */
3571 : GEN
3572 63603 : nfcyclotomicunits(GEN nf, GEN zu)
3573 : {
3574 63603 : long n = itos(gel(zu, 1)), n2, lP, i, a;
3575 : GEN z, fa, P, E, L, mz, powz;
3576 63603 : if (n <= 6) return cgetg(1, t_VEC);
3577 :
3578 1897 : z = algtobasis(nf,gel(zu, 2));
3579 1897 : if ((n & 3) == 2) { n = n >> 1; z = ZC_neg(z); } /* ensure n != 2 (mod 4) */
3580 1897 : n2 = n/2;
3581 1897 : mz = zk_multable(nf, z); /* multiplication by z */
3582 1897 : powz = cgetg(n2, t_VEC); gel(powz,1) = z;
3583 6237 : for (i = 2; i < n2; i++) gel(powz,i) = ZM_ZC_mul(mz, gel(powz,i-1));
3584 : /* powz[i] = z^i */
3585 :
3586 1897 : L = vectrunc_init(n);
3587 1897 : fa = factoru(n);
3588 1897 : P = gel(fa,1); lP = lg(P);
3589 1897 : E = gel(fa,2);
3590 4578 : for (i = 1; i < lP; i++)
3591 : { /* second kind */
3592 2681 : long p = P[i], k = E[i], pk = upowuu(p,k), pk2 = (pk-1) / 2;
3593 2681 : GEN u = gen_1;
3594 4935 : for (a = 2; a <= pk2; a++)
3595 : {
3596 2254 : u = nfadd(nf, u, gel(powz, (n/pk) * (a-1))); /* = (Z^a-1)/(Z-1) */
3597 2254 : if (a % p) vectrunc_append(L, u);
3598 : }
3599 : }
3600 6104 : if (lP > 2) for (a = 1; a < n2; a++)
3601 : { /* first kind, when n not a prime power */
3602 : ulong p;
3603 4207 : if (a > 1 && (n % a == 0 || uisprimepower(n/ugcd(a,n), &p))) continue;
3604 1848 : vectrunc_append(L, nfadd(nf, gel(powz, a), gen_m1));
3605 : }
3606 1897 : return L;
3607 : }
3608 : static void
3609 63603 : add_cyclotomic_units(GEN nf, GEN zu, RELCACHE_t *cache, FB_t *F)
3610 : {
3611 63603 : pari_sp av = avma;
3612 63603 : GEN L = nfcyclotomicunits(nf, zu);
3613 63603 : long i, l = lg(L);
3614 63603 : if (l > 1)
3615 : {
3616 1897 : GEN R = zero_Flv(F->KC);
3617 5901 : for(i = 1; i < l; i++) add_rel(cache, F, R, F->KC+1, gel(L,i), 0);
3618 : }
3619 63603 : set_avma(av);
3620 63603 : }
3621 :
3622 : static GEN
3623 128545 : trim_list(FB_t *F)
3624 : {
3625 128545 : pari_sp av = avma;
3626 128545 : GEN v, L_jid = F->L_jid, minidx = F->minidx, present = zero_Flv(F->KC);
3627 128545 : long i, j, imax = minss(lg(L_jid), F->KC + 1);
3628 :
3629 128545 : v = cgetg(imax, t_VECSMALL);
3630 2297400 : for (i = j = 1; i < imax; i++)
3631 : {
3632 2168855 : long k = minidx[ L_jid[i] ];
3633 2168855 : if (!present[k]) { v[j++] = L_jid[i]; present[k] = 1; }
3634 : }
3635 128545 : setlg(v, j); return gerepileuptoleaf(av, v);
3636 : }
3637 :
3638 : static void
3639 8342 : try_elt(RELCACHE_t *cache, FB_t *F, GEN nf, GEN x, FACT *fact)
3640 : {
3641 8342 : pari_sp av = avma;
3642 : GEN R, Nx;
3643 8342 : long nz, tx = typ(x);
3644 :
3645 8342 : if (tx == t_INT || tx == t_FRAC) return;
3646 8205 : if (tx != t_COL) x = algtobasis(nf, x);
3647 8205 : if (RgV_isscalar(x)) return;
3648 8205 : x = Q_primpart(x);
3649 8205 : Nx = nfnorm(nf, x);
3650 8205 : if (!can_factor(F, nf, NULL, x, Nx, fact)) return;
3651 :
3652 : /* smooth element */
3653 8205 : R = set_fact(F, fact, NULL, &nz);
3654 : /* make sure we get maximal rank first, then allow all relations */
3655 8205 : (void) add_rel(cache, F, R, nz, x, 0);
3656 8205 : set_avma(av);
3657 : }
3658 :
3659 : static void
3660 46187 : matenlarge(GEN C, long h)
3661 : {
3662 46187 : GEN _0 = zerocol(h);
3663 : long i;
3664 4087247 : for (i = lg(C); --i; ) gel(C,i) = shallowconcat(gel(C,i), _0);
3665 46187 : }
3666 :
3667 : /* E = floating point embeddings */
3668 : static GEN
3669 46187 : matbotidembs(RELCACHE_t *cache, GEN E)
3670 : {
3671 46187 : long w = cache->last - cache->chk, h = cache->last - cache->base;
3672 46187 : long j, d = h - w, hE = nbrows(E);
3673 46187 : GEN y = cgetg(w+1,t_MAT), _0 = zerocol(h);
3674 192806 : for (j = 1; j <= w; j++)
3675 : {
3676 146619 : GEN c = shallowconcat(gel(E,j), _0);
3677 146619 : if (d + j >= 1) gel(c, d + j + hE) = gen_1;
3678 146619 : gel(y,j) = c;
3679 : }
3680 46187 : return y;
3681 : }
3682 : static GEN
3683 62042 : matbotid(RELCACHE_t *cache)
3684 : {
3685 62042 : long w = cache->last - cache->chk, h = cache->last - cache->base;
3686 62042 : long j, d = h - w;
3687 62042 : GEN y = cgetg(w+1,t_MAT);
3688 834826 : for (j = 1; j <= w; j++)
3689 : {
3690 772784 : GEN c = zerocol(h);
3691 772784 : if (d + j >= 1) gel(c, d + j) = gen_1;
3692 772784 : gel(y,j) = c;
3693 : }
3694 62042 : return y;
3695 : }
3696 :
3697 : static long
3698 141 : myprecdbl(long prec, GEN C)
3699 : {
3700 141 : long p = prec2nbits(prec) < 1280? precdbl(prec): (long)(prec * 1.5);
3701 141 : if (C) p = maxss(p, minss(3*p, prec + nbits2extraprec(gexpo(C))));
3702 141 : return p;
3703 : }
3704 :
3705 : static GEN
3706 54289 : _nfnewprec(GEN nf, long prec, long *isclone)
3707 : {
3708 54289 : GEN NF = gclone(nfnewprec_shallow(nf, prec));
3709 54289 : if (*isclone) gunclone(nf);
3710 54289 : *isclone = 1; return NF;
3711 : }
3712 :
3713 : /* Nrelid = nb relations per ideal, possibly 0. If flag is set, keep data in
3714 : * algebraic form. */
3715 : GEN
3716 64687 : Buchall_param(GEN P, double cbach, double cbach2, long Nrelid, long flag, long prec)
3717 : {
3718 : pari_timer T;
3719 64687 : pari_sp av0 = avma, av, av2;
3720 : long PREC, N, R1, R2, RU, low, high, LIMC0, LIMC, LIMC2, LIMCMAX, zc, i;
3721 64687 : long LIMres, bit = 0, flag_nfinit = 0;
3722 64687 : long nreldep, sfb_trials, need, old_need, precdouble = 0, TRIES = 0;
3723 64687 : long nfisclone = 0;
3724 : long done_small, small_fail, fail_limit, squash_index, small_norm_prec;
3725 : double LOGD, LOGD2, lim;
3726 64687 : GEN computed = NULL, fu = NULL, zu, nf, M_sn, D, A, W, R, h, Ce, PERM;
3727 : GEN small_multiplier, auts, cyclic, embs, SUnits;
3728 : GEN res, L, invhr, B, C, lambda, dep, clg1, clg2, Vbase;
3729 64687 : const char *precpb = NULL;
3730 : nfmaxord_t nfT;
3731 : RELCACHE_t cache;
3732 : FB_t F;
3733 : GRHcheck_t GRHcheck;
3734 : FACT *fact;
3735 :
3736 64687 : if (DEBUGLEVEL) timer_start(&T);
3737 64687 : P = get_nfpol(P, &nf);
3738 64677 : if (nf)
3739 3514 : D = nf_get_disc(nf);
3740 : else
3741 : {
3742 61163 : nfinit_basic(&nfT, P);
3743 61168 : D = nfT.dK;
3744 61168 : if (!ZX_is_monic(nfT.T0))
3745 : {
3746 14 : pari_warn(warner,"nonmonic polynomial in bnfinit, using polredbest");
3747 14 : flag_nfinit = nf_RED;
3748 : }
3749 : }
3750 64682 : PREC = maxss(DEFAULTPREC, prec);
3751 64682 : N = degpol(P);
3752 64683 : if (N <= 1)
3753 : {
3754 1127 : if (!nf) nf = nfinit_complete(&nfT, flag_nfinit, PREC);
3755 1127 : return gerepilecopy(av0, Buchall_deg1(nf));
3756 : }
3757 63556 : D = absi_shallow(D);
3758 63556 : LOGD = dbllog2(D) * M_LN2;
3759 63555 : LOGD2 = LOGD*LOGD;
3760 63555 : LIMCMAX = (long)(12.*LOGD2);
3761 : /* In small_norm, LLL reduction produces v0 in I such that
3762 : * T2(v0) <= (4/3)^((n-1)/2) NI^(2/n) disc(K)^(1/n)
3763 : * We consider v with T2(v) <= BMULT * T2(v0)
3764 : * Hence Nv <= ((4/3)^((n-1)/2) * BMULT / n)^(n/2) NI sqrt(disc(K)).
3765 : * NI <= LIMCMAX^2 */
3766 63555 : if (nf) PREC = maxss(PREC, nf_get_prec(nf));
3767 63555 : PREC = maxss(PREC, nbits2prec((long)(LOGD2 * 0.02) + N*N));
3768 63555 : if (DEBUGLEVEL) err_printf("PREC = %ld\n", PREC);
3769 63555 : small_norm_prec = nbits2prec( BITS_IN_LONG +
3770 63555 : (N/2. * ((N-1)/2.*log(4./3) + log(BMULT/(double)N))
3771 63555 : + 2*log((double) LIMCMAX) + LOGD/2) / M_LN2 ); /*enough to compute norms*/
3772 63555 : if (small_norm_prec > PREC) PREC = small_norm_prec;
3773 63555 : if (!nf)
3774 60217 : nf = nfinit_complete(&nfT, flag_nfinit, PREC);
3775 3338 : else if (nf_get_prec(nf) < PREC)
3776 199 : nf = nfnewprec_shallow(nf, PREC);
3777 63558 : M_sn = nf_get_M(nf);
3778 63558 : if (PREC > small_norm_prec) M_sn = gprec_w(M_sn, small_norm_prec);
3779 :
3780 63558 : zu = nfrootsof1(nf);
3781 63557 : gel(zu,2) = nf_to_scalar_or_alg(nf, gel(zu,2));
3782 :
3783 63558 : nf_get_sign(nf, &R1, &R2); RU = R1+R2;
3784 63558 : auts = automorphism_matrices(nf, &cyclic);
3785 63557 : F.embperm = automorphism_perms(nf_get_M(nf), auts, cyclic, R1, R2, N);
3786 63556 : if (DEBUGLEVEL)
3787 : {
3788 0 : timer_printf(&T, "nfinit & nfrootsof1");
3789 0 : err_printf("%s bnf: R1 = %ld, R2 = %ld\nD = %Ps\n",
3790 : flag? "Algebraic": "Floating point", R1,R2, D);
3791 : }
3792 63556 : if (LOGD < 20.)
3793 : { /* tiny disc, Minkowski may be smaller than Bach */
3794 62114 : lim = exp(-N + R2 * log(4/M_PI) + LOGD/2) * sqrt(2*M_PI*N);
3795 62114 : if (lim < 3) lim = 3;
3796 : }
3797 : else /* to be ignored */
3798 1442 : lim = -1;
3799 63556 : if (cbach > 12.) {
3800 0 : if (cbach2 < cbach) cbach2 = cbach;
3801 0 : cbach = 12.;
3802 : }
3803 63556 : if (cbach < 0.)
3804 0 : pari_err_DOMAIN("Buchall","Bach constant","<",gen_0,dbltor(cbach));
3805 :
3806 63556 : cache.base = NULL; F.subFB = NULL; F.LP = NULL; SUnits = Ce = NULL;
3807 63556 : init_GRHcheck(&GRHcheck, N, R1, LOGD);
3808 63559 : high = low = LIMC0 = maxss((long)(cbach2*LOGD2), 1);
3809 309948 : while (!GRHchk(nf, &GRHcheck, high)) { low = high; high *= 2; }
3810 246434 : while (high - low > 1)
3811 : {
3812 182874 : long test = (low+high)/2;
3813 182874 : if (GRHchk(nf, &GRHcheck, test)) high = test; else low = test;
3814 : }
3815 63560 : LIMC2 = (high == LIMC0+1 && GRHchk(nf, &GRHcheck, LIMC0))? LIMC0: high;
3816 63560 : if (LIMC2 > LIMCMAX) LIMC2 = LIMCMAX;
3817 : /* Assuming GRH, {P, NP <= LIMC2} generate Cl(K) */
3818 63560 : if (DEBUGLEVEL) err_printf("LIMC2 = %ld\n", LIMC2);
3819 63560 : LIMC0 = (long)(cbach*LOGD2); /* initial value for LIMC */
3820 63560 : LIMC = cbach? LIMC0: LIMC2; /* use {P, NP <= LIMC} as a factorbase */
3821 63560 : LIMC = maxss(LIMC, nthideal(&GRHcheck, nf, N));
3822 63560 : if (DEBUGLEVEL) timer_printf(&T, "computing Bach constant");
3823 63560 : LIMres = primeneeded(N, R1, R2, LOGD);
3824 63560 : cache_prime_dec(&GRHcheck, LIMres, nf);
3825 : /* invhr ~ 2^r1 (2pi)^r2 / sqrt(D) w * Res(zeta_K, s=1) = 1 / hR */
3826 127120 : invhr = gmul(gdiv(gmul2n(powru(mppi(DEFAULTPREC), R2), RU),
3827 63560 : mulri(gsqrt(D,DEFAULTPREC),gel(zu,1))),
3828 : compute_invres(&GRHcheck, LIMres));
3829 63560 : if (DEBUGLEVEL) timer_printf(&T, "computing inverse of hR");
3830 63560 : av = avma;
3831 :
3832 65780 : START:
3833 65780 : if (DEBUGLEVEL) timer_start(&T);
3834 65780 : if (TRIES) LIMC = bnf_increase_LIMC(LIMC,LIMCMAX);
3835 65780 : if (DEBUGLEVEL && LIMC > LIMC0)
3836 0 : err_printf("%s*** Bach constant: %f\n", TRIES?"\n":"", LIMC/LOGD2);
3837 65780 : if (cache.base)
3838 : {
3839 : REL_t *rel;
3840 23556 : for (i = 1, rel = cache.base + 1; rel < cache.last; rel++)
3841 23513 : if (rel->m) i++;
3842 43 : computed = cgetg(i, t_VEC);
3843 23556 : for (i = 1, rel = cache.base + 1; rel < cache.last; rel++)
3844 23513 : if (rel->m) gel(computed, i++) = rel->m;
3845 43 : computed = gclone(computed); delete_cache(&cache);
3846 : }
3847 65780 : TRIES++; set_avma(av);
3848 65780 : if (F.LP) delete_FB(&F);
3849 65780 : if (LIMC2 < LIMC) LIMC2 = LIMC;
3850 65780 : if (DEBUGLEVEL) { err_printf("LIMC = %ld, LIMC2 = %ld\n",LIMC,LIMC2); }
3851 :
3852 65780 : FBgen(&F, nf, N, LIMC, LIMC2, &GRHcheck);
3853 65775 : if (!F.KC) goto START;
3854 65775 : av = avma;
3855 65775 : subFBgen(&F,auts,cyclic,lim < 0? LIMC2: mindd(lim,LIMC2),MINSFB);
3856 65780 : if (lg(F.subFB) == 1) goto START;
3857 63603 : if (DEBUGLEVEL)
3858 0 : timer_printf(&T, "factorbase (#subFB = %ld) and ideal permutations",
3859 0 : lg(F.subFB)-1);
3860 :
3861 63603 : fact = (FACT*)stack_malloc((F.KC+1)*sizeof(FACT));
3862 63602 : PERM = leafcopy(F.perm); /* to be restored in case of precision increase */
3863 63603 : cache.basis = zero_Flm_copy(F.KC,F.KC);
3864 63603 : small_multiplier = zero_Flv(F.KC);
3865 63603 : done_small = small_fail = squash_index = zc = sfb_trials = nreldep = 0;
3866 63603 : fail_limit = F.KC + 1;
3867 63603 : W = A = R = NULL;
3868 63603 : av2 = avma;
3869 63603 : init_rel(&cache, &F, RELSUP + RU-1);
3870 63603 : old_need = need = cache.end - cache.last;
3871 63603 : add_cyclotomic_units(nf, zu, &cache, &F);
3872 63603 : if (DEBUGLEVEL) err_printf("\n");
3873 63603 : cache.end = cache.last + need;
3874 :
3875 63603 : if (computed)
3876 : {
3877 8385 : for (i = 1; i < lg(computed); i++)
3878 8342 : try_elt(&cache, &F, nf, gel(computed, i), fact);
3879 43 : gunclone(computed);
3880 43 : if (DEBUGLEVEL && i > 1)
3881 0 : timer_printf(&T, "including already computed relations");
3882 43 : need = 0;
3883 : }
3884 :
3885 : do
3886 : {
3887 : GEN Ar, C0;
3888 : do
3889 : {
3890 128711 : pari_sp av4 = avma;
3891 128711 : if (need > 0)
3892 : {
3893 128545 : long oneed = cache.end - cache.last;
3894 : /* Test below can be true if small_norm did not find enough linearly
3895 : * dependent relations */
3896 128545 : if (need < oneed) need = oneed;
3897 128545 : pre_allocate(&cache, need+lg(auts)-1+(R ? lg(W)-1 : 0));
3898 128545 : cache.end = cache.last + need;
3899 128545 : F.L_jid = trim_list(&F);
3900 : }
3901 128711 : if (need > 0 && Nrelid > 0 && (done_small <= F.KC+1 || A) &&
3902 127531 : small_fail <= fail_limit &&
3903 127531 : cache.last < cache.base + 2*F.KC+2*RU+RELSUP /* heuristic */)
3904 : {
3905 114837 : long j, k, LIE = (R && lg(W) > 1 && (done_small % 2));
3906 114837 : REL_t *last = cache.last;
3907 114837 : pari_sp av3 = avma;
3908 : GEN p0;
3909 114837 : if (LIE)
3910 : { /* We have full rank for class group and unit. The following tries to
3911 : * improve the prime group lattice by looking for relations involving
3912 : * the primes generating the class group. */
3913 3196 : long n = lg(W)-1; /* need n relations to squash the class group */
3914 3196 : F.L_jid = vecslice(F.perm, 1, n);
3915 3196 : cache.end = cache.last + n;
3916 : /* Lie to the add_rel subsystem: pretend we miss relations involving
3917 : * the primes generating the class group (and only those). */
3918 3196 : cache.missing = n;
3919 14892 : for ( ; n > 0; n--) mael(cache.basis, F.perm[n], F.perm[n]) = 0;
3920 : }
3921 114837 : j = done_small % (F.KC+1);
3922 114837 : if (j == 0) p0 = NULL;
3923 : else
3924 : {
3925 49946 : p0 = gel(F.LP, j);
3926 49946 : if (!A)
3927 : { /* Prevent considering both P_iP_j and P_jP_i in small_norm */
3928 : /* Not all elements end up in F.L_jid (eliminated by hnfspec/add or
3929 : * by trim_list): keep track of which ideals are being considered
3930 : * at each run. */
3931 18020 : long mj = small_multiplier[j];
3932 254575 : for (i = k = 1; i < lg(F.L_jid); i++)
3933 236555 : if (F.L_jid[i] > mj)
3934 : {
3935 228871 : small_multiplier[F.L_jid[i]] = j;
3936 228871 : F.L_jid[k++] = F.L_jid[i];
3937 : }
3938 18020 : setlg(F.L_jid, k);
3939 : }
3940 : }
3941 114837 : if (lg(F.L_jid) > 1)
3942 114669 : small_norm(&cache, &F, nf, Nrelid, M_sn, fact, p0);
3943 114837 : F.L_jid = F.perm; set_avma(av3);
3944 114837 : if (!A && cache.last != last) small_fail = 0; else small_fail++;
3945 114837 : if (LIE)
3946 : { /* restore add_rel subsystem: undo above lie */
3947 3196 : long n = lg(W) - 1;
3948 14892 : for ( ; n > 0; n--) mael(cache.basis, F.perm[n], F.perm[n]) = 1;
3949 3196 : cache.missing = 0;
3950 : }
3951 114837 : cache.end = cache.last;
3952 114837 : done_small++;
3953 114837 : need = F.sfb_chg = 0;
3954 : }
3955 128711 : if (need > 0)
3956 : { /* Random relations */
3957 13708 : if (++nreldep > F.MAXDEPSIZESFB) {
3958 157 : if (++sfb_trials > SFB_MAX && LIMC < LIMCMAX/6) goto START;
3959 128 : F.sfb_chg = sfb_INCREASE;
3960 128 : nreldep = 0;
3961 : }
3962 13551 : else if (!(nreldep % F.MAXDEPSFB))
3963 2272 : F.sfb_chg = sfb_CHANGE;
3964 13679 : if (F.sfb_chg && !subFB_change(&F)) goto START;
3965 13665 : rnd_rel(&cache, &F, nf, fact);
3966 13665 : F.L_jid = F.perm;
3967 : }
3968 128668 : if (DEBUGLEVEL) timer_start(&T);
3969 128668 : if (precpb)
3970 : {
3971 : REL_t *rel;
3972 148 : if (DEBUGLEVEL)
3973 : {
3974 0 : char str[64]; sprintf(str,"Buchall_param (%s)",precpb);
3975 0 : pari_warn(warnprec,str,PREC);
3976 : }
3977 148 : nf = _nfnewprec(nf, PREC, &nfisclone);
3978 148 : precdouble++; precpb = NULL;
3979 :
3980 148 : if (flag)
3981 : { /* recompute embs only, no need to redo HNF */
3982 105 : long j, le = lg(embs), lC = lg(C);
3983 105 : GEN E, M = nf_get_M(nf);
3984 105 : set_avma(av4);
3985 32540 : for (rel = cache.base+1, i = 1; i < le; i++,rel++)
3986 32435 : gel(embs,i) = rel_embed(rel, &F, embs, i, M, RU, R1, PREC);
3987 105 : E = RgM_ZM_mul(embs, rowslice(C, RU+1, nbrows(C)));
3988 32540 : for (j = 1; j < lC; j++)
3989 148511 : for (i = 1; i <= RU; i++) gcoeff(C,i,j) = gcoeff(E,i,j);
3990 105 : av4 = avma;
3991 : }
3992 : else
3993 : { /* recompute embs + HNF */
3994 10532 : for(i = 1; i < lg(PERM); i++) F.perm[i] = PERM[i];
3995 43 : cache.chk = cache.base;
3996 43 : W = NULL;
3997 : }
3998 148 : if (DEBUGLEVEL) timer_printf(&T, "increasing accuracy");
3999 : }
4000 128668 : set_avma(av4);
4001 128668 : if (cache.chk != cache.last)
4002 : { /* Reduce relation matrices */
4003 114042 : long l = cache.last - cache.chk + 1, j;
4004 114042 : GEN mat = cgetg(l, t_MAT);
4005 : REL_t *rel;
4006 :
4007 1089744 : for (j=1,rel = cache.chk + 1; j < l; rel++,j++) gel(mat,j) = rel->R;
4008 114042 : if (!flag || W)
4009 : {
4010 52000 : embs = get_embs(&F, &cache, nf, embs, PREC);
4011 52000 : if (DEBUGLEVEL && timer_get(&T) > 1)
4012 0 : timer_printf(&T, "floating point embeddings");
4013 : }
4014 114042 : if (!W)
4015 : { /* never reduced before */
4016 63646 : C = flag? matbotid(&cache): embs;
4017 63646 : W = hnfspec_i(mat, F.perm, &dep, &B, &C, F.subFB ? lg(F.subFB)-1:0);
4018 63646 : if (DEBUGLEVEL)
4019 0 : timer_printf(&T, "hnfspec [%ld x %ld]", lg(F.perm)-1, l-1);
4020 63646 : if (flag)
4021 : {
4022 62042 : PREC += nbits2extraprec(gexpo(C));
4023 62042 : if (nf_get_prec(nf) < PREC) nf = _nfnewprec(nf, PREC, &nfisclone);
4024 62042 : embs = get_embs(&F, &cache, nf, embs, PREC);
4025 62040 : C = vconcat(RgM_ZM_mul(embs, C), C);
4026 : }
4027 63646 : if (DEBUGLEVEL)
4028 0 : timer_printf(&T, "hnfspec floating points");
4029 : }
4030 : else
4031 : {
4032 50396 : long k = lg(embs);
4033 50396 : GEN E = vecslice(embs, k-l+1,k-1);
4034 50396 : if (flag)
4035 : {
4036 46187 : E = matbotidembs(&cache, E);
4037 46187 : matenlarge(C, cache.last - cache.chk);
4038 : }
4039 50396 : W = hnfadd_i(W, F.perm, &dep, &B, &C, mat, E);
4040 50396 : if (DEBUGLEVEL)
4041 0 : timer_printf(&T, "hnfadd (%ld + %ld)", l-1, lg(dep)-1);
4042 : }
4043 114042 : gerepileall(av2, 5, &W,&C,&B,&dep,&embs);
4044 114042 : cache.chk = cache.last;
4045 : }
4046 14626 : else if (!W)
4047 : {
4048 0 : need = old_need;
4049 0 : F.L_jid = vecslice(F.perm, 1, need);
4050 0 : continue;
4051 : }
4052 128668 : need = F.KC - (lg(W)-1) - (lg(B)-1);
4053 128668 : if (!need && cache.missing)
4054 : { /* The test above will never be true except if 27449|class number.
4055 : * Ensure that if we have maximal rank for the ideal lattice, then
4056 : * cache.missing == 0. */
4057 14 : for (i = 1; cache.missing; i++)
4058 7 : if (!mael(cache.basis, i, i))
4059 : {
4060 : long j;
4061 7 : cache.missing--; mael(cache.basis, i, i) = 1;
4062 427 : for (j = i+1; j <= F.KC; j++) mael(cache.basis, j, i) = 0;
4063 : }
4064 : }
4065 128668 : zc = (lg(C)-1) - (lg(B)-1) - (lg(W)-1);
4066 128668 : if (RU-1-zc > 0) need = minss(need + RU-1-zc, F.KC); /* for units */
4067 128668 : if (need)
4068 : { /* dependent rows */
4069 18434 : F.L_jid = vecslice(F.perm, 1, need);
4070 18434 : vecsmall_sort(F.L_jid);
4071 18434 : if (need != old_need) { nreldep = 0; old_need = need; }
4072 : }
4073 : else
4074 : { /* If the relation lattice is too small, check will be > 1 and we will
4075 : * do a new run of small_norm/rnd_rel asking for 1 relation. This often
4076 : * gives a relation involving L_jid[1]. We rotate the first element of
4077 : * L_jid in order to increase the probability of finding relations that
4078 : * increases the lattice. */
4079 110234 : long j, n = lg(W) - 1;
4080 110234 : if (n > 1 && squash_index % n)
4081 : {
4082 7601 : F.L_jid = leafcopy(F.perm);
4083 41678 : for (j = 1; j <= n; j++)
4084 34077 : F.L_jid[j] = F.perm[1 + (j + squash_index - 1) % n];
4085 : }
4086 : else
4087 102633 : F.L_jid = F.perm;
4088 110234 : squash_index++;
4089 : }
4090 : }
4091 128668 : while (need);
4092 :
4093 110234 : if (!A)
4094 : {
4095 63609 : small_fail = old_need = 0;
4096 63609 : fail_limit = maxss(F.KC / FAIL_DIVISOR, MINFAIL);
4097 : }
4098 110234 : A = vecslice(C, 1, zc); /* cols corresponding to units */
4099 110234 : if (flag) A = rowslice(A, 1, RU);
4100 110234 : Ar = real_i(A);
4101 110234 : R = compute_multiple_of_R(Ar, RU, N, &need, &bit, &lambda);
4102 110233 : if (need < old_need) small_fail = 0;
4103 : #if 0 /* A good idea if we are indeed stuck but needs tuning */
4104 : /* we have computed way more relations than should be necessary */
4105 : if (TRIES < 3 && LIMC < LIMCMAX / 24 &&
4106 : cache.last - cache.base > 10 * F.KC) goto START;
4107 : #endif
4108 110233 : old_need = need;
4109 110233 : if (!lambda)
4110 77 : { precpb = "bestappr"; PREC = myprecdbl(PREC, flag? C: NULL); continue; }
4111 110156 : if (!R)
4112 : { /* not full rank for units */
4113 37105 : if (!need)
4114 1 : { precpb = "regulator"; PREC = myprecdbl(PREC, flag? C: NULL); }
4115 37105 : continue;
4116 : }
4117 73051 : h = ZM_det_triangular(W);
4118 73051 : if (DEBUGLEVEL) err_printf("\n#### Tentative class number: %Ps\n", h);
4119 73051 : i = compute_R(lambda, mulir(h,invhr), &L, &R);
4120 73052 : if (DEBUGLEVEL)
4121 : {
4122 0 : err_printf("\n");
4123 0 : timer_printf(&T, "computing regulator and check");
4124 : }
4125 73052 : switch(i)
4126 : {
4127 9422 : case fupb_RELAT:
4128 9422 : need = 1; /* not enough relations */
4129 9422 : continue;
4130 63 : case fupb_PRECI: /* prec problem unless we cheat on Bach constant */
4131 63 : if ((precdouble&7) == 7 && LIMC<=LIMCMAX/6) goto START;
4132 63 : precpb = "compute_R"; PREC = myprecdbl(PREC, flag? C: NULL);
4133 63 : continue;
4134 : }
4135 : /* DONE */
4136 :
4137 63567 : if (F.KCZ2 > F.KCZ)
4138 : {
4139 7 : if (F.sfb_chg && !subFB_change(&F)) goto START;
4140 7 : if (!be_honest(&F, nf, auts, fact)) goto START;
4141 7 : if (DEBUGLEVEL) timer_printf(&T, "to be honest");
4142 : }
4143 63567 : F.KCZ2 = 0; /* be honest only once */
4144 :
4145 : /* fundamental units */
4146 : {
4147 63567 : GEN AU, CU, U, v = extract_full_lattice(L); /* L may be large */
4148 63567 : CU = NULL;
4149 63567 : if (v) { A = vecpermute(A, v); L = vecpermute(L, v); }
4150 : /* arch. components of fund. units */
4151 63567 : U = ZM_lll(L, 0.99, LLL_IM);
4152 63566 : U = ZM_mul(U, lll(RgM_ZM_mul(real_i(A), U)));
4153 63565 : if (DEBUGLEVEL) timer_printf(&T, "units LLL");
4154 63565 : AU = RgM_ZM_mul(A, U);
4155 63565 : A = cleanarchunit(AU, N, NULL, PREC);
4156 63567 : if (!A || lg(A) < RU || expo(gsub(get_regulator(A), R)) > -1)
4157 : {
4158 6 : long add = nbits2extraprec( gexpo(AU) + 64 ) - gprecision(AU);
4159 6 : long t = maxss((PREC-2) * 0.15, add);
4160 6 : if (!A && DEBUGLEVEL) err_printf("### Incorrect units lognorm");
4161 6 : precpb = "cleanarch"; PREC += maxss(t, EXTRAPRECWORD); continue;
4162 : }
4163 63561 : if (flag)
4164 : {
4165 62006 : long l = lgcols(C) - RU;
4166 : REL_t *rel;
4167 62006 : SUnits = cgetg(l, t_COL);
4168 964116 : for (rel = cache.base+1, i = 1; i < l; i++,rel++)
4169 902111 : set_rel_alpha(rel, auts, SUnits, i);
4170 62005 : if (RU > 1)
4171 : {
4172 47340 : GEN c = v? vecpermute(C,v): vecslice(C,1,zc);
4173 47340 : CU = ZM_mul(rowslice(c, RU+1, nbrows(c)), U);
4174 : }
4175 : }
4176 63559 : if (DEBUGLEVEL) err_printf("\n#### Computing fundamental units\n");
4177 63560 : fu = getfu(nf, &A, CU? &U: NULL, PREC);
4178 63561 : CU = CU? ZM_mul(CU, U): cgetg(1, t_MAT);
4179 63561 : if (DEBUGLEVEL) timer_printf(&T, "getfu");
4180 63561 : Ce = vecslice(C, zc+1, lg(C)-1);
4181 63561 : if (flag) SUnits = mkvec4(SUnits, CU, rowslice(Ce, RU+1, nbrows(Ce)),
4182 : utoipos(LIMC));
4183 : }
4184 : /* class group generators */
4185 63561 : if (flag) Ce = rowslice(Ce, 1, RU);
4186 63559 : C0 = Ce; Ce = cleanarch(Ce, N, NULL, PREC);
4187 63561 : if (!Ce) {
4188 1 : long add = nbits2extraprec( gexpo(C0) + 64 ) - gprecision(C0);
4189 1 : precpb = "cleanarch"; PREC += maxss(add, 1);
4190 : }
4191 63561 : if (DEBUGLEVEL) timer_printf(&T, "cleanarch");
4192 110234 : } while (need || precpb);
4193 :
4194 63560 : Vbase = vecpermute(F.LP, F.perm);
4195 63560 : if (!fu) fu = cgetg(1, t_MAT);
4196 63560 : if (!SUnits) SUnits = gen_1;
4197 63560 : clg1 = class_group_gen(nf,W,Ce,Vbase,PREC, &clg2);
4198 63560 : res = mkvec5(clg1, R, SUnits, zu, fu);
4199 63559 : res = buchall_end(nf,res,clg2,W,B,A,Ce,Vbase);
4200 63559 : delete_FB(&F);
4201 63560 : res = gerepilecopy(av0, res);
4202 63560 : if (flag) obj_insert_shallow(res, MATAL, cgetg(1,t_VEC));
4203 63560 : if (nfisclone) gunclone(nf);
4204 63560 : delete_cache(&cache);
4205 63560 : free_GRHcheck(&GRHcheck);
4206 63560 : return res;
4207 : }
|
