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 maxtry_FACT = 500;
34 : /* rnd_rel */
35 : static const long RND_REL_RELPID = 1;
36 : /* random relations */
37 : static const long MINSFB = 3;
38 : static const long SFB_MAX = 3;
39 : static const long DEPSIZESFBMULT = 16;
40 : static const long DEPSFBDIV = 10;
41 : /* add_rel_i */
42 : static const ulong mod_p = 27449UL;
43 : /* be_honest */
44 : static const long maxtry_HONEST = 50;
45 :
46 : typedef struct FACT {
47 : long pr, ex;
48 : } FACT;
49 :
50 : typedef struct subFB_t {
51 : GEN subFB;
52 : struct subFB_t *old;
53 : } subFB_t;
54 :
55 : /* a factor base contains only noninert primes
56 : * KC = # of P in factor base (p <= n, NP <= n2)
57 : * KC2= # of P assumed to generate class group (NP <= n2)
58 : *
59 : * KCZ = # of rational primes under ideals counted by KC
60 : * KCZ2= same for KC2 */
61 :
62 : typedef struct FB_t {
63 : GEN FB; /* FB[i] = i-th rational prime used in factor base */
64 : GEN LP; /* vector of all prime ideals in FB, by increasing norm */
65 : GEN LV; /* LV[p] = vector of P|p, NP <= n2
66 : * isclone() is set for LV[p] iff all P|p are in FB
67 : * LV[i], i not prime or i > n2, is undefined! */
68 : GEN iLP; /* iLP[p] = i such that LV[p] = [LP[i],...] */
69 : GEN L_jid; /* indexes of "useful" prime ideals for rnd_rel */
70 : long KC, KCZ, KCZ2;
71 : GEN prodZ; /* product of the primes in KCZ*/
72 : GEN subFB; /* LP o subFB = part of FB used to build random relations */
73 : int sfb_chg; /* need to change subFB ? */
74 : GEN perm; /* permutation of LP used to represent relations [updated by
75 : hnfspec/hnfadd: dense rows come first] */
76 : GEN idealperm; /* permutation of ideals under field automorphisms */
77 : GEN minidx; /* minidx[i] min ideal in orbit of LP[i] under field autom */
78 : subFB_t *allsubFB; /* all subFB's used */
79 : GEN embperm; /* permutations of the complex embeddings */
80 : long MAXDEPSIZESFB; /* # trials before increasing subFB */
81 : long MAXDEPSFB; /* MAXDEPSIZESFB / DEPSFBDIV, # trials befor rotating subFB */
82 : double ballvol;
83 : } FB_t;
84 :
85 : enum { sfb_CHANGE = 1, sfb_INCREASE = 2 };
86 :
87 : typedef struct REL_t {
88 : GEN R; /* relation vector as t_VECSMALL; clone */
89 : long nz; /* index of first nonzero elt in R (hash) */
90 : GEN m; /* pseudo-minimum yielding the relation; clone */
91 : long relorig; /* relation this one is an image of */
92 : long relaut; /* automorphim used to compute this relation from the original */
93 : GEN emb; /* archimedean embeddings */
94 : GEN junk[2]; /*make sure sizeof(struct) is a power of two.*/
95 : } REL_t;
96 :
97 : typedef struct RELCACHE_t {
98 : REL_t *chk; /* last checkpoint */
99 : REL_t *base; /* first rel found */
100 : REL_t *last; /* last rel found so far */
101 : REL_t *end; /* target for last relation. base <= last <= end */
102 : size_t len; /* number of rels pre-allocated in base */
103 : long relsup; /* how many linearly dependent relations we allow */
104 : GEN basis; /* mod p basis (generating family actually) */
105 : ulong missing; /* missing vectors in generating family above */
106 : } RELCACHE_t;
107 :
108 : typedef struct FP_t {
109 : double **q, *v, *y, *z;
110 : GEN x;
111 : } FP_t;
112 :
113 : static void
114 0 : wr_rel(GEN e)
115 : {
116 0 : long i, l = lg(e);
117 0 : for (i = 1; i < l; i++)
118 0 : if (e[i]) err_printf("%ld^%ld ",i,e[i]);
119 0 : }
120 : static void
121 0 : dbg_newrel(RELCACHE_t *cache)
122 : {
123 0 : if (DEBUGLEVEL > 1)
124 : {
125 0 : err_printf("\n++++ cglob = %ld\nrel = ", cache->last - cache->base);
126 0 : wr_rel(cache->last->R);
127 0 : err_printf("\n");
128 : }
129 : else
130 0 : err_printf("%ld ", cache->last - cache->base);
131 0 : }
132 :
133 : static void
134 63861 : delete_cache(RELCACHE_t *M)
135 : {
136 : REL_t *rel;
137 1047745 : for (rel = M->base+1; rel <= M->last; rel++)
138 : {
139 983886 : gunclone(rel->R);
140 983886 : if (rel->m) gunclone(rel->m);
141 : }
142 63859 : pari_free((void*)M->base); M->base = NULL;
143 63861 : }
144 :
145 : static void
146 66037 : delete_FB(FB_t *F)
147 : {
148 : subFB_t *s, *sold;
149 132814 : for (s = F->allsubFB; s; s = sold) { sold = s->old; pari_free(s); }
150 66038 : gunclone(F->minidx);
151 66038 : gunclone(F->idealperm);
152 66038 : }
153 :
154 : static void
155 63861 : reallocate(RELCACHE_t *M, long len)
156 : {
157 63861 : M->len = len;
158 63861 : if (!M->base)
159 63861 : M->base = (REL_t*)pari_malloc((len+1) * sizeof(REL_t));
160 : else
161 : {
162 0 : size_t l = M->last - M->base, c = M->chk - M->base, e = M->end - M->base;
163 0 : pari_realloc_ip((void**)&M->base, (len+1) * sizeof(REL_t));
164 0 : M->last = M->base + l;
165 0 : M->chk = M->base + c;
166 0 : M->end = M->base + e;
167 : }
168 63861 : }
169 :
170 : #define pr_get_smallp(pr) gel(pr,1)[2]
171 :
172 : /* don't take P|p all other Q|p are already there */
173 : static int
174 306145 : bad_subFB(FB_t *F, long t)
175 : {
176 306145 : GEN LP, P = gel(F->LP,t);
177 306145 : long p = pr_get_smallp(P);
178 306145 : LP = gel(F->LV,p);
179 306145 : return (isclone(LP) && t == F->iLP[p] + lg(LP)-1);
180 : }
181 :
182 : static void
183 66778 : assign_subFB(FB_t *F, GEN yes, long iyes)
184 : {
185 66778 : long i, lv = sizeof(subFB_t) + iyes*sizeof(long); /* for struct + GEN */
186 66778 : subFB_t *s = (subFB_t *)pari_malloc(lv);
187 66778 : s->subFB = (GEN)&s[1];
188 66778 : s->old = F->allsubFB; F->allsubFB = s;
189 287667 : for (i = 0; i < iyes; i++) s->subFB[i] = yes[i];
190 66778 : F->subFB = s->subFB;
191 66778 : F->MAXDEPSIZESFB = (iyes-1) * DEPSIZESFBMULT;
192 66778 : F->MAXDEPSFB = F->MAXDEPSIZESFB / DEPSFBDIV;
193 66778 : }
194 :
195 : /* Determine the permutation of the ideals made by each field automorphism */
196 : static GEN
197 66038 : FB_aut_perm(FB_t *F, GEN auts, GEN cyclic)
198 : {
199 66038 : long i, j, m, KC = F->KC, nauts = lg(auts)-1;
200 66038 : GEN minidx, perm = zero_Flm_copy(KC, nauts);
201 :
202 66038 : if (!nauts) { F->minidx = gclone(identity_zv(KC)); return cgetg(1,t_MAT); }
203 41755 : minidx = zero_Flv(KC);
204 90933 : for (m = 1; m < lg(cyclic); m++)
205 : {
206 49179 : GEN thiscyc = gel(cyclic, m);
207 49179 : long k0 = thiscyc[1];
208 49179 : GEN aut = gel(auts, k0), permk0 = gel(perm, k0), ppermk;
209 49179 : i = 1;
210 211534 : while (i <= KC)
211 : {
212 162356 : pari_sp av2 = avma;
213 162356 : GEN seen = zero_Flv(KC), P = gel(F->LP, i);
214 162356 : long imin = i, p, f, l;
215 162356 : p = pr_get_smallp(P);
216 162356 : f = pr_get_f(P);
217 : do
218 : {
219 477650 : if (++i > KC) break;
220 428472 : P = gel(F->LP, i);
221 : }
222 428472 : while (p == pr_get_smallp(P) && f == pr_get_f(P));
223 639990 : for (j = imin; j < i; j++)
224 : {
225 477637 : GEN img = ZM_ZC_mul(aut, pr_get_gen(gel(F->LP, j)));
226 1663121 : for (l = imin; l < i; l++)
227 1663121 : if (!seen[l] && ZC_prdvd(img, gel(F->LP, l)))
228 : {
229 477634 : seen[l] = 1; permk0[j] = l; break;
230 : }
231 : }
232 162353 : set_avma(av2);
233 : }
234 68365 : for (ppermk = permk0, i = 2; i < lg(thiscyc); i++)
235 : {
236 19187 : GEN permk = gel(perm, thiscyc[i]);
237 384196 : for (j = 1; j <= KC; j++) permk[j] = permk0[ppermk[j]];
238 19187 : ppermk = permk;
239 : }
240 : }
241 308917 : for (j = 1; j <= KC; j++)
242 : {
243 267163 : if (minidx[j]) continue;
244 128529 : minidx[j] = j;
245 358693 : for (i = 1; i <= nauts; i++) minidx[coeff(perm, j, i)] = j;
246 : }
247 41754 : F->minidx = gclone(minidx); return perm;
248 : }
249 :
250 : /* set subFB.
251 : * Fill F->perm (if != NULL): primes ideals sorted by increasing norm (except
252 : * the ones in subFB come first [dense rows for hnfspec]) */
253 : static void
254 66038 : subFBgen(FB_t *F, GEN auts, GEN cyclic, double PROD, long minsFB)
255 : {
256 : GEN y, perm, yes, no;
257 66038 : long i, j, k, iyes, ino, lv = F->KC + 1;
258 : double prod;
259 : pari_sp av;
260 :
261 66038 : F->LP = cgetg(lv, t_VEC);
262 66038 : F->L_jid = F->perm = cgetg(lv, t_VECSMALL);
263 66038 : av = avma;
264 66038 : y = cgetg(lv,t_COL); /* Norm P */
265 311347 : for (k=0, i=1; i <= F->KCZ; i++)
266 : {
267 245309 : GEN LP = gel(F->LV,F->FB[i]);
268 245309 : long l = lg(LP);
269 709392 : for (j = 1; j < l; j++)
270 : {
271 464084 : GEN P = gel(LP,j);
272 464084 : k++;
273 464084 : gel(y,k) = pr_norm(P);
274 464083 : gel(F->LP,k) = P;
275 : }
276 : }
277 : /* perm sorts LP by increasing norm */
278 66038 : perm = indexsort(y);
279 66038 : no = cgetg(lv, t_VECSMALL); ino = 1;
280 66038 : yes = cgetg(lv, t_VECSMALL); iyes = 1;
281 66038 : prod = 1.0;
282 302383 : for (i = 1; i < lv; i++)
283 : {
284 272402 : long t = perm[i];
285 272402 : if (bad_subFB(F, t)) { no[ino++] = t; continue; }
286 :
287 152369 : yes[iyes++] = t;
288 152369 : prod *= (double)itos(gel(y,t));
289 152369 : if (iyes > minsFB && prod > PROD) break;
290 : }
291 66038 : setlg(yes, iyes);
292 218407 : for (j=1; j<iyes; j++) F->perm[j] = yes[j];
293 186071 : for (i=1; i<ino; i++, j++) F->perm[j] = no[i];
294 257729 : for ( ; j<lv; j++) F->perm[j] = perm[j];
295 66038 : F->allsubFB = NULL;
296 66038 : F->idealperm = gclone(FB_aut_perm(F, auts, cyclic));
297 66038 : if (iyes) assign_subFB(F, yes, iyes);
298 66038 : set_avma(av);
299 66038 : }
300 : static int
301 26202 : subFB_change(FB_t *F)
302 : {
303 26202 : long i, iyes, minsFB, lv = F->KC + 1, l = lg(F->subFB)-1;
304 26202 : pari_sp av = avma;
305 26202 : GEN yes, L_jid = F->L_jid, present = zero_zv(lv-1);
306 :
307 26200 : switch (F->sfb_chg)
308 : {
309 14 : case sfb_INCREASE: minsFB = l + 1; break;
310 26186 : default: minsFB = l; break;
311 : }
312 :
313 26200 : yes = cgetg(minsFB+1, t_VECSMALL); iyes = 1;
314 26200 : if (L_jid)
315 : {
316 33637 : for (i = 1; i < lg(L_jid); i++)
317 : {
318 33477 : long l = L_jid[i];
319 33477 : if (bad_subFB(F, l)) continue;
320 31234 : yes[iyes++] = l;
321 31234 : present[l] = 1;
322 31234 : if (iyes > minsFB) break;
323 : }
324 : }
325 0 : else i = 1;
326 26201 : if (iyes <= minsFB)
327 : {
328 343 : for ( ; i < lv; i++)
329 : {
330 273 : long l = F->perm[i];
331 273 : if (present[l] || bad_subFB(F, l)) continue;
332 109 : yes[iyes++] = l;
333 109 : if (iyes > minsFB) break;
334 : }
335 159 : if (i == lv) return 0;
336 : }
337 26131 : if (zv_equal(F->subFB, yes))
338 : {
339 25390 : if (DEBUGLEVEL) err_printf("\n*** NOT Changing sub factor base\n");
340 : }
341 : else
342 : {
343 740 : if (DEBUGLEVEL) err_printf("\n*** Changing sub factor base\n");
344 740 : assign_subFB(F, yes, iyes);
345 : }
346 26130 : F->sfb_chg = 0; return gc_bool(av, 1);
347 : }
348 :
349 : /* make sure enough room to store n more relations */
350 : static void
351 121000 : pre_allocate(RELCACHE_t *cache, size_t n)
352 : {
353 121000 : size_t len = (cache->last - cache->base) + n;
354 121000 : if (len >= cache->len) reallocate(cache, len << 1);
355 121000 : }
356 :
357 : void
358 134152 : init_GRHcheck(GRHcheck_t *S, long N, long R1, double LOGD)
359 : {
360 134152 : const double c1 = M_PI*M_PI/2;
361 134152 : const double c2 = 3.663862376709;
362 134152 : const double c3 = 3.801387092431; /* Euler + log(8*Pi)*/
363 134152 : S->clone = 0;
364 134152 : S->cN = R1*c2 + N*c1;
365 134152 : S->cD = LOGD - N*c3 - R1*M_PI/2;
366 134152 : S->maxprimes = 16000; /* sufficient for LIMC=176081*/
367 134152 : S->primes = (GRHprime_t*)pari_malloc(S->maxprimes*sizeof(*S->primes));
368 134153 : S->nprimes = 0;
369 134153 : S->limp = 0;
370 134153 : u_forprime_init(&S->P, 2, ULONG_MAX);
371 134153 : }
372 :
373 : void
374 134154 : free_GRHcheck(GRHcheck_t *S)
375 : {
376 134154 : if (S->clone)
377 : {
378 63791 : long i = S->nprimes;
379 : GRHprime_t *pr;
380 7547859 : for (pr = S->primes, i = S->nprimes; i > 0; pr++, i--) gunclone(pr->dec);
381 : }
382 134159 : pari_free(S->primes);
383 134154 : }
384 :
385 : int
386 1528633 : GRHok(GRHcheck_t *S, double L, double SA, double SB)
387 : {
388 1528633 : return (S->cD + (S->cN + 2*SB) / L - 2*SA < -1e-8);
389 : }
390 :
391 : /* Return factorization pattern of p: [f,n], where n[i] primes of
392 : * residue degree f[i] */
393 : static GEN
394 7481746 : get_fs(GEN nf, GEN P, GEN index, ulong p)
395 : {
396 : long j, k, f, n, l;
397 : GEN fs, ns;
398 :
399 7481746 : if (umodiu(index, p))
400 : { /* easy case: p does not divide index */
401 7443514 : GEN F = Flx_degfact(ZX_to_Flx(P,p), p);
402 7444331 : fs = gel(F,1); l = lg(fs);
403 : }
404 : else
405 : {
406 38053 : GEN F = idealprimedec(nf, utoipos(p));
407 38072 : l = lg(F);
408 38072 : fs = cgetg(l, t_VECSMALL);
409 119295 : for (j = 1; j < l; j++) fs[j] = pr_get_f(gel(F,j));
410 : }
411 7482403 : ns = cgetg(l, t_VECSMALL);
412 7480983 : f = fs[1]; n = 1;
413 13841370 : for (j = 2, k = 1; j < l; j++)
414 6360387 : if (fs[j] == f)
415 4637986 : n++;
416 : else
417 : {
418 1722401 : ns[k] = n; fs[k] = f; k++;
419 1722401 : f = fs[j]; n = 1;
420 : }
421 7480983 : ns[k] = n; fs[k] = f; k++;
422 7480983 : setlg(fs, k);
423 7480483 : setlg(ns, k); return mkvec2(fs,ns);
424 : }
425 :
426 : /* cache data for all rational primes up to the LIM */
427 : static void
428 918001 : cache_prime_dec(GRHcheck_t *S, ulong LIM, GEN nf)
429 : {
430 918001 : pari_sp av = avma;
431 : GRHprime_t *pr;
432 : GEN index, P;
433 : double nb;
434 :
435 918001 : if (S->limp >= LIM) return;
436 328539 : S->clone = 1;
437 328539 : nb = primepi_upper_bound((double)LIM); /* #{p <= LIM} <= nb */
438 328553 : GRH_ensure(S, nb+1); /* room for one extra prime */
439 328551 : P = nf_get_pol(nf);
440 328549 : index = nf_get_index(nf);
441 328546 : for (pr = S->primes + S->nprimes;;)
442 7153422 : {
443 7481968 : ulong p = u_forprime_next(&(S->P));
444 7481774 : pr->p = p;
445 7481774 : pr->logp = log((double)p);
446 7481774 : pr->dec = gclone(get_fs(nf, P, index, p));
447 7481949 : S->nprimes++;
448 7481949 : pr++;
449 7481949 : set_avma(av);
450 : /* store up to nextprime(LIM) included */
451 7481970 : if (p >= LIM) { S->limp = p; break; }
452 : }
453 : }
454 :
455 : static double
456 2251176 : tailresback(long R1, long R2, double rK, long C, double C2, double C3, double r1K, double r2K, double logC, double logC2, double logC3)
457 : {
458 2251176 : const double rQ = 1.83787706641;
459 2251176 : const double r1Q = 1.98505372441;
460 2251176 : const double r2Q = 1.07991541347;
461 4502352 : return fabs((R1+R2-1)*(12*logC3+4*logC2-9*logC-6)/(2*C*logC3)
462 2251176 : + (rK-rQ)*(6*logC2 + 5*logC + 2)/(C*logC3)
463 2251176 : - R2*(6*logC2+11*logC+6)/(C2*logC2)
464 2251176 : - 2*(r1K-r1Q)*(3*logC2 + 4*logC + 2)/(C2*logC3)
465 2251176 : + (R1+R2-1)*(12*logC3+40*logC2+45*logC+18)/(6*C3*logC3)
466 2251176 : + (r2K-r2Q)*(2*logC2 + 3*logC + 2)/(C3*logC3));
467 : }
468 :
469 : static double
470 1125590 : tailres(long R1, long R2, double al2K, double rKm, double rKM, double r1Km,
471 : double r1KM, double r2Km, double r2KM, double C, long i)
472 : {
473 : /* C >= 3*2^i, lower bound for eint1(log(C)/2) */
474 : /* for(i=0,30,print(eint1(log(3*2^i)/2))) */
475 : static double tab[] = {
476 : 0.50409264803,
477 : 0.26205336997,
478 : 0.14815491171,
479 : 0.08770540561,
480 : 0.05347651832,
481 : 0.03328934284,
482 : 0.02104510690,
483 : 0.01346475900,
484 : 0.00869778586,
485 : 0.00566279855,
486 : 0.00371111950,
487 : 0.00244567837,
488 : 0.00161948049,
489 : 0.00107686891,
490 : 0.00071868750,
491 : 0.00048119961,
492 : 0.00032312188,
493 : 0.00021753772,
494 : 0.00014679818,
495 : 9.9272855581E-5,
496 : 6.7263969995E-5,
497 : 4.5656812967E-5,
498 : 3.1041124593E-5,
499 : 2.1136011590E-5,
500 : 1.4411645381E-5,
501 : 9.8393304088E-6,
502 : 6.7257395409E-6,
503 : 4.6025878272E-6,
504 : 3.1529719271E-6,
505 : 2.1620490021E-6,
506 : 1.4839266071E-6
507 : };
508 1125590 : const double logC = log(C), logC2 = logC*logC, logC3 = logC*logC2;
509 1125590 : const double C2 = C*C, C3 = C*C2;
510 1125590 : double E1 = i >30? 0: tab[i];
511 1125590 : return al2K*((33*logC2+22*logC+8)/(8*logC3*sqrt(C))+15*E1/16)
512 1125590 : + maxdd(tailresback(rKm,r1KM,r2Km, C,C2,C3,R1,R2,logC,logC2,logC3),
513 1125595 : tailresback(rKM,r1Km,r2KM, C,C2,C3,R1,R2,logC,logC2,logC3))/2
514 1125595 : + ((R1+R2-1)*4*C+R2)*(C2+6*logC)/(4*C2*C2*logC2);
515 : }
516 :
517 : static long
518 63791 : primeneeded(long N, long R1, long R2, double LOGD)
519 : {
520 63791 : const double lim = 0.25; /* should be log(2)/2 == 0.34657... */
521 63791 : const double al2K = 0.3526*LOGD - 0.8212*N + 4.5007;
522 63791 : const double rKm = -1.0155*LOGD + 2.1042*N - 8.3419;
523 63791 : const double rKM = -0.5 *LOGD + 1.2076*N + 1;
524 63791 : const double r1Km = - LOGD + 1.4150*N;
525 63791 : const double r1KM = - LOGD + 1.9851*N;
526 63791 : const double r2Km = - LOGD + 0.9151*N;
527 63791 : const double r2KM = - LOGD + 1.0800*N;
528 63791 : long Cmin = 3, Cmax = 3, i = 0;
529 572266 : while (tailres(R1, R2, al2K, rKm, rKM, r1Km, r1KM, r2Km, r2KM, Cmax, i) > lim)
530 : {
531 508475 : Cmin = Cmax;
532 508475 : Cmax *= 2;
533 508475 : i++;
534 : }
535 63788 : i--;
536 617132 : while (Cmax - Cmin > 1)
537 : {
538 553342 : long t = (Cmin + Cmax)/2;
539 553342 : if (tailres(R1, R2, al2K, rKm, rKM, r1Km, r1KM, r2Km, r2KM, t, i) > lim)
540 342813 : Cmin = t;
541 : else
542 210531 : Cmax = t;
543 : }
544 63790 : return Cmax;
545 : }
546 :
547 : /* ~ 1 / Res(s = 1, zeta_K) */
548 : static GEN
549 63791 : compute_invres(GRHcheck_t *S, long LIMC)
550 : {
551 63791 : pari_sp av = avma;
552 63791 : double loginvres = 0.;
553 : GRHprime_t *pr;
554 : long i;
555 63791 : double logLIMC = log((double)LIMC);
556 63791 : double logLIMC2 = logLIMC*logLIMC, denc;
557 : double c0, c1, c2;
558 63791 : denc = 1/(pow((double)LIMC, 3.) * logLIMC * logLIMC2);
559 63791 : c2 = ( logLIMC2 + 3 * logLIMC / 2 + 1) * denc;
560 63791 : denc *= LIMC;
561 63791 : c1 = (3 * logLIMC2 + 4 * logLIMC + 2) * denc;
562 63791 : denc *= LIMC;
563 63791 : c0 = (3 * logLIMC2 + 5 * logLIMC / 2 + 1) * denc;
564 7491302 : for (pr = S->primes, i = S->nprimes; i > 0; pr++, i--)
565 : {
566 : GEN dec, fs, ns;
567 : long addpsi;
568 : double addpsi1, addpsi2;
569 7483392 : double logp = pr->logp, NPk;
570 7483392 : long j, k, limp = logLIMC/logp;
571 7483392 : ulong p = pr->p, p2 = p*p;
572 7483392 : if (limp < 1) break;
573 7427511 : dec = pr->dec;
574 7427511 : fs = gel(dec, 1); ns = gel(dec, 2);
575 7427511 : loginvres += 1./p;
576 : /* NB: limp = 1 nearly always and limp > 2 for very few primes */
577 8789009 : for (k=2, NPk = p; k <= limp; k++) { NPk *= p; loginvres += 1/(k * NPk); }
578 7427511 : addpsi = limp;
579 7427511 : addpsi1 = p *(pow((double)p , (double)limp)-1)/(p -1);
580 7427511 : addpsi2 = p2*(pow((double)p2, (double)limp)-1)/(p2-1);
581 7427511 : j = lg(fs);
582 16566149 : while (--j > 0)
583 : {
584 : long f, nb, kmax;
585 : double NP, NP2, addinvres;
586 9138638 : f = fs[j]; if (f > limp) continue;
587 3972901 : nb = ns[j];
588 3972901 : NP = pow((double)p, (double)f);
589 3972901 : addinvres = 1/NP;
590 3972901 : kmax = limp / f;
591 4847967 : for (k=2, NPk = NP; k <= kmax; k++) { NPk *= NP; addinvres += 1/(k*NPk); }
592 3972901 : NP2 = NP*NP;
593 3972901 : loginvres -= nb * addinvres;
594 3972901 : addpsi -= nb * f * kmax;
595 3972901 : addpsi1 -= nb*(f*NP *(pow(NP ,(double)kmax)-1)/(NP -1));
596 3972901 : addpsi2 -= nb*(f*NP2*(pow(NP2,(double)kmax)-1)/(NP2-1));
597 : }
598 7427511 : loginvres -= (addpsi*c0 - addpsi1*c1 + addpsi2*c2)*logp;
599 : }
600 63791 : return gerepileuptoleaf(av, mpexp(dbltor(loginvres)));
601 : }
602 :
603 : static long
604 63791 : nthideal(GRHcheck_t *S, GEN nf, long n)
605 : {
606 63791 : pari_sp av = avma;
607 63791 : GEN P = nf_get_pol(nf);
608 63791 : ulong p = 0, *vecN = (ulong*)const_vecsmall(n, LONG_MAX);
609 63791 : long i, N = poldegree(P, -1);
610 63790 : for (i = 0; ; i++)
611 229737 : {
612 : GRHprime_t *pr;
613 : GEN fs;
614 293527 : cache_prime_dec(S, p+1, nf);
615 293528 : pr = S->primes + i;
616 293528 : fs = gel(pr->dec, 1);
617 293528 : p = pr->p;
618 293528 : if (fs[1] != N)
619 : {
620 196984 : GEN ns = gel(pr->dec, 2);
621 196984 : long k, l, j = lg(fs);
622 441650 : while (--j > 0)
623 : {
624 244666 : ulong NP = upowuu(p, fs[j]);
625 : long nf;
626 244666 : if (!NP) continue;
627 751060 : for (k = 1; k <= n; k++) if (vecN[k] > NP) break;
628 244274 : if (k > n) continue;
629 : /* vecN[k] <= NP */
630 158230 : nf = ns[j]; /*#{primes of norme NP} = nf, insert them here*/
631 354287 : for (l = k+nf; l <= n; l++) vecN[l] = vecN[l-nf];
632 399711 : for (l = 0; l < nf && k+l <= n; l++) vecN[k+l] = NP;
633 363908 : while (l <= k) vecN[l++] = NP;
634 : }
635 : }
636 293528 : if (p > vecN[n]) break;
637 : }
638 63791 : return gc_long(av, vecN[n]);
639 : }
640 :
641 : /* volume of unit ball in R^n: \pi^{n/2} / \Gamma(n/2 + 1) */
642 : static double
643 66037 : ballvol(long n)
644 : {
645 66037 : double v = odd(n)? 2: 1;
646 150894 : for (; n > 1; n -= 2) v *= (2 * M_PI) / n;
647 66037 : return v;
648 : }
649 :
650 : /* Compute FB, LV, iLP + KC*. Reset perm
651 : * C2: bound for norm of tested prime ideals (includes be_honest())
652 : * C1: bound for p, such that P|p (NP <= C2) used to build relations */
653 : static void
654 66038 : FBgen(FB_t *F, GEN nf, long N, ulong C1, ulong C2, GRHcheck_t *S)
655 : {
656 : GRHprime_t *pr;
657 : long i, ip;
658 : GEN prim;
659 66038 : const double L = log((double)C2 + 0.5);
660 :
661 66038 : cache_prime_dec(S, C2, nf);
662 66038 : pr = S->primes;
663 66038 : F->sfb_chg = 0;
664 66038 : F->FB = cgetg(C2+1, t_VECSMALL);
665 66038 : F->iLP = cgetg(C2+1, t_VECSMALL);
666 66038 : F->LV = zerovec(C2);
667 :
668 66038 : prim = icopy(gen_1);
669 66038 : i = ip = 0;
670 66038 : F->KC = F->KCZ = 0;
671 435014 : for (;; pr++) /* p <= C2 */
672 435014 : {
673 501052 : ulong p = pr->p;
674 : long k, l, m;
675 : GEN LP, nb, f;
676 :
677 501052 : if (!F->KC && p > C1) { F->KCZ = i; F->KC = ip; }
678 501052 : if (p > C2) break;
679 :
680 463805 : if (DEBUGLEVEL>1) err_printf(" %ld",p);
681 :
682 463805 : f = gel(pr->dec, 1); nb = gel(pr->dec, 2);
683 463805 : if (f[1] == N)
684 : {
685 145905 : if (p == C2) break;
686 137365 : continue; /* p inert */
687 : }
688 317900 : l = (long)(L/pr->logp); /* p^f <= C2 <=> f <= l */
689 579893 : for (k=0, m=1; m < lg(f) && f[m]<=l; m++) k += nb[m];
690 317900 : if (!k)
691 : { /* too inert to appear in FB */
692 72581 : if (p == C2) break;
693 71944 : continue;
694 : }
695 245319 : prim[2] = p; LP = idealprimedec_limit_f(nf,prim, l);
696 : /* keep noninert ideals with Norm <= C2 */
697 245319 : if (m == lg(f)) setisclone(LP); /* flag it: all prime divisors in FB */
698 245319 : F->FB[++i]= p;
699 245319 : gel(F->LV,p) = LP;
700 245319 : F->iLP[p] = ip; ip += k;
701 245319 : if (p == C2) break;
702 : }
703 66038 : if (!F->KC) { F->KCZ = i; F->KC = ip; }
704 : /* Note F->KC > 0 otherwise GRHchk is false */
705 66038 : setlg(F->FB, F->KCZ+1); F->KCZ2 = i;
706 66038 : F->prodZ = zv_prod_Z(F->FB);
707 66037 : if (DEBUGLEVEL>1)
708 : {
709 0 : err_printf("\n");
710 0 : if (DEBUGLEVEL>6)
711 : {
712 0 : err_printf("########## FACTORBASE ##########\n\n");
713 0 : err_printf("KC2=%ld, KC=%ld, KCZ=%ld, KCZ2=%ld\n",
714 : ip, F->KC, F->KCZ, F->KCZ2);
715 0 : for (i=1; i<=F->KCZ; i++) err_printf("++ LV[%ld] = %Ps",i,gel(F->LV,F->FB[i]));
716 : }
717 : }
718 66037 : F->perm = NULL; F->L_jid = NULL;
719 66037 : F->ballvol = ballvol(nf_get_degree(nf));
720 66037 : }
721 :
722 : static int
723 494661 : GRHchk(GEN nf, GRHcheck_t *S, ulong LIMC)
724 : {
725 494661 : double logC = log((double)LIMC), SA = 0, SB = 0;
726 494661 : GRHprime_t *pr = S->primes;
727 :
728 494661 : cache_prime_dec(S, LIMC, nf);
729 494660 : for (pr = S->primes;; pr++)
730 3049484 : {
731 3544144 : ulong p = pr->p;
732 : GEN dec, fs, ns;
733 : double logCslogp;
734 : long j;
735 :
736 3544144 : if (p > LIMC) break;
737 3155412 : dec = pr->dec; fs = gel(dec, 1); ns = gel(dec,2);
738 3155412 : logCslogp = logC/pr->logp;
739 4966435 : for (j = 1; j < lg(fs); j++)
740 : {
741 3883614 : long f = fs[j], M, nb;
742 : double logNP, q, A, B;
743 3883614 : if (f > logCslogp) break;
744 1811027 : logNP = f * pr->logp;
745 1811027 : q = 1/sqrt((double)upowuu(p, f));
746 1811023 : A = logNP * q; B = logNP * A; M = (long)(logCslogp/f);
747 1811023 : if (M > 1)
748 : {
749 375533 : double inv1_q = 1 / (1-q);
750 375533 : A *= (1 - pow(q, (double)M)) * inv1_q;
751 375533 : B *= (1 - pow(q, (double)M)*(M+1 - M*q)) * inv1_q * inv1_q;
752 : }
753 1811023 : nb = ns[j];
754 1811023 : SA += nb * A;
755 1811023 : SB += nb * B;
756 : }
757 3155408 : if (p == LIMC) break;
758 : }
759 494656 : return GRHok(S, logC, SA, SB);
760 : }
761 :
762 : /* SMOOTH IDEALS */
763 : static void
764 9118786 : store(long i, long e, FACT *fact)
765 : {
766 9118786 : ++fact[0].pr;
767 9118786 : fact[fact[0].pr].pr = i; /* index */
768 9118786 : fact[fact[0].pr].ex = e; /* exponent */
769 9118786 : }
770 :
771 : /* divide out m by all P|p, k = v_p(Nm) */
772 : static int
773 2276 : divide_p_elt(GEN LP, long ip, long k, GEN m, FACT *fact)
774 : {
775 2276 : long j, l = lg(LP);
776 2996 : for (j=1; j<l; j++)
777 : {
778 2996 : GEN P = gel(LP,j);
779 2996 : long v = ZC_nfval(m, P);
780 2996 : if (!v) continue;
781 2638 : store(ip + j, v, fact); /* v = v_P(m) > 0 */
782 2638 : k -= v * pr_get_f(P);
783 2638 : if (!k) return 1;
784 : }
785 0 : return 0;
786 : }
787 : /* divide out I by all P|p, k = v_p(NI) */
788 : static int
789 163556 : divide_p_id(GEN LP, long ip, long k, GEN nf, GEN I, FACT *fact)
790 : {
791 163556 : long j, l = lg(LP);
792 245361 : for (j=1; j<l; j++)
793 : {
794 237499 : GEN P = gel(LP,j);
795 237499 : long v = idealval(nf,I, P);
796 237500 : if (!v) continue;
797 159194 : store(ip + j, v, fact); /* v = v_P(I) > 0 */
798 159194 : k -= v * pr_get_f(P);
799 159194 : if (!k) return 1;
800 : }
801 7862 : return 0;
802 : }
803 : /* divide out m/I by all P|p, k = v_p(Nm/NI) */
804 : static int
805 5231923 : divide_p_quo(GEN LP, long ip, long k, GEN nf, GEN I, GEN m, FACT *fact)
806 : {
807 5231923 : long j, l = lg(LP);
808 17332433 : for (j=1; j<l; j++)
809 : {
810 17243893 : GEN P = gel(LP,j);
811 17243893 : long v = ZC_nfval(m, P);
812 17242712 : if (!v) continue;
813 7809323 : v -= idealval(nf,I, P);
814 7810675 : if (!v) continue;
815 6649218 : store(ip + j, v, fact); /* v = v_P(m / I) > 0 */
816 6649260 : k -= v * pr_get_f(P);
817 6649199 : if (!k) return 1;
818 : }
819 88540 : return 0;
820 : }
821 :
822 : static int
823 5397753 : divide_p(FB_t *F, long p, long k, GEN nf, GEN I, GEN m, FACT *fact)
824 : {
825 5397753 : GEN LP = gel(F->LV,p);
826 5397753 : long ip = F->iLP[p];
827 5397753 : if (!m) return divide_p_id (LP,ip,k,nf,I,fact);
828 5234197 : if (!I) return divide_p_elt(LP,ip,k,m,fact);
829 5231921 : return divide_p_quo(LP,ip,k,nf,I,m,fact);
830 : }
831 :
832 : /* Let x = m if I == NULL,
833 : * I if m == NULL,
834 : * m/I otherwise.
835 : * Can we factor the integral primitive ideal x ? |N| = Norm x > 0 */
836 : static long
837 18001195 : can_factor(FB_t *F, GEN nf, GEN I, GEN m, GEN N, FACT *fact)
838 : {
839 : GEN f, p, e;
840 : long i, l;
841 18001195 : fact[0].pr = 0;
842 18001195 : if (is_pm1(N)) return 1;
843 17041586 : if (!is_pm1(Z_ppo(N, F->prodZ))) return 0;
844 2715909 : f = absZ_factor(N); p = gel(f,1); e = gel(f,2); l = lg(p);
845 8017634 : for (i = 1; i < l; i++)
846 5397747 : if (!divide_p(F, itou(gel(p,i)), itou(gel(e,i)), nf, I, m, fact))
847 : {
848 96176 : if (DEBUGLEVEL > 1) err_printf(".");
849 96176 : return 0;
850 : }
851 2619887 : return 1;
852 : }
853 :
854 : /* can we factor m/I ? [m in I from idealpseudomin_nonscalar], NI = norm I */
855 : static long
856 16659064 : factorgen(FB_t *F, GEN nf, GEN I, GEN NI, GEN m, FACT *fact)
857 : {
858 : long e;
859 16659064 : GEN Nm = embed_norm(RgM_RgC_mul(nf_get_M(nf),m), nf_get_r1(nf));
860 16659119 : GEN N = grndtoi(NI? divri(Nm, NI): Nm, &e); /* ~ N(m/I) */
861 16659010 : if (e > -32)
862 : {
863 0 : if (DEBUGLEVEL > 1) err_printf("+");
864 0 : return 0;
865 : }
866 16659010 : return can_factor(F, nf, I, m, N, fact);
867 : }
868 :
869 : /* FUNDAMENTAL UNITS */
870 :
871 : /* a, y real. Return (Re(x) + a) + I * (Im(x) % y) */
872 : static GEN
873 6593865 : addRe_modIm(GEN x, GEN a, GEN y, GEN iy)
874 : {
875 : GEN z;
876 6593865 : if (typ(x) == t_COMPLEX)
877 : {
878 4687688 : GEN re, im = modRr_i(gel(x,2), y, iy);
879 4687665 : if (!im) return NULL;
880 4687665 : re = gadd(gel(x,1), a);
881 4687612 : z = gequal0(im)? re: mkcomplex(re, im);
882 : }
883 : else
884 1906177 : z = gadd(x, a);
885 6593797 : return z;
886 : }
887 : static GEN
888 201903 : modIm(GEN x, GEN y, GEN iy)
889 : {
890 201903 : if (typ(x) == t_COMPLEX)
891 : {
892 189237 : GEN im = modRr_i(gel(x,2), y, iy);
893 189236 : if (!im) return NULL;
894 189236 : x = gequal0(im)? gel(x,1): mkcomplex(gel(x,1), im);
895 : }
896 201898 : return x;
897 : }
898 :
899 : /* clean archimedean components. ipi = 2^n / pi (n arbitrary); its
900 : * exponent may be modified */
901 : static GEN
902 2927672 : cleanarch(GEN x, long N, GEN ipi, long prec)
903 : {
904 : long i, l, R1, RU;
905 2927672 : GEN s, y = cgetg_copy(x, &l);
906 :
907 2927672 : if (!ipi) ipi = invr(mppi(prec));
908 2927669 : if (typ(x) == t_MAT)
909 : {
910 525718 : for (i = 1; i < l; i++)
911 461778 : if (!(gel(y,i) = cleanarch(gel(x,i), N, ipi, prec))) return NULL;
912 63940 : return y;
913 : }
914 2863725 : RU = l-1; R1 = (RU<<1) - N;
915 2863725 : s = gdivgs(RgV_sum(real_i(x)), -N); /* -log |norm(x)| / N */
916 2863711 : i = 1;
917 2863711 : if (R1)
918 : {
919 2385193 : GEN pi2 = Pi2n(1, prec);
920 2385194 : setexpo(ipi, -3); /* 1/(2pi) */
921 7344162 : for (; i <= R1; i++)
922 4959003 : if (!(gel(y,i) = addRe_modIm(gel(x,i), s, pi2, ipi))) return NULL;
923 : }
924 2863677 : if (i <= RU)
925 : {
926 1075394 : GEN pi4 = Pi2n(2, prec), s2 = gmul2n(s, 1);
927 1075404 : setexpo(ipi, -4); /* 1/(4pi) */
928 2710309 : for (; i <= RU; i++)
929 1634872 : if (!(gel(y,i) = addRe_modIm(gel(x,i), s2, pi4, ipi))) return NULL;
930 : }
931 2863720 : return y;
932 : }
933 : GEN
934 194923 : nf_cxlog_normalize(GEN nf, GEN x, long prec)
935 : {
936 194923 : long N = nf_get_degree(nf);
937 194923 : return cleanarch(x, N, NULL, prec);
938 : }
939 :
940 : /* clean unit archimedean components. ipi = 2^n / pi (n arbitrary); its
941 : * exponent may be modified */
942 : static GEN
943 132976 : cleanarchunit(GEN x, long N, GEN ipi, long prec)
944 : {
945 : long i, l, R1, RU;
946 132976 : GEN y = cgetg_copy(x, &l);
947 :
948 132976 : if (!ipi) ipi = invr(mppi(prec));
949 132978 : if (typ(x) == t_MAT)
950 : {
951 132969 : for (i = 1; i < l; i++)
952 69180 : if (!(gel(y,i) = cleanarchunit(gel(x,i), N, ipi, prec))) return NULL;
953 63789 : return y;
954 : }
955 69180 : if (gexpo(RgV_sum(real_i(x))) > -10) return NULL;
956 69172 : RU = l-1; R1 = (RU<<1) - N;
957 69172 : i = 1;
958 69172 : if (R1)
959 : {
960 54675 : GEN pi2 = Pi2n(1, prec);
961 54676 : setexpo(ipi, -3); /* 1/(2pi) */
962 185859 : for (; i <= R1; i++)
963 131186 : if (!(gel(y,i) = modIm(gel(x,i), pi2, ipi))) return NULL;
964 : }
965 69170 : if (i <= RU)
966 : {
967 34468 : GEN pi4 = Pi2n(2, prec);
968 34468 : setexpo(ipi, -4); /* 1/(4pi) */
969 105183 : for (; i <= RU; i++)
970 70714 : if (!(gel(y,i) = modIm(gel(x,i), pi4, ipi))) return NULL;
971 : }
972 69171 : return y;
973 : }
974 :
975 : static GEN
976 389 : not_given(long reason)
977 : {
978 389 : if (DEBUGLEVEL)
979 0 : switch(reason)
980 : {
981 0 : case fupb_LARGE:
982 0 : pari_warn(warner,"fundamental units too large, not given");
983 0 : break;
984 0 : case fupb_PRECI:
985 0 : pari_warn(warner,"insufficient precision for fundamental units, not given");
986 0 : break;
987 : }
988 389 : return NULL;
989 : }
990 :
991 : /* check whether exp(x) will 1) get too big (real(x) large), 2) require
992 : * large accuracy for argument reduction (imag(x) large) */
993 : static long
994 2686610 : expbitprec(GEN x, long *e)
995 : {
996 : GEN re, im;
997 2686610 : if (typ(x) != t_COMPLEX) re = x;
998 : else
999 : {
1000 1670807 : im = gel(x,2); *e = maxss(*e, expo(im) + 5 - bit_prec(im));
1001 1670807 : re = gel(x,1);
1002 : }
1003 2686610 : return (expo(re) <= 20);
1004 :
1005 : }
1006 : static long
1007 1167050 : RgC_expbitprec(GEN x)
1008 : {
1009 1167050 : long l = lg(x), i, e = - (long)HIGHEXPOBIT;
1010 3651290 : for (i = 1; i < l; i++)
1011 2484814 : if (!expbitprec(gel(x,i), &e)) return LONG_MAX;
1012 1166476 : return e;
1013 : }
1014 : static long
1015 48566 : RgM_expbitprec(GEN x)
1016 : {
1017 48566 : long i, j, I, J, e = - (long)HIGHEXPOBIT;
1018 48566 : RgM_dimensions(x, &I,&J);
1019 117670 : for (j = 1; j <= J; j++)
1020 270900 : for (i = 1; i <= I; i++)
1021 201796 : if (!expbitprec(gcoeff(x,i,j), &e)) return LONG_MAX;
1022 48503 : return e;
1023 : }
1024 :
1025 : static GEN
1026 1379 : FlxqX_chinese_unit(GEN X, GEN U, GEN invzk, GEN D, GEN T, ulong p)
1027 : {
1028 1379 : long i, lU = lg(U), lX = lg(X), d = lg(invzk)-1;
1029 1379 : GEN M = cgetg(lU, t_MAT);
1030 1379 : if (D)
1031 : {
1032 1274 : D = Flv_inv(D, p);
1033 69756 : for (i = 1; i < lX; i++)
1034 68482 : if (uel(D, i) != 1)
1035 56624 : gel(X,i) = Flx_Fl_mul(gel(X,i), uel(D,i), p);
1036 : }
1037 3877 : for (i = 1; i < lU; i++)
1038 : {
1039 2498 : GEN H = FlxqV_factorback(X, gel(U, i), T, p);
1040 2499 : gel(M, i) = Flm_Flc_mul(invzk, Flx_to_Flv(H, d), p);
1041 : }
1042 1379 : return M;
1043 : }
1044 :
1045 : static GEN
1046 274 : chinese_unit_slice(GEN A, GEN U, GEN B, GEN D, GEN C, GEN P, GEN *mod)
1047 : {
1048 274 : pari_sp av = avma;
1049 274 : long i, n = lg(P)-1, v = varn(C);
1050 : GEN H, T;
1051 274 : if (n == 1)
1052 : {
1053 0 : ulong p = uel(P,1);
1054 0 : GEN a = ZXV_to_FlxV(A, p), b = ZM_to_Flm(B, p), c = ZX_to_Flx(C, p);
1055 0 : GEN d = D ? ZV_to_Flv(D, p): NULL;
1056 0 : GEN Hp = FlxqX_chinese_unit(a, U, b, d, c, p);
1057 0 : H = gerepileupto(av, Flm_to_ZM(Hp));
1058 0 : *mod = utoi(p);
1059 0 : return H;
1060 : }
1061 274 : T = ZV_producttree(P);
1062 274 : A = ZXC_nv_mod_tree(A, P, T, v);
1063 274 : B = ZM_nv_mod_tree(B, P, T);
1064 274 : D = D ? ZV_nv_mod_tree(D, P, T): NULL;
1065 274 : C = ZX_nv_mod_tree(C, P, T);
1066 :
1067 274 : H = cgetg(n+1, t_VEC);
1068 1653 : for(i=1; i <= n; i++)
1069 : {
1070 1379 : ulong p = P[i];
1071 1379 : GEN a = gel(A,i), b = gel(B,i), c = gel(C,i), d = D ? gel(D,i): NULL;
1072 1379 : gel(H,i) = FlxqX_chinese_unit(a, U, b, d, c, p);
1073 : }
1074 274 : H = nmV_chinese_center_tree_seq(H, P, T, ZV_chinesetree(P, T));
1075 274 : *mod = gmael(T, lg(T)-1, 1); return gc_all(av, 2, &H, mod);
1076 : }
1077 :
1078 : GEN
1079 274 : chinese_unit_worker(GEN P, GEN A, GEN U, GEN B, GEN D, GEN C)
1080 : {
1081 274 : GEN V = cgetg(3, t_VEC);
1082 274 : gel(V,1) = chinese_unit_slice(A, U, B, isintzero(D) ? NULL: D, C, P, &gel(V,2));
1083 274 : return V;
1084 : }
1085 :
1086 : /* Let x = \prod X[i]^E[i] = u, return u.
1087 : * If dX != NULL, X[i] = nX[i] / dX[i] where nX[i] is a ZX, dX[i] in Z */
1088 : static GEN
1089 94 : chinese_unit(GEN nf, GEN nX, GEN dX, GEN U, ulong bnd)
1090 : {
1091 94 : pari_sp av = avma;
1092 94 : GEN f = nf_get_index(nf), T = nf_get_pol(nf), invzk = nf_get_invzk(nf);
1093 : GEN H, mod;
1094 : forprime_t S;
1095 94 : GEN worker = snm_closure(is_entry("_chinese_unit_worker"),
1096 : mkcol5(nX, U, invzk, dX? dX: gen_0, T));
1097 94 : init_modular_big(&S);
1098 94 : H = gen_crt("chinese_units", worker, &S, f, bnd, 0, &mod, nmV_chinese_center, FpM_center);
1099 94 : settyp(H, t_VEC); return gerepilecopy(av, H);
1100 : }
1101 :
1102 : /* *pE a ZM */
1103 : static void
1104 164 : ZM_remove_unused(GEN *pE, GEN *pX)
1105 : {
1106 164 : long j, k, l = lg(*pX);
1107 164 : GEN E = *pE, v = cgetg(l, t_VECSMALL);
1108 16395 : for (j = k = 1; j < l; j++)
1109 16231 : if (!ZMrow_equal0(E, j)) v[k++] = j;
1110 164 : if (k < l)
1111 : {
1112 164 : setlg(v, k);
1113 164 : *pX = vecpermute(*pX,v);
1114 164 : *pE = rowpermute(E,v);
1115 : }
1116 164 : }
1117 :
1118 : /* s = -log|norm(x)|/N */
1119 : static GEN
1120 1236218 : fixarch(GEN x, GEN s, long R1)
1121 : {
1122 : long i, l;
1123 1236218 : GEN y = cgetg_copy(x, &l);
1124 3422530 : for (i = 1; i <= R1; i++) gel(y,i) = gadd(s, gel(x,i));
1125 1737190 : for ( ; i < l; i++) gel(y,i) = gadd(s, gmul2n(gel(x,i),-1));
1126 1236222 : return y;
1127 : }
1128 :
1129 : static GEN
1130 63791 : getfu(GEN nf, GEN *ptA, GEN *ptU, long prec)
1131 : {
1132 63791 : GEN U, y, matep, A, T = nf_get_pol(nf), M = nf_get_M(nf);
1133 63791 : long e, j, R1, RU, N = degpol(T);
1134 :
1135 63791 : R1 = nf_get_r1(nf); RU = (N+R1) >> 1;
1136 63791 : if (RU == 1) return cgetg(1,t_VEC);
1137 :
1138 48566 : A = *ptA;
1139 48566 : matep = cgetg(RU,t_MAT);
1140 117739 : for (j = 1; j < RU; j++)
1141 : {
1142 69174 : GEN Aj = gel(A,j), s = gdivgs(RgV_sum(real_i(Aj)), -N);
1143 69171 : gel(matep,j) = fixarch(Aj, s, R1);
1144 : }
1145 48565 : U = lll(real_i(matep));
1146 48565 : if (lg(U) < RU) return not_given(fupb_PRECI);
1147 48565 : if (ptU) { *ptU = U; *ptA = A = RgM_ZM_mul(A,U); }
1148 48564 : y = RgM_ZM_mul(matep,U);
1149 48566 : e = RgM_expbitprec(y);
1150 48566 : if (e >= 0) return not_given(e == LONG_MAX? fupb_LARGE: fupb_PRECI);
1151 48503 : if (prec <= 0) prec = gprecision(A);
1152 48503 : y = RgM_solve_realimag(M, gexp(y,prec));
1153 48503 : if (!y) return not_given(fupb_PRECI);
1154 48503 : y = grndtoi(y, &e); if (e >= 0) return not_given(fupb_PRECI);
1155 48185 : settyp(y, t_VEC);
1156 :
1157 48185 : if (!ptU) *ptA = A = RgM_ZM_mul(A, U);
1158 116625 : for (j = 1; j < RU; j++)
1159 : { /* y[i] are hopefully unit generators. Normalize: smallest T2 norm */
1160 68448 : GEN u = gel(y,j), v = zk_inv(nf, u);
1161 68448 : if (!v || !is_pm1(Q_denom(v)) || ZV_isscalar(u))
1162 8 : return not_given(fupb_PRECI);
1163 68440 : if (gcmp(RgC_fpnorml2(v,DEFAULTPREC), RgC_fpnorml2(u,DEFAULTPREC)) < 0)
1164 : {
1165 29480 : gel(A,j) = RgC_neg(gel(A,j));
1166 29480 : if (ptU) gel(U,j) = ZC_neg(gel(U,j));
1167 29480 : u = v;
1168 : }
1169 68439 : gel(y,j) = nf_to_scalar_or_alg(nf, u);
1170 : }
1171 48177 : return y;
1172 : }
1173 :
1174 : static void
1175 0 : err_units() { pari_err_PREC("makeunits [cannot get units, use bnfinit(,1)]"); }
1176 :
1177 : /* bound for log2 |sigma(u)|, sigma complex embedding, u fundamental unit
1178 : * attached to bnf_get_logfu */
1179 : static double
1180 94 : log2fubound(GEN bnf)
1181 : {
1182 94 : GEN LU = bnf_get_logfu(bnf);
1183 94 : long i, j, l = lg(LU), r1 = nf_get_r1(bnf_get_nf(bnf));
1184 94 : double e = 0.0;
1185 330 : for (j = 1; j < l; j++)
1186 : {
1187 236 : GEN u = gel(LU,j);
1188 624 : for (i = 1; i <= r1; i++)
1189 : {
1190 388 : GEN E = real_i(gel(u,i));
1191 388 : e = maxdd(e, gtodouble(E));
1192 : }
1193 842 : for ( ; i <= l; i++)
1194 : {
1195 606 : GEN E = real_i(gel(u,i));
1196 606 : e = maxdd(e, gtodouble(E) / 2);
1197 : }
1198 : }
1199 94 : return e / M_LN2;
1200 : }
1201 : /* bound for log2(|RgM_solve_realimag(M, y)|_oo / |y|_oo)*/
1202 : static double
1203 94 : log2Mbound(GEN nf)
1204 : {
1205 94 : GEN G = nf_get_G(nf), D = nf_get_disc(nf);
1206 94 : long r2 = nf_get_r2(nf), l = lg(G), i;
1207 94 : double e, d = dbllog2(D)/2 - r2 * M_LN2; /* log2 |det(split_realimag(M))| */
1208 94 : e = log2(nf_get_degree(nf));
1209 535 : for (i = 2; i < l; i++) e += dbllog2(gnorml2(gel(G,i))); /* Hadamard bound */
1210 94 : return e / 2 - d;
1211 : }
1212 :
1213 : static GEN
1214 94 : vec_chinese_units(GEN bnf)
1215 : {
1216 94 : GEN nf = bnf_get_nf(bnf), SUnits = bnf_get_sunits(bnf);
1217 94 : double bnd = ceil(log2Mbound(nf) + log2fubound(bnf));
1218 94 : GEN X, dX, Y, U, f = nf_get_index(nf);
1219 94 : long j, l, v = nf_get_varn(nf);
1220 94 : if (!SUnits) err_units(); /* no compact units */
1221 94 : Y = gel(SUnits,1);
1222 94 : U = gel(SUnits,2);
1223 94 : ZM_remove_unused(&U, &Y); l = lg(Y); X = cgetg(l, t_VEC);
1224 94 : if (is_pm1(f)) f = dX = NULL; else dX = cgetg(l, t_VEC);
1225 5142 : for (j = 1; j < l; j++)
1226 : {
1227 5048 : GEN t = nf_to_scalar_or_alg(nf, gel(Y,j));
1228 5048 : if (f)
1229 : {
1230 : GEN den;
1231 4202 : t = Q_remove_denom(t, &den);
1232 4202 : gel(dX,j) = den ? den: gen_1;
1233 : }
1234 5048 : gel(X,j) = typ(t) == t_INT? scalarpol_shallow(t,v): t;
1235 : }
1236 94 : if (dblexpo(bnd) >= BITS_IN_LONG)
1237 0 : pari_err_OVERFLOW("vec_chinese_units [units too large]");
1238 94 : return chinese_unit(nf, X, dX, U, (ulong)bnd);
1239 : }
1240 :
1241 : static GEN
1242 24901 : makeunits(GEN bnf)
1243 : {
1244 24901 : GEN nf = bnf_get_nf(bnf), fu = bnf_get_fu_nocheck(bnf);
1245 24901 : GEN tu = nf_to_scalar_or_basis(nf, bnf_get_tuU(bnf));
1246 24901 : fu = (typ(fu) == t_MAT)? vec_chinese_units(bnf): matalgtobasis(nf, fu);
1247 24901 : return vec_prepend(fu, tu);
1248 : }
1249 :
1250 : /*******************************************************************/
1251 : /* */
1252 : /* PRINCIPAL IDEAL ALGORITHM (DISCRETE LOG) */
1253 : /* */
1254 : /*******************************************************************/
1255 :
1256 : /* G: prime ideals, E: vector of nonnegative exponents.
1257 : * C = possible extra prime (^1) or NULL
1258 : * Return Norm (product) */
1259 : static GEN
1260 69 : get_norm_fact_primes(GEN G, GEN E, GEN C)
1261 : {
1262 69 : pari_sp av=avma;
1263 69 : GEN N = gen_1, P, p;
1264 69 : long i, c = lg(E);
1265 69 : for (i=1; i<c; i++)
1266 : {
1267 0 : GEN ex = gel(E,i);
1268 0 : long s = signe(ex);
1269 0 : if (!s) continue;
1270 :
1271 0 : P = gel(G,i); p = pr_get_p(P);
1272 0 : N = mulii(N, powii(p, mului(pr_get_f(P), ex)));
1273 : }
1274 69 : if (C) N = mulii(N, pr_norm(C));
1275 69 : return gerepileuptoint(av, N);
1276 : }
1277 :
1278 : /* gen: HNF ideals */
1279 : static GEN
1280 1161436 : get_norm_fact(GEN gen, GEN ex, GEN *pd)
1281 : {
1282 1161436 : long i, c = lg(ex);
1283 : GEN d,N,I,e,n,ne,de;
1284 1161436 : d = N = gen_1;
1285 1458012 : for (i=1; i<c; i++)
1286 296576 : if (signe(gel(ex,i)))
1287 : {
1288 175464 : I = gel(gen,i); e = gel(ex,i); n = ZM_det_triangular(I);
1289 175464 : ne = powii(n,e);
1290 175463 : de = equalii(n, gcoeff(I,1,1))? ne: powii(gcoeff(I,1,1), e);
1291 175463 : N = mulii(N, ne);
1292 175464 : d = mulii(d, de);
1293 : }
1294 1161436 : *pd = d; return N;
1295 : }
1296 :
1297 : static GEN
1298 1322346 : get_pr_lists(GEN FB, long N, int list_pr)
1299 : {
1300 : GEN pr, L;
1301 1322346 : long i, l = lg(FB), p, pmax;
1302 :
1303 1322346 : pmax = 0;
1304 9183963 : for (i=1; i<l; i++)
1305 : {
1306 7861617 : pr = gel(FB,i); p = pr_get_smallp(pr);
1307 7861617 : if (p > pmax) pmax = p;
1308 : }
1309 1322346 : L = const_vec(pmax, NULL);
1310 1322347 : if (list_pr)
1311 : {
1312 0 : for (i=1; i<l; i++)
1313 : {
1314 0 : pr = gel(FB,i); p = pr_get_smallp(pr);
1315 0 : if (!L[p]) gel(L,p) = vectrunc_init(N+1);
1316 0 : vectrunc_append(gel(L,p), pr);
1317 : }
1318 0 : for (p=1; p<=pmax; p++)
1319 0 : if (L[p]) gen_sort_inplace(gel(L,p), (void*)&cmp_prime_over_p,
1320 : &cmp_nodata, NULL);
1321 : }
1322 : else
1323 : {
1324 9183966 : for (i=1; i<l; i++)
1325 : {
1326 7861618 : pr = gel(FB,i); p = pr_get_smallp(pr);
1327 7861618 : if (!L[p]) gel(L,p) = vecsmalltrunc_init(N+1);
1328 7861618 : vecsmalltrunc_append(gel(L,p), i);
1329 : }
1330 : }
1331 1322348 : return L;
1332 : }
1333 :
1334 : /* recover FB, LV, iLP, KCZ from Vbase */
1335 : static GEN
1336 1322346 : recover_partFB(FB_t *F, GEN Vbase, long N)
1337 : {
1338 1322346 : GEN FB, LV, iLP, L = get_pr_lists(Vbase, N, 0);
1339 1322347 : long l = lg(L), p, ip, i;
1340 :
1341 1322347 : i = ip = 0;
1342 1322347 : FB = cgetg(l, t_VECSMALL);
1343 1322347 : iLP= cgetg(l, t_VECSMALL);
1344 1322347 : LV = cgetg(l, t_VEC);
1345 20036340 : for (p = 2; p < l; p++)
1346 : {
1347 18713991 : if (!L[p]) continue;
1348 4304130 : FB[++i] = p;
1349 4304130 : gel(LV,p) = vecpermute(Vbase, gel(L,p));
1350 4304130 : iLP[p]= ip; ip += lg(gel(L,p))-1;
1351 : }
1352 1322349 : F->KCZ = i;
1353 1322349 : F->KC = ip;
1354 1322349 : F->FB = FB; setlg(FB, i+1);
1355 1322349 : F->prodZ = zv_prod_Z(F->FB);
1356 1322347 : F->LV = LV;
1357 1322347 : F->iLP= iLP; return L;
1358 : }
1359 :
1360 : /* add v^e to factorization */
1361 : static void
1362 2831404 : add_to_fact(long v, long e, FACT *fact)
1363 : {
1364 2831404 : long i, n = fact[0].pr;
1365 9577140 : for (i=1; i<=n; i++)
1366 7269388 : if (fact[i].pr == v) { fact[i].ex += e; return; }
1367 2307752 : store(v, e, fact);
1368 : }
1369 : static void
1370 0 : inv_fact(FACT *fact)
1371 : {
1372 0 : long i, n = fact[0].pr;
1373 0 : for (i=1; i<=n; i++) fact[i].ex = -fact[i].ex;
1374 0 : }
1375 :
1376 : /* L (small) list of primes above the same p including pr. Return pr index */
1377 : static int
1378 3349 : pr_index(GEN L, GEN pr)
1379 : {
1380 3349 : long j, l = lg(L);
1381 3349 : GEN al = pr_get_gen(pr);
1382 3349 : for (j=1; j<l; j++)
1383 3349 : if (ZV_equal(al, pr_get_gen(gel(L,j)))) return j;
1384 0 : pari_err_BUG("codeprime");
1385 : return 0; /* LCOV_EXCL_LINE */
1386 : }
1387 :
1388 : static long
1389 3349 : Vbase_to_FB(FB_t *F, GEN pr)
1390 : {
1391 3349 : long p = pr_get_smallp(pr);
1392 3349 : return F->iLP[p] + pr_index(gel(F->LV,p), pr);
1393 : }
1394 :
1395 : /* x, y 2 extended ideals whose first component is an integral HNF and second
1396 : * a famat */
1397 : static GEN
1398 3522 : idealHNF_mulred(GEN nf, GEN x, GEN y)
1399 : {
1400 3522 : GEN A = idealHNF_mul(nf, gel(x,1), gel(y,1));
1401 3522 : GEN F = famat_mul_shallow(gel(x,2), gel(y,2));
1402 3522 : return idealred(nf, mkvec2(A, F));
1403 : }
1404 : /* idealred(x * pr^n), n > 0 is small, x extended ideal. Reduction in order to
1405 : * avoid prec pb: don't let id become too large as lgsub increases */
1406 : static GEN
1407 4488 : idealmulpowprime2(GEN nf, GEN x, GEN pr, ulong n)
1408 : {
1409 4488 : GEN A = idealmulpowprime(nf, gel(x,1), pr, utoipos(n));
1410 4488 : return mkvec2(A, gel(x,2));
1411 : }
1412 : static GEN
1413 65539 : init_famat(GEN x) { return mkvec2(x, trivial_fact()); }
1414 : /* optimized idealfactorback + reduction; z = init_famat() */
1415 : static GEN
1416 28812 : genback(GEN z, GEN nf, GEN P, GEN E)
1417 : {
1418 28812 : long i, l = lg(E);
1419 28812 : GEN I = NULL;
1420 76704 : for (i = 1; i < l; i++)
1421 47892 : if (signe(gel(E,i)))
1422 : {
1423 : GEN J;
1424 32334 : gel(z,1) = gel(P,i);
1425 32334 : J = idealpowred(nf, z, gel(E,i));
1426 32334 : I = I? idealHNF_mulred(nf, I, J): J;
1427 : }
1428 28812 : return I; /* != NULL since a generator */
1429 : }
1430 :
1431 : static GEN
1432 1204348 : SPLIT_i(FB_t *F, GEN nf, GEN G, GEN x, GEN xred, GEN Nx, FACT *fact)
1433 : {
1434 1204348 : pari_sp av = avma;
1435 1204348 : GEN L = idealpseudominvec(xred, G);
1436 1204346 : long k, l = lg(L);
1437 1287299 : for(k = 1; k < l; k++)
1438 1271372 : if (factorgen(F, nf, x, Nx, gel(L,k), fact)) return gel(L,k);
1439 15927 : return gc_NULL(av);
1440 : }
1441 : /* return famat y (principal ideal) such that y / x is smooth [wrt Vbase] */
1442 : static GEN
1443 1338691 : SPLIT(FB_t *F, GEN nf, GEN x, GEN Vbase, FACT *fact)
1444 : {
1445 1338691 : GEN vecG, ex, y, x0, Nx = ZM_det_triangular(x);
1446 : long nbtest_lim, nbtest, i, j, ru, lgsub;
1447 : pari_sp av;
1448 :
1449 : /* try without reduction if x is small */
1450 2677195 : if (expi(gcoeff(x,1,1)) < 100 &&
1451 1488775 : can_factor(F, nf, x, NULL, Nx, fact)) return NULL;
1452 1188420 : if ((y = SPLIT_i(F, nf, nf_get_roundG(nf), x, x, Nx, fact))) return y;
1453 :
1454 : /* reduce in various directions */
1455 8796 : ru = lg(nf_get_roots(nf));
1456 8796 : vecG = cgetg(ru, t_VEC);
1457 14345 : for (j=1; j<ru; j++)
1458 : {
1459 12597 : gel(vecG,j) = nf_get_Gtwist1(nf, j);
1460 12597 : if ((y = SPLIT_i(F, nf, gel(vecG,j), x, x, Nx, fact))) return y;
1461 : }
1462 :
1463 : /* tough case, multiply by random products */
1464 1748 : lgsub = 3; nbtest = 1; nbtest_lim = 4;
1465 1748 : ex = cgetg(lgsub, t_VECSMALL);
1466 1748 : x0 = init_famat(x);
1467 : for(;;)
1468 594 : {
1469 2342 : GEN Ired, I, NI, id = x0;
1470 2342 : av = avma;
1471 2342 : if (DEBUGLEVEL>2) err_printf("# ideals tried = %ld\n",nbtest);
1472 7138 : for (i=1; i<lgsub; i++)
1473 : {
1474 4796 : ex[i] = random_bits(RANDOM_BITS);
1475 4796 : if (ex[i]) id = idealmulpowprime2(nf, id, gel(Vbase,i), ex[i]);
1476 : }
1477 2342 : if (id == x0) continue;
1478 : /* I^(-1) * \prod Vbase[i]^ex[i] = (id[2]) / x */
1479 :
1480 2321 : I = gel(id,1); NI = ZM_det_triangular(I);
1481 2321 : if (can_factor(F, nf, I, NULL, NI, fact))
1482 : {
1483 0 : inv_fact(fact); /* I^(-1) */
1484 0 : for (i=1; i<lgsub; i++)
1485 0 : if (ex[i]) add_to_fact(Vbase_to_FB(F,gel(Vbase,i)), ex[i], fact);
1486 0 : return gel(id,2);
1487 : }
1488 2321 : Ired = ru == 2? I: ZM_lll(I, 0.99, LLL_INPLACE);
1489 3903 : for (j=1; j<ru; j++)
1490 3330 : if ((y = SPLIT_i(F, nf, gel(vecG,j), I, Ired, NI, fact)))
1491 : {
1492 5265 : for (i=1; i<lgsub; i++)
1493 3517 : if (ex[i]) add_to_fact(Vbase_to_FB(F,gel(Vbase,i)), ex[i], fact);
1494 1748 : return famat_mul_shallow(gel(id,2), y);
1495 : }
1496 573 : set_avma(av);
1497 573 : if (++nbtest > nbtest_lim)
1498 : {
1499 21 : nbtest = 0;
1500 21 : if (++lgsub < minss(8, lg(Vbase)-1))
1501 : {
1502 21 : nbtest_lim <<= 1;
1503 21 : ex = cgetg(lgsub, t_VECSMALL);
1504 : }
1505 0 : else nbtest_lim = LONG_MAX; /* don't increase further */
1506 21 : if (DEBUGLEVEL>2) err_printf("SPLIT: increasing factor base [%ld]\n",lgsub);
1507 : }
1508 : }
1509 : }
1510 :
1511 : INLINE GEN
1512 1327258 : bnf_get_W(GEN bnf) { return gel(bnf,1); }
1513 : INLINE GEN
1514 2644582 : bnf_get_B(GEN bnf) { return gel(bnf,2); }
1515 : INLINE GEN
1516 2679097 : bnf_get_C(GEN bnf) { return gel(bnf,4); }
1517 : INLINE GEN
1518 1322365 : bnf_get_vbase(GEN bnf) { return gel(bnf,5); }
1519 : INLINE GEN
1520 1322283 : bnf_get_Ur(GEN bnf) { return gmael(bnf,9,1); }
1521 : INLINE GEN
1522 271837 : bnf_get_ga(GEN bnf) { return gmael(bnf,9,2); }
1523 : INLINE GEN
1524 276793 : bnf_get_GD(GEN bnf) { return gmael(bnf,9,3); }
1525 :
1526 : /* Return y (as an elt of K or a t_MAT representing an elt in Z[K])
1527 : * such that x / (y) is smooth and store the exponents of its factorization
1528 : * on g_W and g_B in Wex / Bex; return NULL for y = 1 */
1529 : static GEN
1530 1322282 : split_ideal(GEN bnf, GEN x, GEN *pWex, GEN *pBex)
1531 : {
1532 1322282 : GEN L, y, Vbase = bnf_get_vbase(bnf);
1533 1322282 : GEN Wex, W = bnf_get_W(bnf);
1534 1322282 : GEN Bex, B = bnf_get_B(bnf);
1535 : long p, j, i, l, nW, nB;
1536 : FACT *fact;
1537 : FB_t F;
1538 :
1539 1322282 : L = recover_partFB(&F, Vbase, lg(x)-1);
1540 1322284 : fact = (FACT*)stack_malloc((F.KC+1)*sizeof(FACT));
1541 1322283 : y = SPLIT(&F, bnf_get_nf(bnf), x, Vbase, fact);
1542 1322282 : nW = lg(W)-1; *pWex = Wex = zero_zv(nW);
1543 1322281 : nB = lg(B)-1; *pBex = Bex = zero_zv(nB); l = lg(F.FB);
1544 1322282 : p = j = 0; /* -Wall */
1545 1969909 : for (i = 1; i <= fact[0].pr; i++)
1546 : { /* decode index C = ip+j --> (p,j) */
1547 647627 : long a, b, t, C = fact[i].pr;
1548 1827490 : for (t = 1; t < l; t++)
1549 : {
1550 1754129 : long q = F.FB[t], k = C - F.iLP[q];
1551 1754129 : if (k <= 0) break;
1552 1179863 : p = q;
1553 1179863 : j = k;
1554 : }
1555 647627 : a = gel(L, p)[j];
1556 647627 : b = a - nW;
1557 647627 : if (b <= 0) Wex[a] = y? -fact[i].ex: fact[i].ex;
1558 494230 : else Bex[b] = y? -fact[i].ex: fact[i].ex;
1559 : }
1560 1322282 : return y;
1561 : }
1562 :
1563 : GEN
1564 1039916 : init_red_mod_units(GEN bnf, long prec)
1565 : {
1566 1039916 : GEN s = gen_0, p1,s1,mat, logfu = bnf_get_logfu(bnf);
1567 1039916 : long i,j, RU = lg(logfu);
1568 :
1569 1039916 : if (RU == 1) return NULL;
1570 1039916 : mat = cgetg(RU,t_MAT);
1571 2358269 : for (j=1; j<RU; j++)
1572 : {
1573 1318352 : p1 = cgetg(RU+1,t_COL); gel(mat,j) = p1;
1574 1318352 : s1 = gen_0;
1575 3264772 : for (i=1; i<RU; i++)
1576 : {
1577 1946420 : gel(p1,i) = real_i(gcoeff(logfu,i,j));
1578 1946420 : s1 = mpadd(s1, mpsqr(gel(p1,i)));
1579 : }
1580 1318352 : gel(p1,RU) = gen_0; if (mpcmp(s1,s) > 0) s = s1;
1581 : }
1582 1039917 : s = gsqrt(gmul2n(s,RU),prec);
1583 1039917 : if (expo(s) < 27) s = utoipos(1UL << 27);
1584 1039916 : return mkvec2(mat, s);
1585 : }
1586 :
1587 : /* z computed above. Return unit exponents that would reduce col (arch) */
1588 : GEN
1589 1039917 : red_mod_units(GEN col, GEN z)
1590 : {
1591 : long i,RU;
1592 : GEN x,mat,N2;
1593 :
1594 1039917 : if (!z) return NULL;
1595 1039917 : mat= gel(z,1);
1596 1039917 : N2 = gel(z,2);
1597 1039917 : RU = lg(mat); x = cgetg(RU+1,t_COL);
1598 2358267 : for (i=1; i<RU; i++) gel(x,i) = real_i(gel(col,i));
1599 1039916 : gel(x,RU) = N2;
1600 1039916 : x = lll(shallowconcat(mat,x));
1601 1039917 : if (typ(x) != t_MAT || lg(x) <= RU) return NULL;
1602 1039917 : x = gel(x,RU);
1603 1039917 : if (signe(gel(x,RU)) < 0) x = gneg_i(x);
1604 1039917 : if (!gequal1(gel(x,RU))) pari_err_BUG("red_mod_units");
1605 1039917 : setlg(x,RU); return x;
1606 : }
1607 :
1608 : static GEN
1609 2128093 : add(GEN a, GEN t) { return a = a? RgC_add(a,t): t; }
1610 :
1611 : /* [x] archimedian components, A column vector. return [x] A */
1612 : static GEN
1613 1986742 : act_arch(GEN A, GEN x)
1614 : {
1615 : GEN a;
1616 1986742 : long i,l = lg(A), tA = typ(A);
1617 1986742 : if (tA == t_MAT)
1618 : { /* assume lg(x) >= l */
1619 191658 : a = cgetg(l, t_MAT);
1620 281630 : for (i=1; i<l; i++) gel(a,i) = act_arch(gel(A,i), x);
1621 191656 : return a;
1622 : }
1623 1795084 : if (l==1) return cgetg(1, t_COL);
1624 1795084 : a = NULL;
1625 1795084 : if (tA == t_VECSMALL)
1626 : {
1627 6833980 : for (i=1; i<l; i++)
1628 : {
1629 5672545 : long c = A[i];
1630 5672545 : if (c) a = add(a, gmulsg(c, gel(x,i)));
1631 : }
1632 : }
1633 : else
1634 : { /* A a t_COL of t_INT. Assume lg(A)==lg(x) */
1635 1384004 : for (i=1; i<l; i++)
1636 : {
1637 750361 : GEN c = gel(A,i);
1638 750361 : if (signe(c)) a = add(a, gmul(c, gel(x,i)));
1639 : }
1640 : }
1641 1795078 : return a? a: zerocol(lgcols(x)-1);
1642 : }
1643 : /* act_arch(matdiagonal(v), x) */
1644 : static GEN
1645 63887 : diagact_arch(GEN v, GEN x)
1646 : {
1647 63887 : long i, l = lg(v);
1648 63887 : GEN a = cgetg(l, t_MAT);
1649 92769 : for (i = 1; i < l; i++) gel(a,i) = gmul(gel(x,i), gel(v,i));
1650 63887 : return a;
1651 : }
1652 :
1653 : static long
1654 1340433 : prec_arch(GEN bnf)
1655 : {
1656 1340433 : GEN a = bnf_get_C(bnf);
1657 1340433 : long i, l = lg(a), prec;
1658 :
1659 1340433 : for (i=1; i<l; i++)
1660 1340349 : if ( (prec = gprecision(gel(a,i))) ) return prec;
1661 84 : return DEFAULTPREC;
1662 : }
1663 :
1664 : static long
1665 3831 : needed_bitprec(GEN x)
1666 : {
1667 3831 : long i, e = 0, l = lg(x);
1668 22386 : for (i = 1; i < l; i++)
1669 : {
1670 18555 : GEN c = gel(x,i);
1671 18555 : long f = gexpo(c) - gprecision(c);
1672 18555 : if (f > e) e = f;
1673 : }
1674 3831 : return e;
1675 : }
1676 :
1677 : /* col = archimedian components of x, Nx its norm, dx a multiple of its
1678 : * denominator. Return x or NULL (fail) */
1679 : GEN
1680 1167045 : isprincipalarch(GEN bnf, GEN col, GEN kNx, GEN e, GEN dx, long *pe)
1681 : {
1682 : GEN nf, x, y, logfu, s, M;
1683 1167045 : long N, prec = gprecision(col);
1684 1167045 : bnf = checkbnf(bnf); nf = bnf_get_nf(bnf); M = nf_get_M(nf);
1685 1167044 : if (!prec) prec = prec_arch(bnf);
1686 1167044 : *pe = 128;
1687 1167044 : logfu = bnf_get_logfu(bnf);
1688 1167044 : N = nf_get_degree(nf);
1689 1167044 : if (!(col = cleanarch(col,N,NULL,prec))) return NULL;
1690 1167044 : if (lg(col) > 2)
1691 : { /* reduce mod units */
1692 1039916 : GEN u, z = init_red_mod_units(bnf,prec);
1693 1039917 : if (!(u = red_mod_units(col,z))) return NULL;
1694 1039917 : col = RgC_add(col, RgM_RgC_mul(logfu, u));
1695 1039916 : if (!(col = cleanarch(col,N,NULL,prec))) return NULL;
1696 : }
1697 1167044 : s = divru(mulir(e, glog(kNx,prec)), N);
1698 1167048 : col = fixarch(col, s, nf_get_r1(nf));
1699 1167050 : if (RgC_expbitprec(col) >= 0) return NULL;
1700 1166476 : col = gexp(col, prec);
1701 : /* d.alpha such that x = alpha \prod gj^ej */
1702 1166473 : x = RgM_solve_realimag(M,col); if (!x) return NULL;
1703 1166475 : x = RgC_Rg_mul(x, dx);
1704 1166474 : y = grndtoi(x, pe);
1705 1166471 : if (*pe > -5) { *pe = needed_bitprec(x); return NULL; }
1706 1162640 : return RgC_Rg_div(y, dx);
1707 : }
1708 :
1709 : /* y = C \prod g[i]^e[i] ? */
1710 : static int
1711 1158567 : fact_ok(GEN nf, GEN y, GEN C, GEN g, GEN e)
1712 : {
1713 1158567 : pari_sp av = avma;
1714 1158567 : long i, c = lg(e);
1715 1158567 : GEN z = C? C: gen_1;
1716 1435668 : for (i=1; i<c; i++)
1717 277101 : if (signe(gel(e,i))) z = idealmul(nf, z, idealpow(nf, gel(g,i), gel(e,i)));
1718 1158567 : if (typ(z) != t_MAT) z = idealhnf_shallow(nf,z);
1719 1158567 : if (typ(y) != t_MAT) y = idealhnf_shallow(nf,y);
1720 1158567 : return gc_bool(av, ZM_equal(y,z));
1721 : }
1722 : static GEN
1723 1322286 : ZV_divrem(GEN A, GEN B, GEN *pR)
1724 : {
1725 1322286 : long i, l = lg(A);
1726 1322286 : GEN Q = cgetg(l, t_COL), R = cgetg(l, t_COL);
1727 1828588 : for (i = 1; i < l; i++) gel(Q,i) = truedvmdii(gel(A,i), gel(B,i), &gel(R,i));
1728 1322285 : *pR = R; return Q;
1729 : }
1730 :
1731 : static GEN
1732 1322283 : Ur_ZC_mul(GEN bnf, GEN v)
1733 : {
1734 1322283 : GEN w, U = bnf_get_Ur(bnf);
1735 1322283 : long i, l = lg(bnf_get_cyc(bnf)); /* may be < lgcols(U) */
1736 :
1737 1322283 : w = cgetg(l, t_COL);
1738 1828587 : for (i = 1; i < l; i++) gel(w,i) = ZMrow_ZC_mul(U, v, i);
1739 1322286 : return w;
1740 : }
1741 :
1742 : static GEN
1743 7283 : ZV_mul(GEN x, GEN y)
1744 : {
1745 7283 : long i, l = lg(x);
1746 7283 : GEN z = cgetg(l, t_COL);
1747 31780 : for (i = 1; i < l; i++) gel(z,i) = mulii(gel(x,i), gel(y,i));
1748 7283 : return z;
1749 : }
1750 : static int
1751 1158017 : dump_gen(GEN SUnits, GEN x, long flag)
1752 : {
1753 : GEN d;
1754 : long e;
1755 1158017 : if (!(flag & nf_GENMAT) || !SUnits) return 0;
1756 266413 : e = gexpo(gel(SUnits,2)); if (e > 64) return 0; /* U large */
1757 266318 : x = Q_remove_denom(x, &d);
1758 266318 : return (d && expi(d) > 32) || gexpo(x) > 32;
1759 : }
1760 :
1761 : /* assume x in HNF; cf class_group_gen for notations. Return NULL iff
1762 : * flag & nf_FORCE and computation of principal ideal generator fails */
1763 : static GEN
1764 1338645 : isprincipalall(GEN bnf, GEN x, long *pprec, long flag)
1765 : {
1766 : GEN xar, Wex, Bex, gen, xc, col, A, Q, R, UA, SUnits;
1767 1338645 : GEN C = bnf_get_C(bnf), nf = bnf_get_nf(bnf), cyc = bnf_get_cyc(bnf);
1768 : long nB, nW, e;
1769 :
1770 1338644 : if (lg(cyc) == 1 && !(flag & (nf_GEN|nf_GENMAT|nf_GEN_IF_PRINCIPAL)))
1771 4732 : return cgetg(1,t_COL);
1772 1333912 : if (lg(x) == 2)
1773 : { /* nf = Q */
1774 84 : col = gel(x,1);
1775 84 : if (flag & nf_GENMAT) col = Q_to_famat(gel(col,1));
1776 84 : return (flag & nf_GEN_IF_PRINCIPAL)? col: mkvec2(cgetg(1,t_COL), col);
1777 : }
1778 :
1779 1333828 : x = Q_primitive_part(x, &xc);
1780 1333825 : if (equali1(gcoeff(x,1,1))) /* trivial ideal */
1781 : {
1782 11543 : R = zerocol(lg(cyc)-1);
1783 11543 : if (!(flag & (nf_GEN|nf_GENMAT|nf_GEN_IF_PRINCIPAL))) return R;
1784 11494 : if (flag & nf_GEN_IF_PRINCIPAL)
1785 6468 : return scalarcol_shallow(xc? xc: gen_1, nf_get_degree(nf));
1786 5026 : if (flag & nf_GENMAT)
1787 2191 : col = xc? Q_to_famat(xc): trivial_fact();
1788 : else
1789 2835 : col = scalarcol_shallow(xc? xc: gen_1, nf_get_degree(nf));
1790 5026 : return mkvec2(R, col);
1791 : }
1792 1322282 : xar = split_ideal(bnf, x, &Wex, &Bex);
1793 : /* x = g_W Wex + g_B Bex + [xar] = g_W (Wex - B*Bex) + [xar] + [C_B]Bex */
1794 1322282 : A = zc_to_ZC(Wex); nB = lg(Bex)-1;
1795 1322282 : if (nB) A = ZC_sub(A, ZM_zc_mul(bnf_get_B(bnf), Bex));
1796 1322283 : UA = Ur_ZC_mul(bnf, A);
1797 1322286 : Q = ZV_divrem(UA, cyc, &R);
1798 : /* g_W (Wex - B*Bex) = G Ur A - [ga]A = G R + [GD]Q - [ga]A
1799 : * Finally: x = G R + [xar] + [C_B]Bex + [GD]Q - [ga]A */
1800 1322284 : if (!(flag & (nf_GEN|nf_GENMAT|nf_GEN_IF_PRINCIPAL))) return R;
1801 1162058 : if ((flag & nf_GEN_IF_PRINCIPAL) && !ZV_equal0(R)) return gen_0;
1802 :
1803 1162051 : nW = lg(Wex)-1;
1804 1162051 : gen = bnf_get_gen(bnf);
1805 1162052 : col = NULL;
1806 1162052 : SUnits = bnf_get_sunits(bnf);
1807 1162051 : if (lg(R) == 1
1808 272452 : || abscmpiu(gel(R,vecindexmax(R)), 4 * (*pprec)) < 0)
1809 : { /* q = N (x / prod gj^ej) = N(alpha), denom(alpha) | d */
1810 1161436 : GEN d, q = gdiv(ZM_det_triangular(x), get_norm_fact(gen, R, &d));
1811 1161436 : col = xar? nf_cxlog(nf, xar, *pprec): NULL;
1812 1161436 : if (nB) col = add(col, act_arch(Bex, nW? vecslice(C,nW+1,lg(C)-1): C));
1813 1161436 : if (nW) col = add(col, RgC_sub(act_arch(Q, bnf_get_GD(bnf)),
1814 : act_arch(A, bnf_get_ga(bnf))));
1815 1161435 : col = isprincipalarch(bnf, col, q, gen_1, d, &e);
1816 1161434 : if (col && (dump_gen(SUnits, col, flag)
1817 1158018 : || !fact_ok(nf,x, col,gen,R))) col = NULL;
1818 : }
1819 1162047 : if (!col && (flag & nf_GENMAT))
1820 : {
1821 8024 : if (SUnits)
1822 : {
1823 7535 : GEN X = gel(SUnits,1), U = gel(SUnits,2), C = gel(SUnits,3);
1824 7535 : GEN v = gel(bnf,9), Ge = gel(v,4), M1 = gel(v,5), M2 = gel(v,6);
1825 7535 : GEN z = NULL, F = NULL;
1826 7535 : if (nB)
1827 : {
1828 7535 : GEN C2 = nW? vecslice(C, nW+1, lg(C)-1): C;
1829 7535 : z = ZM_zc_mul(C2, Bex);
1830 : }
1831 7535 : if (nW)
1832 : { /* [GD]Q - [ga]A = ([X]M1 - [Ge]D) Q - ([X]M2 - [Ge]Ur) A */
1833 7283 : GEN C1 = vecslice(C, 1, nW);
1834 7283 : GEN v = ZC_sub(ZM_ZC_mul(M1,Q), ZM_ZC_mul(M2,A));
1835 7283 : z = add(z, ZM_ZC_mul(C1, v));
1836 7283 : F = famat_reduce(famatV_factorback(Ge, ZC_sub(UA, ZV_mul(cyc,Q))));
1837 7283 : if (lgcols(F) == 1) F = NULL;
1838 : }
1839 : /* reduce modulo units and Q^* */
1840 7535 : if (lg(U) != 1) z = ZC_sub(z, ZM_ZC_mul(U, RgM_Babai(U,z)));
1841 7535 : col = mkmat2(X, z);
1842 7535 : if (F) col = famat_mul_shallow(col, F);
1843 7535 : col = famat_remove_trivial(col);
1844 7535 : if (xar) col = famat_mul_shallow(col, xar);
1845 : }
1846 489 : else if (!ZV_equal0(R))
1847 : { /* in case isprincipalfact calls bnfinit() due to prec trouble...*/
1848 483 : GEN y = isprincipalfact(bnf, x, gen, ZC_neg(R), flag);
1849 483 : if (typ(y) != t_VEC) return y;
1850 483 : col = gel(y,2);
1851 : }
1852 : }
1853 1162047 : if (col)
1854 : { /* add back missing content */
1855 1161957 : if (typ(col) == t_MAT)
1856 8018 : { if (xc) col = famat_mul_shallow(col, xc); }
1857 1153939 : else if (flag & nf_GENMAT)
1858 : {
1859 : GEN c;
1860 1140240 : if (RgV_isscalar(col))
1861 3654 : col = Q_to_famat(mul_content(xc, gel(col,1)));
1862 : else
1863 : {
1864 1136585 : col = Q_primitive_part(col, &c);
1865 1136584 : col = to_famat_shallow(col, gen_1);
1866 1136585 : xc = mul_content(xc, c);
1867 1136585 : if (xc) col = famat_mul(col, Q_to_famat(xc));
1868 : }
1869 : }
1870 : else
1871 13699 : { if (xc) col = RgC_Rg_mul(col,xc); }
1872 : }
1873 : else
1874 : {
1875 90 : if (e < 0) e = 0;
1876 90 : *pprec += nbits2extraprec(e + 128);
1877 90 : if (flag & nf_FORCE)
1878 : {
1879 76 : if (DEBUGLEVEL)
1880 0 : pari_warn(warner,"precision too low for generators, e = %ld",e);
1881 76 : return NULL;
1882 : }
1883 14 : pari_warn(warner,"precision too low for generators, not given");
1884 14 : col = cgetg(1, t_COL);
1885 : }
1886 1161970 : return (flag & nf_GEN_IF_PRINCIPAL)? col: mkvec2(R, col);
1887 : }
1888 :
1889 : static GEN
1890 462861 : triv_gen(GEN bnf, GEN x, long flag)
1891 : {
1892 462861 : pari_sp av = avma;
1893 462861 : GEN nf = bnf_get_nf(bnf);
1894 : long c;
1895 462861 : if (flag & nf_GEN_IF_PRINCIPAL)
1896 : {
1897 7 : if (!(flag & nf_GENMAT)) return algtobasis(nf,x);
1898 7 : x = nf_to_scalar_or_basis(nf,x);
1899 7 : if (typ(x) == t_INT && is_pm1(x)) return trivial_fact();
1900 0 : return gerepilecopy(av, to_famat_shallow(x, gen_1));
1901 : }
1902 462854 : c = lg(bnf_get_cyc(bnf)) - 1;
1903 462854 : if (flag & nf_GENMAT)
1904 453250 : retmkvec2(zerocol(c), to_famat_shallow(algtobasis(nf,x), gen_1));
1905 9604 : if (flag & nf_GEN)
1906 28 : retmkvec2(zerocol(c), algtobasis(nf,x));
1907 9576 : return zerocol(c);
1908 : }
1909 :
1910 : GEN
1911 1769270 : bnfisprincipal0(GEN bnf,GEN x,long flag)
1912 : {
1913 1769270 : pari_sp av = avma;
1914 : GEN c, nf;
1915 : long pr;
1916 :
1917 1769270 : bnf = checkbnf(bnf);
1918 1769271 : nf = bnf_get_nf(bnf);
1919 1769271 : switch( idealtyp(&x, NULL) )
1920 : {
1921 57400 : case id_PRINCIPAL:
1922 57400 : if (gequal0(x)) pari_err_DOMAIN("bnfisprincipal","ideal","=",gen_0,x);
1923 57400 : return triv_gen(bnf, x, flag);
1924 1688169 : case id_PRIME:
1925 1688169 : if (pr_is_inert(x)) return triv_gen(bnf, pr_get_p(x), flag);
1926 1282715 : x = pr_hnf(nf, x);
1927 1282714 : break;
1928 23702 : case id_MAT:
1929 23702 : if (lg(x)==1) pari_err_DOMAIN("bnfisprincipal","ideal","=",gen_0,x);
1930 23702 : if (nf_get_degree(nf) != lg(x)-1)
1931 0 : pari_err_TYPE("idealtyp [dimension != degree]", x);
1932 : }
1933 1306415 : pr = prec_arch(bnf); /* precision of unit matrix */
1934 1306416 : c = getrand();
1935 : for (;;)
1936 6 : {
1937 1306423 : pari_sp av1 = avma;
1938 1306423 : GEN y = isprincipalall(bnf,x,&pr,flag);
1939 1306416 : if (y) return gerepilecopy(av, y);
1940 :
1941 6 : if (DEBUGLEVEL) pari_warn(warnprec,"isprincipal",pr);
1942 6 : set_avma(av1); bnf = bnfnewprec_shallow(bnf,pr); setrand(c);
1943 : }
1944 : }
1945 : GEN
1946 174534 : isprincipal(GEN bnf,GEN x) { return bnfisprincipal0(bnf,x,0); }
1947 :
1948 : /* FIXME: OBSOLETE */
1949 : GEN
1950 0 : isprincipalgen(GEN bnf,GEN x)
1951 0 : { return bnfisprincipal0(bnf,x,nf_GEN); }
1952 : GEN
1953 0 : isprincipalforce(GEN bnf,GEN x)
1954 0 : { return bnfisprincipal0(bnf,x,nf_FORCE); }
1955 : GEN
1956 0 : isprincipalgenforce(GEN bnf,GEN x)
1957 0 : { return bnfisprincipal0(bnf,x,nf_GEN | nf_FORCE); }
1958 :
1959 : /* lg(u) > 1 */
1960 : static int
1961 105 : RgV_is1(GEN u) { return isint1(gel(u,1)) && RgV_isscalar(u); }
1962 : static GEN
1963 32152 : add_principal_part(GEN nf, GEN u, GEN v, long flag)
1964 : {
1965 32152 : if (flag & nf_GENMAT)
1966 14540 : return (typ(u) == t_COL && RgV_is1(u))? v: famat_mul_shallow(v,u);
1967 : else
1968 17612 : return nfmul(nf, v, u);
1969 : }
1970 :
1971 : #if 0
1972 : /* compute C prod P[i]^e[i], e[i] >=0 for all i. C may be NULL (omitted)
1973 : * e destroyed ! */
1974 : static GEN
1975 : expand(GEN nf, GEN C, GEN P, GEN e)
1976 : {
1977 : long i, l = lg(e), done = 1;
1978 : GEN id = C;
1979 : for (i=1; i<l; i++)
1980 : {
1981 : GEN ei = gel(e,i);
1982 : if (signe(ei))
1983 : {
1984 : if (mod2(ei)) id = id? idealmul(nf, id, gel(P,i)): gel(P,i);
1985 : ei = shifti(ei,-1);
1986 : if (signe(ei)) done = 0;
1987 : gel(e,i) = ei;
1988 : }
1989 : }
1990 : if (id != C) id = idealred(nf, id);
1991 : if (done) return id;
1992 : return idealmulred(nf, id, idealsqr(nf, expand(nf,id,P,e)));
1993 : }
1994 : /* C is an extended ideal, possibly with C[1] = NULL */
1995 : static GEN
1996 : expandext(GEN nf, GEN C, GEN P, GEN e)
1997 : {
1998 : long i, l = lg(e), done = 1;
1999 : GEN A = gel(C,1);
2000 : for (i=1; i<l; i++)
2001 : {
2002 : GEN ei = gel(e,i);
2003 : if (signe(ei))
2004 : {
2005 : if (mod2(ei)) A = A? idealmul(nf, A, gel(P,i)): gel(P,i);
2006 : ei = shifti(ei,-1);
2007 : if (signe(ei)) done = 0;
2008 : gel(e,i) = ei;
2009 : }
2010 : }
2011 : if (A == gel(C,1))
2012 : A = C;
2013 : else
2014 : A = idealred(nf, mkvec2(A, gel(C,2)));
2015 : if (done) return A;
2016 : return idealmulred(nf, A, idealsqr(nf, expand(nf,A,P,e)));
2017 : }
2018 : #endif
2019 :
2020 : static GEN
2021 0 : expand(GEN nf, GEN C, GEN P, GEN e)
2022 : {
2023 0 : long i, l = lg(e);
2024 0 : GEN B, A = C;
2025 0 : for (i=1; i<l; i++) /* compute prod P[i]^e[i] */
2026 0 : if (signe(gel(e,i)))
2027 : {
2028 0 : B = idealpowred(nf, gel(P,i), gel(e,i));
2029 0 : A = A? idealmulred(nf,A,B): B;
2030 : }
2031 0 : return A;
2032 : }
2033 : static GEN
2034 32173 : expandext(GEN nf, GEN C, GEN P, GEN e)
2035 : {
2036 32173 : long i, l = lg(e);
2037 32173 : GEN B, A = gel(C,1), C1 = A;
2038 95178 : for (i=1; i<l; i++) /* compute prod P[i]^e[i] */
2039 63005 : if (signe(gel(e,i)))
2040 : {
2041 35195 : gel(C,1) = gel(P,i);
2042 35195 : B = idealpowred(nf, C, gel(e,i));
2043 35195 : A = A? idealmulred(nf,A,B): B;
2044 : }
2045 32173 : return A == C1? C: A;
2046 : }
2047 :
2048 : /* isprincipal for C * \prod P[i]^e[i] (C omitted if NULL) */
2049 : GEN
2050 32173 : isprincipalfact(GEN bnf, GEN C, GEN P, GEN e, long flag)
2051 : {
2052 32173 : const long gen = flag & (nf_GEN|nf_GENMAT|nf_GEN_IF_PRINCIPAL);
2053 : long prec;
2054 32173 : pari_sp av = avma;
2055 32173 : GEN C0, Cext, c, id, nf = bnf_get_nf(bnf);
2056 :
2057 32173 : if (gen)
2058 : {
2059 14547 : Cext = (flag & nf_GENMAT)? trivial_fact()
2060 32173 : : mkpolmod(gen_1,nf_get_pol(nf));
2061 32173 : C0 = mkvec2(C, Cext);
2062 32173 : id = expandext(nf, C0, P, e);
2063 : } else {
2064 0 : Cext = NULL;
2065 0 : C0 = C;
2066 0 : id = expand(nf, C, P, e);
2067 : }
2068 32173 : if (id == C0) /* e = 0 */
2069 : {
2070 12477 : if (!C) return bnfisprincipal0(bnf, gen_1, flag);
2071 12463 : switch(typ(C))
2072 : {
2073 7 : case t_INT: case t_FRAC: case t_POL: case t_POLMOD: case t_COL:
2074 7 : return triv_gen(bnf, C, flag);
2075 : }
2076 12456 : C = idealhnf_shallow(nf,C);
2077 : }
2078 : else
2079 : {
2080 19696 : if (gen) { C = gel(id,1); Cext = gel(id,2); } else C = id;
2081 : }
2082 32152 : prec = prec_arch(bnf);
2083 32152 : c = getrand();
2084 : for (;;)
2085 70 : {
2086 32222 : pari_sp av1 = avma;
2087 32222 : GEN y = isprincipalall(bnf, C, &prec, flag);
2088 32222 : if (y)
2089 : {
2090 32152 : if (flag & nf_GEN_IF_PRINCIPAL)
2091 : {
2092 20818 : if (typ(y) == t_INT) return gc_NULL(av);
2093 20818 : y = add_principal_part(nf, y, Cext, flag);
2094 : }
2095 : else
2096 : {
2097 11334 : GEN u = gel(y,2);
2098 11334 : if (!gen || typ(y) != t_VEC) return gerepileupto(av,y);
2099 11334 : if (lg(u) != 1) gel(y,2) = add_principal_part(nf, u, Cext, flag);
2100 : }
2101 32151 : return gerepilecopy(av, y);
2102 : }
2103 70 : if (DEBUGLEVEL) pari_warn(warnprec,"isprincipal",prec);
2104 70 : set_avma(av1); bnf = bnfnewprec_shallow(bnf,prec); setrand(c);
2105 : }
2106 : }
2107 : GEN
2108 0 : isprincipalfact_or_fail(GEN bnf, GEN C, GEN P, GEN e)
2109 : {
2110 0 : const long flag = nf_GENMAT|nf_FORCE;
2111 : long prec;
2112 0 : pari_sp av = avma;
2113 0 : GEN u, y, id, C0, Cext, nf = bnf_get_nf(bnf);
2114 :
2115 0 : Cext = trivial_fact();
2116 0 : C0 = mkvec2(C, Cext);
2117 0 : id = expandext(nf, C0, P, e);
2118 0 : if (id == C0) /* e = 0 */
2119 0 : C = idealhnf_shallow(nf,C);
2120 : else {
2121 0 : C = gel(id,1); Cext = gel(id,2);
2122 : }
2123 0 : prec = prec_arch(bnf);
2124 0 : y = isprincipalall(bnf, C, &prec, flag);
2125 0 : if (!y) return gc_utoipos(av, prec);
2126 0 : u = gel(y,2);
2127 0 : if (lg(u) != 1) gel(y,2) = add_principal_part(nf, u, Cext, flag);
2128 0 : return gerepilecopy(av, y);
2129 : }
2130 :
2131 : GEN
2132 148888 : nfsign_from_logarch(GEN LA, GEN invpi, GEN archp)
2133 : {
2134 148888 : long l = lg(archp), i;
2135 148888 : GEN y = cgetg(l, t_VECSMALL);
2136 148890 : pari_sp av = avma;
2137 :
2138 279575 : for (i=1; i<l; i++)
2139 : {
2140 130684 : GEN c = ground( gmul(imag_i(gel(LA,archp[i])), invpi) );
2141 130684 : y[i] = mpodd(c)? 1: 0;
2142 : }
2143 148891 : set_avma(av); return y;
2144 : }
2145 :
2146 : GEN
2147 227144 : nfsign_tu(GEN bnf, GEN archp)
2148 : {
2149 : long n;
2150 227144 : if (bnf_get_tuN(bnf) != 2) return cgetg(1, t_VECSMALL);
2151 160044 : n = archp? lg(archp) - 1: nf_get_r1(bnf_get_nf(bnf));
2152 160044 : return const_vecsmall(n, 1);
2153 : }
2154 : GEN
2155 228320 : nfsign_fu(GEN bnf, GEN archp)
2156 : {
2157 228320 : GEN invpi, y, A = bnf_get_logfu(bnf), nf = bnf_get_nf(bnf);
2158 228318 : long j = 1, RU = lg(A);
2159 :
2160 228318 : if (!archp) archp = identity_perm( nf_get_r1(nf) );
2161 228318 : invpi = invr( mppi(nf_get_prec(nf)) );
2162 228311 : y = cgetg(RU,t_MAT);
2163 377113 : for (j = 1; j < RU; j++)
2164 148790 : gel(y,j) = nfsign_from_logarch(gel(A,j), invpi, archp);
2165 228323 : return y;
2166 : }
2167 : GEN
2168 35 : nfsign_units(GEN bnf, GEN archp, int add_zu)
2169 : {
2170 35 : GEN sfu = nfsign_fu(bnf, archp);
2171 35 : return add_zu? vec_prepend(sfu, nfsign_tu(bnf, archp)): sfu;
2172 : }
2173 :
2174 : /* obsolete */
2175 : GEN
2176 7 : signunits(GEN bnf)
2177 : {
2178 : pari_sp av;
2179 : GEN S, y, nf;
2180 : long i, j, r1, r2;
2181 :
2182 7 : bnf = checkbnf(bnf); nf = bnf_get_nf(bnf);
2183 7 : nf_get_sign(nf, &r1,&r2);
2184 7 : S = zeromatcopy(r1, r1+r2-1); av = avma;
2185 7 : y = nfsign_fu(bnf, NULL);
2186 14 : for (j = 1; j < lg(y); j++)
2187 : {
2188 7 : GEN Sj = gel(S,j), yj = gel(y,j);
2189 21 : for (i = 1; i <= r1; i++) gel(Sj,i) = yj[i]? gen_m1: gen_1;
2190 : }
2191 7 : set_avma(av); return S;
2192 : }
2193 :
2194 : static GEN
2195 720930 : get_log_embed(REL_t *rel, GEN M, long RU, long R1, long prec)
2196 : {
2197 720930 : GEN arch, C, z = rel->m;
2198 : long i;
2199 720930 : arch = typ(z) == t_COL? RgM_RgC_mul(M, z): const_col(nbrows(M), z);
2200 720918 : C = cgetg(RU+1, t_COL); arch = glog(arch, prec);
2201 1632720 : for (i=1; i<=R1; i++) gel(C,i) = gel(arch,i);
2202 1559562 : for ( ; i<=RU; i++) gel(C,i) = gmul2n(gel(arch,i), 1);
2203 720916 : return C;
2204 : }
2205 : static GEN
2206 1007657 : rel_embed(REL_t *rel, FB_t *F, GEN embs, long ind, GEN M, long RU, long R1,
2207 : long prec)
2208 : {
2209 : GEN C, D, perm;
2210 : long i, n;
2211 1007657 : if (!rel->relaut) return get_log_embed(rel, M, RU, R1, prec);
2212 : /* image of another relation by automorphism */
2213 286728 : C = gel(embs, ind - rel->relorig);
2214 286728 : perm = gel(F->embperm, rel->relaut);
2215 286728 : D = cgetg_copy(C, &n);
2216 1197424 : for (i = 1; i < n; i++)
2217 : {
2218 910696 : long v = perm[i];
2219 910696 : gel(D,i) = (v > 0)? gel(C,v): conj_i(gel(C,-v));
2220 : }
2221 286728 : return D;
2222 : }
2223 : static GEN
2224 120963 : get_embs(FB_t *F, RELCACHE_t *cache, GEN nf, GEN embs, long PREC)
2225 : {
2226 120963 : long ru, j, k, l = cache->last - cache->chk + 1, r1 = nf_get_r1(nf);
2227 120963 : GEN M = nf_get_M(nf), nembs = cgetg(cache->last - cache->base+1, t_MAT);
2228 : REL_t *rel;
2229 :
2230 1484050 : for (k = 1; k <= cache->chk - cache->base; k++) gel(nembs,k) = gel(embs,k);
2231 120963 : embs = nembs; ru = nbrows(M);
2232 1116033 : for (j=1,rel = cache->chk + 1; j < l; rel++,j++,k++)
2233 995079 : gel(embs,k) = rel_embed(rel, F, embs, k, M, ru, r1, PREC);
2234 120954 : return embs;
2235 : }
2236 : static void
2237 939043 : set_rel_alpha(REL_t *rel, GEN auts, GEN vA, long ind)
2238 : {
2239 : GEN u;
2240 939043 : if (!rel->relaut)
2241 669637 : u = rel->m;
2242 : else
2243 269406 : u = ZM_ZC_mul(gel(auts, rel->relaut), gel(vA, ind - rel->relorig));
2244 939040 : gel(vA, ind) = u;
2245 939040 : }
2246 : static GEN
2247 2240846 : set_fact(FB_t *F, FACT *fact, GEN e, long *pnz)
2248 : {
2249 2240846 : long i, n = fact[0].pr, nz = F->KC + 1;
2250 2240846 : GEN c = zero_Flv(F->KC);
2251 10496731 : for (i = 1; i <= n; i++)
2252 : {
2253 8255886 : long p = fact[i].pr;
2254 8255886 : if (p < nz) nz = p;
2255 8255886 : c[p] = fact[i].ex;
2256 : }
2257 2240845 : if (e)
2258 : {
2259 113681 : long l = lg(e);
2260 331574 : for (i = 1; i < l; i++)
2261 217893 : if (e[i]) { long v = F->subFB[i]; c[v] += e[i]; if (v < nz) nz = v; }
2262 : }
2263 2240845 : *pnz = nz; return c;
2264 : }
2265 :
2266 : /* Is cols already in the cache ? bs = index of first non zero coeff in cols
2267 : * General check for colinearity useless since exceedingly rare */
2268 : static int
2269 2885792 : already_known(RELCACHE_t *cache, long bs, GEN cols)
2270 : {
2271 : REL_t *r;
2272 2885792 : long l = lg(cols);
2273 214345705 : for (r = cache->last; r > cache->base; r--)
2274 211942740 : if (bs == r->nz)
2275 : {
2276 34000294 : GEN coll = r->R;
2277 34000294 : long b = bs;
2278 121524714 : while (b < l && cols[b] == coll[b]) b++;
2279 34000294 : if (b == l) return 1;
2280 : }
2281 2402965 : return 0;
2282 : }
2283 :
2284 : /* Add relation R to cache, nz = index of first non zero coeff in R.
2285 : * If relation is a linear combination of the previous ones, return 0.
2286 : * Otherwise, update basis and return > 0. Compute mod p (much faster)
2287 : * so some kernel vector might not be genuine. */
2288 : static int
2289 2889921 : add_rel_i(RELCACHE_t *cache, GEN R, long nz, GEN m, long orig, long aut, REL_t **relp, long in_rnd_rel)
2290 : {
2291 2889921 : long i, k, n = lg(R)-1;
2292 :
2293 2889921 : if (nz == n+1) { k = 0; goto ADD_REL; }
2294 2885785 : if (already_known(cache, nz, R)) return -1;
2295 2403009 : if (cache->last >= cache->base + cache->len) return 0;
2296 2403009 : if (DEBUGLEVEL>6)
2297 : {
2298 0 : err_printf("adding vector = %Ps\n",R);
2299 0 : err_printf("generators =\n%Ps\n", cache->basis);
2300 : }
2301 2403036 : if (cache->missing)
2302 : {
2303 2011004 : GEN a = leafcopy(R), basis = cache->basis;
2304 2011002 : k = lg(a);
2305 124188375 : do --k; while (!a[k]);
2306 7057004 : while (k)
2307 : {
2308 5512139 : GEN c = gel(basis, k);
2309 5512139 : if (c[k])
2310 : {
2311 5046002 : long ak = a[k];
2312 261937235 : for (i=1; i < k; i++) if (c[i]) a[i] = (a[i] + ak*(mod_p-c[i])) % mod_p;
2313 5046002 : a[k] = 0;
2314 128663020 : do --k; while (!a[k]); /* k cannot go below 0: codeword is a sentinel */
2315 : }
2316 : else
2317 : {
2318 466137 : ulong invak = Fl_inv(uel(a,k), mod_p);
2319 : /* Cleanup a */
2320 13776321 : for (i = k; i-- > 1; )
2321 : {
2322 13310185 : long j, ai = a[i];
2323 13310185 : c = gel(basis, i);
2324 13310185 : if (!ai || !c[i]) continue;
2325 236201 : ai = mod_p-ai;
2326 4447895 : for (j = 1; j < i; j++) if (c[j]) a[j] = (a[j] + ai*c[j]) % mod_p;
2327 236201 : a[i] = 0;
2328 : }
2329 : /* Insert a/a[k] as k-th column */
2330 466136 : c = gel(basis, k);
2331 13776313 : for (i = 1; i<k; i++) if (a[i]) c[i] = (a[i] * invak) % mod_p;
2332 466136 : c[k] = 1; a = c;
2333 : /* Cleanup above k */
2334 13577537 : for (i = k+1; i<n; i++)
2335 : {
2336 : long j, ck;
2337 13111401 : c = gel(basis, i);
2338 13111401 : ck = c[k];
2339 13111401 : if (!ck) continue;
2340 2728357 : ck = mod_p-ck;
2341 99167083 : for (j = 1; j < k; j++) if (a[j]) c[j] = (c[j] + ck*a[j]) % mod_p;
2342 2728357 : c[k] = 0;
2343 : }
2344 466136 : cache->missing--;
2345 466136 : break;
2346 : }
2347 : }
2348 : }
2349 : else
2350 392032 : k = (cache->last - cache->base) + 1;
2351 2403033 : if (k || cache->relsup > 0 || (m && in_rnd_rel))
2352 : {
2353 : REL_t *rel;
2354 :
2355 979730 : ADD_REL:
2356 983866 : rel = ++cache->last;
2357 983866 : if (!k && cache->relsup && nz < n+1)
2358 : {
2359 121224 : cache->relsup--;
2360 121224 : k = (rel - cache->base) + cache->missing;
2361 : }
2362 983866 : rel->R = gclone(R);
2363 983865 : rel->m = m ? gclone(m) : NULL;
2364 983870 : rel->nz = nz;
2365 983870 : if (aut)
2366 : {
2367 284132 : rel->relorig = (rel - cache->base) - orig;
2368 284132 : rel->relaut = aut;
2369 : }
2370 : else
2371 699738 : rel->relaut = 0;
2372 983870 : if (relp) *relp = rel;
2373 983870 : if (DEBUGLEVEL) dbg_newrel(cache);
2374 : }
2375 2407166 : return k;
2376 : }
2377 :
2378 : /* m a t_INT or primitive t_COL */
2379 : static int
2380 2412711 : add_rel(RELCACHE_t *cache, FB_t *F, GEN R, long nz, GEN m, long in_rnd_rel)
2381 : {
2382 : REL_t *rel;
2383 : long k, l, reln;
2384 2412711 : const long lauts = lg(F->idealperm), KC = F->KC;
2385 :
2386 2412711 : k = add_rel_i(cache, R, nz, m, 0, 0, &rel, in_rnd_rel);
2387 2412778 : if (k > 0 && typ(m) != t_INT)
2388 : {
2389 527393 : GEN Rl = cgetg(KC+1, t_VECSMALL);
2390 527392 : reln = rel - cache->base;
2391 1004615 : for (l = 1; l < lauts; l++)
2392 : {
2393 477219 : GEN perml = gel(F->idealperm, l);
2394 477219 : long i, nzl = perml[nz];
2395 :
2396 20345985 : for (i = 1; i <= KC; i++) Rl[i] = 0;
2397 18153472 : for (i = nz; i <= KC; i++)
2398 17676253 : if (R[i])
2399 : {
2400 1270744 : long v = perml[i];
2401 :
2402 1270744 : if (v < nzl) nzl = v;
2403 1270744 : Rl[v] = R[i];
2404 : }
2405 477219 : (void)add_rel_i(cache, Rl, nzl, NULL, reln, l, NULL, in_rnd_rel);
2406 : }
2407 : }
2408 2412781 : return k;
2409 : }
2410 :
2411 : INLINE void
2412 28016740 : step(GEN x, double *y, GEN inc, long k)
2413 : {
2414 28016740 : if (!y[k])
2415 2199020 : x[k]++; /* leading coeff > 0 */
2416 : else
2417 : {
2418 25817720 : long i = inc[k];
2419 25817720 : x[k] += i;
2420 25817720 : inc[k] = (i > 0)? -1-i: 1-i;
2421 : }
2422 28016740 : }
2423 :
2424 : static double
2425 264009 : Fincke_Pohst_bound(double T, GEN r)
2426 : {
2427 264009 : pari_sp av = avma;
2428 264009 : GEN zT = dbltor(T * T), p = gmael(r,1,1), B = real_1(DEFAULTPREC);
2429 264000 : long i, n = lg(r)-1;
2430 : double g;
2431 606014 : for (i = 2; i <= n; i++)
2432 : {
2433 606010 : p = gmul(p, gmael(r,i,i));
2434 606022 : B = sqrtnr(gmul(zT,p), i);
2435 606004 : if (i == n || cmprr(B, gmael(r,i+1,i+1)) < 0) break;
2436 : }
2437 263995 : if (!gisdouble(B,&g)) return gc_double(av, 0.);
2438 264001 : return gc_double(av, rtodbl(B));
2439 : }
2440 :
2441 : static void
2442 1971620 : fact_update(GEN R, FB_t *F, long ipr, GEN c)
2443 : {
2444 1971620 : GEN pr = gel(F->LP,ipr), p = pr_get_p(pr);
2445 1971620 : long v = Z_lval(c, itou(p));
2446 1971623 : if (v) R[ipr] -= pr_get_e(pr) * v;
2447 1971623 : }
2448 :
2449 : static long
2450 264012 : Fincke_Pohst_ideal(RELCACHE_t *cache, FB_t *F, GEN nf, GEN I, GEN NI,
2451 : FACT *fact, long Nrelid, FP_t *fp, GEN rex, long jid, long jid0, long e0,
2452 : long *Nsmall, long *Nfact)
2453 : {
2454 : pari_sp av;
2455 264012 : GEN G = nf_get_G(nf), G0 = nf_get_roundG(nf), r, u, gx, cgx, inc, ideal;
2456 264010 : long prec = nf_get_prec(nf), N = nf_get_degree(nf);
2457 264008 : long j, k, skipfirst, relid = 0, try_factor = 0;
2458 264008 : long try_elt = 0, maxtry_ELEMENT = 4*maxtry_FACT*maxtry_FACT;
2459 : double BOUND, B1, B2;
2460 :
2461 264008 : inc = const_vecsmall(N, 1);
2462 264006 : u = ZM_lll(ZM_mul(G0, I), 0.99, LLL_IM);
2463 264012 : ideal = ZM_mul(I,u); /* approximate T2-LLL reduction */
2464 263981 : r = gaussred_from_QR(RgM_mul(G, ideal), prec); /* Cholesky for T2 | ideal */
2465 264012 : if (!r) pari_err_BUG("small_norm (precision too low)");
2466 :
2467 1172719 : for (k=1; k<=N; k++)
2468 : {
2469 908711 : if (!gisdouble(gcoeff(r,k,k),&(fp->v[k]))) return 0;
2470 2704122 : for (j=1; j<k; j++) if (!gisdouble(gcoeff(r,j,k),&(fp->q[j][k]))) return 0;
2471 908711 : if (DEBUGLEVEL>3) err_printf("v[%ld]=%.4g ",k,fp->v[k]);
2472 : }
2473 264008 : B1 = fp->v[1]; /* T2(ideal[1]) */
2474 264008 : B2 = fp->v[2] + B1 * fp->q[1][2] * fp->q[1][2]; /* T2(ideal[2]) */
2475 264008 : skipfirst = ZV_isscalar(gel(ideal,1));
2476 264009 : BOUND = maxdd(2*B2, Fincke_Pohst_bound(4 * maxtry_FACT / F->ballvol, r));
2477 264006 : if (DEBUGLEVEL>1)
2478 : {
2479 0 : if (DEBUGLEVEL>3) err_printf("\n");
2480 0 : err_printf("BOUND = %.4g\n",BOUND);
2481 : }
2482 :
2483 264006 : k = N; fp->y[N] = fp->z[N] = 0; fp->x[N] = 0;
2484 18458398 : for (av = avma;; set_avma(av), step(fp->x,fp->y,inc,k))
2485 18193906 : {
2486 : GEN R;
2487 : long nz;
2488 : do
2489 : { /* look for primitive element of small norm, cf minim00 */
2490 23290864 : int fl = 0;
2491 : double p;
2492 23290864 : if (k > 1)
2493 : {
2494 5097038 : long l = k-1;
2495 5097038 : fp->z[l] = 0;
2496 45467786 : for (j=k; j<=N; j++) fp->z[l] += fp->q[l][j]*fp->x[j];
2497 5097038 : p = (double)fp->x[k] + fp->z[k];
2498 5097038 : fp->y[l] = fp->y[k] + p*p*fp->v[k];
2499 5097038 : if (l <= skipfirst && !fp->y[1]) fl = 1;
2500 5097038 : fp->x[l] = (long)floor(-fp->z[l] + 0.5);
2501 5097038 : k = l;
2502 : }
2503 4457604 : for(;; step(fp->x,fp->y,inc,k))
2504 : {
2505 27748420 : if (!fl)
2506 : {
2507 27708927 : if (++try_elt > maxtry_ELEMENT) goto END_Fincke_Pohst_ideal;
2508 27706519 : p = (double)fp->x[k] + fp->z[k];
2509 27706519 : if (fp->y[k] + p*p*fp->v[k] <= BOUND) break;
2510 :
2511 5365055 : step(fp->x,fp->y,inc,k);
2512 :
2513 5365298 : p = (double)fp->x[k] + fp->z[k];
2514 5365298 : if (fp->y[k] + p*p*fp->v[k] <= BOUND) break;
2515 : }
2516 4460012 : fl = 0; inc[k] = 1;
2517 4460012 : if (++k > N) goto END_Fincke_Pohst_ideal;
2518 : }
2519 23288651 : } while (k > 1);
2520 :
2521 : /* element complete */
2522 33335397 : if (zv_content(fp->x) !=1) continue; /* not primitive */
2523 15427583 : gx = ZM_zc_mul(ideal,fp->x);
2524 15427544 : if (ZV_isscalar(gx)) continue;
2525 15550974 : if (++try_factor > maxtry_FACT) break;
2526 :
2527 15387676 : if (DEBUGLEVEL && Nsmall) (*Nsmall)++;
2528 15387676 : if (!factorgen(F,nf,I,NI,gx,fact)) continue;
2529 2337580 : if (!Nrelid) return 1;
2530 2239274 : if (jid == jid0)
2531 27261 : add_to_fact(jid, 1 + e0, fact);
2532 : else
2533 : {
2534 2212013 : add_to_fact(jid, 1, fact);
2535 2212229 : if (jid0) add_to_fact(jid0, e0, fact);
2536 : }
2537 :
2538 : /* smooth element */
2539 2239490 : R = set_fact(F, fact, rex, &nz);
2540 2239491 : cgx = Z_content(gx);
2541 2239443 : if (cgx)
2542 : { /* relatively rare, compute relation attached to gx/cgx */
2543 482409 : long i, n = fact[0].pr;
2544 482409 : gx = Q_div_to_int(gx, cgx);
2545 2404076 : for (i = 1; i <= n; i++) fact_update(R, F, fact[i].pr, cgx);
2546 482410 : if (rex)
2547 : {
2548 33070 : long l = lg(rex);
2549 108808 : for (i = 1; i < l; i++)
2550 75738 : if (rex[i])
2551 : {
2552 73818 : long t, ipr = F->subFB[i];
2553 236867 : for (t = 1; t <= n; t++)
2554 186915 : if (fact[t].pr == ipr) break;
2555 73818 : if (t > n) fact_update(R, F, ipr, cgx);
2556 : }
2557 : }
2558 : }
2559 2239444 : if (DEBUGLEVEL && Nfact) (*Nfact)++;
2560 : /* make sure we get maximal rank first, then allow all relations */
2561 2239444 : if (add_rel(cache, F, R, nz, gx, rex? 1: 0) <= 0)
2562 : { /* probably Q-dependent from previous ones: forget it */
2563 1712508 : if (DEBUGLEVEL>1) err_printf("*");
2564 1712516 : continue;
2565 : }
2566 527000 : if (cache->last >= cache->end) return 1; /* we have enough */
2567 428708 : if (++relid == Nrelid) break;
2568 : }
2569 165706 : END_Fincke_Pohst_ideal:
2570 165706 : return 0;
2571 : }
2572 :
2573 : static void
2574 66048 : small_norm(RELCACHE_t *cache, FB_t *F, GEN nf, long Nrelid, FACT *fact, long j0)
2575 : {
2576 66048 : const long N = nf_get_degree(nf);
2577 : FP_t fp;
2578 : pari_sp av;
2579 66048 : GEN L_jid = F->L_jid, Np0 = NULL, p0 = j0? gel(F->LP,j0): NULL;
2580 66048 : long Nsmall, Nfact, n = lg(L_jid), e0 = 0;
2581 : pari_timer T;
2582 :
2583 66048 : if (DEBUGLEVEL)
2584 : {
2585 0 : timer_start(&T);
2586 0 : err_printf("#### Look for %ld relations in %ld ideals (small_norm)\n",
2587 0 : cache->end - cache->last, lg(L_jid)-1);
2588 0 : if (p0) err_printf("Look in p0 = %Ps\n", vecslice(p0,1,4));
2589 : }
2590 66048 : Nsmall = Nfact = 0;
2591 66048 : minim_alloc(N+1, &fp.q, &fp.x, &fp.y, &fp.z, &fp.v);
2592 66048 : if (p0)
2593 : {
2594 26197 : GEN n = pr_norm(p0);
2595 26197 : e0 = logint0(sqri(pr_norm(veclast(F->LP))), n, NULL);
2596 26197 : p0 = idealpows(nf, p0, e0);
2597 26197 : Np0 = powiu(n,e0);
2598 : }
2599 163408 : for (av = avma; --n; set_avma(av))
2600 : {
2601 163012 : long j = L_jid[n];
2602 163012 : GEN id = gel(F->LP, j), Nid;
2603 163012 : if (DEBUGLEVEL>1)
2604 0 : err_printf("\n*** Ideal no %ld: %Ps\n", j, vecslice(id,1,4));
2605 163012 : if (p0)
2606 : {
2607 31887 : if (j == j0)
2608 : { /* avoid trivial relation */
2609 3685 : long e = pr_get_e(id);
2610 3685 : if ((e0 + 1) % e == 0 && e * pr_get_f(id) == N) continue;
2611 : }
2612 31206 : Nid = mulii(Np0, pr_norm(id)); id = idealmul(nf, p0, id);
2613 : }
2614 : else
2615 131125 : { Nid = pr_norm(id); id = pr_hnf(nf, id);}
2616 162331 : if (Fincke_Pohst_ideal(cache, F, nf, id, Nid, fact, Nrelid, &fp,
2617 65652 : NULL, j, j0, e0, &Nsmall, &Nfact)) break;
2618 : }
2619 66048 : if (DEBUGLEVEL && Nsmall)
2620 : {
2621 0 : if (DEBUGLEVEL == 1)
2622 0 : { if (Nfact) err_printf("\n"); }
2623 : else
2624 0 : err_printf(" \nnb. fact./nb. small norm = %ld/%ld = %.3f\n",
2625 0 : Nfact,Nsmall,((double)Nfact)/Nsmall);
2626 0 : if (timer_get(&T)>1) timer_printf(&T,"small_norm");
2627 : }
2628 66048 : }
2629 :
2630 : static GEN
2631 54883 : get_random_ideal(FB_t *F, GEN nf, GEN ex)
2632 : {
2633 54883 : long i, l = lg(ex);
2634 : for (;;)
2635 1054 : {
2636 55937 : GEN I = NULL;
2637 156278 : for (i = 1; i < l; i++)
2638 100340 : if ((ex[i] = random_bits(RANDOM_BITS)))
2639 : {
2640 96174 : GEN pr = gel(F->LP, F->subFB[i]), e = utoipos(ex[i]);
2641 96173 : I = I? idealmulpowprime(nf, I, pr, e): idealpow(nf, pr, e);
2642 : }
2643 55938 : if (I && !ZM_isscalar(I,NULL)) return I; /* != (n)Z_K */
2644 : }
2645 : }
2646 :
2647 : static void
2648 54882 : rnd_rel(RELCACHE_t *cache, FB_t *F, GEN nf, FACT *fact)
2649 : {
2650 : pari_timer T;
2651 54882 : GEN L_jid = F->L_jid, R, NR, ex;
2652 54882 : long i, l = lg(L_jid), Nfact = 0;
2653 : FP_t fp;
2654 : pari_sp av;
2655 :
2656 54882 : if (DEBUGLEVEL) {
2657 0 : timer_start(&T);
2658 0 : err_printf("#### Look for %ld relations in %ld ideals (rnd_rel)\n",
2659 0 : cache->end - cache->last, l-1);
2660 : }
2661 54882 : ex = cgetg(lg(F->subFB), t_VECSMALL);
2662 54883 : R = get_random_ideal(F, nf, ex); /* random product from subFB */
2663 54883 : NR = ZM_det_triangular(R);
2664 54883 : minim_alloc(nf_get_degree(nf)+1, &fp.q, &fp.x, &fp.y, &fp.z, &fp.v);
2665 123910 : for (av = avma, i = 1; i < l; i++, set_avma(av))
2666 : { /* try P[j] * base */
2667 101668 : long j = L_jid[i];
2668 101668 : GEN P = gel(F->LP, j), Nid = mulii(NR, pr_norm(P));
2669 101659 : if (DEBUGLEVEL>1) err_printf("\n*** Ideal %ld: %Ps\n", j, vecslice(P,1,4));
2670 101659 : if (Fincke_Pohst_ideal(cache, F, nf, idealHNF_mul(nf, R, P), Nid, fact,
2671 32640 : RND_REL_RELPID, &fp, ex, j, 0, 0, NULL, &Nfact)) break;
2672 : }
2673 54882 : if (DEBUGLEVEL)
2674 : {
2675 0 : if (Nfact) err_printf("\n");
2676 0 : if (timer_get(&T)>=0) timer_printf(&T,"rnd_rel");
2677 : }
2678 54882 : }
2679 :
2680 : static GEN
2681 63789 : automorphism_perms(GEN M, GEN auts, GEN cyclic, long r1, long r2, long N)
2682 : {
2683 63789 : long L = lgcols(M), lauts = lg(auts), lcyc = lg(cyclic), i, j, l, m;
2684 63789 : GEN Mt, perms = cgetg(lauts, t_VEC);
2685 : pari_sp av;
2686 :
2687 128013 : for (l = 1; l < lauts; l++) gel(perms, l) = cgetg(L, t_VECSMALL);
2688 63789 : av = avma;
2689 63789 : Mt = shallowtrans(gprec_w(M, LOWDEFAULTPREC));
2690 63790 : Mt = shallowconcat(Mt, conj_i(vecslice(Mt, r1+1, r1+r2)));
2691 110634 : for (l = 1; l < lcyc; l++)
2692 : {
2693 46844 : GEN thiscyc = gel(cyclic, l), thisperm, perm, prev, Nt;
2694 46844 : long k = thiscyc[1];
2695 :
2696 46844 : Nt = RgM_mul(shallowtrans(gel(auts, k)), Mt);
2697 46845 : perm = gel(perms, k);
2698 154081 : for (i = 1; i < L; i++)
2699 : {
2700 107238 : GEN v = gel(Nt, i), minD;
2701 107238 : minD = gnorml2(gsub(v, gel(Mt, 1)));
2702 107236 : perm[i] = 1;
2703 566871 : for (j = 2; j <= N; j++)
2704 : {
2705 459635 : GEN D = gnorml2(gsub(v, gel(Mt, j)));
2706 459630 : if (gcmp(D, minD) < 0) { minD = D; perm[i] = j >= L ? r2-j : j; }
2707 : }
2708 : }
2709 65456 : for (prev = perm, m = 2; m < lg(thiscyc); m++, prev = thisperm)
2710 : {
2711 18613 : thisperm = gel(perms, thiscyc[m]);
2712 94360 : for (i = 1; i < L; i++)
2713 : {
2714 75747 : long pp = labs(prev[i]);
2715 75747 : thisperm[i] = prev[i] < 0 ? -perm[pp] : perm[pp];
2716 : }
2717 : }
2718 : }
2719 63790 : set_avma(av); return perms;
2720 : }
2721 :
2722 : /* Determine the field automorphisms as matrices on the integral basis */
2723 : static GEN
2724 63852 : automorphism_matrices(GEN nf, GEN *cycp)
2725 : {
2726 63852 : pari_sp av = avma;
2727 63852 : GEN auts = galoisconj(nf, NULL), mats, cyclic, cyclicidx;
2728 63852 : long nauts = lg(auts)-1, i, j, k, l;
2729 :
2730 63852 : cyclic = cgetg(nauts+1, t_VEC);
2731 63852 : cyclicidx = zero_Flv(nauts);
2732 98067 : for (l = 1; l <= nauts; l++)
2733 : {
2734 98067 : GEN aut = gel(auts, l);
2735 98067 : if (gequalX(aut)) { swap(gel(auts, l), gel(auts, nauts)); break; }
2736 : }
2737 : /* trivial automorphism is last */
2738 191956 : for (l = 1; l <= nauts; l++) gel(auts, l) = algtobasis(nf, gel(auts, l));
2739 : /* Compute maximal cyclic subgroups */
2740 128102 : for (l = nauts; --l > 0; ) if (!cyclicidx[l])
2741 : {
2742 48349 : GEN elt = gel(auts, l), aut = elt, cyc = cgetg(nauts+1, t_VECSMALL);
2743 48348 : cyc[1] = cyclicidx[l] = l; j = 1;
2744 : do
2745 : {
2746 67503 : elt = galoisapply(nf, elt, aut);
2747 219182 : for (k = 1; k <= nauts; k++) if (gequal(elt, gel(auts, k))) break;
2748 67502 : cyclicidx[k] = l; cyc[++j] = k;
2749 : }
2750 67502 : while (k != nauts);
2751 48347 : setlg(cyc, j);
2752 48347 : gel(cyclic, l) = cyc;
2753 : }
2754 128098 : for (i = j = 1; i < nauts; i++)
2755 64247 : if (cyclicidx[i] == i) cyclic[j++] = cyclic[i];
2756 63851 : setlg(cyclic, j);
2757 63852 : mats = cgetg(nauts, t_VEC);
2758 110722 : while (--j > 0)
2759 : {
2760 46870 : GEN cyc = gel(cyclic, j);
2761 46870 : long id = cyc[1];
2762 46870 : GEN M, Mi, aut = gel(auts, id);
2763 :
2764 46870 : gel(mats, id) = Mi = M = nfgaloismatrix(nf, aut);
2765 65482 : for (i = 2; i < lg(cyc); i++) gel(mats, cyc[i]) = Mi = ZM_mul(Mi, M);
2766 : }
2767 63852 : gerepileall(av, 2, &mats, &cyclic);
2768 63853 : if (cycp) *cycp = cyclic;
2769 63853 : return mats;
2770 : }
2771 :
2772 : /* vP a list of maximal ideals above the same p from idealprimedec: f(P/p) is
2773 : * increasing; 1 <= j <= #vP; orbit a zc of length <= #vP; auts a vector of
2774 : * automorphisms in ZM form.
2775 : * Set orbit[i] = 1 for all vP[i], i >= j, in the orbit of pr = vP[j] wrt auts.
2776 : * N.B.1 orbit need not be initialized to 0: useful to incrementally run
2777 : * through successive orbits
2778 : * N.B.2 i >= j, so primes with index < j will be missed; run incrementally
2779 : * starting from j = 1 ! */
2780 : static void
2781 11865 : pr_orbit_fill(GEN orbit, GEN auts, GEN vP, long j)
2782 : {
2783 11865 : GEN pr = gel(vP,j), gen = pr_get_gen(pr);
2784 11865 : long i, l = lg(auts), J = lg(orbit), f = pr_get_f(pr);
2785 11865 : orbit[j] = 1;
2786 23730 : for (i = 1; i < l; i++)
2787 : {
2788 11865 : GEN g = ZM_ZC_mul(gel(auts,i), gen);
2789 : long k;
2790 11886 : for (k = j+1; k < J; k++)
2791 : {
2792 35 : GEN prk = gel(vP,k);
2793 35 : if (pr_get_f(prk) > f) break; /* f(P[k]) increases with k */
2794 : /* don't check that e matches: (almost) always 1 ! */
2795 35 : if (!orbit[k] && ZC_prdvd(g, prk)) { orbit[k] = 1; break; }
2796 : }
2797 : }
2798 11865 : }
2799 : /* remark: F->KCZ changes if be_honest() fails */
2800 : static int
2801 7 : be_honest(FB_t *F, GEN nf, GEN auts, FACT *fact)
2802 : {
2803 : long i, iz, nbtest;
2804 7 : long lgsub = lg(F->subFB), KCZ0 = F->KCZ, N = nf_get_degree(nf);
2805 : FP_t fp;
2806 : pari_sp av;
2807 :
2808 7 : if (DEBUGLEVEL) {
2809 0 : err_printf("Be honest for %ld primes from %ld to %ld\n", F->KCZ2 - F->KCZ,
2810 0 : F->FB[ F->KCZ+1 ], F->FB[ F->KCZ2 ]);
2811 : }
2812 7 : minim_alloc(N+1, &fp.q, &fp.x, &fp.y, &fp.z, &fp.v);
2813 7 : if (lg(auts) == 1) auts = NULL;
2814 7 : av = avma;
2815 14 : for (iz=F->KCZ+1; iz<=F->KCZ2; iz++, set_avma(av))
2816 : {
2817 7 : long p = F->FB[iz];
2818 7 : GEN pr_orbit, P = gel(F->LV,p);
2819 7 : long j, J = lg(P); /* > 1 */
2820 : /* the P|p, NP > C2 are assumed in subgroup generated by FB + last P
2821 : * with NP <= C2 is unramified --> check all but last */
2822 7 : if (pr_get_e(gel(P,J-1)) == 1) J--;
2823 7 : if (J == 1) continue;
2824 7 : if (DEBUGLEVEL>1) err_printf("%ld ", p);
2825 7 : pr_orbit = auts? zero_zv(J-1): NULL;
2826 28 : for (j = 1; j < J; j++)
2827 : {
2828 : GEN Nid, id, id0;
2829 21 : if (pr_orbit)
2830 : {
2831 21 : if (pr_orbit[j]) continue;
2832 : /* discard all primes in automorphism orbit simultaneously */
2833 14 : pr_orbit_fill(pr_orbit, auts, P, j);
2834 : }
2835 14 : id = id0 = pr_hnf(nf,gel(P,j));
2836 14 : Nid = pr_norm(gel(P,j));
2837 14 : for (nbtest=0;;)
2838 : {
2839 14 : if (Fincke_Pohst_ideal(NULL, F, nf, id, Nid, fact, 0, &fp,
2840 14 : NULL, 0, 0, 0, NULL, NULL)) break;
2841 0 : if (++nbtest > maxtry_HONEST)
2842 : {
2843 0 : if (DEBUGLEVEL)
2844 0 : pari_warn(warner,"be_honest() failure on prime %Ps\n", gel(P,j));
2845 0 : return 0;
2846 : }
2847 : /* occurs at most once in the whole function */
2848 0 : for (i = 1, id = id0; i < lgsub; i++)
2849 : {
2850 0 : long ex = random_bits(RANDOM_BITS);
2851 0 : if (ex)
2852 : {
2853 0 : GEN pr = gel(F->LP, F->subFB[i]);
2854 0 : id = idealmulpowprime(nf, id, pr, utoipos(ex));
2855 : }
2856 : }
2857 0 : if (!equali1(gcoeff(id,N,N))) id = Q_primpart(id);
2858 0 : if (expi(gcoeff(id,1,1)) > 100) id = idealred(nf, id);
2859 0 : Nid = ZM_det_triangular(id);
2860 : }
2861 : }
2862 7 : F->KCZ++; /* SUCCESS, "enlarge" factorbase */
2863 : }
2864 7 : F->KCZ = KCZ0; return gc_bool(av,1);
2865 : }
2866 :
2867 : /* all primes with N(P) <= BOUND factor on factorbase ? */
2868 : void
2869 63 : bnftestprimes(GEN bnf, GEN BOUND)
2870 : {
2871 63 : pari_sp av0 = avma, av;
2872 63 : ulong count = 0;
2873 63 : GEN auts, p, nf = bnf_get_nf(bnf), Vbase = bnf_get_vbase(bnf);
2874 63 : GEN fb = gen_sort_shallow(Vbase, (void*)&cmp_prime_ideal, cmp_nodata);
2875 63 : ulong pmax = pr_get_smallp(veclast(fb)); /*largest p in factorbase*/
2876 : forprime_t S;
2877 : FACT *fact;
2878 : FB_t F;
2879 :
2880 63 : (void)recover_partFB(&F, Vbase, nf_get_degree(nf));
2881 63 : fact = (FACT*)stack_malloc((F.KC+1)*sizeof(FACT));
2882 63 : forprime_init(&S, gen_2, BOUND);
2883 63 : auts = automorphism_matrices(nf, NULL);
2884 63 : if (lg(auts) == 1) auts = NULL;
2885 63 : av = avma;
2886 37604 : while (( p = forprime_next(&S) ))
2887 : {
2888 : GEN pr_orbit, vP;
2889 : long j, J;
2890 :
2891 37541 : if (DEBUGLEVEL == 1 && ++count > 1000)
2892 : {
2893 0 : err_printf("passing p = %Ps / %Ps\n", p, BOUND);
2894 0 : count = 0;
2895 : }
2896 37541 : set_avma(av);
2897 37541 : vP = idealprimedec_limit_norm(nf, p, BOUND);
2898 37541 : J = lg(vP);
2899 : /* if last is unramified, all P|p in subgroup generated by FB: skip last */
2900 37541 : if (J > 1 && pr_get_e(gel(vP,J-1)) == 1) J--;
2901 37541 : if (J == 1) continue;
2902 14525 : if (DEBUGLEVEL>1) err_printf("*** p = %Ps\n",p);
2903 14525 : pr_orbit = auts? zero_zv(J-1): NULL;
2904 31549 : for (j = 1; j < J; j++)
2905 : {
2906 17024 : GEN P = gel(vP,j);
2907 17024 : long k = 0;
2908 17024 : if (pr_orbit)
2909 : {
2910 11858 : if (pr_orbit[j]) continue;
2911 : /* discard all primes in automorphism orbit simultaneously */
2912 11851 : pr_orbit_fill(pr_orbit, auts, vP, j);
2913 : }
2914 17017 : if (abscmpiu(p, pmax) > 0 || !(k = tablesearch(fb, P, &cmp_prime_ideal)))
2915 16408 : (void)SPLIT(&F, nf, pr_hnf(nf,P), Vbase, fact);
2916 17017 : if (DEBUGLEVEL>1)
2917 : {
2918 0 : err_printf(" Testing P = %Ps\n",P);
2919 0 : if (k) err_printf(" #%ld in factor base\n",k);
2920 0 : else err_printf(" is %Ps\n", isprincipal(bnf,P));
2921 : }
2922 : }
2923 : }
2924 63 : set_avma(av0);
2925 63 : }
2926 :
2927 : /* A t_MAT of complex floats, in fact reals. Extract a submatrix B
2928 : * whose columns are definitely nonzero, i.e. gexpo(A[j]) >= -2
2929 : *
2930 : * If possible precision problem (t_REAL 0 with large exponent), set
2931 : * *precpb to 1 */
2932 : static GEN
2933 91654 : clean_cols(GEN A, int *precpb)
2934 : {
2935 91654 : long l = lg(A), h, i, j, k;
2936 : GEN B;
2937 91654 : *precpb = 0;
2938 91654 : if (l == 1) return A;
2939 91654 : h = lgcols(A);;
2940 91654 : B = cgetg(l, t_MAT);
2941 1068162 : for (i = k = 1; i < l; i++)
2942 : {
2943 976508 : GEN Ai = gel(A,i);
2944 976508 : int non0 = 0;
2945 4246351 : for (j = 1; j < h; j++)
2946 : {
2947 3269843 : GEN c = gel(Ai,j);
2948 3269843 : if (gexpo(c) >= -2)
2949 : {
2950 1959261 : if (gequal0(c)) *precpb = 1; else non0 = 1;
2951 : }
2952 : }
2953 976508 : if (non0) gel(B, k++) = Ai;
2954 : }
2955 91654 : setlg(B, k); return B;
2956 : }
2957 :
2958 : static long
2959 557795 : compute_multiple_of_R_pivot(GEN X, GEN x0/*unused*/, long ix, GEN c)
2960 : {
2961 557795 : GEN x = gel(X,ix);
2962 557795 : long i, k = 0, ex = - (long)HIGHEXPOBIT, lx = lg(x);
2963 : (void)x0;
2964 2753901 : for (i=1; i<lx; i++)
2965 2196106 : if (!c[i] && !gequal0(gel(x,i)))
2966 : {
2967 703180 : long e = gexpo(gel(x,i));
2968 703181 : if (e > ex) { ex = e; k = i; }
2969 : }
2970 557795 : return (k && ex > -32)? k: lx;
2971 : }
2972 :
2973 : /* Ar = (log |sigma_i(u_j)|) for units (u_j) found so far;
2974 : * RU = R1+R2 = target rank for unit matrix, after adding [1 x r1, 2 x r2];
2975 : * N = field degree, need = unit rank defect;
2976 : * L = NULL (prec problem) or B^(-1) * A with approximate rational entries
2977 : * (as t_REAL), B a submatrix of A, with (probably) maximal rank RU */
2978 : static GEN
2979 120270 : compute_multiple_of_R(GEN Ar, long RU, long N, long *pneed, long *bit, GEN *ptL)
2980 : {
2981 : GEN T, d, mdet, Im_mdet, kR, L;
2982 120270 : long i, j, r, R1 = 2*RU - N;
2983 : int precpb;
2984 120270 : pari_sp av = avma;
2985 :
2986 120270 : if (RU == 1) { *ptL = zeromat(0, lg(Ar)-1); return gen_1; }
2987 :
2988 91654 : if (DEBUGLEVEL) err_printf("\n#### Computing regulator multiple\n");
2989 91654 : mdet = clean_cols(Ar, &precpb);
2990 : /* will cause precision to increase on later failure, but we may succeed! */
2991 91654 : *ptL = precpb? NULL: gen_1;
2992 91654 : T = cgetg(RU+1,t_COL);
2993 228054 : for (i=1; i<=R1; i++) gel(T,i) = gen_1;
2994 192150 : for ( ; i<=RU; i++) gel(T,i) = gen_2;
2995 91654 : mdet = shallowconcat(T, mdet); /* det(Span(mdet)) = N * R */
2996 :
2997 : /* could be using indexrank(), but need custom "get_pivot" function */
2998 91654 : d = RgM_pivots(mdet, NULL, &r, &compute_multiple_of_R_pivot);
2999 : /* # of independent columns = target rank ? */
3000 91653 : if (lg(mdet)-1 - r != RU)
3001 : {
3002 24574 : if (DEBUGLEVEL)
3003 0 : err_printf("Units matrix target rank = %ld < %ld\n",lg(mdet)-1 - r, RU);
3004 24574 : *pneed = RU - (lg(mdet)-1-r); return gc_NULL(av);
3005 : }
3006 :
3007 67079 : Im_mdet = cgetg(RU+1, t_MAT); /* extract independent columns */
3008 : /* N.B: d[1] = 1, corresponding to T above */
3009 67078 : gel(Im_mdet, 1) = T;
3010 252560 : for (i = j = 2; i <= RU; j++)
3011 185482 : if (d[j]) gel(Im_mdet, i++) = gel(mdet,j);
3012 :
3013 : /* integral multiple of R: the cols we picked form a Q-basis, they have an
3014 : * index in the full lattice. First column is T */
3015 67078 : kR = divru(det2(Im_mdet), N);
3016 : /* R > 0.2 uniformly */
3017 67077 : if (!signe(kR) || expo(kR) < -3)
3018 : {
3019 0 : if (DEBUGLEVEL) err_printf("Regulator is zero.\n");
3020 0 : *pneed = 0; return gc_NULL(av);
3021 : }
3022 67077 : d = det2(rowslice(vecslice(Im_mdet, 2, RU), 2, RU));
3023 67079 : setabssign(d); setabssign(kR);
3024 67079 : if (gexpo(gsub(d,kR)) - gexpo(d) > -20) { *ptL = NULL; return gc_NULL(av); }
3025 67075 : L = RgM_inv(Im_mdet);
3026 : /* estimate # of correct bits in result */
3027 67078 : if (!L || (*bit = -gexpo(RgM_Rg_sub_shallow(RgM_mul(L,Im_mdet), gen_1))) < 16)
3028 10 : { *ptL = NULL; return gc_NULL(av); }
3029 :
3030 67067 : *ptL = RgM_mul(rowslice(L,2,RU), Ar); /* approximate rational entries */
3031 67068 : return gc_all(av,2, &kR, ptL);
3032 : }
3033 :
3034 : /* leave small integer n as is, convert huge n to t_REAL (for readability) */
3035 : static GEN
3036 0 : i2print(GEN n)
3037 0 : { return lgefint(n) <= DEFAULTPREC? n: itor(n,LOWDEFAULTPREC); }
3038 :
3039 : static long
3040 95542 : bad_check(GEN c)
3041 : {
3042 95542 : long ec = gexpo(c);
3043 95542 : if (DEBUGLEVEL) err_printf("\n ***** check = %.28Pg\n",c);
3044 : /* safe check for c < 0.75 : avoid underflow in gtodouble() */
3045 95543 : if (ec < -1 || (ec == -1 && gtodouble(c) < 0.75)) return fupb_PRECI;
3046 : /* safe check for c > 1.3 : avoid overflow */
3047 95543 : if (ec > 0 || (ec == 0 && gtodouble(c) > 1.3)) return fupb_RELAT;
3048 63798 : return fupb_NONE;
3049 : }
3050 : /* Input:
3051 : * lambda = approximate rational entries: coords of units found so far on a
3052 : * sublattice of maximal rank (sublambda)
3053 : * *ptkR = regulator of sublambda = multiple of regulator of lambda
3054 : * Compute R = true regulator of lambda.
3055 : *
3056 : * If c := Rz ~ 1, by Dirichlet's formula, then lambda is the full group of
3057 : * units AND the full set of relations for the class group has been computed.
3058 : * In fact z is a very rough approximation and we only expect 0.75 < Rz < 1.3
3059 : *
3060 : * Output: *ptkR = R, *ptL = numerator(units) (in terms of lambda) */
3061 : static long
3062 95604 : compute_R(GEN lambda, GEN z, GEN *ptL, GEN *ptkR)
3063 : {
3064 95604 : pari_sp av = avma;
3065 95604 : long bit, r, reason, RU = lg(lambda) == 1? 1: lgcols(lambda);
3066 : GEN L, H, D, den, R, c;
3067 :
3068 95604 : *ptL = NULL;
3069 95604 : if (RU == 1) { *ptkR = gen_1; *ptL = lambda; return bad_check(z); }
3070 66988 : D = gmul2n(mpmul(*ptkR,z), 1); /* bound for denom(lambda) */
3071 66987 : if (expo(D) < 0 && rtodbl(D) < 0.95) return fupb_PRECI;
3072 66987 : L = bestappr(lambda,D);
3073 66988 : if (lg(L) == 1)
3074 : {
3075 0 : if (DEBUGLEVEL) err_printf("truncation error in bestappr\n");
3076 0 : return fupb_PRECI;
3077 : }
3078 66988 : den = Q_denom(L);
3079 66989 : if (mpcmp(den,D) > 0)
3080 : {
3081 21 : if (DEBUGLEVEL) err_printf("D = %Ps\nden = %Ps\n",D, i2print(den));
3082 21 : return fupb_PRECI;
3083 : }
3084 66966 : bit = -gexpo(gsub(L, lambda)); /* input accuracy */
3085 66968 : L = Q_muli_to_int(L, den);
3086 66964 : if (gexpo(L) + expi(den) > bit - 32)
3087 : {
3088 41 : if (DEBUGLEVEL) err_printf("dubious bestappr; den = %Ps\n", i2print(den));
3089 41 : return fupb_PRECI;
3090 : }
3091 66926 : H = ZM_hnf(L); r = lg(H)-1;
3092 66927 : if (!r || r != nbrows(H))
3093 0 : R = gen_0; /* wrong rank */
3094 : else
3095 66927 : R = gmul(*ptkR, gdiv(ZM_det_triangular(H), powiu(den, r)));
3096 : /* R = tentative regulator; regulator > 0.2 uniformly */
3097 66926 : if (gexpo(R) < -3) {
3098 0 : if (DEBUGLEVEL) err_printf("\n#### Tentative regulator: %.28Pg\n", R);
3099 0 : return gc_long(av, fupb_PRECI);
3100 : }
3101 66925 : c = gmul(R,z); /* should be n (= 1 if we are done) */
3102 66926 : if (DEBUGLEVEL) err_printf("\n#### Tentative regulator: %.28Pg\n", R);
3103 66926 : if ((reason = bad_check(c))) return gc_long(av, reason);
3104 48573 : *ptkR = R; *ptL = L; return fupb_NONE;
3105 : }
3106 : static GEN
3107 63887 : get_clg2(GEN cyc, GEN Ga, GEN C, GEN Ur, GEN Ge, GEN M1, GEN M2)
3108 : {
3109 63887 : GEN GD = gsub(act_arch(M1, C), diagact_arch(cyc, Ga));
3110 63885 : GEN ga = gsub(act_arch(M2, C), act_arch(Ur, Ga));
3111 63887 : return mkvecn(6, Ur, ga, GD, Ge, M1, M2);
3112 : }
3113 : /* compute class group (clg1) + data for isprincipal (clg2) */
3114 : static GEN
3115 63791 : class_group_gen(GEN nf,GEN W,GEN C,GEN Vbase,long prec, GEN *pclg2)
3116 : {
3117 : GEN M1, M2, z, G, Ga, Ge, cyc, X, Y, D, U, V, Ur, Ui, Uir;
3118 : long j, l;
3119 :
3120 63791 : D = ZM_snfall(W,&U,&V); /* UWV=D, D diagonal, G = g Ui (G=new gens, g=old) */
3121 63790 : Ui = ZM_inv(U, NULL);
3122 63790 : l = lg(D); cyc = cgetg(l, t_VEC); /* elementary divisors */
3123 92602 : for (j = 1; j < l; j++)
3124 : {
3125 30409 : gel(cyc,j) = gcoeff(D,j,j); /* strip useless components */
3126 30409 : if (is_pm1(gel(cyc,j))) break;
3127 : }
3128 63790 : l = j;
3129 63790 : Ur = ZM_hnfdivrem(U, D, &Y);
3130 63790 : Uir = ZM_hnfdivrem(Ui,W, &X);
3131 : /* {x} = logarithmic embedding of x (arch. component)
3132 : * NB: [J,z] = idealred(I) --> I = y J, with {y} = - z
3133 : * G = g Uir - {Ga}, Uir = Ui + WX
3134 : * g = G Ur - {ga}, Ur = U + DY */
3135 63791 : G = cgetg(l,t_VEC);
3136 63791 : Ga= cgetg(l,t_MAT);
3137 63791 : Ge= cgetg(l,t_COL);
3138 63791 : z = init_famat(NULL);
3139 92603 : for (j = 1; j < l; j++)
3140 : {
3141 28812 : GEN I = genback(z, nf, Vbase, gel(Uir,j));
3142 28812 : gel(G,j) = gel(I,1); /* generator, order cyc[j] */
3143 28812 : gel(Ge,j)= gel(I,2);
3144 28812 : gel(Ga,j)= nf_cxlog(nf, gel(I,2), prec);
3145 28812 : if (!gel(Ga,j)) pari_err_PREC("class_group_gen");
3146 : }
3147 : /* {ga} = {GD}Y + G U - g = {GD}Y - {Ga} U + gW X U
3148 : = gW (X Ur + V Y) - {Ga}Ur */
3149 63791 : M2 = ZM_add(ZM_mul(X,Ur), ZM_mul(V,Y));
3150 63790 : setlg(cyc,l); setlg(V,l); setlg(D,l);
3151 : /* G D =: {GD} = g (Ui + W X) D - {Ga}D = g W (V + X D) - {Ga}D
3152 : * NB: Ui D = W V. gW is given by (first l-1 cols of) C */
3153 63790 : M1 = ZM_add(V, ZM_mul(X,D));
3154 63790 : *pclg2 = get_clg2(cyc, Ga, C, Ur, Ge, M1, M2);
3155 63790 : return mkvec3(ZV_prod(cyc), cyc, G);
3156 : }
3157 :
3158 : /* compute principal ideals corresponding to (gen[i]^cyc[i]) */
3159 : static GEN
3160 4956 : makecycgen(GEN bnf)
3161 : {
3162 4956 : GEN cyc = bnf_get_cyc(bnf), gen = bnf_get_gen(bnf), nf = bnf_get_nf(bnf);
3163 4956 : GEN h, y, GD = bnf_get_GD(bnf), W = bnf_get_W(bnf); /* HNF */
3164 4956 : GEN Sunits = bnf_get_sunits(bnf);
3165 4956 : GEN X = Sunits? gel(Sunits,1): NULL, C = Sunits? gel(Sunits,3): NULL;
3166 : long e, i, l;
3167 :
3168 4956 : if (DEBUGLEVEL) pari_warn(warner,"completing bnf (building cycgen)");
3169 4956 : h = cgetg_copy(gen, &l);
3170 11613 : for (i = 1; i < l; i++)
3171 : {
3172 6657 : GEN gi = gel(gen,i), ci = gel(cyc,i);
3173 6657 : if (X && equalii(ci, gcoeff(W,i,i)))
3174 : {
3175 : long j;
3176 8590 : for (j = i+1; j < l; j++)
3177 3213 : if (signe(gcoeff(W,i,j))) break;
3178 5551 : if (j == i) { gel(h,i) = mkmat2(X, gel(C,i)); continue; }
3179 : }
3180 6657 : if (abscmpiu(ci, 5) < 0)
3181 : {
3182 5544 : GEN N = ZM_det_triangular(gi);
3183 5544 : y = isprincipalarch(bnf,gel(GD,i), N, ci, gen_1, &e);
3184 5544 : if (y && fact_ok(nf,y,NULL,mkvec(gi),mkvec(ci)))
3185 : {
3186 4556 : gel(h,i) = to_famat_shallow(y,gen_1);
3187 4556 : continue;
3188 : }
3189 : }
3190 2101 : y = isprincipalfact(bnf, NULL, mkvec(gi), mkvec(ci), nf_GENMAT|nf_FORCE);
3191 2101 : gel(h,i) = gel(y,2);
3192 : }
3193 4956 : return h;
3194 : }
3195 :
3196 : static GEN
3197 69 : get_y(GEN bnf, GEN W, GEN B, GEN C, GEN pFB, long j)
3198 : {
3199 69 : GEN y, nf = bnf_get_nf(bnf);
3200 69 : long e, lW = lg(W)-1;
3201 69 : GEN ex = (j<=lW)? gel(W,j): gel(B,j-lW);
3202 69 : GEN P = (j<=lW)? NULL: gel(pFB,j);
3203 69 : if (C)
3204 : { /* archimedean embeddings known: cheap trial */
3205 69 : GEN Nx = get_norm_fact_primes(pFB, ex, P);
3206 69 : y = isprincipalarch(bnf,gel(C,j), Nx,gen_1, gen_1, &e);
3207 69 : if (y && fact_ok(nf,y,P,pFB,ex)) return y;
3208 : }
3209 0 : y = isprincipalfact_or_fail(bnf, P, pFB, ex);
3210 0 : return typ(y) == t_INT? y: gel(y,2);
3211 : }
3212 : /* compute principal ideals corresponding to bnf relations */
3213 : static GEN
3214 20 : makematal(GEN bnf)
3215 : {
3216 20 : GEN W = bnf_get_W(bnf), B = bnf_get_B(bnf), C = bnf_get_C(bnf);
3217 : GEN pFB, ma, retry;
3218 20 : long lma, j, prec = 0;
3219 :
3220 20 : if (DEBUGLEVEL) pari_warn(warner,"completing bnf (building matal)");
3221 20 : lma=lg(W)+lg(B)-1;
3222 20 : pFB = bnf_get_vbase(bnf);
3223 20 : ma = cgetg(lma,t_VEC);
3224 20 : retry = vecsmalltrunc_init(lma);
3225 89 : for (j=lma-1; j>0; j--)
3226 : {
3227 69 : pari_sp av = avma;
3228 69 : GEN y = get_y(bnf, W, B, C, pFB, j);
3229 69 : if (typ(y) == t_INT)
3230 : {
3231 0 : long E = itos(y);
3232 0 : if (DEBUGLEVEL>1) err_printf("\n%ld done later at prec %ld\n",j,E);
3233 0 : set_avma(av);
3234 0 : vecsmalltrunc_append(retry, j);
3235 0 : if (E > prec) prec = E;
3236 : }
3237 : else
3238 : {
3239 69 : if (DEBUGLEVEL>1) err_printf("%ld ",j);
3240 69 : gel(ma,j) = gerepileupto(av,y);
3241 : }
3242 : }
3243 20 : if (prec)
3244 : {
3245 0 : long k, l = lg(retry);
3246 0 : GEN y, nf = bnf_get_nf(bnf);
3247 0 : if (DEBUGLEVEL) pari_warn(warnprec,"makematal",prec);
3248 0 : nf = nfnewprec_shallow(nf,prec);
3249 0 : bnf = Buchall(nf, nf_FORCE, prec);
3250 0 : if (DEBUGLEVEL) err_printf("makematal, adding missing entries:");
3251 0 : for (k=1; k<l; k++)
3252 : {
3253 0 : pari_sp av = avma;
3254 0 : long j = retry[k];
3255 0 : y = get_y(bnf,W,B,NULL, pFB, j);
3256 0 : if (typ(y) == t_INT) pari_err_PREC("makematal");
3257 0 : if (DEBUGLEVEL>1) err_printf("%ld ",j);
3258 0 : gel(ma,j) = gerepileupto(av,y);
3259 : }
3260 : }
3261 20 : if (DEBUGLEVEL>1) err_printf("\n");
3262 20 : return ma;
3263 : }
3264 :
3265 : enum { MATAL = 1, CYCGEN, UNITS };
3266 : GEN
3267 26726 : bnf_build_cycgen(GEN bnf)
3268 26726 : { return obj_checkbuild(bnf, CYCGEN, &makecycgen); }
3269 : GEN
3270 20 : bnf_build_matalpha(GEN bnf)
3271 20 : { return obj_checkbuild(bnf, MATAL, &makematal); }
3272 : GEN
3273 32055 : bnf_build_units(GEN bnf)
3274 32055 : { return obj_checkbuild(bnf, UNITS, &makeunits); }
3275 :
3276 : /* return fu in compact form if available; in terms of a fixed basis
3277 : * of S-units */
3278 : GEN
3279 70 : bnf_compactfu_mat(GEN bnf)
3280 : {
3281 70 : GEN X, U, SUnits = bnf_get_sunits(bnf);
3282 70 : if (!SUnits) return NULL;
3283 70 : X = gel(SUnits,1);
3284 70 : U = gel(SUnits,2); ZM_remove_unused(&U, &X);
3285 70 : return mkvec2(X, U);
3286 : }
3287 : /* return fu in compact form if available; individually as famat */
3288 : GEN
3289 37191 : bnf_compactfu(GEN bnf)
3290 : {
3291 37191 : GEN fu, X, U, SUnits = bnf_get_sunits(bnf);
3292 : long i, l;
3293 37191 : if (!SUnits) return NULL;
3294 36960 : X = gel(SUnits,1);
3295 36960 : U = gel(SUnits,2); l = lg(U); fu = cgetg(l, t_VEC);
3296 60333 : for (i = 1; i < l; i++)
3297 23373 : gel(fu,i) = famat_remove_trivial(mkmat2(X, gel(U,i)));
3298 36960 : return fu;
3299 : }
3300 : /* return expanded fu if available */
3301 : GEN
3302 263983 : bnf_has_fu(GEN bnf)
3303 : {
3304 263983 : GEN fu = obj_check(bnf, UNITS);
3305 263981 : if (fu) return vecsplice(fu, 1);
3306 263186 : fu = bnf_get_fu_nocheck(bnf);
3307 263186 : return (typ(fu) == t_MAT)? NULL: fu;
3308 : }
3309 : /* return expanded fu if available; build if cheap */
3310 : GEN
3311 263704 : bnf_build_cheapfu(GEN bnf)
3312 : {
3313 : GEN fu, SUnits;
3314 263704 : if ((fu = bnf_has_fu(bnf))) return fu;
3315 142 : if ((SUnits = bnf_get_sunits(bnf)))
3316 : {
3317 142 : pari_sp av = avma;
3318 142 : long e = gexpo(real_i(bnf_get_logfu(bnf)));
3319 142 : set_avma(av); if (e < 13) return vecsplice(bnf_build_units(bnf), 1);
3320 : }
3321 77 : return NULL;
3322 : }
3323 :
3324 : static GEN
3325 48662 : get_regulator(GEN A)
3326 : {
3327 48662 : pari_sp av = avma;
3328 : GEN R;
3329 :
3330 48662 : if (lg(A) == 1) return gen_1;
3331 48655 : R = det( rowslice(real_i(A), 1, lgcols(A)-2) );
3332 48656 : setabssign(R); return gerepileuptoleaf(av, R);
3333 : }
3334 :
3335 : /* return corrected archimedian components for elts of x (vector)
3336 : * (= log(sigma_i(x)) - log(|Nx|) / [K:Q]) */
3337 : static GEN
3338 40 : get_archclean(GEN nf, GEN x, long prec, int units)
3339 : {
3340 40 : long k, N, l = lg(x);
3341 40 : GEN M = cgetg(l, t_MAT);
3342 :
3343 40 : if (l == 1) return M;
3344 26 : N = nf_get_degree(nf);
3345 114 : for (k = 1; k < l; k++)
3346 : {
3347 88 : pari_sp av = avma;
3348 88 : GEN c = nf_cxlog(nf, gel(x,k), prec);
3349 88 : if (!c || (!units && !(c = cleanarch(c, N, NULL,prec)))) return NULL;
3350 88 : gel(M,k) = gerepilecopy(av, c);
3351 : }
3352 26 : return M;
3353 : }
3354 : static void
3355 77 : Sunits_archclean(GEN nf, GEN Sunits, GEN *pmun, GEN *pC, long prec)
3356 : {
3357 77 : GEN ipi, M, X = gel(Sunits,1), U = gel(Sunits,2), G = gel(Sunits,3);
3358 77 : long k, N = nf_get_degree(nf), l = lg(X);
3359 :
3360 77 : M = cgetg(l, t_MAT);
3361 3640 : for (k = 1; k < l; k++)
3362 3563 : if (!(gel(M,k) = nf_cxlog(nf, gel(X,k), prec))) return;
3363 77 : ipi = invr(mppi(prec));
3364 77 : *pmun = cleanarch(RgM_ZM_mul(M, U), N, ipi, prec); /* not cleanarchunit ! */
3365 77 : if (*pmun) *pC = cleanarch(RgM_ZM_mul(M, G), N, ipi, prec);
3366 : }
3367 :
3368 : GEN
3369 97 : bnfnewprec_shallow(GEN bnf, long prec)
3370 : {
3371 97 : GEN nf0 = bnf_get_nf(bnf), nf, v, fu, matal, y, A, C;
3372 97 : GEN Sunits = bnf_get_sunits(bnf), Ur, Ga, Ge, M1, M2;
3373 97 : long r1, r2, prec0 = prec;
3374 :
3375 97 : nf_get_sign(nf0, &r1, &r2);
3376 97 : if (Sunits)
3377 : {
3378 77 : fu = matal = NULL;
3379 77 : prec += nbits2extraprec(gexpo(Sunits));
3380 : }
3381 : else
3382 : {
3383 20 : fu = bnf_build_units(bnf);
3384 20 : fu = vecslice(fu, 2, lg(fu)-1);
3385 20 : if (r1 + r2 > 1) {
3386 13 : long e = gexpo(bnf_get_logfu(bnf)) + 1 - TWOPOTBITS_IN_LONG;
3387 13 : if (e >= 0) prec += nbits2extraprec(e);
3388 : }
3389 20 : matal = bnf_build_matalpha(bnf);
3390 : }
3391 :
3392 97 : if (DEBUGLEVEL && prec0 != prec) pari_warn(warnprec,"bnfnewprec",prec);
3393 97 : for(C = NULL;;)
3394 0 : {
3395 97 : pari_sp av = avma;
3396 97 : nf = nfnewprec_shallow(nf0,prec);
3397 97 : if (Sunits)
3398 77 : Sunits_archclean(nf, Sunits, &A, &C, prec);
3399 : else
3400 : {
3401 20 : A = get_archclean(nf, fu, prec, 1);
3402 20 : if (A) C = get_archclean(nf, matal, prec, 0);
3403 : }
3404 97 : if (C) break;
3405 0 : set_avma(av); prec = precdbl(prec);
3406 0 : if (DEBUGLEVEL) pari_warn(warnprec,"bnfnewprec(extra)",prec);
3407 : }
3408 97 : y = leafcopy(bnf);
3409 97 : gel(y,3) = A;
3410 97 : gel(y,4) = C;
3411 97 : gel(y,7) = nf;
3412 97 : gel(y,8) = v = leafcopy(gel(bnf,8));
3413 97 : gel(v,2) = get_regulator(A);
3414 97 : v = gel(bnf,9);
3415 97 : if (lg(v) < 7) pari_err_TYPE("bnfnewprec [obsolete bnf format]", bnf);
3416 97 : Ur = gel(v,1);
3417 97 : Ge = gel(v,4);
3418 97 : Ga = nfV_cxlog(nf, Ge, prec);
3419 97 : M1 = gel(v,5);
3420 97 : M2 = gel(v,6);
3421 97 : gel(y,9) = get_clg2(bnf_get_cyc(bnf), Ga, C, Ur, Ge, M1, M2);
3422 97 : return y;
3423 : }
3424 : GEN
3425 7 : bnfnewprec(GEN bnf, long prec)
3426 : {
3427 7 : pari_sp av = avma;
3428 7 : return gerepilecopy(av, bnfnewprec_shallow(checkbnf(bnf), prec));
3429 : }
3430 :
3431 : GEN
3432 0 : bnrnewprec_shallow(GEN bnr, long prec)
3433 : {
3434 0 : GEN y = cgetg(7,t_VEC);
3435 : long i;
3436 0 : gel(y,1) = bnfnewprec_shallow(bnr_get_bnf(bnr), prec);
3437 0 : for (i=2; i<7; i++) gel(y,i) = gel(bnr,i);
3438 0 : return y;
3439 : }
3440 : GEN
3441 7 : bnrnewprec(GEN bnr, long prec)
3442 : {
3443 7 : GEN y = cgetg(7,t_VEC);
3444 : long i;
3445 7 : checkbnr(bnr);
3446 7 : gel(y,1) = bnfnewprec(bnr_get_bnf(bnr), prec);
3447 42 : for (i=2; i<7; i++) gel(y,i) = gcopy(gel(bnr,i));
3448 7 : return y;
3449 : }
3450 :
3451 : static GEN
3452 64966 : buchall_end(GEN nf,GEN res, GEN clg2, GEN W, GEN B, GEN A, GEN C,GEN Vbase)
3453 : {
3454 64966 : GEN z = obj_init(9, 3);
3455 64966 : gel(z,1) = W;
3456 64966 : gel(z,2) = B;
3457 64966 : gel(z,3) = A;
3458 64966 : gel(z,4) = C;
3459 64966 : gel(z,5) = Vbase;
3460 64966 : gel(z,6) = gen_0;
3461 64966 : gel(z,7) = nf;
3462 64966 : gel(z,8) = res;
3463 64966 : gel(z,9) = clg2;
3464 64966 : return z;
3465 : }
3466 :
3467 : GEN
3468 2611 : bnfinit0(GEN P, long flag, GEN data, long prec)
3469 : {
3470 2611 : double c1 = 0., c2 = 0.;
3471 2611 : long fl, relpid = BNF_RELPID;
3472 :
3473 2611 : if (data)
3474 : {
3475 21 : long lx = lg(data);
3476 21 : if (typ(data) != t_VEC || lx > 5) pari_err_TYPE("bnfinit",data);
3477 21 : switch(lx)
3478 : {
3479 0 : case 4: relpid = itos(gel(data,3));
3480 14 : case 3: c2 = gtodouble(gel(data,2));
3481 21 : case 2: c1 = gtodouble(gel(data,1));
3482 : }
3483 : }
3484 2611 : switch(flag)
3485 : {
3486 1764 : case 2:
3487 1764 : case 0: fl = 0; break;
3488 847 : case 1: fl = nf_FORCE; break;
3489 0 : default: pari_err_FLAG("bnfinit");
3490 : return NULL; /* LCOV_EXCL_LINE */
3491 : }
3492 2611 : return Buchall_param(P, c1, c2, relpid, fl, prec);
3493 : }
3494 : GEN
3495 62366 : Buchall(GEN P, long flag, long prec)
3496 62366 : { return Buchall_param(P, 0., 0., BNF_RELPID, flag & nf_FORCE, prec); }
3497 :
3498 : static GEN
3499 1176 : Buchall_deg1(GEN nf)
3500 : {
3501 1176 : GEN v = cgetg(1,t_VEC), m = cgetg(1,t_MAT);
3502 1176 : GEN res, W, A, B, C, Vbase = cgetg(1,t_COL);
3503 1176 : GEN fu = v, R = gen_1, zu = mkvec2(gen_2, gen_m1);
3504 1176 : GEN clg1 = mkvec3(gen_1,v,v), clg2 = mkvecn(6, m,m,m,v,m,m);
3505 :
3506 1176 : W = A = B = C = m; res = mkvec5(clg1, R, gen_1, zu, fu);
3507 1176 : return buchall_end(nf,res,clg2,W,B,A,C,Vbase);
3508 : }
3509 :
3510 : /* return (small set of) indices of columns generating the same lattice as x.
3511 : * Assume HNF(x) is inexpensive (few rows, many columns).
3512 : * Dichotomy approach since interesting columns may be at the very end */
3513 : GEN
3514 63798 : extract_full_lattice(GEN x)
3515 : {
3516 63798 : long dj, j, k, l = lg(x);
3517 : GEN h, h2, H, v;
3518 :
3519 63798 : if (l < 200) return NULL; /* not worth it */
3520 :
3521 3 : v = vecsmalltrunc_init(l);
3522 3 : H = ZM_hnf(x);
3523 3 : h = cgetg(1, t_MAT);
3524 3 : dj = 1;
3525 135 : for (j = 1; j < l; )
3526 : {
3527 135 : pari_sp av = avma;
3528 135 : long lv = lg(v);
3529 :
3530 474 : for (k = 0; k < dj; k++) v[lv+k] = j+k;
3531 135 : setlg(v, lv + dj);
3532 135 : h2 = ZM_hnf(vecpermute(x, v));
3533 135 : if (ZM_equal(h, h2))
3534 : { /* these dj columns can be eliminated */
3535 54 : set_avma(av); setlg(v, lv);
3536 54 : j += dj;
3537 54 : if (j >= l) break;
3538 54 : dj <<= 1;
3539 54 : if (j + dj >= l) { dj = (l - j) >> 1; if (!dj) dj = 1; }
3540 : }
3541 81 : else if (dj > 1)
3542 : { /* at least one interesting column, try with first half of this set */
3543 54 : set_avma(av); setlg(v, lv);
3544 54 : dj >>= 1; /* > 0 */
3545 : }
3546 : else
3547 : { /* this column should be kept */
3548 27 : if (ZM_equal(h2, H)) break;
3549 24 : h = h2; j++;
3550 : }
3551 : }
3552 3 : return v;
3553 : }
3554 :
3555 : static void
3556 63861 : init_rel(RELCACHE_t *cache, FB_t *F, long add_need)
3557 : {
3558 63861 : const long n = F->KC + add_need; /* expected # of needed relations */
3559 : long i, j, k, p;
3560 : GEN c, P;
3561 : GEN R;
3562 :
3563 63861 : if (DEBUGLEVEL) err_printf("KCZ = %ld, KC = %ld, n = %ld\n", F->KCZ,F->KC,n);
3564 63861 : reallocate(cache, 10*n + 50); /* make room for lots of relations */
3565 63861 : cache->chk = cache->base;
3566 63861 : cache->end = cache->base + n;
3567 63861 : cache->relsup = add_need;
3568 63861 : cache->last = cache->base;
3569 63861 : cache->missing = lg(cache->basis) - 1;
3570 304201 : for (i = 1; i <= F->KCZ; i++)
3571 : { /* trivial relations (p) = prod P^e */
3572 240342 : p = F->FB[i]; P = gel(F->LV,p);
3573 240342 : if (!isclone(P)) continue;
3574 :
3575 : /* all prime divisors in FB */
3576 167870 : c = zero_Flv(F->KC); k = F->iLP[p];
3577 167871 : R = c; c += k;
3578 535823 : for (j = lg(P)-1; j; j--) c[j] = pr_get_e(gel(P,j));
3579 167871 : add_rel(cache, F, R, k+1, pr_get_p(gel(P,1)), 0);
3580 : }
3581 63859 : }
3582 :
3583 : /* Let z = \zeta_n in nf. List of not-obviously-dependent generators for
3584 : * cyclotomic units modulo torsion in Q(z) [independent when n a prime power]:
3585 : * - z^a - 1, n/(a,n) not a prime power, a \nmid n unless a=1, 1 <= a < n/2
3586 : * - (Z^a - 1)/(Z - 1), p^k || n, Z = z^{n/p^k}, (p,a) = 1, 1 < a <= (p^k-1)/2
3587 : */
3588 : GEN
3589 63860 : nfcyclotomicunits(GEN nf, GEN zu)
3590 : {
3591 63860 : long n = itos(gel(zu, 1)), n2, lP, i, a;
3592 : GEN z, fa, P, E, L, mz, powz;
3593 63860 : if (n <= 6) return cgetg(1, t_VEC);
3594 :
3595 1911 : z = algtobasis(nf,gel(zu, 2));
3596 1911 : if ((n & 3) == 2) { n = n >> 1; z = ZC_neg(z); } /* ensure n != 2 (mod 4) */
3597 1911 : n2 = n/2;
3598 1911 : mz = zk_multable(nf, z); /* multiplication by z */
3599 1911 : powz = cgetg(n2, t_VEC); gel(powz,1) = z;
3600 6286 : for (i = 2; i < n2; i++) gel(powz,i) = ZM_ZC_mul(mz, gel(powz,i-1));
3601 : /* powz[i] = z^i */
3602 :
3603 1911 : L = vectrunc_init(n);
3604 1911 : fa = factoru(n);
3605 1911 : P = gel(fa,1); lP = lg(P);
3606 1911 : E = gel(fa,2);
3607 4613 : for (i = 1; i < lP; i++)
3608 : { /* second kind */
3609 2702 : long p = P[i], k = E[i], pk = upowuu(p,k), pk2 = (pk-1) / 2;
3610 2702 : GEN u = gen_1;
3611 4970 : for (a = 2; a <= pk2; a++)
3612 : {
3613 2268 : u = nfadd(nf, u, gel(powz, (n/pk) * (a-1))); /* = (Z^a-1)/(Z-1) */
3614 2268 : if (a % p) vectrunc_append(L, u);
3615 : }
3616 : }
3617 6160 : if (lP > 2) for (a = 1; a < n2; a++)
3618 : { /* first kind, when n not a prime power */
3619 : ulong p;
3620 4249 : if (a > 1 && (n % a == 0 || uisprimepower(n/ugcd(a,n), &p))) continue;
3621 1869 : vectrunc_append(L, nfadd(nf, gel(powz, a), gen_m1));
3622 : }
3623 1911 : return L;
3624 : }
3625 : static void
3626 63860 : add_cyclotomic_units(GEN nf, GEN zu, RELCACHE_t *cache, FB_t *F)
3627 : {
3628 63860 : pari_sp av = avma;
3629 63860 : GEN L = nfcyclotomicunits(nf, zu);
3630 63860 : long i, l = lg(L);
3631 63860 : if (l > 1)
3632 : {
3633 1911 : GEN R = zero_Flv(F->KC);
3634 5950 : for(i = 1; i < l; i++) add_rel(cache, F, R, F->KC+1, gel(L,i), 0);
3635 : }
3636 63860 : set_avma(av);
3637 63860 : }
3638 :
3639 : static GEN
3640 121000 : trim_list(FB_t *F)
3641 : {
3642 121000 : pari_sp av = avma;
3643 121000 : GEN v, L_jid = F->L_jid, minidx = F->minidx, present = zero_Flv(F->KC);
3644 121001 : long i, j, imax = minss(lg(L_jid), F->KC + 1);
3645 :
3646 121001 : v = cgetg(imax, t_VECSMALL);
3647 1314402 : for (i = j = 1; i < imax; i++)
3648 : {
3649 1193400 : long k = minidx[ L_jid[i] ];
3650 1193400 : if (!present[k]) { v[j++] = L_jid[i]; present[k] = 1; }
3651 : }
3652 121002 : setlg(v, j); return gerepileuptoleaf(av, v);
3653 : }
3654 :
3655 : /* x t_INT or primitive ZC */
3656 : static void
3657 1581 : try_elt(RELCACHE_t *cache, FB_t *F, GEN nf, GEN x, FACT *fact)
3658 : {
3659 1581 : pari_sp av = avma;
3660 : long nz;
3661 : GEN R;
3662 :
3663 1581 : if (typ(x) == t_INT /* 2nd path can't fail */
3664 1581 : || !can_factor(F, nf, NULL, x, nfnorm(nf, x), fact)) return;
3665 : /* smooth element */
3666 1361 : R = set_fact(F, fact, NULL, &nz);
3667 : /* make sure we get maximal rank first, then allow all relations */
3668 1361 : (void)add_rel(cache, F, R, nz, x, 0);
3669 1361 : set_avma(av);
3670 : }
3671 :
3672 : static void
3673 54267 : matenlarge(GEN C, long h)
3674 : {
3675 54267 : GEN _0 = zerocol(h);
3676 : long i;
3677 1232124 : for (i = lg(C); --i; ) gel(C,i) = shallowconcat(gel(C,i), _0);
3678 54265 : }
3679 :
3680 : /* E = floating point embeddings */
3681 : static GEN
3682 54265 : matbotidembs(RELCACHE_t *cache, GEN E)
3683 : {
3684 54265 : long w = cache->last - cache->chk, h = cache->last - cache->base;
3685 54265 : long j, d = h - w, hE = nbrows(E);
3686 54265 : GEN y = cgetg(w+1,t_MAT), _0 = zerocol(h);
3687 211913 : for (j = 1; j <= w; j++)
3688 : {
3689 157646 : GEN c = shallowconcat(gel(E,j), _0);
3690 157646 : if (d + j >= 1) gel(c, d + j + hE) = gen_1;
3691 157646 : gel(y,j) = c;
3692 : }
3693 54267 : return y;
3694 : }
3695 : static GEN
3696 62251 : matbotid(RELCACHE_t *cache)
3697 : {
3698 62251 : long w = cache->last - cache->chk, h = cache->last - cache->base;
3699 62251 : long j, d = h - w;
3700 62251 : GEN y = cgetg(w+1,t_MAT);
3701 846169 : for (j = 1; j <= w; j++)
3702 : {
3703 783918 : GEN c = zerocol(h);
3704 783918 : if (d + j >= 1) gel(c, d + j) = gen_1;
3705 783918 : gel(y,j) = c;
3706 : }
3707 62251 : return y;
3708 : }
3709 :
3710 : static long
3711 73 : myprecdbl(long prec, GEN C)
3712 : {
3713 73 : long p = prec < 1280? precdbl(prec): (long)(prec * 1.5);
3714 73 : if (C) p = maxss(p, minss(3*p, prec + nbits2extraprec(gexpo(C))));
3715 73 : return p;
3716 : }
3717 :
3718 : static GEN
3719 57730 : _nfnewprec(GEN nf, long prec, long *isclone)
3720 : {
3721 57730 : GEN NF = gclone(nfnewprec_shallow(nf, prec));
3722 57730 : if (*isclone) gunclone(nf);
3723 57730 : *isclone = 1; return NF;
3724 : }
3725 :
3726 : /* Nrelid = nb relations per ideal, possibly 0. If flag is set, keep data in
3727 : * algebraic form. */
3728 : GEN
3729 64977 : Buchall_param(GEN P, double cbach, double cbach2, long Nrelid, long flag, long prec)
3730 : {
3731 : pari_timer T;
3732 64977 : pari_sp av0 = avma, av, av2;
3733 : long PREC, N, R1, R2, RU, low, high, LIMC0, LIMC, LIMC2, LIMCMAX, zc, i;
3734 64977 : long LIMres, bit = 0, flag_nfinit = 0, nfisclone = 0;
3735 64977 : long nreldep, sfb_trials, need, old_need, precdouble = 0, TRIES = 0;
3736 : long done_small, small_fail, fail_limit, squash_index;
3737 : double LOGD, LOGD2, lim;
3738 64977 : GEN computed = NULL, fu = NULL, zu, nf, D, A, W, R, h, Ce, PERM;
3739 : GEN small_multiplier, auts, cyclic, embs, SUnits;
3740 : GEN res, L, invhr, B, C, lambda, dep, clg1, clg2, Vbase;
3741 64977 : const char *precpb = NULL;
3742 64977 : REL_t *old_cache = NULL;
3743 : nfmaxord_t nfT;
3744 : RELCACHE_t cache;
3745 : FB_t F;
3746 : GRHcheck_t GRHcheck;
3747 : FACT *fact;
3748 :
3749 64977 : if (DEBUGLEVEL) timer_start(&T);
3750 64977 : P = get_nfpol(P, &nf);
3751 64963 : if (degpol(P)==2) Nrelid = 0;
3752 64964 : if (nf)
3753 3640 : D = nf_get_disc(nf);
3754 : else
3755 : {
3756 61324 : nfinit_basic(&nfT, P);
3757 61326 : D = nfT.dK;
3758 61326 : if (!ZX_is_monic(nfT.T0))
3759 : {
3760 14 : pari_warn(warner,"nonmonic polynomial in bnfinit, using polredbest");
3761 14 : flag_nfinit = nf_RED;
3762 : }
3763 : }
3764 64966 : PREC = maxss(DEFAULTPREC, prec);
3765 64966 : N = degpol(P);
3766 64966 : if (N <= 1)
3767 : {
3768 1176 : if (!nf) nf = nfinit_complete(&nfT, flag_nfinit, PREC);
3769 1176 : return gerepilecopy(av0, Buchall_deg1(nf));
3770 : }
3771 63790 : D = absi_shallow(D);
3772 63789 : LOGD = dbllog2(D) * M_LN2;
3773 63789 : LOGD2 = LOGD*LOGD;
3774 63789 : LIMCMAX = (long)(4.*LOGD2);
3775 63789 : if (nf) PREC = maxss(PREC, nf_get_prec(nf));
3776 63789 : PREC = maxss(PREC, nbits2prec((long)(LOGD2 * 0.02) + N*N));
3777 63789 : if (DEBUGLEVEL) err_printf("PREC = %ld\n", PREC);
3778 63789 : if (!nf)
3779 60345 : nf = nfinit_complete(&nfT, flag_nfinit, PREC);
3780 3444 : else if (nf_get_prec(nf) < PREC)
3781 161 : nf = nfnewprec_shallow(nf, PREC);
3782 63791 : zu = nfrootsof1(nf);
3783 63791 : gel(zu,2) = nf_to_scalar_or_alg(nf, gel(zu,2));
3784 :
3785 63789 : nf_get_sign(nf, &R1, &R2); RU = R1+R2;
3786 63789 : auts = automorphism_matrices(nf, &cyclic);
3787 63790 : F.embperm = automorphism_perms(nf_get_M(nf), auts, cyclic, R1, R2, N);
3788 63790 : if (DEBUGLEVEL)
3789 : {
3790 0 : timer_printf(&T, "nfinit & nfrootsof1");
3791 0 : err_printf("%s bnf: R1 = %ld, R2 = %ld\nD = %Ps\n",
3792 : flag? "Algebraic": "Floating point", R1,R2, D);
3793 : }
3794 63790 : if (LOGD < 20.)
3795 : { /* tiny disc, Minkowski may be smaller than Bach */
3796 62341 : lim = exp(-N + R2 * log(4/M_PI) + LOGD/2) * sqrt(2*M_PI*N);
3797 62341 : if (lim < 3) lim = 3;
3798 : }
3799 : else /* to be ignored */
3800 1449 : lim = -1;
3801 63790 : if (cbach > 12.) {
3802 0 : if (cbach2 < cbach) cbach2 = cbach;
3803 0 : cbach = 12.;
3804 : }
3805 63790 : if (cbach < 0.)
3806 0 : pari_err_DOMAIN("Buchall","Bach constant","<",gen_0,dbltor(cbach));
3807 :
3808 63790 : cache.base = NULL; F.subFB = NULL; F.LP = NULL; SUnits = Ce = NULL;
3809 63790 : init_GRHcheck(&GRHcheck, N, R1, LOGD);
3810 63791 : high = low = LIMC0 = maxss((long)(cbach2*LOGD2), 1);
3811 311104 : while (!GRHchk(nf, &GRHcheck, high)) { low = high; high *= 2; }
3812 247353 : while (high - low > 1)
3813 : {
3814 183568 : long test = (low+high)/2;
3815 183568 : if (GRHchk(nf, &GRHcheck, test)) high = test; else low = test;
3816 : }
3817 63785 : LIMC2 = (high == LIMC0+1 && GRHchk(nf, &GRHcheck, LIMC0))? LIMC0: high;
3818 63785 : if (LIMC2 > LIMCMAX) LIMC2 = LIMCMAX;
3819 : /* Assuming GRH, {P, NP <= LIMC2} generate Cl(K) */
3820 63785 : if (DEBUGLEVEL) err_printf("LIMC2 = %ld\n", LIMC2);
3821 63791 : LIMC0 = (long)(cbach*LOGD2); /* initial value for LIMC */
3822 63791 : LIMC = cbach? LIMC0: LIMC2; /* use {P, NP <= LIMC} as a factorbase */
3823 63791 : LIMC = maxss(LIMC, nthideal(&GRHcheck, nf, N));
3824 63791 : if (DEBUGLEVEL) timer_printf(&T, "computing Bach constant");
3825 63791 : LIMres = primeneeded(N, R1, R2, LOGD);
3826 63791 : cache_prime_dec(&GRHcheck, LIMres, nf);
3827 : /* invhr ~ 2^r1 (2pi)^r2 / sqrt(D) w * Res(zeta_K, s=1) = 1 / hR */
3828 127581 : invhr = gmul(gdiv(gmul2n(powru(mppi(DEFAULTPREC), R2), RU),
3829 63790 : mulri(gsqrt(D,DEFAULTPREC),gel(zu,1))),
3830 : compute_invres(&GRHcheck, LIMres));
3831 63791 : if (DEBUGLEVEL) timer_printf(&T, "computing inverse of hR");
3832 63791 : av = avma;
3833 :
3834 66038 : START:
3835 66038 : if (DEBUGLEVEL) timer_start(&T);
3836 66038 : if (TRIES) LIMC = bnf_increase_LIMC(LIMC,LIMCMAX);
3837 66038 : if (DEBUGLEVEL && LIMC > LIMC0)
3838 0 : err_printf("%s*** Bach constant: %f\n", TRIES?"\n":"", LIMC/LOGD2);
3839 66038 : if (cache.base)
3840 : {
3841 : REL_t *rel;
3842 3733 : for (i = 1, rel = cache.base + 1; rel < cache.last; rel++)
3843 3663 : if (rel->m) i++;
3844 70 : computed = cgetg(i, t_VEC);
3845 3733 : for (i = 1, rel = cache.base + 1; rel < cache.last; rel++)
3846 3663 : if (rel->m) gel(computed, i++) = rel->m;
3847 70 : computed = gclone(computed); delete_cache(&cache);
3848 : }
3849 66038 : TRIES++; set_avma(av);
3850 66038 : if (F.LP) delete_FB(&F);
3851 66038 : if (LIMC2 < LIMC) LIMC2 = LIMC;
3852 66038 : if (DEBUGLEVEL) { err_printf("LIMC = %ld, LIMC2 = %ld\n",LIMC,LIMC2); }
3853 :
3854 66038 : FBgen(&F, nf, N, LIMC, LIMC2, &GRHcheck);
3855 66037 : if (!F.KC) goto START;
3856 66037 : av = avma;
3857 66037 : subFBgen(&F,auts,cyclic,lim < 0? LIMC2: mindd(lim,LIMC2),MINSFB);
3858 66038 : if (lg(F.subFB) == 1) goto START;
3859 63861 : if (DEBUGLEVEL)
3860 0 : timer_printf(&T, "factorbase (#subFB = %ld) and ideal permutations",
3861 0 : lg(F.subFB)-1);
3862 :
3863 63861 : fact = (FACT*)stack_malloc((F.KC+1)*sizeof(FACT));
3864 63861 : PERM = leafcopy(F.perm); /* to be restored in case of precision increase */
3865 63861 : cache.basis = zero_Flm_copy(F.KC,F.KC);
3866 63861 : small_multiplier = zero_Flv(F.KC);
3867 63861 : done_small = small_fail = squash_index = zc = sfb_trials = nreldep = 0;
3868 63861 : fail_limit = F.KC + 1;
3869 63861 : W = A = R = NULL;
3870 63861 : av2 = avma;
3871 63861 : init_rel(&cache, &F, RELSUP + RU-1);
3872 63860 : old_need = need = cache.end - cache.last;
3873 63860 : add_cyclotomic_units(nf, zu, &cache, &F);
3874 63860 : if (DEBUGLEVEL) err_printf("\n");
3875 63860 : cache.end = cache.last + need;
3876 :
3877 63860 : if (computed)
3878 : {
3879 1651 : for (i = 1; i < lg(computed); i++)
3880 1581 : try_elt(&cache, &F, nf, gel(computed, i), fact);
3881 70 : gunclone(computed);
3882 70 : if (DEBUGLEVEL && i > 1)
3883 0 : timer_printf(&T, "including already computed relations");
3884 70 : need = 0;
3885 : }
3886 :
3887 : do
3888 : {
3889 : GEN Ar, C0;
3890 : do
3891 : {
3892 121150 : pari_sp av4 = avma;
3893 121150 : if (need > 0)
3894 : {
3895 121000 : long oneed = cache.end - cache.last;
3896 : /* Test below can be true if small_norm did not find enough linearly
3897 : * dependent relations */
3898 121000 : if (need < oneed) need = oneed;
3899 121000 : pre_allocate(&cache, need+lg(auts)-1+(R ? lg(W)-1 : 0));
3900 121000 : cache.end = cache.last + need;
3901 121000 : F.L_jid = trim_list(&F);
3902 : }
3903 121152 : if (need > 0 && Nrelid > 0 && (done_small <= F.KC+1 || A) &&
3904 69177 : small_fail <= fail_limit &&
3905 69177 : cache.last < cache.base + 2*F.KC+2*RU+RELSUP /* heuristic */)
3906 : {
3907 66048 : long j, k, LIE = (R && lg(W) > 1 && (done_small % 2));
3908 66048 : REL_t *last = cache.last;
3909 66048 : pari_sp av3 = avma;
3910 66048 : if (LIE)
3911 : { /* We have full rank for class group and unit. The following tries to
3912 : * improve the prime group lattice by looking for relations involving
3913 : * the primes generating the class group. */
3914 3388 : long n = lg(W)-1; /* need n relations to squash the class group */
3915 3388 : F.L_jid = vecslice(F.perm, 1, n);
3916 3388 : cache.end = cache.last + n;
3917 : /* Lie to the add_rel subsystem: pretend we miss relations involving
3918 : * the primes generating the class group (and only those). */
3919 3388 : cache.missing = n;
3920 10554 : for ( ; n > 0; n--) mael(cache.basis, F.perm[n], F.perm[n]) = 0;
3921 : }
3922 66048 : j = done_small % (F.KC+1);
3923 66048 : if (j && !A)
3924 : { /* Prevent considering both P_iP_j and P_jP_i in small_norm */
3925 : /* Not all elements end up in F.L_jid (eliminated by hnfspec/add or
3926 : * by trim_list): keep track of which ideals are being considered
3927 : * at each run. */
3928 407 : long mj = small_multiplier[j];
3929 6359 : for (i = k = 1; i < lg(F.L_jid); i++)
3930 5952 : if (F.L_jid[i] > mj)
3931 : {
3932 5952 : small_multiplier[F.L_jid[i]] = j;
3933 5952 : F.L_jid[k++] = F.L_jid[i];
3934 : }
3935 407 : setlg(F.L_jid, k);
3936 : }
3937 66048 : if (lg(F.L_jid) > 1) small_norm(&cache, &F, nf, Nrelid, fact, j);
3938 66048 : F.L_jid = F.perm; set_avma(av3);
3939 66048 : if (!A && cache.last != last) small_fail = 0; else small_fail++;
3940 66048 : if (LIE)
3941 : { /* restore add_rel subsystem: undo above lie */
3942 3388 : long n = lg(W) - 1;
3943 10554 : for ( ; n > 0; n--) mael(cache.basis, F.perm[n], F.perm[n]) = 1;
3944 3388 : cache.missing = 0;
3945 : }
3946 66048 : cache.end = cache.last;
3947 66048 : done_small++;
3948 66048 : need = F.sfb_chg = 0;
3949 : }
3950 121152 : if (need > 0)
3951 : { /* Random relations */
3952 54954 : if (++nreldep > F.MAXDEPSIZESFB) {
3953 14 : if (++sfb_trials > SFB_MAX && LIMC < LIMCMAX/2) goto START;
3954 14 : F.sfb_chg = sfb_INCREASE;
3955 14 : nreldep = 0;
3956 : }
3957 54940 : else if (!(nreldep % F.MAXDEPSFB))
3958 26188 : F.sfb_chg = sfb_CHANGE;
3959 54954 : if (F.sfb_chg && !subFB_change(&F)) goto START;
3960 54882 : rnd_rel(&cache, &F, nf, fact);
3961 54884 : F.L_jid = F.perm;
3962 : }
3963 121082 : if (DEBUGLEVEL) timer_start(&T);
3964 121082 : if (precpb)
3965 : {
3966 : REL_t *rel;
3967 80 : if (DEBUGLEVEL)
3968 : {
3969 0 : char str[64]; sprintf(str,"Buchall_param (%s)",precpb);
3970 0 : pari_warn(warnprec,str,PREC);
3971 : }
3972 80 : nf = _nfnewprec(nf, PREC, &nfisclone);
3973 80 : precdouble++; precpb = NULL;
3974 :
3975 80 : if (flag)
3976 : { /* recompute embs only, no need to redo HNF */
3977 38 : long j, le = lg(embs), lC = lg(C);
3978 38 : GEN E, M = nf_get_M(nf);
3979 38 : set_avma(av4);
3980 12611 : for (rel = cache.base+1, i = 1; i < le; i++,rel++)
3981 12573 : gel(embs,i) = rel_embed(rel, &F, embs, i, M, RU, R1, PREC);
3982 38 : E = RgM_ZM_mul(embs, rowslice(C, RU+1, nbrows(C)));
3983 12611 : for (j = 1; j < lC; j++)
3984 65595 : for (i = 1; i <= RU; i++) gcoeff(C,i,j) = gcoeff(E,i,j);
3985 38 : av4 = avma;
3986 : }
3987 : else
3988 : { /* recompute embs + HNF */
3989 10318 : for(i = 1; i < lg(PERM); i++) F.perm[i] = PERM[i];
3990 42 : cache.chk = cache.base;
3991 42 : W = NULL;
3992 : }
3993 80 : if (DEBUGLEVEL) timer_printf(&T, "increasing accuracy");
3994 : }
3995 121082 : set_avma(av4);
3996 121082 : if (cache.chk != cache.last)
3997 : { /* Reduce relation matrices */
3998 120963 : long l = cache.last - cache.chk + 1, j;
3999 120963 : GEN mat = cgetg(l, t_MAT);
4000 : REL_t *rel;
4001 :
4002 1116081 : for (j=1,rel = cache.chk + 1; j < l; rel++,j++) gel(mat,j) = rel->R;
4003 120963 : if (!flag || W)
4004 : {
4005 58712 : embs = get_embs(&F, &cache, nf, embs, PREC);
4006 58710 : if (DEBUGLEVEL && timer_get(&T) > 1)
4007 0 : timer_printf(&T, "floating point embeddings");
4008 : }
4009 120961 : if (!W)
4010 : { /* never reduced before */
4011 63903 : C = flag? matbotid(&cache): embs;
4012 63902 : W = hnfspec_i(mat, F.perm, &dep, &B, &C, F.subFB ? lg(F.subFB)-1:0);
4013 63903 : if (DEBUGLEVEL)
4014 0 : timer_printf(&T, "hnfspec [%ld x %ld]", lg(F.perm)-1, l-1);
4015 63903 : if (flag)
4016 : {
4017 62251 : PREC += nbits2extraprec(gexpo(C));
4018 62251 : if (nf_get_prec(nf) < PREC) nf = _nfnewprec(nf, PREC, &nfisclone);
4019 62251 : embs = get_embs(&F, &cache, nf, embs, PREC);
4020 62250 : C = vconcat(RgM_ZM_mul(embs, C), C);
4021 : }
4022 63903 : if (DEBUGLEVEL)
4023 0 : timer_printf(&T, "hnfspec floating points");
4024 : }
4025 : else
4026 : {
4027 57058 : long k = lg(embs);
4028 57058 : GEN E = vecslice(embs, k-l+1,k-1);
4029 57058 : if (flag)
4030 : {
4031 54265 : E = matbotidembs(&cache, E);
4032 54267 : matenlarge(C, cache.last - cache.chk);
4033 : }
4034 57060 : W = hnfadd_i(W, F.perm, &dep, &B, &C, mat, E);
4035 57060 : if (DEBUGLEVEL)
4036 0 : timer_printf(&T, "hnfadd (%ld + %ld)", l-1, lg(dep)-1);
4037 : }
4038 120963 : gerepileall(av2, 5, &W,&C,&B,&dep,&embs);
4039 120963 : cache.chk = cache.last;
4040 : }
4041 119 : else if (!W)
4042 : {
4043 0 : need = old_need;
4044 0 : F.L_jid = vecslice(F.perm, 1, need);
4045 0 : continue;
4046 : }
4047 121082 : need = F.KC - (lg(W)-1) - (lg(B)-1);
4048 121082 : if (!need && cache.missing)
4049 : { /* The test above will never be true except if 27449|class number.
4050 : * Ensure that if we have maximal rank for the ideal lattice, then
4051 : * cache.missing == 0. */
4052 14 : for (i = 1; cache.missing; i++)
4053 7 : if (!mael(cache.basis, i, i))
4054 : {
4055 : long j;
4056 7 : cache.missing--; mael(cache.basis, i, i) = 1;
4057 427 : for (j = i+1; j <= F.KC; j++) mael(cache.basis, j, i) = 0;
4058 : }
4059 : }
4060 121082 : zc = (lg(C)-1) - (lg(B)-1) - (lg(W)-1);
4061 121082 : if (RU-1-zc > 0) need = minss(need + RU-1-zc, F.KC); /* for units */
4062 121082 : if (need)
4063 : { /* dependent rows */
4064 812 : F.L_jid = vecslice(F.perm, 1, need);
4065 812 : vecsmall_sort(F.L_jid);
4066 812 : if (need != old_need) { nreldep = 0; old_need = need; }
4067 : }
4068 : else
4069 : { /* If the relation lattice is too small, check will be > 1 and we will
4070 : * do a new run of small_norm/rnd_rel asking for 1 relation. This often
4071 : * gives a relation involving L_jid[1]. We rotate the first element of
4072 : * L_jid in order to increase the probability of finding relations that
4073 : * increases the lattice. */
4074 120270 : long j, n = lg(W) - 1;
4075 120270 : if (n > 1 && squash_index % n)
4076 : {
4077 9110 : F.L_jid = leafcopy(F.perm);
4078 36819 : for (j = 1; j <= n; j++)
4079 27709 : F.L_jid[j] = F.perm[1 + (j + squash_index - 1) % n];
4080 : }
4081 : else
4082 111160 : F.L_jid = F.perm;
4083 120270 : squash_index++;
4084 : }
4085 : }
4086 121082 : while (need);
4087 :
4088 120270 : if (!A)
4089 : {
4090 63868 : small_fail = old_need = 0;
4091 63868 : fail_limit = maxss(F.KC / FAIL_DIVISOR, MINFAIL);
4092 : }
4093 120270 : A = vecslice(C, 1, zc); /* cols corresponding to units */
4094 120270 : if (flag) A = rowslice(A, 1, RU);
4095 120270 : Ar = real_i(A);
4096 120270 : R = compute_multiple_of_R(Ar, RU, N, &need, &bit, &lambda);
4097 120269 : if (need < old_need) small_fail = 0;
4098 : #if 0 /* A good idea if we are indeed stuck but needs tuning */
4099 : /* we have computed way more relations than should be necessary */
4100 : if (TRIES < 3 && LIMC < LIMCMAX / 8 &&
4101 : cache.last - cache.base > 10 * F.KC) goto START;
4102 : #endif
4103 120269 : old_need = need;
4104 120269 : if (!lambda)
4105 11 : { precpb = "bestappr"; PREC = myprecdbl(PREC, flag? C: NULL); continue; }
4106 120258 : if (!R)
4107 : { /* not full rank for units */
4108 24574 : if (!need)
4109 0 : { precpb = "regulator"; PREC = myprecdbl(PREC, flag? C: NULL); }
4110 24574 : continue;
4111 : }
4112 95684 : if (cache.last==old_cache) { need=1; continue; }
4113 95605 : old_cache = cache.last;
4114 95605 : h = ZM_det_triangular(W);
4115 95605 : if (DEBUGLEVEL) err_printf("\n#### Tentative class number: %Ps\n", h);
4116 95605 : i = compute_R(lambda, mulir(h,invhr), &L, &R);
4117 95604 : if (DEBUGLEVEL)
4118 : {
4119 0 : err_printf("\n");
4120 0 : timer_printf(&T, "computing regulator and check");
4121 : }
4122 95605 : switch(i)
4123 : {
4124 31745 : case fupb_RELAT:
4125 31745 : need = 1; /* not enough relations */
4126 31745 : continue;
4127 62 : case fupb_PRECI: /* prec problem unless we cheat on Bach constant */
4128 62 : if ((precdouble&7) == 7 && LIMC <= LIMCMAX/2) goto START;
4129 62 : precpb = "compute_R"; PREC = myprecdbl(PREC, flag? C: NULL);
4130 62 : continue;
4131 : }
4132 : /* DONE */
4133 :
4134 63798 : if (F.KCZ2 > F.KCZ)
4135 : {
4136 7 : if (F.sfb_chg && !subFB_change(&F)) goto START;
4137 7 : if (!be_honest(&F, nf, auts, fact)) goto START;
4138 7 : if (DEBUGLEVEL) timer_printf(&T, "to be honest");
4139 : }
4140 63798 : F.KCZ2 = 0; /* be honest only once */
4141 :
4142 : /* fundamental units */
4143 : {
4144 63798 : GEN AU, CU, U, v = extract_full_lattice(L); /* L may be large */
4145 63798 : CU = NULL;
4146 63798 : if (v) { A = vecpermute(A, v); L = vecpermute(L, v); }
4147 : /* arch. components of fund. units */
4148 63798 : U = ZM_lll(L, 0.99, LLL_IM);
4149 63798 : U = ZM_mul(U, lll(RgM_ZM_mul(real_i(A), U)));
4150 63798 : if (DEBUGLEVEL) timer_printf(&T, "units LLL");
4151 63798 : AU = RgM_ZM_mul(A, U);
4152 63797 : A = cleanarchunit(AU, N, NULL, PREC);
4153 63797 : if (RU > 1 /* if there are fund units, test we have correct regulator */
4154 48572 : && (!A || lg(A) < RU || expo(subrr(get_regulator(A), R)) > -1))
4155 7 : {
4156 8 : long add = nbits2extraprec( gexpo(AU) + 64 ) - gprecision(AU);
4157 7 : long t = maxss(PREC * 0.15, add);
4158 7 : if (!A && DEBUGLEVEL) err_printf("### Incorrect units lognorm");
4159 7 : precpb = "cleanarch"; PREC += maxss(t, EXTRAPREC64); continue;
4160 : }
4161 63790 : if (flag)
4162 : {
4163 62194 : long l = lgcols(C) - RU;
4164 : REL_t *rel;
4165 62194 : SUnits = cgetg(l, t_COL);
4166 1001235 : for (rel = cache.base+1, i = 1; i < l; i++,rel++)
4167 939040 : set_rel_alpha(rel, auts, SUnits, i);
4168 62195 : if (RU > 1)
4169 : {
4170 47495 : GEN c = v? vecpermute(C,v): vecslice(C,1,zc);
4171 47495 : CU = ZM_mul(rowslice(c, RU+1, nbrows(c)), U);
4172 : }
4173 : }
4174 63791 : if (DEBUGLEVEL) err_printf("\n#### Computing fundamental units\n");
4175 63791 : fu = getfu(nf, &A, CU? &U: NULL, PREC);
4176 63791 : CU = CU? ZM_mul(CU, U): cgetg(1, t_MAT);
4177 63790 : if (DEBUGLEVEL) timer_printf(&T, "getfu");
4178 63790 : Ce = vecslice(C, zc+1, lg(C)-1);
4179 63790 : if (flag) SUnits = mkvec4(SUnits, CU, rowslice(Ce, RU+1, nbrows(Ce)),
4180 : utoipos(LIMC));
4181 : }
4182 : /* class group generators */
4183 63791 : if (flag) Ce = rowslice(Ce, 1, RU);
4184 63791 : C0 = Ce; Ce = cleanarch(Ce, N, NULL, PREC);
4185 63791 : if (!Ce) {
4186 0 : long add = nbits2extraprec( gexpo(C0) + 64 ) - gprecision(C0);
4187 0 : precpb = "cleanarch"; PREC += maxss(add, 1);
4188 : }
4189 63791 : if (DEBUGLEVEL) timer_printf(&T, "cleanarch");
4190 120269 : } while (need || precpb);
4191 :
4192 63791 : Vbase = vecpermute(F.LP, F.perm);
4193 63791 : if (!fu) fu = cgetg(1, t_MAT);
4194 63791 : if (!SUnits) SUnits = gen_1;
4195 63791 : clg1 = class_group_gen(nf,W,Ce,Vbase,PREC, &clg2);
4196 63790 : res = mkvec5(clg1, R, SUnits, zu, fu);
4197 63790 : res = buchall_end(nf,res,clg2,W,B,A,Ce,Vbase);
4198 63790 : delete_FB(&F);
4199 63791 : res = gerepilecopy(av0, res);
4200 63791 : if (flag) obj_insert_shallow(res, MATAL, cgetg(1,t_VEC));
4201 63791 : if (nfisclone) gunclone(nf);
4202 63791 : delete_cache(&cache);
4203 63791 : free_GRHcheck(&GRHcheck);
4204 63791 : return res;
4205 : }
|
