Line data Source code
1 : /* Copyright (C) 2000 The PARI group.
2 :
3 : This file is part of the PARI/GP package.
4 :
5 : PARI/GP is free software; you can redistribute it and/or modify it under the
6 : terms of the GNU General Public License as published by the Free Software
7 : Foundation; either version 2 of the License, or (at your option) any later
8 : version. It is distributed in the hope that it will be useful, but WITHOUT
9 : ANY WARRANTY WHATSOEVER.
10 :
11 : Check the License for details. You should have received a copy of it, along
12 : with the package; see the file 'COPYING'. If not, write to the Free Software
13 : Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301 USA. */
14 :
15 : /*******************************************************************/
16 : /* */
17 : /* BASIC NF OPERATIONS */
18 : /* */
19 : /*******************************************************************/
20 : #include "pari.h"
21 : #include "paripriv.h"
22 :
23 : #define DEBUGLEVEL DEBUGLEVEL_nf
24 :
25 : /*******************************************************************/
26 : /* */
27 : /* OPERATIONS OVER NUMBER FIELD ELEMENTS. */
28 : /* represented as column vectors over the integral basis */
29 : /* */
30 : /*******************************************************************/
31 : static GEN
32 34086405 : get_tab(GEN nf, long *N)
33 : {
34 34086405 : GEN tab = (typ(nf) == t_MAT)? nf: gel(nf,9);
35 34086405 : *N = nbrows(tab); return tab;
36 : }
37 :
38 : /* x != 0, y t_INT. Return x * y (not memory clean if x = 1) */
39 : static GEN
40 1110247613 : _mulii(GEN x, GEN y) {
41 1772600725 : return is_pm1(x)? (signe(x) < 0)? negi(y): y
42 1772475877 : : mulii(x, y);
43 : }
44 :
45 : GEN
46 21375 : tablemul_ei_ej(GEN M, long i, long j)
47 : {
48 : long N;
49 21375 : GEN tab = get_tab(M, &N);
50 21375 : tab += (i-1)*N; return gel(tab,j);
51 : }
52 :
53 : /* Outputs x.ei, where ei is the i-th elt of the algebra basis.
54 : * x an RgV of correct length and arbitrary content (polynomials, scalars...).
55 : * M is the multiplication table ei ej = sum_k M_k^(i,j) ek */
56 : GEN
57 11648 : tablemul_ei(GEN M, GEN x, long i)
58 : {
59 : long j, k, N;
60 : GEN v, tab;
61 :
62 11648 : if (i==1) return gcopy(x);
63 11648 : tab = get_tab(M, &N);
64 11648 : if (typ(x) != t_COL) { v = zerocol(N); gel(v,i) = gcopy(x); return v; }
65 11648 : tab += (i-1)*N; v = cgetg(N+1,t_COL);
66 : /* wi . x = [ sum_j tab[k,j] x[j] ]_k */
67 79044 : for (k=1; k<=N; k++)
68 : {
69 67396 : pari_sp av = avma;
70 67396 : GEN s = gen_0;
71 476448 : for (j=1; j<=N; j++)
72 : {
73 409052 : GEN c = gcoeff(tab,k,j);
74 409052 : if (!gequal0(c)) s = gadd(s, gmul(c, gel(x,j)));
75 : }
76 67396 : gel(v,k) = gerepileupto(av,s);
77 : }
78 11648 : return v;
79 : }
80 : /* as tablemul_ei, assume x a ZV of correct length */
81 : GEN
82 24564420 : zk_ei_mul(GEN nf, GEN x, long i)
83 : {
84 : long j, k, N;
85 : GEN v, tab;
86 :
87 24564420 : if (i==1) return ZC_copy(x);
88 24564406 : tab = get_tab(nf, &N); tab += (i-1)*N;
89 24564420 : v = cgetg(N+1,t_COL);
90 185029469 : for (k=1; k<=N; k++)
91 : {
92 160467513 : pari_sp av = avma;
93 160467513 : GEN s = gen_0;
94 2305455800 : for (j=1; j<=N; j++)
95 : {
96 2145275523 : GEN c = gcoeff(tab,k,j);
97 2145275523 : if (signe(c)) s = addii(s, _mulii(c, gel(x,j)));
98 : }
99 160180277 : gel(v,k) = gerepileuptoint(av, s);
100 : }
101 24561956 : return v;
102 : }
103 :
104 : /* table of multiplication by wi in R[w1,..., wN] */
105 : GEN
106 38358 : ei_multable(GEN TAB, long i)
107 : {
108 : long k,N;
109 38358 : GEN m, tab = get_tab(TAB, &N);
110 38358 : tab += (i-1)*N;
111 38358 : m = cgetg(N+1,t_MAT);
112 149290 : for (k=1; k<=N; k++) gel(m,k) = gel(tab,k);
113 38358 : return m;
114 : }
115 :
116 : GEN
117 10204547 : zk_multable(GEN nf, GEN x)
118 : {
119 10204547 : long i, l = lg(x);
120 10204547 : GEN mul = cgetg(l,t_MAT);
121 10204366 : gel(mul,1) = x; /* assume w_1 = 1 */
122 34413373 : for (i=2; i<l; i++) gel(mul,i) = zk_ei_mul(nf,x,i);
123 10202048 : return mul;
124 : }
125 : GEN
126 2566 : multable(GEN M, GEN x)
127 : {
128 : long i, N;
129 : GEN mul;
130 2566 : if (typ(x) == t_MAT) return x;
131 0 : M = get_tab(M, &N);
132 0 : if (typ(x) != t_COL) return scalarmat(x, N);
133 0 : mul = cgetg(N+1,t_MAT);
134 0 : gel(mul,1) = x; /* assume w_1 = 1 */
135 0 : for (i=2; i<=N; i++) gel(mul,i) = tablemul_ei(M,x,i);
136 0 : return mul;
137 : }
138 :
139 : /* x integral in nf; table of multiplication by x in ZK = Z[w1,..., wN].
140 : * Return a t_INT if x is scalar, and a ZM otherwise */
141 : GEN
142 4788682 : zk_scalar_or_multable(GEN nf, GEN x)
143 : {
144 4788682 : long tx = typ(x);
145 4788682 : if (tx == t_MAT || tx == t_INT) return x;
146 4683963 : x = nf_to_scalar_or_basis(nf, x);
147 4683814 : return (typ(x) == t_COL)? zk_multable(nf, x): x;
148 : }
149 :
150 : GEN
151 21406 : nftrace(GEN nf, GEN x)
152 : {
153 21406 : pari_sp av = avma;
154 21406 : nf = checknf(nf);
155 21406 : x = nf_to_scalar_or_basis(nf, x);
156 21390 : x = (typ(x) == t_COL)? RgV_dotproduct(x, gel(nf_get_Tr(nf),1))
157 21411 : : gmulgu(x, nf_get_degree(nf));
158 21411 : return gerepileupto(av, x);
159 : }
160 : GEN
161 791 : rnfelttrace(GEN rnf, GEN x)
162 : {
163 791 : pari_sp av = avma;
164 791 : checkrnf(rnf);
165 784 : x = rnfeltabstorel(rnf, x);
166 616 : x = (typ(x) == t_POLMOD)? rnfeltdown(rnf, gtrace(x))
167 693 : : gmulgu(x, rnf_get_degree(rnf));
168 693 : return gerepileupto(av, x);
169 : }
170 :
171 : static GEN
172 35 : famatQ_to_famatZ(GEN fa)
173 : {
174 35 : GEN E, F, Q, P = gel(fa,1);
175 35 : long i, j, l = lg(P);
176 35 : if (l == 1 || RgV_is_ZV(P)) return fa;
177 7 : Q = cgetg(2*l, t_COL);
178 7 : F = cgetg(2*l, t_COL); E = gel(fa, 2);
179 35 : for (i = j = 1; i < l; i++)
180 : {
181 28 : GEN p = gel(P,i);
182 28 : if (typ(p) == t_INT)
183 14 : { gel(Q, j) = p; gel(F, j) = gel(E, i); j++; }
184 : else
185 : {
186 14 : gel(Q, j) = gel(p,1); gel(F, j) = gel(E, i); j++;
187 14 : gel(Q, j) = gel(p,2); gel(F, j) = negi(gel(E, i)); j++;
188 : }
189 : }
190 7 : setlg(Q, j); setlg(F, j); return mkmat2(Q, F);
191 : }
192 : static GEN
193 35 : famat_cba(GEN fa)
194 : {
195 35 : GEN Q, F, P = gel(fa, 1), E = gel(fa, 2);
196 35 : long i, j, lQ, l = lg(P);
197 35 : if (l == 1) return fa;
198 28 : Q = ZV_cba(P); lQ = lg(Q);
199 28 : F = cgetg(lQ, t_COL);
200 77 : for (j = 1; j < lQ; j++)
201 : {
202 49 : GEN v = gen_0, q = gel(Q,j);
203 49 : if (!equali1(q))
204 203 : for (i = 1; i < l; i++)
205 : {
206 161 : long e = Z_pval(gel(P,i), q);
207 161 : if (e) v = addii(v, muliu(gel(E,i), e));
208 : }
209 49 : gel(F, j) = v;
210 : }
211 28 : return mkmat2(Q, F);
212 : }
213 : static long
214 35 : famat_sign(GEN fa)
215 : {
216 35 : GEN P = gel(fa,1), E = gel(fa,2);
217 35 : long i, l = lg(P), s = 1;
218 126 : for (i = 1; i < l; i++)
219 91 : if (signe(gel(P,i)) < 0 && mpodd(gel(E,i))) s = -s;
220 35 : return s;
221 : }
222 : static GEN
223 35 : famat_abs(GEN fa)
224 : {
225 35 : GEN Q, P = gel(fa,1);
226 : long i, l;
227 35 : Q = cgetg_copy(P, &l);
228 126 : for (i = 1; i < l; i++) gel(Q,i) = absi_shallow(gel(P,i));
229 35 : return mkmat2(Q, gel(fa,2));
230 : }
231 :
232 : /* assume nf is a genuine nf, fa a famat */
233 : static GEN
234 35 : famat_norm(GEN nf, GEN fa)
235 : {
236 35 : pari_sp av = avma;
237 35 : GEN G, g = gel(fa,1);
238 : long i, l, s;
239 :
240 35 : G = cgetg_copy(g, &l);
241 112 : for (i = 1; i < l; i++) gel(G,i) = nfnorm(nf, gel(g,i));
242 35 : fa = mkmat2(G, gel(fa,2));
243 35 : fa = famatQ_to_famatZ(fa);
244 35 : s = famat_sign(fa);
245 35 : fa = famat_reduce(famat_abs(fa));
246 35 : fa = famat_cba(fa);
247 35 : g = factorback(fa);
248 35 : return gerepileupto(av, s < 0? gneg(g): g);
249 : }
250 : GEN
251 100113 : nfnorm(GEN nf, GEN x)
252 : {
253 100113 : pari_sp av = avma;
254 : GEN c, den;
255 : long n;
256 100113 : nf = checknf(nf);
257 100113 : n = nf_get_degree(nf);
258 100113 : if (typ(x) == t_MAT) return famat_norm(nf, x);
259 100078 : x = nf_to_scalar_or_basis(nf, x);
260 100077 : if (typ(x)!=t_COL)
261 12316 : return gerepileupto(av, gpowgs(x, n));
262 87761 : x = nf_to_scalar_or_alg(nf, Q_primitive_part(x, &c));
263 87761 : x = Q_remove_denom(x, &den);
264 87761 : x = ZX_resultant_all(nf_get_pol(nf), x, den, 0);
265 87762 : return gerepileupto(av, c ? gmul(x, gpowgs(c, n)): x);
266 : }
267 :
268 : static GEN
269 70 : to_RgX(GEN P, long vx)
270 : {
271 70 : return varn(P) == vx ? P: scalarpol_shallow(P, vx);
272 : }
273 :
274 : GEN
275 238 : rnfeltnorm(GEN rnf, GEN x)
276 : {
277 238 : pari_sp av = avma;
278 : GEN nf, pol;
279 : long v;
280 238 : checkrnf(rnf);
281 231 : v = rnf_get_varn(rnf);
282 231 : x = liftpol_shallow(rnfeltabstorel(rnf, x));
283 140 : nf = rnf_get_nf(rnf); pol = rnf_get_pol(rnf);
284 280 : x = (typ(x) == t_POL)
285 70 : ? rnfeltdown(rnf, nfX_resultant(nf,pol,to_RgX(x,v)))
286 140 : : gpowgs(x, rnf_get_degree(rnf));
287 140 : return gerepileupto(av, x);
288 : }
289 :
290 : /* x + y in nf */
291 : GEN
292 19123661 : nfadd(GEN nf, GEN x, GEN y)
293 : {
294 19123661 : pari_sp av = avma;
295 : GEN z;
296 :
297 19123661 : nf = checknf(nf);
298 19123661 : x = nf_to_scalar_or_basis(nf, x);
299 19123661 : y = nf_to_scalar_or_basis(nf, y);
300 19123661 : if (typ(x) != t_COL)
301 15375795 : { z = (typ(y) == t_COL)? RgC_Rg_add(y, x): gadd(x,y); }
302 : else
303 3747866 : { z = (typ(y) == t_COL)? RgC_add(x, y): RgC_Rg_add(x, y); }
304 19123660 : return gerepileupto(av, z);
305 : }
306 : /* x - y in nf */
307 : GEN
308 1329204 : nfsub(GEN nf, GEN x, GEN y)
309 : {
310 1329204 : pari_sp av = avma;
311 : GEN z;
312 :
313 1329204 : nf = checknf(nf);
314 1329204 : x = nf_to_scalar_or_basis(nf, x);
315 1329204 : y = nf_to_scalar_or_basis(nf, y);
316 1329204 : if (typ(x) != t_COL)
317 940909 : { z = (typ(y) == t_COL)? Rg_RgC_sub(x,y): gsub(x,y); }
318 : else
319 388295 : { z = (typ(y) == t_COL)? RgC_sub(x,y): RgC_Rg_sub(x,y); }
320 1329204 : return gerepileupto(av, z);
321 : }
322 :
323 : /* product of ZC x,y in (true) nf; ( sum_i x_i sum_j y_j m^{i,j}_k )_k */
324 : static GEN
325 4633068 : nfmuli_ZC(GEN nf, GEN x, GEN y)
326 : {
327 : long i, j, k, N;
328 4633068 : GEN TAB = get_tab(nf, &N), v = cgetg(N+1,t_COL);
329 :
330 25139467 : for (k = 1; k <= N; k++)
331 : {
332 20506445 : pari_sp av = avma;
333 20506445 : GEN s, TABi = TAB;
334 20506445 : if (k == 1)
335 4633053 : s = mulii(gel(x,1),gel(y,1));
336 : else
337 15873173 : s = addii(mulii(gel(x,1),gel(y,k)),
338 15873392 : mulii(gel(x,k),gel(y,1)));
339 158450040 : for (i=2; i<=N; i++)
340 : {
341 137951043 : GEN t, xi = gel(x,i);
342 137951043 : TABi += N;
343 137951043 : if (!signe(xi)) continue;
344 :
345 68267568 : t = NULL;
346 869900737 : for (j=2; j<=N; j++)
347 : {
348 801639574 : GEN p1, c = gcoeff(TABi, k, j); /* m^{i,j}_k */
349 801639574 : if (!signe(c)) continue;
350 236511579 : p1 = _mulii(c, gel(y,j));
351 236522112 : t = t? addii(t, p1): p1;
352 : }
353 68261163 : if (t) s = addii(s, mulii(xi, t));
354 : }
355 20498997 : gel(v,k) = gerepileuptoint(av,s);
356 : }
357 4633022 : return v;
358 : }
359 : static int
360 50817039 : is_famat(GEN x) { return typ(x) == t_MAT && lg(x) == 3; }
361 : /* product of x and y in nf */
362 : GEN
363 25955023 : nfmul(GEN nf, GEN x, GEN y)
364 : {
365 : GEN z;
366 25955023 : pari_sp av = avma;
367 :
368 25955023 : if (x == y) return nfsqr(nf,x);
369 :
370 21874335 : nf = checknf(nf);
371 21874335 : if (is_famat(x) || is_famat(y)) return famat_mul(x, y);
372 21874005 : x = nf_to_scalar_or_basis(nf, x);
373 21874005 : y = nf_to_scalar_or_basis(nf, y);
374 21874006 : if (typ(x) != t_COL)
375 : {
376 15848049 : if (isintzero(x)) return gen_0;
377 10622877 : z = (typ(y) == t_COL)? RgC_Rg_mul(y, x): gmul(x,y); }
378 : else
379 : {
380 6025957 : if (typ(y) != t_COL)
381 : {
382 4196233 : if (isintzero(y)) return gen_0;
383 1334536 : z = RgC_Rg_mul(x, y);
384 : }
385 : else
386 : {
387 : GEN dx, dy;
388 1829724 : x = Q_remove_denom(x, &dx);
389 1829728 : y = Q_remove_denom(y, &dy);
390 1829728 : z = nfmuli_ZC(nf,x,y);
391 1829726 : dx = mul_denom(dx,dy);
392 1829727 : if (dx) z = ZC_Z_div(z, dx);
393 : }
394 : }
395 13787135 : return gerepileupto(av, z);
396 : }
397 : /* square of ZC x in nf */
398 : static GEN
399 4819652 : nfsqri_ZC(GEN nf, GEN x)
400 : {
401 : long i, j, k, N;
402 4819652 : GEN TAB = get_tab(nf, &N), v = cgetg(N+1,t_COL);
403 :
404 29379275 : for (k = 1; k <= N; k++)
405 : {
406 24559704 : pari_sp av = avma;
407 24559704 : GEN s, TABi = TAB;
408 24559704 : if (k == 1)
409 4819709 : s = sqri(gel(x,1));
410 : else
411 19739995 : s = shifti(mulii(gel(x,1),gel(x,k)), 1);
412 217565606 : for (i=2; i<=N; i++)
413 : {
414 193029185 : GEN p1, c, t, xi = gel(x,i);
415 193029185 : TABi += N;
416 193029185 : if (!signe(xi)) continue;
417 :
418 68462330 : c = gcoeff(TABi, k, i);
419 68462330 : t = signe(c)? _mulii(c,xi): NULL;
420 616399514 : for (j=i+1; j<=N; j++)
421 : {
422 547937166 : c = gcoeff(TABi, k, j);
423 547937166 : if (!signe(c)) continue;
424 225625690 : p1 = _mulii(c, shifti(gel(x,j),1));
425 225633432 : t = t? addii(t, p1): p1;
426 : }
427 68462348 : if (t) s = addii(s, mulii(xi, t));
428 : }
429 24536421 : gel(v,k) = gerepileuptoint(av,s);
430 : }
431 4819571 : return v;
432 : }
433 : /* square of x in nf */
434 : GEN
435 6108071 : nfsqr(GEN nf, GEN x)
436 : {
437 6108071 : pari_sp av = avma;
438 : GEN z;
439 :
440 6108071 : nf = checknf(nf);
441 6108074 : if (is_famat(x)) return famat_sqr(x);
442 6108073 : x = nf_to_scalar_or_basis(nf, x);
443 6108073 : if (typ(x) != t_COL) z = gsqr(x);
444 : else
445 : {
446 : GEN dx;
447 244167 : x = Q_remove_denom(x, &dx);
448 244168 : z = nfsqri_ZC(nf,x);
449 244174 : if (dx) z = RgC_Rg_div(z, sqri(dx));
450 : }
451 6108080 : return gerepileupto(av, z);
452 : }
453 :
454 : /* x a ZC, v a t_COL of ZC/Z */
455 : GEN
456 208061 : zkC_multable_mul(GEN v, GEN x)
457 : {
458 208061 : long i, l = lg(v);
459 208061 : GEN y = cgetg(l, t_COL);
460 805222 : for (i = 1; i < l; i++)
461 : {
462 597161 : GEN c = gel(v,i);
463 597161 : if (typ(c)!=t_COL) {
464 0 : if (!isintzero(c)) c = ZC_Z_mul(gel(x,1), c);
465 : } else {
466 597161 : c = ZM_ZC_mul(x,c);
467 597161 : if (ZV_isscalar(c)) c = gel(c,1);
468 : }
469 597161 : gel(y,i) = c;
470 : }
471 208061 : return y;
472 : }
473 :
474 : GEN
475 55786 : nfC_multable_mul(GEN v, GEN x)
476 : {
477 55786 : long i, l = lg(v);
478 55786 : GEN y = cgetg(l, t_COL);
479 378225 : for (i = 1; i < l; i++)
480 : {
481 322439 : GEN c = gel(v,i);
482 322439 : if (typ(c)!=t_COL) {
483 269574 : if (!isintzero(c)) c = RgC_Rg_mul(gel(x,1), c);
484 : } else {
485 52865 : c = RgM_RgC_mul(x,c);
486 52865 : if (QV_isscalar(c)) c = gel(c,1);
487 : }
488 322439 : gel(y,i) = c;
489 : }
490 55786 : return y;
491 : }
492 :
493 : GEN
494 196876 : nfC_nf_mul(GEN nf, GEN v, GEN x)
495 : {
496 : long tx;
497 : GEN y;
498 :
499 196876 : x = nf_to_scalar_or_basis(nf, x);
500 196876 : tx = typ(x);
501 196876 : if (tx != t_COL)
502 : {
503 : long l, i;
504 149273 : if (tx == t_INT)
505 : {
506 140342 : long s = signe(x);
507 140342 : if (!s) return zerocol(lg(v)-1);
508 132936 : if (is_pm1(x)) return s > 0? leafcopy(v): RgC_neg(v);
509 : }
510 48078 : l = lg(v); y = cgetg(l, t_COL);
511 346503 : for (i=1; i < l; i++)
512 : {
513 298425 : GEN c = gel(v,i);
514 298425 : if (typ(c) != t_COL) c = gmul(c, x); else c = RgC_Rg_mul(c, x);
515 298425 : gel(y,i) = c;
516 : }
517 48078 : return y;
518 : }
519 : else
520 : {
521 : GEN dx;
522 47603 : x = zk_multable(nf, Q_remove_denom(x,&dx));
523 47603 : y = nfC_multable_mul(v, x);
524 47603 : return dx? RgC_Rg_div(y, dx): y;
525 : }
526 : }
527 : static GEN
528 10756 : mulbytab(GEN M, GEN c)
529 10756 : { return typ(c) == t_COL? RgM_RgC_mul(M,c): RgC_Rg_mul(gel(M,1), c); }
530 : GEN
531 2566 : tablemulvec(GEN M, GEN x, GEN v)
532 : {
533 : long l, i;
534 : GEN y;
535 :
536 2566 : if (typ(x) == t_COL && RgV_isscalar(x))
537 : {
538 0 : x = gel(x,1);
539 0 : return typ(v) == t_POL? RgX_Rg_mul(v,x): RgV_Rg_mul(v,x);
540 : }
541 2566 : x = multable(M, x); /* multiplication table by x */
542 2566 : y = cgetg_copy(v, &l);
543 2566 : if (typ(v) == t_POL)
544 : {
545 2566 : y[1] = v[1];
546 13322 : for (i=2; i < l; i++) gel(y,i) = mulbytab(x, gel(v,i));
547 2566 : y = normalizepol(y);
548 : }
549 : else
550 : {
551 0 : for (i=1; i < l; i++) gel(y,i) = mulbytab(x, gel(v,i));
552 : }
553 2566 : return y;
554 : }
555 :
556 : GEN
557 1255814 : zkmultable_capZ(GEN mx) { return Q_denom(zkmultable_inv(mx)); }
558 : GEN
559 1413240 : zkmultable_inv(GEN mx) { return ZM_gauss(mx, col_ei(lg(mx)-1,1)); }
560 : /* nf a true nf, x a ZC */
561 : GEN
562 157427 : zk_inv(GEN nf, GEN x) { return zkmultable_inv(zk_multable(nf,x)); }
563 :
564 : /* inverse of x in nf */
565 : GEN
566 83674 : nfinv(GEN nf, GEN x)
567 : {
568 83674 : pari_sp av = avma;
569 : GEN z;
570 :
571 83674 : nf = checknf(nf);
572 83674 : if (is_famat(x)) return famat_inv(x);
573 83674 : x = nf_to_scalar_or_basis(nf, x);
574 83674 : if (typ(x) == t_COL)
575 : {
576 : GEN d;
577 35942 : x = Q_remove_denom(x, &d);
578 35942 : z = zk_inv(nf, x);
579 35942 : if (d) z = RgC_Rg_mul(z, d);
580 : }
581 : else
582 47732 : z = ginv(x);
583 83674 : return gerepileupto(av, z);
584 : }
585 :
586 : /* quotient of x and y in nf */
587 : GEN
588 34315 : nfdiv(GEN nf, GEN x, GEN y)
589 : {
590 34315 : pari_sp av = avma;
591 : GEN z;
592 :
593 34315 : nf = checknf(nf);
594 34315 : if (is_famat(x) || is_famat(y)) return famat_div(x,y);
595 34224 : y = nf_to_scalar_or_basis(nf, y);
596 34224 : if (typ(y) != t_COL)
597 : {
598 21945 : x = nf_to_scalar_or_basis(nf, x);
599 21945 : z = (typ(x) == t_COL)? RgC_Rg_div(x, y): gdiv(x,y);
600 : }
601 : else
602 : {
603 : GEN d;
604 12279 : y = Q_remove_denom(y, &d);
605 12279 : z = nfmul(nf, x, zk_inv(nf,y));
606 12279 : if (d) z = typ(z) == t_COL? RgC_Rg_mul(z, d): gmul(z, d);
607 : }
608 34224 : return gerepileupto(av, z);
609 : }
610 :
611 : /* product of INTEGERS (t_INT or ZC) x and y in (true) nf */
612 : GEN
613 4131708 : nfmuli(GEN nf, GEN x, GEN y)
614 : {
615 4131708 : if (typ(x) == t_INT) return (typ(y) == t_COL)? ZC_Z_mul(y, x): mulii(x,y);
616 3025313 : if (typ(y) == t_INT) return ZC_Z_mul(x, y);
617 2803305 : return nfmuli_ZC(nf, x, y);
618 : }
619 : GEN
620 4575420 : nfsqri(GEN nf, GEN x)
621 4575420 : { return (typ(x) == t_INT)? sqri(x): nfsqri_ZC(nf, x); }
622 :
623 : /* both x and y are RgV */
624 : GEN
625 0 : tablemul(GEN TAB, GEN x, GEN y)
626 : {
627 : long i, j, k, N;
628 : GEN s, v;
629 0 : if (typ(x) != t_COL) return gmul(x, y);
630 0 : if (typ(y) != t_COL) return gmul(y, x);
631 0 : N = lg(x)-1;
632 0 : v = cgetg(N+1,t_COL);
633 0 : for (k=1; k<=N; k++)
634 : {
635 0 : pari_sp av = avma;
636 0 : GEN TABi = TAB;
637 0 : if (k == 1)
638 0 : s = gmul(gel(x,1),gel(y,1));
639 : else
640 0 : s = gadd(gmul(gel(x,1),gel(y,k)),
641 0 : gmul(gel(x,k),gel(y,1)));
642 0 : for (i=2; i<=N; i++)
643 : {
644 0 : GEN t, xi = gel(x,i);
645 0 : TABi += N;
646 0 : if (gequal0(xi)) continue;
647 :
648 0 : t = NULL;
649 0 : for (j=2; j<=N; j++)
650 : {
651 0 : GEN p1, c = gcoeff(TABi, k, j); /* m^{i,j}_k */
652 0 : if (gequal0(c)) continue;
653 0 : p1 = gmul(c, gel(y,j));
654 0 : t = t? gadd(t, p1): p1;
655 : }
656 0 : if (t) s = gadd(s, gmul(xi, t));
657 : }
658 0 : gel(v,k) = gerepileupto(av,s);
659 : }
660 0 : return v;
661 : }
662 : GEN
663 49428 : tablesqr(GEN TAB, GEN x)
664 : {
665 : long i, j, k, N;
666 : GEN s, v;
667 :
668 49428 : if (typ(x) != t_COL) return gsqr(x);
669 49428 : N = lg(x)-1;
670 49428 : v = cgetg(N+1,t_COL);
671 :
672 352154 : for (k=1; k<=N; k++)
673 : {
674 302726 : pari_sp av = avma;
675 302726 : GEN TABi = TAB;
676 302726 : if (k == 1)
677 49428 : s = gsqr(gel(x,1));
678 : else
679 253298 : s = gmul2n(gmul(gel(x,1),gel(x,k)), 1);
680 1926274 : for (i=2; i<=N; i++)
681 : {
682 1623548 : GEN p1, c, t, xi = gel(x,i);
683 1623548 : TABi += N;
684 1623548 : if (gequal0(xi)) continue;
685 :
686 416606 : c = gcoeff(TABi, k, i);
687 416606 : t = !gequal0(c)? gmul(c,xi): NULL;
688 1663370 : for (j=i+1; j<=N; j++)
689 : {
690 1246764 : c = gcoeff(TABi, k, j);
691 1246764 : if (gequal0(c)) continue;
692 647626 : p1 = gmul(gmul2n(c,1), gel(x,j));
693 647626 : t = t? gadd(t, p1): p1;
694 : }
695 416606 : if (t) s = gadd(s, gmul(xi, t));
696 : }
697 302726 : gel(v,k) = gerepileupto(av,s);
698 : }
699 49428 : return v;
700 : }
701 :
702 : static GEN
703 197737 : _mul(void *data, GEN x, GEN y) { return nfmuli((GEN)data,x,y); }
704 : static GEN
705 713226 : _sqr(void *data, GEN x) { return nfsqri((GEN)data,x); }
706 :
707 : /* Compute z^n in nf, left-shift binary powering */
708 : GEN
709 793391 : nfpow(GEN nf, GEN z, GEN n)
710 : {
711 793391 : pari_sp av = avma;
712 : long s;
713 : GEN x, cx;
714 :
715 793391 : if (typ(n)!=t_INT) pari_err_TYPE("nfpow",n);
716 793391 : nf = checknf(nf);
717 793394 : s = signe(n); if (!s) return gen_1;
718 793394 : if (is_famat(z)) return famat_pow(z, n);
719 732732 : x = nf_to_scalar_or_basis(nf, z);
720 732730 : if (typ(x) != t_COL) return powgi(x,n);
721 616085 : if (s < 0)
722 : { /* simplified nfinv */
723 : GEN d;
724 41040 : x = Q_remove_denom(x, &d);
725 41040 : x = zk_inv(nf, x);
726 41040 : x = primitive_part(x, &cx);
727 41040 : cx = mul_content(cx, d);
728 41040 : n = negi(n);
729 : }
730 : else
731 575045 : x = primitive_part(x, &cx);
732 616074 : x = gen_pow_i(x, n, (void*)nf, _sqr, _mul);
733 616081 : if (cx)
734 45441 : x = gerepileupto(av, gmul(x, powgi(cx, n)));
735 : else
736 570640 : x = gerepilecopy(av, x);
737 616086 : return x;
738 : }
739 : /* Compute z^n in nf, left-shift binary powering */
740 : GEN
741 223007 : nfpow_u(GEN nf, GEN z, ulong n)
742 : {
743 223007 : pari_sp av = avma;
744 : GEN x, cx;
745 :
746 223007 : if (!n) return gen_1;
747 223007 : x = nf_to_scalar_or_basis(nf, z);
748 223007 : if (typ(x) != t_COL) return gpowgs(x,n);
749 186760 : x = primitive_part(x, &cx);
750 186759 : x = gen_powu_i(x, n, (void*)nf, _sqr, _mul);
751 186760 : if (cx)
752 : {
753 27506 : x = gmul(x, powgi(cx, utoipos(n)));
754 27506 : return gerepileupto(av,x);
755 : }
756 159254 : return gerepilecopy(av, x);
757 : }
758 :
759 : long
760 56 : nfissquare(GEN nf, GEN z, GEN *px)
761 : {
762 56 : pari_sp av = avma;
763 56 : long v = fetch_var_higher();
764 : GEN R;
765 56 : nf = checknf(nf);
766 56 : if (nf_get_degree(nf) == 1)
767 : {
768 21 : z = algtobasis(nf, z);
769 21 : if (!issquareall(gel(z,1), px)) return gc_long(av, 0);
770 14 : if (px) *px = gerepileupto(av, *px); else set_avma(av);
771 14 : return 1;
772 : }
773 35 : z = nf_to_scalar_or_alg(nf, z);
774 35 : R = nfroots(nf, deg2pol_shallow(gen_m1, gen_0, z, v));
775 35 : delete_var(); if (lg(R) == 1) return gc_long(av, 0);
776 28 : if (px) *px = gerepilecopy(av, nf_to_scalar_or_basis(nf, gel(R,1)));
777 14 : else set_avma(av);
778 28 : return 1;
779 : }
780 :
781 : long
782 6406 : nfispower(GEN nf, GEN z, long n, GEN *px)
783 : {
784 6406 : pari_sp av = avma;
785 6406 : long v = fetch_var_higher();
786 : GEN R;
787 6406 : nf = checknf(nf);
788 6406 : if (nf_get_degree(nf) == 1)
789 : {
790 329 : z = algtobasis(nf, z);
791 329 : if (!ispower(gel(z,1), stoi(n), px)) return gc_long(av, 0);
792 147 : if (px) *px = gerepileupto(av, *px); else set_avma(av);
793 147 : return 1;
794 : }
795 6077 : if (n <= 0)
796 0 : pari_err_DOMAIN("nfeltispower","exponent","<=",gen_0,stoi(n));
797 6077 : z = nf_to_scalar_or_alg(nf, z);
798 6077 : if (n==1)
799 : {
800 0 : if (px) *px = gerepilecopy(av, z);
801 0 : return 1;
802 : }
803 6077 : R = nfroots(nf, gsub(pol_xn(n, v), z));
804 6077 : delete_var(); if (lg(R) == 1) return gc_long(av, 0);
805 1428 : if (px) *px = gerepilecopy(av, nf_to_scalar_or_basis(nf, gel(R,1)));
806 1414 : else set_avma(av);
807 1428 : return 1;
808 : }
809 :
810 : static GEN
811 56 : idmulred(void *nf, GEN x, GEN y) { return idealmulred((GEN) nf, x, y); }
812 : static GEN
813 413 : idpowred(void *nf, GEN x, GEN n) { return idealpowred((GEN) nf, x, n); }
814 : static GEN
815 71003 : idmul(void *nf, GEN x, GEN y) { return idealmul((GEN) nf, x, y); }
816 : static GEN
817 86851 : idpow(void *nf, GEN x, GEN n) { return idealpow((GEN) nf, x, n); }
818 : GEN
819 85210 : idealfactorback(GEN nf, GEN L, GEN e, int red)
820 : {
821 85210 : nf = checknf(nf);
822 85210 : if (red) return gen_factorback(L, e, (void*)nf, &idmulred, &idpowred, NULL);
823 84853 : if (!e && typ(L) == t_MAT && lg(L) == 3) { e = gel(L,2); L = gel(L,1); }
824 84853 : if (is_vec_t(typ(L)) && RgV_is_prV(L))
825 : { /* don't use gen_factorback since *= pr^v can be done more efficiently */
826 64350 : pari_sp av = avma;
827 64350 : long i, l = lg(L);
828 : GEN a;
829 64350 : if (!e) e = const_vec(l-1, gen_1);
830 61494 : else switch(typ(e))
831 : {
832 7518 : case t_VECSMALL: e = zv_to_ZV(e); break;
833 53976 : case t_VEC: case t_COL:
834 53976 : if (!RgV_is_ZV(e))
835 0 : pari_err_TYPE("factorback [not an exponent vector]", e);
836 53976 : break;
837 0 : default: pari_err_TYPE("idealfactorback", e);
838 : }
839 64350 : if (l != lg(e))
840 0 : pari_err_TYPE("factorback [not an exponent vector]", e);
841 64350 : if (l == 1 || ZV_equal0(e)) return gc_const(av, gen_1);
842 22993 : a = idealpow(nf, gel(L,1), gel(e,1));
843 232922 : for (i = 2; i < l; i++)
844 209929 : if (signe(gel(e,i))) a = idealmulpowprime(nf, a, gel(L,i), gel(e,i));
845 22993 : return gerepileupto(av, a);
846 : }
847 20503 : return gen_factorback(L, e, (void*)nf, &idmul, &idpow, NULL);
848 : }
849 : static GEN
850 297694 : eltmul(void *nf, GEN x, GEN y) { return nfmul((GEN) nf, x, y); }
851 : static GEN
852 434604 : eltpow(void *nf, GEN x, GEN n) { return nfpow((GEN) nf, x, n); }
853 : GEN
854 264947 : nffactorback(GEN nf, GEN L, GEN e)
855 264947 : { return gen_factorback(L, e, (void*)checknf(nf), &eltmul, &eltpow, NULL); }
856 :
857 : static GEN
858 3052804 : _nf_red(void *E, GEN x) { (void)E; return gcopy(x); }
859 :
860 : static GEN
861 12617129 : _nf_add(void *E, GEN x, GEN y) { return nfadd((GEN)E,x,y); }
862 :
863 : static GEN
864 743222 : _nf_neg(void *E, GEN x) { (void)E; return gneg(x); }
865 :
866 : static GEN
867 15128873 : _nf_mul(void *E, GEN x, GEN y) { return nfmul((GEN)E,x,y); }
868 :
869 : static GEN
870 51916 : _nf_inv(void *E, GEN x) { return nfinv((GEN)E,x); }
871 :
872 : static GEN
873 10524 : _nf_s(void *E, long x) { (void)E; return stoi(x); }
874 :
875 : static const struct bb_field nf_field={_nf_red,_nf_add,_nf_mul,_nf_neg,
876 : _nf_inv,&gequal0,_nf_s };
877 :
878 222449 : const struct bb_field *get_nf_field(void **E, GEN nf)
879 222449 : { *E = (void*)nf; return &nf_field; }
880 :
881 : GEN
882 14 : nfM_det(GEN nf, GEN M)
883 : {
884 : void *E;
885 14 : const struct bb_field *S = get_nf_field(&E, nf);
886 14 : return gen_det(M, E, S);
887 : }
888 : GEN
889 10510 : nfM_inv(GEN nf, GEN M)
890 : {
891 : void *E;
892 10510 : const struct bb_field *S = get_nf_field(&E, nf);
893 10510 : return gen_Gauss(M, matid(lg(M)-1), E, S);
894 : }
895 :
896 : GEN
897 0 : nfM_ker(GEN nf, GEN M)
898 : {
899 : void *E;
900 0 : const struct bb_field *S = get_nf_field(&E, nf);
901 0 : return gen_ker(M, 0, E, S);
902 : }
903 :
904 : GEN
905 10090 : nfM_mul(GEN nf, GEN A, GEN B)
906 : {
907 : void *E;
908 10090 : const struct bb_field *S = get_nf_field(&E, nf);
909 10090 : return gen_matmul(A, B, E, S);
910 : }
911 : GEN
912 201835 : nfM_nfC_mul(GEN nf, GEN A, GEN B)
913 : {
914 : void *E;
915 201835 : const struct bb_field *S = get_nf_field(&E, nf);
916 201835 : return gen_matcolmul(A, B, E, S);
917 : }
918 :
919 : /* valuation of integral x (ZV), with resp. to prime ideal pr */
920 : long
921 37148244 : ZC_nfvalrem(GEN x, GEN pr, GEN *newx)
922 : {
923 37148244 : pari_sp av = avma;
924 : long i, v, l;
925 37148244 : GEN r, y, p = pr_get_p(pr), mul = pr_get_tau(pr);
926 :
927 : /* p inert */
928 37148442 : if (typ(mul) == t_INT) return newx? ZV_pvalrem(x, p, newx):ZV_pval(x, p);
929 36166398 : y = cgetg_copy(x, &l); /* will hold the new x */
930 36166506 : x = leafcopy(x);
931 51731431 : for(v=0;; v++)
932 : {
933 172363673 : for (i=1; i<l; i++)
934 : { /* is (x.b)[i] divisible by p ? */
935 156793886 : gel(y,i) = dvmdii(ZMrow_ZC_mul(mul,x,i),p,&r);
936 156795594 : if (r != gen_0) { if (newx) *newx = x; return v; }
937 : }
938 15569787 : swap(x, y);
939 15569787 : if (!newx && (v & 0xf) == 0xf) v += pr_get_e(pr) * ZV_pvalrem(x, p, &x);
940 15569787 : if (gc_needed(av,1))
941 : {
942 0 : if(DEBUGMEM>1) pari_warn(warnmem,"ZC_nfvalrem, v >= %ld", v);
943 0 : gerepileall(av, 2, &x, &y);
944 : }
945 : }
946 : }
947 : long
948 32996467 : ZC_nfval(GEN x, GEN P)
949 32996467 : { return ZC_nfvalrem(x, P, NULL); }
950 :
951 : /* v_P(x) != 0, x a ZV. Simpler version of ZC_nfvalrem */
952 : int
953 1242964 : ZC_prdvd(GEN x, GEN P)
954 : {
955 1242964 : pari_sp av = avma;
956 : long i, l;
957 1242964 : GEN p = pr_get_p(P), mul = pr_get_tau(P);
958 1242978 : if (typ(mul) == t_INT) return ZV_Z_dvd(x, p);
959 1242607 : l = lg(x);
960 5037922 : for (i=1; i<l; i++)
961 4524940 : if (!dvdii(ZMrow_ZC_mul(mul,x,i), p)) return gc_bool(av,0);
962 512982 : return gc_bool(av,1);
963 : }
964 :
965 : int
966 294 : pr_equal(GEN P, GEN Q)
967 : {
968 294 : GEN gQ, p = pr_get_p(P);
969 294 : long e = pr_get_e(P), f = pr_get_f(P), n;
970 294 : if (!equalii(p, pr_get_p(Q)) || e != pr_get_e(Q) || f != pr_get_f(Q))
971 273 : return 0;
972 21 : gQ = pr_get_gen(Q); n = lg(gQ)-1;
973 21 : if (2*e*f > n) return 1; /* room for only one such pr */
974 14 : return ZV_equal(pr_get_gen(P), gQ) || ZC_prdvd(gQ, P);
975 : }
976 :
977 : GEN
978 420658 : famat_nfvalrem(GEN nf, GEN x, GEN pr, GEN *py)
979 : {
980 420658 : pari_sp av = avma;
981 420658 : GEN P = gel(x,1), E = gel(x,2), V = gen_0, y = NULL;
982 420658 : long l = lg(P), simplify = 0, i;
983 420658 : if (py) { *py = gen_1; y = cgetg(l, t_COL); }
984 :
985 2355914 : for (i = 1; i < l; i++)
986 : {
987 1935256 : GEN e = gel(E,i);
988 : long v;
989 1935256 : if (!signe(e))
990 : {
991 7 : if (py) gel(y,i) = gen_1;
992 7 : simplify = 1; continue;
993 : }
994 1935249 : v = nfvalrem(nf, gel(P,i), pr, py? &gel(y,i): NULL);
995 1935249 : if (v == LONG_MAX) { set_avma(av); if (py) *py = gen_0; return mkoo(); }
996 1935249 : V = addmulii(V, stoi(v), e);
997 : }
998 420658 : if (!py) V = gerepileuptoint(av, V);
999 : else
1000 : {
1001 42 : y = mkmat2(y, gel(x,2));
1002 42 : if (simplify) y = famat_remove_trivial(y);
1003 42 : gerepileall(av, 2, &V, &y); *py = y;
1004 : }
1005 420658 : return V;
1006 : }
1007 : long
1008 3360703 : nfval(GEN nf, GEN x, GEN pr)
1009 : {
1010 3360703 : pari_sp av = avma;
1011 : long w, e;
1012 : GEN cx, p;
1013 :
1014 3360703 : if (gequal0(x)) return LONG_MAX;
1015 3358796 : nf = checknf(nf);
1016 3358788 : checkprid(pr);
1017 3358811 : p = pr_get_p(pr);
1018 3358805 : e = pr_get_e(pr);
1019 3358805 : x = nf_to_scalar_or_basis(nf, x);
1020 3358687 : if (typ(x) != t_COL) return e*Q_pval(x,p);
1021 1546863 : x = Q_primitive_part(x, &cx);
1022 1546952 : w = ZC_nfval(x,pr);
1023 1546874 : if (cx) w += e*Q_pval(cx,p);
1024 1546894 : return gc_long(av,w);
1025 : }
1026 :
1027 : /* want to write p^v = uniformizer^(e*v) * z^v, z coprime to pr */
1028 : /* z := tau^e / p^(e-1), algebraic integer coprime to pr; return z^v */
1029 : static GEN
1030 940082 : powp(GEN nf, GEN pr, long v)
1031 : {
1032 : GEN b, z;
1033 : long e;
1034 940082 : if (!v) return gen_1;
1035 412678 : b = pr_get_tau(pr);
1036 412678 : if (typ(b) == t_INT) return gen_1;
1037 115297 : e = pr_get_e(pr);
1038 115297 : z = gel(b,1);
1039 115297 : if (e != 1) z = gdiv(nfpow_u(nf, z, e), powiu(pr_get_p(pr),e-1));
1040 115297 : if (v < 0) { v = -v; z = nfinv(nf, z); }
1041 115297 : if (v != 1) z = nfpow_u(nf, z, v);
1042 115297 : return z;
1043 : }
1044 : long
1045 3721552 : nfvalrem(GEN nf, GEN x, GEN pr, GEN *py)
1046 : {
1047 3721552 : pari_sp av = avma;
1048 : long w, e;
1049 : GEN cx, p, t;
1050 :
1051 3721552 : if (!py) return nfval(nf,x,pr);
1052 1773165 : if (gequal0(x)) { *py = gen_0; return LONG_MAX; }
1053 1773109 : nf = checknf(nf);
1054 1773108 : checkprid(pr);
1055 1773109 : p = pr_get_p(pr);
1056 1773109 : e = pr_get_e(pr);
1057 1773108 : x = nf_to_scalar_or_basis(nf, x);
1058 1773108 : if (typ(x) != t_COL) {
1059 527177 : w = Q_pvalrem(x,p, py);
1060 527177 : if (!w) { *py = gerepilecopy(av, x); return 0; }
1061 321068 : *py = gerepileupto(av, gmul(powp(nf, pr, w), *py));
1062 321068 : return e*w;
1063 : }
1064 1245931 : x = Q_primitive_part(x, &cx);
1065 1245927 : w = ZC_nfvalrem(x,pr, py);
1066 1245923 : if (cx)
1067 : {
1068 619014 : long v = Q_pvalrem(cx,p, &t);
1069 619014 : *py = nfmul(nf, *py, gmul(powp(nf,pr,v), t));
1070 619014 : *py = gerepileupto(av, *py);
1071 619014 : w += e*v;
1072 : }
1073 : else
1074 626909 : *py = gerepilecopy(av, *py);
1075 1245938 : return w;
1076 : }
1077 : GEN
1078 14952 : gpnfvalrem(GEN nf, GEN x, GEN pr, GEN *py)
1079 : {
1080 : long v;
1081 14952 : if (is_famat(x)) return famat_nfvalrem(nf, x, pr, py);
1082 14945 : v = nfvalrem(nf,x,pr,py);
1083 14945 : return v == LONG_MAX? mkoo(): stoi(v);
1084 : }
1085 :
1086 : /* true nf */
1087 : GEN
1088 103302 : coltoalg(GEN nf, GEN x)
1089 : {
1090 103302 : return mkpolmod( nf_to_scalar_or_alg(nf, x), nf_get_pol(nf) );
1091 : }
1092 :
1093 : GEN
1094 154115 : basistoalg(GEN nf, GEN x)
1095 : {
1096 : GEN T;
1097 :
1098 154115 : nf = checknf(nf);
1099 154115 : switch(typ(x))
1100 : {
1101 97163 : case t_COL: {
1102 97163 : pari_sp av = avma;
1103 97163 : return gerepilecopy(av, coltoalg(nf, x));
1104 : }
1105 32935 : case t_POLMOD:
1106 32935 : T = nf_get_pol(nf);
1107 32935 : if (!RgX_equal_var(T,gel(x,1)))
1108 0 : pari_err_MODULUS("basistoalg", T,gel(x,1));
1109 32935 : return gcopy(x);
1110 1820 : case t_POL:
1111 1820 : T = nf_get_pol(nf);
1112 1820 : if (varn(T) != varn(x)) pari_err_VAR("basistoalg",x,T);
1113 1820 : retmkpolmod(RgX_rem(x, T), ZX_copy(T));
1114 22197 : case t_INT:
1115 : case t_FRAC:
1116 22197 : T = nf_get_pol(nf);
1117 22197 : retmkpolmod(gcopy(x), ZX_copy(T));
1118 0 : default:
1119 0 : pari_err_TYPE("basistoalg",x);
1120 : return NULL; /* LCOV_EXCL_LINE */
1121 : }
1122 : }
1123 :
1124 : /* true nf, x a t_POL */
1125 : static GEN
1126 4254328 : pol_to_scalar_or_basis(GEN nf, GEN x)
1127 : {
1128 4254328 : GEN T = nf_get_pol(nf);
1129 4254312 : long l = lg(x);
1130 4254312 : if (varn(x) != varn(T)) pari_err_VAR("nf_to_scalar_or_basis", x,T);
1131 4254242 : if (l >= lg(T)) { x = RgX_rem(x, T); l = lg(x); }
1132 4254242 : if (l == 2) return gen_0;
1133 3225102 : if (l == 3)
1134 : {
1135 788178 : x = gel(x,2);
1136 788178 : if (!is_rational_t(typ(x))) pari_err_TYPE("nf_to_scalar_or_basis",x);
1137 788171 : return x;
1138 : }
1139 2436924 : return poltobasis(nf,x);
1140 : }
1141 : /* Assume nf is a genuine nf. */
1142 : GEN
1143 123309167 : nf_to_scalar_or_basis(GEN nf, GEN x)
1144 : {
1145 123309167 : switch(typ(x))
1146 : {
1147 80513618 : case t_INT: case t_FRAC:
1148 80513618 : return x;
1149 223050 : case t_POLMOD:
1150 223050 : x = checknfelt_mod(nf,x,"nf_to_scalar_or_basis");
1151 222999 : switch(typ(x))
1152 : {
1153 36070 : case t_INT: case t_FRAC: return x;
1154 186929 : case t_POL: return pol_to_scalar_or_basis(nf,x);
1155 : }
1156 0 : break;
1157 4067404 : case t_POL: return pol_to_scalar_or_basis(nf,x);
1158 38508140 : case t_COL:
1159 38508140 : if (lg(x)-1 != nf_get_degree(nf)) break;
1160 38508006 : return QV_isscalar(x)? gel(x,1): x;
1161 : }
1162 54 : pari_err_TYPE("nf_to_scalar_or_basis",x);
1163 : return NULL; /* LCOV_EXCL_LINE */
1164 : }
1165 : /* Let x be a polynomial with coefficients in Q or nf. Return the same
1166 : * polynomial with coefficients expressed as vectors (on the integral basis).
1167 : * No consistency checks, not memory-clean. */
1168 : GEN
1169 24328 : RgX_to_nfX(GEN nf, GEN x)
1170 : {
1171 : long i, l;
1172 24328 : GEN y = cgetg_copy(x, &l); y[1] = x[1];
1173 205838 : for (i=2; i<l; i++) gel(y,i) = nf_to_scalar_or_basis(nf, gel(x,i));
1174 24328 : return y;
1175 : }
1176 :
1177 : /* Assume nf is a genuine nf. */
1178 : GEN
1179 3584872 : nf_to_scalar_or_alg(GEN nf, GEN x)
1180 : {
1181 3584872 : switch(typ(x))
1182 : {
1183 102399 : case t_INT: case t_FRAC:
1184 102399 : return x;
1185 0 : case t_POLMOD:
1186 0 : x = checknfelt_mod(nf,x,"nf_to_scalar_or_alg");
1187 0 : if (typ(x) != t_POL) return x;
1188 : /* fall through */
1189 : case t_POL:
1190 : {
1191 4655 : GEN T = nf_get_pol(nf);
1192 4655 : long l = lg(x);
1193 4655 : if (varn(x) != varn(T)) pari_err_VAR("nf_to_scalar_or_alg", x,T);
1194 4655 : if (l >= lg(T)) { x = RgX_rem(x, T); l = lg(x); }
1195 4655 : if (l == 2) return gen_0;
1196 4655 : if (l == 3) return gel(x,2);
1197 3171 : return x;
1198 : }
1199 3477776 : case t_COL:
1200 : {
1201 : GEN dx;
1202 3477776 : if (lg(x)-1 != nf_get_degree(nf)) break;
1203 6902448 : if (QV_isscalar(x)) return gel(x,1);
1204 3424565 : x = Q_remove_denom(x, &dx);
1205 3424635 : x = RgV_RgC_mul(nf_get_zkprimpart(nf), x);
1206 3424717 : dx = mul_denom(dx, nf_get_zkden(nf));
1207 3424692 : return gdiv(x,dx);
1208 : }
1209 : }
1210 42 : pari_err_TYPE("nf_to_scalar_or_alg",x);
1211 : return NULL; /* LCOV_EXCL_LINE */
1212 : }
1213 :
1214 : /* gmul(A, RgX_to_RgC(x)), A t_MAT of compatible dimensions */
1215 : GEN
1216 1337 : RgM_RgX_mul(GEN A, GEN x)
1217 : {
1218 1337 : long i, l = lg(x)-1;
1219 : GEN z;
1220 1337 : if (l == 1) return zerocol(nbrows(A));
1221 1337 : z = gmul(gel(x,2), gel(A,1));
1222 2541 : for (i = 2; i < l; i++)
1223 1204 : if (!gequal0(gel(x,i+1))) z = gadd(z, gmul(gel(x,i+1), gel(A,i)));
1224 1337 : return z;
1225 : }
1226 : GEN
1227 9409161 : ZM_ZX_mul(GEN A, GEN x)
1228 : {
1229 9409161 : long i, l = lg(x)-1;
1230 : GEN z;
1231 9409161 : if (l == 1) return zerocol(nbrows(A));
1232 9408027 : z = ZC_Z_mul(gel(A,1), gel(x,2));
1233 29743240 : for (i = 2; i < l ; i++)
1234 20338761 : if (signe(gel(x,i+1))) z = ZC_add(z, ZC_Z_mul(gel(A,i), gel(x,i+1)));
1235 9404479 : return z;
1236 : }
1237 : /* x a t_POL, nf a genuine nf. No garbage collecting. No check. */
1238 : GEN
1239 8814662 : poltobasis(GEN nf, GEN x)
1240 : {
1241 8814662 : GEN d, T = nf_get_pol(nf);
1242 8814577 : if (varn(x) != varn(T)) pari_err_VAR( "poltobasis", x,T);
1243 8814444 : if (degpol(x) >= degpol(T)) x = RgX_rem(x,T);
1244 8814401 : x = Q_remove_denom(x, &d);
1245 8814655 : if (!RgX_is_ZX(x)) pari_err_TYPE("poltobasis",x);
1246 8814633 : x = ZM_ZX_mul(nf_get_invzk(nf), x);
1247 8812339 : if (d) x = RgC_Rg_div(x, d);
1248 8812492 : return x;
1249 : }
1250 :
1251 : GEN
1252 913567 : algtobasis(GEN nf, GEN x)
1253 : {
1254 : pari_sp av;
1255 :
1256 913567 : nf = checknf(nf);
1257 913565 : switch(typ(x))
1258 : {
1259 110852 : case t_POLMOD:
1260 110852 : if (!RgX_equal_var(nf_get_pol(nf),gel(x,1)))
1261 7 : pari_err_MODULUS("algtobasis", nf_get_pol(nf),gel(x,1));
1262 110845 : x = gel(x,2);
1263 110845 : switch(typ(x))
1264 : {
1265 7882 : case t_INT:
1266 7882 : case t_FRAC: return scalarcol(x, nf_get_degree(nf));
1267 102963 : case t_POL:
1268 102963 : av = avma;
1269 102963 : return gerepileupto(av,poltobasis(nf,x));
1270 : }
1271 0 : break;
1272 :
1273 249212 : case t_POL:
1274 249212 : av = avma;
1275 249212 : return gerepileupto(av,poltobasis(nf,x));
1276 :
1277 83058 : case t_COL:
1278 83058 : if (!RgV_is_QV(x)) pari_err_TYPE("nfalgtobasis",x);
1279 83051 : if (lg(x)-1 != nf_get_degree(nf)) pari_err_DIM("nfalgtobasis");
1280 83051 : return gcopy(x);
1281 :
1282 470447 : case t_INT:
1283 470447 : case t_FRAC: return scalarcol(x, nf_get_degree(nf));
1284 : }
1285 0 : pari_err_TYPE("algtobasis",x);
1286 : return NULL; /* LCOV_EXCL_LINE */
1287 : }
1288 :
1289 : GEN
1290 46200 : rnfbasistoalg(GEN rnf,GEN x)
1291 : {
1292 46200 : const char *f = "rnfbasistoalg";
1293 : long lx, i;
1294 46200 : pari_sp av = avma;
1295 : GEN z, nf, R, T;
1296 :
1297 46200 : checkrnf(rnf);
1298 46200 : nf = rnf_get_nf(rnf);
1299 46200 : T = nf_get_pol(nf);
1300 46200 : R = QXQX_to_mod_shallow(rnf_get_pol(rnf), T);
1301 46200 : switch(typ(x))
1302 : {
1303 826 : case t_COL:
1304 826 : z = cgetg_copy(x, &lx);
1305 2478 : for (i=1; i<lx; i++)
1306 : {
1307 1701 : GEN c = nf_to_scalar_or_alg(nf, gel(x,i));
1308 1652 : if (typ(c) == t_POL) c = mkpolmod(c,T);
1309 1652 : gel(z,i) = c;
1310 : }
1311 777 : z = RgV_RgC_mul(gel(rnf_get_zk(rnf),1), z);
1312 714 : return gerepileupto(av, gmodulo(z,R));
1313 :
1314 30555 : case t_POLMOD:
1315 30555 : x = polmod_nffix(f, rnf, x, 0);
1316 30352 : if (typ(x) != t_POL) break;
1317 14079 : retmkpolmod(RgX_copy(x), RgX_copy(R));
1318 1148 : case t_POL:
1319 1148 : if (varn(x) == varn(T)) { RgX_check_QX(x,f); x = gmodulo(x,T); break; }
1320 924 : if (varn(x) == varn(R))
1321 : {
1322 875 : x = RgX_nffix(f,nf_get_pol(nf),x,0);
1323 875 : return gmodulo(x, R);
1324 : }
1325 49 : pari_err_VAR(f, x,R);
1326 : }
1327 30119 : retmkpolmod(scalarpol(x, varn(R)), RgX_copy(R));
1328 : }
1329 :
1330 : GEN
1331 2142 : matbasistoalg(GEN nf,GEN x)
1332 : {
1333 : long i, j, li, lx;
1334 2142 : GEN z = cgetg_copy(x, &lx);
1335 :
1336 2142 : if (lx == 1) return z;
1337 2135 : switch(typ(x))
1338 : {
1339 70 : case t_VEC: case t_COL:
1340 259 : for (i=1; i<lx; i++) gel(z,i) = basistoalg(nf, gel(x,i));
1341 70 : return z;
1342 2065 : case t_MAT: break;
1343 0 : default: pari_err_TYPE("matbasistoalg",x);
1344 : }
1345 2065 : li = lgcols(x);
1346 7735 : for (j=1; j<lx; j++)
1347 : {
1348 5670 : GEN c = cgetg(li,t_COL), xj = gel(x,j);
1349 5670 : gel(z,j) = c;
1350 26817 : for (i=1; i<li; i++) gel(c,i) = basistoalg(nf, gel(xj,i));
1351 : }
1352 2065 : return z;
1353 : }
1354 :
1355 : GEN
1356 29936 : matalgtobasis(GEN nf,GEN x)
1357 : {
1358 : long i, j, li, lx;
1359 29936 : GEN z = cgetg_copy(x, &lx);
1360 :
1361 29936 : if (lx == 1) return z;
1362 29495 : switch(typ(x))
1363 : {
1364 29488 : case t_VEC: case t_COL:
1365 75672 : for (i=1; i<lx; i++) gel(z,i) = algtobasis(nf, gel(x,i));
1366 29488 : return z;
1367 7 : case t_MAT: break;
1368 0 : default: pari_err_TYPE("matalgtobasis",x);
1369 : }
1370 7 : li = lgcols(x);
1371 14 : for (j=1; j<lx; j++)
1372 : {
1373 7 : GEN c = cgetg(li,t_COL), xj = gel(x,j);
1374 7 : gel(z,j) = c;
1375 21 : for (i=1; i<li; i++) gel(c,i) = algtobasis(nf, gel(xj,i));
1376 : }
1377 7 : return z;
1378 : }
1379 : GEN
1380 10763 : RgM_to_nfM(GEN nf,GEN x)
1381 : {
1382 : long i, j, li, lx;
1383 10763 : GEN z = cgetg_copy(x, &lx);
1384 :
1385 10763 : if (lx == 1) return z;
1386 10763 : li = lgcols(x);
1387 80735 : for (j=1; j<lx; j++)
1388 : {
1389 69972 : GEN c = cgetg(li,t_COL), xj = gel(x,j);
1390 69972 : gel(z,j) = c;
1391 460207 : for (i=1; i<li; i++) gel(c,i) = nf_to_scalar_or_basis(nf, gel(xj,i));
1392 : }
1393 10763 : return z;
1394 : }
1395 : GEN
1396 90463 : RgC_to_nfC(GEN nf, GEN x)
1397 648681 : { pari_APPLY_type(t_COL, nf_to_scalar_or_basis(nf, gel(x,i))) }
1398 :
1399 : /* x a t_POLMOD, supposedly in rnf = K[z]/(T), K = Q[y]/(Tnf) */
1400 : GEN
1401 138110 : polmod_nffix(const char *f, GEN rnf, GEN x, int lift)
1402 138110 : { return polmod_nffix2(f, rnf_get_nfpol(rnf), rnf_get_pol(rnf), x,lift); }
1403 : GEN
1404 138201 : polmod_nffix2(const char *f, GEN T, GEN R, GEN x, int lift)
1405 : {
1406 138201 : if (RgX_equal_var(gel(x,1), R))
1407 : {
1408 127386 : x = gel(x,2);
1409 127386 : if (typ(x) == t_POL && varn(x) == varn(R))
1410 : {
1411 97432 : x = RgX_nffix(f, T, x, lift);
1412 97432 : switch(lg(x))
1413 : {
1414 5789 : case 2: return gen_0;
1415 11896 : case 3: return gel(x,2);
1416 : }
1417 79747 : return x;
1418 : }
1419 : }
1420 40769 : return Rg_nffix(f, T, x, lift);
1421 : }
1422 : GEN
1423 1204 : rnfalgtobasis(GEN rnf,GEN x)
1424 : {
1425 1204 : const char *f = "rnfalgtobasis";
1426 1204 : pari_sp av = avma;
1427 : GEN T, R;
1428 :
1429 1204 : checkrnf(rnf);
1430 1204 : R = rnf_get_pol(rnf);
1431 1204 : T = rnf_get_nfpol(rnf);
1432 1204 : switch(typ(x))
1433 : {
1434 49 : case t_COL:
1435 49 : if (lg(x)-1 != rnf_get_degree(rnf)) pari_err_DIM(f);
1436 28 : x = RgV_nffix(f, T, x, 0);
1437 21 : return gerepilecopy(av, x);
1438 :
1439 1071 : case t_POLMOD:
1440 1071 : x = polmod_nffix(f, rnf, x, 0);
1441 1036 : if (typ(x) != t_POL) break;
1442 714 : return gerepileupto(av, RgM_RgX_mul(rnf_get_invzk(rnf), x));
1443 56 : case t_POL:
1444 56 : if (varn(x) == varn(T))
1445 : {
1446 21 : RgX_check_QX(x,f);
1447 14 : if (degpol(x) >= degpol(T)) x = RgX_rem(x,T);
1448 14 : x = mkpolmod(x,T); break;
1449 : }
1450 35 : x = RgX_nffix(f, T, x, 0);
1451 28 : if (degpol(x) >= degpol(R)) x = RgX_rem(x, R);
1452 28 : return gerepileupto(av, RgM_RgX_mul(rnf_get_invzk(rnf), x));
1453 : }
1454 364 : return gerepileupto(av, scalarcol(x, rnf_get_degree(rnf)));
1455 : }
1456 :
1457 : /* Given a and b in nf, gives an algebraic integer y in nf such that a-b.y
1458 : * is "small" */
1459 : GEN
1460 259 : nfdiveuc(GEN nf, GEN a, GEN b)
1461 : {
1462 259 : pari_sp av = avma;
1463 259 : a = nfdiv(nf,a,b);
1464 259 : return gerepileupto(av, ground(a));
1465 : }
1466 :
1467 : /* Given a and b in nf, gives a "small" algebraic integer r in nf
1468 : * of the form a-b.y */
1469 : GEN
1470 259 : nfmod(GEN nf, GEN a, GEN b)
1471 : {
1472 259 : pari_sp av = avma;
1473 259 : GEN p1 = gneg_i(nfmul(nf,b,ground(nfdiv(nf,a,b))));
1474 259 : return gerepileupto(av, nfadd(nf,a,p1));
1475 : }
1476 :
1477 : /* Given a and b in nf, gives a two-component vector [y,r] in nf such
1478 : * that r=a-b.y is "small". */
1479 : GEN
1480 259 : nfdivrem(GEN nf, GEN a, GEN b)
1481 : {
1482 259 : pari_sp av = avma;
1483 259 : GEN p1,z, y = ground(nfdiv(nf,a,b));
1484 :
1485 259 : p1 = gneg_i(nfmul(nf,b,y));
1486 259 : z = cgetg(3,t_VEC);
1487 259 : gel(z,1) = gcopy(y);
1488 259 : gel(z,2) = nfadd(nf,a,p1); return gerepileupto(av, z);
1489 : }
1490 :
1491 : /*************************************************************************/
1492 : /** **/
1493 : /** LOGARITHMIC EMBEDDINGS **/
1494 : /** **/
1495 : /*************************************************************************/
1496 :
1497 : static int
1498 4575635 : low_prec(GEN x)
1499 : {
1500 4575635 : switch(typ(x))
1501 : {
1502 0 : case t_INT: return !signe(x);
1503 4575635 : case t_REAL: return !signe(x) || realprec(x) <= DEFAULTPREC;
1504 0 : default: return 0;
1505 : }
1506 : }
1507 :
1508 : static GEN
1509 23187 : cxlog_1(GEN nf) { return zerocol(lg(nf_get_roots(nf))-1); }
1510 : static GEN
1511 518 : cxlog_m1(GEN nf, long prec)
1512 : {
1513 518 : long i, l = lg(nf_get_roots(nf)), r1 = nf_get_r1(nf);
1514 518 : GEN v = cgetg(l, t_COL), p, P;
1515 518 : p = mppi(prec); P = mkcomplex(gen_0, p);
1516 1197 : for (i = 1; i <= r1; i++) gel(v,i) = P; /* IPi*/
1517 518 : if (i < l) P = gmul2n(P,1);
1518 1106 : for ( ; i < l; i++) gel(v,i) = P; /* 2IPi */
1519 518 : return v;
1520 : }
1521 : static GEN
1522 1696691 : ZC_cxlog(GEN nf, GEN x, long prec)
1523 : {
1524 : long i, l, r1;
1525 : GEN v;
1526 1696691 : x = RgM_RgC_mul(nf_get_M(nf), Q_primpart(x));
1527 1696691 : l = lg(x); r1 = nf_get_r1(nf);
1528 4264774 : for (i = 1; i <= r1; i++)
1529 2568083 : if (low_prec(gel(x,i))) return NULL;
1530 3475607 : for ( ; i < l; i++)
1531 1778929 : if (low_prec(gnorm(gel(x,i)))) return NULL;
1532 1696678 : v = cgetg(l,t_COL);
1533 4264760 : for (i = 1; i <= r1; i++) gel(v,i) = glog(gel(x,i),prec);
1534 3475557 : for ( ; i < l; i++) gel(v,i) = gmul2n(glog(gel(x,i),prec),1);
1535 1696677 : return v;
1536 : }
1537 : static GEN
1538 223407 : famat_cxlog(GEN nf, GEN fa, long prec)
1539 : {
1540 223407 : GEN G, E, y = NULL;
1541 : long i, l;
1542 :
1543 223407 : if (typ(fa) != t_MAT) pari_err_TYPE("famat_cxlog",fa);
1544 223407 : if (lg(fa) == 1) return cxlog_1(nf);
1545 223407 : G = gel(fa,1);
1546 223407 : E = gel(fa,2); l = lg(E);
1547 1104924 : for (i = 1; i < l; i++)
1548 : {
1549 881529 : GEN t, e = gel(E,i), x = nf_to_scalar_or_basis(nf, gel(G,i));
1550 : /* multiplicative arch would be better (save logs), but exponents overflow
1551 : * [ could keep track of expo separately, but not worth it ] */
1552 881529 : switch(typ(x))
1553 : { /* ignore positive rationals */
1554 15980 : case t_FRAC: x = gel(x,1); /* fall through */
1555 269080 : case t_INT: if (signe(x) > 0) continue;
1556 14 : if (!mpodd(e)) continue;
1557 14 : t = cxlog_m1(nf, prec); /* we probably should not reach this line */
1558 14 : break;
1559 612449 : default: /* t_COL */
1560 612449 : t = ZC_cxlog(nf,x,prec); if (!t) return NULL;
1561 612437 : t = RgC_Rg_mul(t, e);
1562 : }
1563 612451 : y = y? RgV_add(y,t): t;
1564 : }
1565 223395 : return y ? y: cxlog_1(nf);
1566 : }
1567 : /* Archimedean components: [e_i Log( sigma_i(X) )], where X = primpart(x),
1568 : * and e_i = 1 (resp 2.) for i <= R1 (resp. > R1) */
1569 : GEN
1570 1308797 : nf_cxlog(GEN nf, GEN x, long prec)
1571 : {
1572 1308797 : if (typ(x) == t_MAT) return famat_cxlog(nf,x,prec);
1573 1085390 : x = nf_to_scalar_or_basis(nf,x);
1574 1085390 : switch(typ(x))
1575 : {
1576 0 : case t_FRAC: x = gel(x,1); /* fall through */
1577 1148 : case t_INT:
1578 1148 : return signe(x) > 0? cxlog_1(nf): cxlog_m1(nf, prec);
1579 1084242 : default:
1580 1084242 : return ZC_cxlog(nf, x, prec);
1581 : }
1582 : }
1583 : GEN
1584 98 : nfV_cxlog(GEN nf, GEN x, long prec)
1585 : {
1586 : long i, l;
1587 98 : GEN v = cgetg_copy(x, &l);
1588 168 : for (i = 1; i < l; i++)
1589 70 : if (!(gel(v,i) = nf_cxlog(nf, gel(x,i), prec))) return NULL;
1590 98 : return v;
1591 : }
1592 :
1593 : static GEN
1594 22099 : scalar_logembed(GEN nf, GEN u, GEN *emb)
1595 : {
1596 : GEN v, logu;
1597 22099 : long i, s = signe(u), RU = lg(nf_get_roots(nf))-1, R1 = nf_get_r1(nf);
1598 :
1599 22099 : if (!s) pari_err_DOMAIN("nflogembed","argument","=",gen_0,u);
1600 22099 : v = cgetg(RU+1, t_COL); logu = logr_abs(u);
1601 24171 : for (i = 1; i <= R1; i++) gel(v,i) = logu;
1602 22099 : if (i <= RU)
1603 : {
1604 21217 : GEN logu2 = shiftr(logu,1);
1605 83335 : for ( ; i <= RU; i++) gel(v,i) = logu2;
1606 : }
1607 22099 : if (emb) *emb = const_col(RU, u);
1608 22099 : return v;
1609 : }
1610 :
1611 : static GEN
1612 1309 : famat_logembed(GEN nf,GEN x,GEN *emb,long prec)
1613 : {
1614 1309 : GEN A, M, T, a, t, g = gel(x,1), e = gel(x,2);
1615 1309 : long i, l = lg(e);
1616 :
1617 1309 : if (l == 1) return scalar_logembed(nf, real_1(prec), emb);
1618 1309 : A = NULL; T = emb? cgetg(l, t_COL): NULL;
1619 1309 : if (emb) *emb = M = mkmat2(T, e);
1620 79591 : for (i = 1; i < l; i++)
1621 : {
1622 78282 : a = nflogembed(nf, gel(g,i), &t, prec);
1623 78282 : if (!a) return NULL;
1624 78282 : a = RgC_Rg_mul(a, gel(e,i));
1625 78282 : A = A? RgC_add(A, a): a;
1626 78282 : if (emb) gel(T,i) = t;
1627 : }
1628 1309 : return A;
1629 : }
1630 :
1631 : /* Get archimedean components: [e_i log( | sigma_i(x) | )], with e_i = 1
1632 : * (resp 2.) for i <= R1 (resp. > R1) and set emb to the embeddings of x.
1633 : * Return NULL if precision problem */
1634 : GEN
1635 116096 : nflogembed(GEN nf, GEN x, GEN *emb, long prec)
1636 : {
1637 : long i, l, r1;
1638 : GEN v, t;
1639 :
1640 116096 : if (typ(x) == t_MAT) return famat_logembed(nf,x,emb,prec);
1641 114787 : x = nf_to_scalar_or_basis(nf,x);
1642 114787 : if (typ(x) != t_COL) return scalar_logembed(nf, gtofp(x,prec), emb);
1643 92688 : x = RgM_RgC_mul(nf_get_M(nf), x);
1644 92687 : l = lg(x); r1 = nf_get_r1(nf); v = cgetg(l,t_COL);
1645 119700 : for (i = 1; i <= r1; i++)
1646 : {
1647 27013 : t = gabs(gel(x,i),prec); if (low_prec(t)) return NULL;
1648 27014 : gel(v,i) = glog(t,prec);
1649 : }
1650 294297 : for ( ; i < l; i++)
1651 : {
1652 201609 : t = gnorm(gel(x,i)); if (low_prec(t)) return NULL;
1653 201610 : gel(v,i) = glog(t,prec);
1654 : }
1655 92688 : if (emb) *emb = x;
1656 92688 : return v;
1657 : }
1658 :
1659 : /*************************************************************************/
1660 : /** **/
1661 : /** REAL EMBEDDINGS **/
1662 : /** **/
1663 : /*************************************************************************/
1664 : static GEN
1665 485747 : sarch_get_cyc(GEN sarch) { return gel(sarch,1); }
1666 : static GEN
1667 700402 : sarch_get_archp(GEN sarch) { return gel(sarch,2); }
1668 : static GEN
1669 172606 : sarch_get_MI(GEN sarch) { return gel(sarch,3); }
1670 : static GEN
1671 172606 : sarch_get_lambda(GEN sarch) { return gel(sarch,4); }
1672 : static GEN
1673 172606 : sarch_get_F(GEN sarch) { return gel(sarch,5); }
1674 :
1675 : /* x not a scalar, true nf, return number of positive roots of char_x */
1676 : static long
1677 1195 : num_positive(GEN nf, GEN x)
1678 : {
1679 1195 : GEN T = nf_get_pol(nf), B, charx;
1680 : long dnf, vnf, N;
1681 1195 : x = nf_to_scalar_or_alg(nf, x); /* not a scalar */
1682 1195 : charx = ZXQ_charpoly(x, T, 0);
1683 1195 : charx = ZX_radical(charx);
1684 1195 : N = degpol(T) / degpol(charx);
1685 : /* real places are unramified ? */
1686 1195 : if (N == 1 || ZX_sturm(charx) * N == nf_get_r1(nf))
1687 1188 : return ZX_sturmpart(charx, mkvec2(gen_0,mkoo())) * N;
1688 : /* painful case, multiply by random square until primitive */
1689 7 : dnf = nf_get_degree(nf);
1690 7 : vnf = varn(T);
1691 7 : B = int2n(10);
1692 : for(;;)
1693 0 : {
1694 7 : GEN y = RgXQ_sqr(random_FpX(dnf, vnf, B), T);
1695 7 : y = RgXQ_mul(x, y, T);
1696 7 : charx = ZXQ_charpoly(y, T, 0);
1697 7 : if (ZX_is_squarefree(charx))
1698 7 : return ZX_sturmpart(charx, mkvec2(gen_0,mkoo())) * N;
1699 : }
1700 : }
1701 :
1702 : /* x a QC: return sigma_k(x) where 1 <= k <= r1+r2; correct but inefficient
1703 : * if x in Q. M = nf_get_M(nf) */
1704 : static GEN
1705 665 : nfembed_i(GEN M, GEN x, long k)
1706 : {
1707 665 : long i, l = lg(M);
1708 665 : GEN z = gel(x,1);
1709 4102 : for (i = 2; i < l; i++) z = gadd(z, gmul(gcoeff(M,k,i), gel(x,i)));
1710 665 : return z;
1711 : }
1712 : GEN
1713 0 : nfembed(GEN nf, GEN x, long k)
1714 : {
1715 0 : pari_sp av = avma;
1716 0 : nf = checknf(nf);
1717 0 : x = nf_to_scalar_or_basis(nf,x);
1718 0 : if (typ(x) != t_COL) return gerepilecopy(av, x);
1719 0 : return gerepileupto(av, nfembed_i(nf_get_M(nf),x,k));
1720 : }
1721 :
1722 : /* x a ZC */
1723 : static GEN
1724 894346 : zk_embed(GEN M, GEN x, long k)
1725 : {
1726 894346 : long i, l = lg(x);
1727 894346 : GEN z = gel(x,1); /* times M[k,1], which is 1 */
1728 2854108 : for (i = 2; i < l; i++) z = mpadd(z, mpmul(gcoeff(M,k,i), gel(x,i)));
1729 894331 : return z;
1730 : }
1731 :
1732 : /* Given floating point approximation z of sigma_k(x), decide its sign
1733 : * [0/+, 1/- and -1 for FAIL] */
1734 : static long
1735 875276 : eval_sign_embed(GEN z)
1736 : {
1737 875276 : if (typ(z) == t_REAL)
1738 : {
1739 875276 : long l = realprec(z);
1740 875276 : if (l <= LOWDEFAULTPREC || (l == LOWDEFAULTPREC + 1 && !z[LOWDEFAULTPREC]))
1741 802 : return -1; /* dubious, fail */
1742 : }
1743 874474 : return (signe(z) < 1)? 1: 0;
1744 : }
1745 : /* return v such that (-1)^v = sign(sigma_k(x)), x primitive ZC */
1746 : static long
1747 760758 : eval_sign(GEN M, GEN x, long k)
1748 760758 : { return eval_sign_embed( zk_embed(M, x, k) ); }
1749 :
1750 : /* check that signs[i..#signs] == s; signs = NULL encodes "totally positive" */
1751 : static int
1752 0 : oksigns(long l, GEN signs, long i, long s)
1753 : {
1754 0 : if (!signs) return s == 0;
1755 0 : for (; i < l; i++)
1756 0 : if (signs[i] != s) return 0;
1757 0 : return 1;
1758 : }
1759 : /* check that signs[i] = s and signs[i+1..#signs] = 1-s */
1760 : static int
1761 0 : oksigns2(long l, GEN signs, long i, long s)
1762 : {
1763 0 : if (!signs) return s == 0 && i == l-1;
1764 0 : return signs[i] == s && oksigns(l, signs, i+1, 1-s);
1765 : }
1766 :
1767 : /* true nf, x a ZC (primitive for efficiency) which is not a scalar; embx its
1768 : * embeddings or NULL */
1769 : static int
1770 96674 : nfchecksigns_i(GEN nf, GEN x, GEN embx, GEN signs, GEN archp)
1771 : {
1772 96674 : long l = lg(archp), i;
1773 96674 : GEN M = nf_get_M(nf), sarch = NULL;
1774 96675 : long np = -1;
1775 149178 : for (i = 1; i < l; i++)
1776 : {
1777 : long s;
1778 115930 : if (embx)
1779 114531 : s = eval_sign_embed(gel(embx,i));
1780 : else
1781 1399 : s = eval_sign(M, x, archp[i]);
1782 : /* 0 / + or 1 / -; -1 for FAIL */
1783 115930 : if (s < 0) /* failure */
1784 : {
1785 0 : long ni, r1 = nf_get_r1(nf);
1786 : GEN xi;
1787 0 : if (np < 0)
1788 : {
1789 0 : np = num_positive(nf, x);
1790 0 : if (np == 0) return oksigns(l, signs, i, 1);
1791 0 : if (np == r1) return oksigns(l, signs, i, 0);
1792 0 : sarch = nfarchstar(nf, NULL, identity_perm(r1));
1793 : }
1794 0 : xi = set_sign_mod_divisor(nf, vecsmall_ei(r1, archp[i]), gen_1, sarch);
1795 0 : xi = Q_primpart(xi);
1796 0 : ni = num_positive(nf, nfmuli(nf,x,xi));
1797 0 : if (ni == 0) return oksigns2(l, signs, i, 0);
1798 0 : if (ni == r1) return oksigns2(l, signs, i, 1);
1799 0 : s = ni < np? 0: 1;
1800 : }
1801 115930 : if (s != (signs? signs[i]: 0)) return 0;
1802 : }
1803 33248 : return 1;
1804 : }
1805 : static void
1806 553 : pl_convert(GEN pl, GEN *psigns, GEN *parchp)
1807 : {
1808 553 : long i, j, l = lg(pl);
1809 553 : GEN signs = cgetg(l, t_VECSMALL);
1810 553 : GEN archp = cgetg(l, t_VECSMALL);
1811 2756 : for (i = j = 1; i < l; i++)
1812 : {
1813 2203 : if (!pl[i]) continue;
1814 1730 : archp[j] = i;
1815 1730 : signs[j] = (pl[i] < 0)? 1: 0;
1816 1730 : j++;
1817 : }
1818 553 : setlg(archp, j); *parchp = archp;
1819 553 : setlg(signs, j); *psigns = signs;
1820 553 : }
1821 : /* pl : requested signs for real embeddings, 0 = no sign constraint */
1822 : int
1823 1085 : nfchecksigns(GEN nf, GEN x, GEN pl)
1824 : {
1825 1085 : pari_sp av = avma;
1826 : GEN signs, archp;
1827 1085 : nf = checknf(nf);
1828 1085 : x = nf_to_scalar_or_basis(nf,x);
1829 1085 : if (typ(x) != t_COL)
1830 : {
1831 532 : long i, l = lg(pl), s = gsigne(x);
1832 1078 : for (i = 1; i < l; i++)
1833 546 : if (pl[i] && pl[i] != s) return gc_bool(av,0);
1834 532 : return gc_bool(av,1);
1835 : }
1836 553 : pl_convert(pl, &signs, &archp);
1837 553 : return gc_bool(av, nfchecksigns_i(nf, x, NULL, signs, archp));
1838 : }
1839 :
1840 : /* signs = NULL: totally positive, else sign[i] = 0 (+) or 1 (-) */
1841 : static GEN
1842 172607 : get_C(GEN lambda, long l, GEN signs)
1843 : {
1844 : long i;
1845 : GEN C, mlambda;
1846 172607 : if (!signs) return const_vec(l-1, lambda);
1847 133631 : C = cgetg(l, t_COL); mlambda = gneg(lambda);
1848 342822 : for (i = 1; i < l; i++) gel(C,i) = signs[i]? mlambda: lambda;
1849 133631 : return C;
1850 : }
1851 : /* signs = NULL: totally positive at archp.
1852 : * Assume that a t_COL x is not a scalar */
1853 : static GEN
1854 286403 : nfsetsigns(GEN nf, GEN signs, GEN x, GEN sarch)
1855 : {
1856 286403 : long i, l = lg(sarch_get_archp(sarch));
1857 : GEN ex;
1858 : /* Is signature already correct ? */
1859 286401 : if (typ(x) != t_COL)
1860 : {
1861 190282 : long s = gsigne(x);
1862 190282 : if (!s) i = 1;
1863 190261 : else if (!signs)
1864 4921 : i = (s < 0)? 1: l;
1865 : else
1866 : {
1867 185340 : s = s < 0? 1: 0;
1868 313231 : for (i = 1; i < l; i++)
1869 236968 : if (signs[i] != s) break;
1870 : }
1871 190282 : ex = (i < l)? const_col(l-1, x): NULL;
1872 : }
1873 : else
1874 : { /* inefficient if x scalar, wrong if x = 0 */
1875 96119 : pari_sp av = avma;
1876 96119 : GEN cex, M = nf_get_M(nf), archp = sarch_get_archp(sarch);
1877 96123 : GEN xp = Q_primitive_part(x,&cex);
1878 96121 : ex = cgetg(l,t_COL);
1879 229714 : for (i = 1; i < l; i++) gel(ex,i) = zk_embed(M,xp,archp[i]);
1880 96121 : if (nfchecksigns_i(nf, xp, ex, signs, archp)) { ex = NULL; set_avma(av); }
1881 63359 : else if (cex) ex = RgC_Rg_mul(ex, cex); /* put back content */
1882 : }
1883 286407 : if (ex)
1884 : { /* If no, fix it */
1885 172606 : GEN MI = sarch_get_MI(sarch), F = sarch_get_F(sarch);
1886 172606 : GEN lambda = sarch_get_lambda(sarch);
1887 172606 : GEN t = RgC_sub(get_C(lambda, l, signs), ex);
1888 172604 : t = grndtoi(RgM_RgC_mul(MI,t), NULL);
1889 172603 : if (lg(F) != 1) t = ZM_ZC_mul(F, t);
1890 172600 : x = typ(x) == t_COL? RgC_add(t, x): RgC_Rg_add(t, x);
1891 : }
1892 286399 : return x;
1893 : }
1894 : /* - true nf
1895 : * - sarch = nfarchstar(nf, F);
1896 : * - x encodes a vector of signs at arch.archp: either a t_VECSMALL
1897 : * (vector of signs as {0,1}-vector), NULL (totally positive at archp),
1898 : * or a nonzero number field element (replaced by its signature at archp);
1899 : * - y is a nonzero number field element
1900 : * Return z = y (mod F) with signs(y, archp) = signs(x) (a {0,1}-vector).
1901 : * Not stack-clean */
1902 : GEN
1903 317886 : set_sign_mod_divisor(GEN nf, GEN x, GEN y, GEN sarch)
1904 : {
1905 317886 : GEN archp = sarch_get_archp(sarch);
1906 317886 : if (lg(archp) == 1) return y;
1907 284629 : if (x && typ(x) != t_VECSMALL) x = nfsign_arch(nf, x, archp);
1908 284629 : return nfsetsigns(nf, x, nf_to_scalar_or_basis(nf,y), sarch);
1909 : }
1910 :
1911 : static GEN
1912 83323 : setsigns_init(GEN nf, GEN archp, GEN F, GEN DATA)
1913 : {
1914 83323 : GEN lambda, Mr = rowpermute(nf_get_M(nf), archp), MI = F? RgM_mul(Mr,F): Mr;
1915 83326 : lambda = gmul2n(matrixnorm(MI,DEFAULTPREC), -1);
1916 83324 : if (typ(lambda) != t_REAL) lambda = gmul(lambda, uutoQ(1001,1000));
1917 83326 : if (lg(archp) < lg(MI))
1918 : {
1919 58937 : GEN perm = gel(indexrank(MI), 2);
1920 58939 : if (!F) F = matid(nf_get_degree(nf));
1921 58939 : MI = vecpermute(MI, perm);
1922 58939 : F = vecpermute(F, perm);
1923 : }
1924 83328 : if (!F) F = cgetg(1,t_MAT);
1925 83328 : MI = RgM_inv(MI);
1926 83328 : return mkvec5(DATA, archp, MI, lambda, F);
1927 : }
1928 : /* F nonzero integral ideal in HNF (or NULL: Z_K), compute elements in 1+F
1929 : * whose sign matrix at archp is identity; archp in 'indices' format */
1930 : GEN
1931 259172 : nfarchstar(GEN nf, GEN F, GEN archp)
1932 : {
1933 259172 : long nba = lg(archp) - 1;
1934 259172 : if (!nba) return mkvec2(cgetg(1,t_VEC), archp);
1935 81554 : if (F && equali1(gcoeff(F,1,1))) F = NULL;
1936 81553 : if (F) F = idealpseudored(F, nf_get_roundG(nf));
1937 81539 : return setsigns_init(nf, archp, F, const_vec(nba, gen_2));
1938 : }
1939 :
1940 : /*************************************************************************/
1941 : /** **/
1942 : /** IDEALCHINESE **/
1943 : /** **/
1944 : /*************************************************************************/
1945 : static int
1946 3407 : isprfact(GEN x)
1947 : {
1948 : long i, l;
1949 : GEN L, E;
1950 3407 : if (typ(x) != t_MAT || lg(x) != 3) return 0;
1951 3407 : L = gel(x,1); l = lg(L);
1952 3407 : E = gel(x,2);
1953 9676 : for(i=1; i<l; i++)
1954 : {
1955 6269 : checkprid(gel(L,i));
1956 6269 : if (typ(gel(E,i)) != t_INT) return 0;
1957 : }
1958 3407 : return 1;
1959 : }
1960 :
1961 : /* initialize projectors mod pr[i]^e[i] for idealchinese */
1962 : static GEN
1963 3407 : pr_init(GEN nf, GEN fa, GEN w, GEN dw)
1964 : {
1965 3407 : GEN U, E, F, FZ, L = gel(fa,1), E0 = gel(fa,2);
1966 3407 : long i, r = lg(L);
1967 :
1968 3407 : if (w && lg(w) != r) pari_err_TYPE("idealchinese", w);
1969 3407 : if (r == 1 && !dw) return cgetg(1,t_VEC);
1970 3400 : E = leafcopy(E0); /* do not destroy fa[2] */
1971 9669 : for (i = 1; i < r; i++)
1972 6269 : if (signe(gel(E,i)) < 0) gel(E,i) = gen_0;
1973 3400 : F = factorbackprime(nf, L, E);
1974 3400 : if (dw)
1975 : {
1976 707 : F = ZM_Z_mul(F, dw);
1977 1638 : for (i = 1; i < r; i++)
1978 : {
1979 931 : GEN pr = gel(L,i);
1980 931 : long e = itos(gel(E0,i)), v = idealval(nf, dw, pr);
1981 931 : if (e >= 0)
1982 924 : gel(E,i) = addiu(gel(E,i), v);
1983 7 : else if (v + e <= 0)
1984 0 : F = idealmulpowprime(nf, F, pr, stoi(-v)); /* coprime to pr */
1985 : else
1986 : {
1987 7 : F = idealmulpowprime(nf, F, pr, stoi(e));
1988 7 : gel(E,i) = stoi(v + e);
1989 : }
1990 : }
1991 : }
1992 3400 : U = cgetg(r, t_VEC);
1993 9669 : for (i = 1; i < r; i++)
1994 : {
1995 : GEN u;
1996 6269 : if (w && gequal0(gel(w,i))) u = gen_0; /* unused */
1997 : else
1998 : {
1999 6192 : GEN pr = gel(L,i), e = gel(E,i), t;
2000 6192 : t = idealdivpowprime(nf,F, pr, e);
2001 6192 : u = hnfmerge_get_1(t, idealpow(nf, pr, e));
2002 6192 : if (!u) pari_err_COPRIME("idealchinese", t,pr);
2003 : }
2004 6269 : gel(U,i) = u;
2005 : }
2006 3400 : FZ = gcoeff(F, 1, 1);
2007 3400 : F = idealpseudored(F, nf_get_roundG(nf));
2008 3400 : return mkvec2(mkvec2(F, FZ), U);
2009 : }
2010 :
2011 : static GEN
2012 2205 : pl_normalize(GEN nf, GEN pl)
2013 : {
2014 2205 : const char *fun = "idealchinese";
2015 2205 : if (lg(pl)-1 != nf_get_r1(nf)) pari_err_TYPE(fun,pl);
2016 2205 : switch(typ(pl))
2017 : {
2018 721 : case t_VEC: RgV_check_ZV(pl,fun); pl = ZV_to_zv(pl);
2019 : /* fall through */
2020 2205 : case t_VECSMALL: break;
2021 0 : default: pari_err_TYPE(fun,pl);
2022 : }
2023 2205 : return pl;
2024 : }
2025 :
2026 : static int
2027 8622 : is_chineseinit(GEN x)
2028 : {
2029 : GEN fa, pl;
2030 : long l;
2031 8622 : if (typ(x) != t_VEC || lg(x)!=3) return 0;
2032 6837 : fa = gel(x,1);
2033 6837 : pl = gel(x,2);
2034 6837 : if (typ(fa) != t_VEC || typ(pl) != t_VEC) return 0;
2035 3288 : l = lg(fa);
2036 3288 : if (l != 1)
2037 : {
2038 : GEN z;
2039 3267 : if (l != 3) return 0;
2040 3267 : z = gel(fa, 1);
2041 3267 : if (typ(z) != t_VEC || lg(z) != 3 || typ(gel(z,1)) != t_MAT
2042 3260 : || typ(gel(z,2)) != t_INT
2043 3260 : || typ(gel(fa,2)) != t_VEC)
2044 7 : return 0;
2045 : }
2046 3281 : l = lg(pl);
2047 3281 : if (l != 1)
2048 : {
2049 560 : if (l != 6 || typ(gel(pl,3)) != t_MAT || typ(gel(pl,1)) != t_VECSMALL
2050 560 : || typ(gel(pl,2)) != t_VECSMALL)
2051 0 : return 0;
2052 : }
2053 3281 : return 1;
2054 : }
2055 :
2056 : /* nf a true 'nf' */
2057 : static GEN
2058 3862 : chineseinit_i(GEN nf, GEN fa, GEN w, GEN dw)
2059 : {
2060 3862 : const char *fun = "idealchineseinit";
2061 3862 : GEN archp = NULL, pl = NULL;
2062 3862 : switch(typ(fa))
2063 : {
2064 2205 : case t_VEC:
2065 2205 : if (is_chineseinit(fa))
2066 : {
2067 0 : if (dw) pari_err_DOMAIN(fun, "denom(y)", "!=", gen_1, w);
2068 0 : return fa;
2069 : }
2070 2205 : if (lg(fa) != 3) pari_err_TYPE(fun, fa);
2071 : /* of the form [x,s] */
2072 2205 : pl = pl_normalize(nf, gel(fa,2));
2073 2205 : fa = gel(fa,1);
2074 2205 : archp = vecsmall01_to_indices(pl);
2075 : /* keep pr_init, reset pl */
2076 2205 : if (is_chineseinit(fa)) { fa = gel(fa,1); break; }
2077 : /* fall through */
2078 : case t_MAT: /* factorization? */
2079 3407 : if (isprfact(fa)) { fa = pr_init(nf, fa, w, dw); break; }
2080 0 : default: pari_err_TYPE(fun,fa);
2081 : }
2082 :
2083 3862 : if (!pl) pl = cgetg(1,t_VEC);
2084 : else
2085 : {
2086 2205 : long r = lg(archp);
2087 2205 : if (r == 1) pl = cgetg(1, t_VEC);
2088 : else
2089 : {
2090 1764 : GEN F = (lg(fa) == 1)? NULL: gmael(fa,1,1), signs = cgetg(r, t_VECSMALL);
2091 : long i;
2092 5096 : for (i = 1; i < r; i++) signs[i] = (pl[archp[i]] < 0)? 1: 0;
2093 1764 : pl = setsigns_init(nf, archp, F, signs);
2094 : }
2095 : }
2096 3862 : return mkvec2(fa, pl);
2097 : }
2098 :
2099 : /* Given a prime ideal factorization x, possibly with 0 or negative exponents,
2100 : * and a vector w of elements of nf, gives b such that
2101 : * v_p(b-w_p)>=v_p(x) for all prime ideals p in the ideal factorization
2102 : * and v_p(b)>=0 for all other p, using the standard proof given in GTM 138. */
2103 : GEN
2104 6688 : idealchinese(GEN nf, GEN x0, GEN w)
2105 : {
2106 6688 : const char *fun = "idealchinese";
2107 6688 : pari_sp av = avma;
2108 6688 : GEN x = x0, x1, x2, s, dw, F;
2109 :
2110 6688 : nf = checknf(nf);
2111 6688 : if (!w) return gerepilecopy(av, chineseinit_i(nf,x,NULL,NULL));
2112 :
2113 4212 : if (typ(w) != t_VEC) pari_err_TYPE(fun,w);
2114 4212 : w = Q_remove_denom(matalgtobasis(nf,w), &dw);
2115 4212 : if (!is_chineseinit(x)) x = chineseinit_i(nf,x,w,dw);
2116 : /* x is a 'chineseinit' */
2117 4212 : x1 = gel(x,1); s = NULL;
2118 4212 : x2 = gel(x,2);
2119 4212 : if (lg(x1) == 1) { F = NULL; dw = NULL; }
2120 : else
2121 : {
2122 4191 : GEN U = gel(x1,2), FZ;
2123 4191 : long i, r = lg(w);
2124 4191 : F = gmael(x1,1,1); FZ = gmael(x1,1,2);
2125 14214 : for (i=1; i<r; i++)
2126 10023 : if (!ZV_equal0(gel(w,i)))
2127 : {
2128 7508 : GEN t = nfmuli(nf, gel(U,i), gel(w,i));
2129 7508 : s = s? ZC_add(s,t): t;
2130 : }
2131 4191 : if (s)
2132 : {
2133 4170 : s = ZC_reducemodmatrix(s, F);
2134 4170 : if (dw && x == x0) /* input was a chineseinit */
2135 : {
2136 7 : dw = modii(dw, FZ);
2137 7 : s = FpC_Fp_mul(s, Fp_inv(dw, FZ), FZ);
2138 7 : dw = NULL;
2139 : }
2140 4170 : if (ZV_isscalar(s)) s = icopy(gel(s,1));
2141 : }
2142 : }
2143 4212 : if (lg(x2) != 1)
2144 : {
2145 1771 : s = nfsetsigns(nf, gel(x2,1), s? s: gen_0, x2);
2146 1771 : if (typ(s) == t_COL && QV_isscalar(s))
2147 : {
2148 287 : s = gel(s,1); if (!dw) s = gcopy(s);
2149 : }
2150 : }
2151 2441 : else if (!s) return gc_const(av, gen_0);
2152 4184 : return gerepileupto(av, dw? gdiv(s, dw): s);
2153 : }
2154 :
2155 : /*************************************************************************/
2156 : /** **/
2157 : /** (Z_K/I)^* **/
2158 : /** **/
2159 : /*************************************************************************/
2160 : GEN
2161 2205 : vecsmall01_to_indices(GEN v)
2162 : {
2163 2205 : long i, k, l = lg(v);
2164 2205 : GEN p = new_chunk(l) + l;
2165 6559 : for (k=1, i=l-1; i; i--)
2166 4354 : if (v[i]) { *--p = i; k++; }
2167 2205 : *--p = evallg(k) | evaltyp(t_VECSMALL);
2168 2205 : set_avma((pari_sp)p); return p;
2169 : }
2170 : GEN
2171 1046340 : vec01_to_indices(GEN v)
2172 : {
2173 : long i, k, l;
2174 : GEN p;
2175 :
2176 1046340 : switch (typ(v))
2177 : {
2178 999854 : case t_VECSMALL: return v;
2179 46486 : case t_VEC: break;
2180 0 : default: pari_err_TYPE("vec01_to_indices",v);
2181 : }
2182 46486 : l = lg(v);
2183 46486 : p = new_chunk(l) + l;
2184 140062 : for (k=1, i=l-1; i; i--)
2185 93576 : if (signe(gel(v,i))) { *--p = i; k++; }
2186 46486 : *--p = evallg(k) | evaltyp(t_VECSMALL);
2187 46486 : set_avma((pari_sp)p); return p;
2188 : }
2189 : GEN
2190 136676 : indices_to_vec01(GEN p, long r)
2191 : {
2192 136676 : long i, l = lg(p);
2193 136676 : GEN v = zerovec(r);
2194 206331 : for (i = 1; i < l; i++) gel(v, p[i]) = gen_1;
2195 136670 : return v;
2196 : }
2197 :
2198 : /* return (column) vector of R1 signatures of x (0 or 1) */
2199 : GEN
2200 999854 : nfsign_arch(GEN nf, GEN x, GEN arch)
2201 : {
2202 999854 : GEN sarch, M, V, archp = vec01_to_indices(arch);
2203 999854 : long i, s, np, n = lg(archp)-1;
2204 : pari_sp av;
2205 :
2206 999854 : if (!n) return cgetg(1,t_VECSMALL);
2207 798256 : if (typ(x) == t_MAT)
2208 : { /* factorisation */
2209 272887 : GEN g = gel(x,1), e = gel(x,2);
2210 272887 : long l = lg(g);
2211 272887 : V = zero_zv(n);
2212 744476 : for (i = 1; i < l; i++)
2213 471591 : if (mpodd(gel(e,i)))
2214 394651 : Flv_add_inplace(V, nfsign_arch(nf,gel(g,i),archp), 2);
2215 272885 : set_avma((pari_sp)V); return V;
2216 : }
2217 525369 : av = avma; V = cgetg(n+1,t_VECSMALL);
2218 525369 : x = nf_to_scalar_or_basis(nf, x);
2219 525369 : switch(typ(x))
2220 : {
2221 135954 : case t_INT:
2222 135954 : s = signe(x);
2223 135954 : if (!s) pari_err_DOMAIN("nfsign_arch","element","=",gen_0,x);
2224 135954 : set_avma(av); return const_vecsmall(n, (s < 0)? 1: 0);
2225 357 : case t_FRAC:
2226 357 : s = signe(gel(x,1));
2227 357 : set_avma(av); return const_vecsmall(n, (s < 0)? 1: 0);
2228 : }
2229 389058 : x = Q_primpart(x); M = nf_get_M(nf); sarch = NULL; np = -1;
2230 1147647 : for (i = 1; i <= n; i++)
2231 : {
2232 759357 : long s = eval_sign(M, x, archp[i]);
2233 759354 : if (s < 0) /* failure */
2234 : {
2235 802 : long ni, r1 = nf_get_r1(nf);
2236 : GEN xi;
2237 802 : if (np < 0)
2238 : {
2239 802 : np = num_positive(nf, x);
2240 802 : if (np == 0) { set_avma(av); return const_vecsmall(n, 1); }
2241 772 : if (np == r1){ set_avma(av); return const_vecsmall(n, 0); }
2242 393 : sarch = nfarchstar(nf, NULL, identity_perm(r1));
2243 : }
2244 393 : xi = set_sign_mod_divisor(nf, vecsmall_ei(r1, archp[i]), gen_1, sarch);
2245 393 : xi = Q_primpart(xi);
2246 393 : ni = num_positive(nf, nfmuli(nf,x,xi));
2247 393 : if (ni == 0) { set_avma(av); V = const_vecsmall(n, 1); V[i] = 0; return V; }
2248 393 : if (ni == r1){ set_avma(av); V = const_vecsmall(n, 0); V[i] = 1; return V; }
2249 34 : s = ni < np? 0: 1;
2250 : }
2251 758586 : V[i] = s;
2252 : }
2253 388290 : set_avma((pari_sp)V); return V;
2254 : }
2255 : static void
2256 6573 : chk_ind(const char *s, long i, long r1)
2257 : {
2258 6573 : if (i <= 0) pari_err_DOMAIN(s, "index", "<=", gen_0, stoi(i));
2259 6559 : if (i > r1) pari_err_DOMAIN(s, "index", ">", utoi(r1), utoi(i));
2260 6524 : }
2261 : static GEN
2262 95753 : parse_embed(GEN ind, long r, const char *f)
2263 : {
2264 : long l, i;
2265 95753 : if (!ind) return identity_perm(r);
2266 4529 : switch(typ(ind))
2267 : {
2268 70 : case t_INT: ind = mkvecsmall(itos(ind)); break;
2269 84 : case t_VEC: case t_COL: ind = vec_to_vecsmall(ind); break;
2270 4375 : case t_VECSMALL: break;
2271 0 : default: pari_err_TYPE(f, ind);
2272 : }
2273 4529 : l = lg(ind);
2274 11053 : for (i = 1; i < l; i++) chk_ind(f, ind[i], r);
2275 4480 : return ind;
2276 : }
2277 : GEN
2278 93940 : nfeltsign(GEN nf, GEN x, GEN ind0)
2279 : {
2280 93940 : pari_sp av = avma;
2281 : long i, l;
2282 : GEN v, ind;
2283 93940 : nf = checknf(nf);
2284 93940 : ind = parse_embed(ind0, nf_get_r1(nf), "nfeltsign");
2285 93919 : l = lg(ind);
2286 93919 : if (is_rational_t(typ(x)))
2287 : { /* nfsign_arch would test this, but avoid converting t_VECSMALL -> t_VEC */
2288 : GEN s;
2289 2135 : switch(gsigne(x))
2290 : {
2291 532 : case -1:s = gen_m1; break;
2292 1596 : case 1: s = gen_1; break;
2293 7 : default: s = gen_0; break;
2294 : }
2295 2135 : set_avma(av);
2296 2135 : return (ind0 && typ(ind0) == t_INT)? s: const_vec(l-1, s);
2297 : }
2298 91784 : v = nfsign_arch(nf, x, ind);
2299 91784 : if (ind0 && typ(ind0) == t_INT) { set_avma(av); return v[1]? gen_m1: gen_1; }
2300 91770 : settyp(v, t_VEC);
2301 259700 : for (i = 1; i < l; i++) gel(v,i) = v[i]? gen_m1: gen_1;
2302 91770 : return gerepileupto(av, v);
2303 : }
2304 :
2305 : /* true nf */
2306 : GEN
2307 294 : nfeltembed_i(GEN *pnf, GEN x, GEN ind0, long prec0)
2308 : {
2309 : long i, e, l, r1, r2, prec, prec1;
2310 294 : GEN v, ind, cx, nf = *pnf;
2311 294 : nf_get_sign(nf,&r1,&r2);
2312 294 : x = nf_to_scalar_or_basis(nf, x);
2313 287 : ind = parse_embed(ind0, r1+r2, "nfeltembed");
2314 280 : l = lg(ind);
2315 280 : if (typ(x) != t_COL)
2316 : {
2317 0 : if (!(ind0 && typ(ind0) == t_INT)) x = const_vec(l-1, x);
2318 0 : return x;
2319 : }
2320 280 : x = Q_primitive_part(x, &cx);
2321 280 : prec1 = prec0; e = gexpo(x);
2322 280 : if (e > 8) prec1 += nbits2extraprec(e);
2323 280 : prec = prec1;
2324 280 : if (nf_get_prec(nf) < prec) nf = nfnewprec_shallow(nf, prec);
2325 280 : v = cgetg(l, t_VEC);
2326 : for(;;)
2327 42 : {
2328 322 : GEN M = nf_get_M(nf);
2329 945 : for (i = 1; i < l; i++)
2330 : {
2331 665 : GEN t = nfembed_i(M, x, ind[i]);
2332 665 : long e = gexpo(t);
2333 665 : if (gequal0(t) || precision(t) < prec0
2334 665 : || (e < 0 && prec < prec1 + nbits2extraprec(-e)) ) break;
2335 623 : if (cx) t = gmul(t, cx);
2336 623 : gel(v,i) = t;
2337 : }
2338 322 : if (i == l) break;
2339 42 : prec = precdbl(prec);
2340 42 : if (DEBUGLEVEL>1) pari_warn(warnprec,"eltnfembed", prec);
2341 42 : *pnf = nf = nfnewprec_shallow(nf, prec);
2342 : }
2343 280 : if (ind0 && typ(ind0) == t_INT) v = gel(v,1);
2344 280 : return v;
2345 : }
2346 : GEN
2347 294 : nfeltembed(GEN nf, GEN x, GEN ind0, long prec0)
2348 : {
2349 294 : pari_sp av = avma; nf = checknf(nf);
2350 294 : return gerepilecopy(av, nfeltembed_i(&nf, x, ind0, prec0));
2351 : }
2352 :
2353 : /* number of distinct roots of sigma(f) */
2354 : GEN
2355 1526 : nfpolsturm(GEN nf, GEN f, GEN ind0)
2356 : {
2357 1526 : pari_sp av = avma;
2358 : long d, l, r1, single;
2359 : GEN ind, u, v, vr1, T, s, t;
2360 :
2361 1526 : nf = checknf(nf); T = nf_get_pol(nf); r1 = nf_get_r1(nf);
2362 1526 : ind = parse_embed(ind0, r1, "nfpolsturm");
2363 1505 : single = ind0 && typ(ind0) == t_INT;
2364 1505 : l = lg(ind);
2365 :
2366 1505 : if (gequal0(f)) pari_err_ROOTS0("nfpolsturm");
2367 1498 : if (typ(f) == t_POL && varn(f) != varn(T))
2368 : {
2369 1477 : f = RgX_nffix("nfpolsturm", T, f,1);
2370 1477 : if (lg(f) == 3) f = NULL;
2371 : }
2372 : else
2373 : {
2374 21 : (void)Rg_nffix("nfpolsturm", T, f, 0);
2375 21 : f = NULL;
2376 : }
2377 1498 : if (!f) { set_avma(av); return single? gen_0: zerovec(l-1); }
2378 1477 : d = degpol(f);
2379 1477 : if (d == 1) { set_avma(av); return single? gen_1: const_vec(l-1,gen_1); }
2380 :
2381 1435 : vr1 = const_vecsmall(l-1, 1);
2382 1435 : u = Q_primpart(f); s = ZV_to_zv(nfeltsign(nf, gel(u,d+2), ind));
2383 1435 : v = RgX_deriv(u); t = odd(d)? leafcopy(s): zv_neg(s);
2384 : for(;;)
2385 182 : {
2386 1617 : GEN r = RgX_neg( Q_primpart(RgX_pseudorem(u, v)) ), sr;
2387 1617 : long i, dr = degpol(r);
2388 1617 : if (dr < 0) break;
2389 1617 : sr = ZV_to_zv(nfeltsign(nf, gel(r,dr+2), ind));
2390 4011 : for (i = 1; i < l; i++)
2391 2394 : if (sr[i] != s[i]) { s[i] = sr[i], vr1[i]--; }
2392 1617 : if (odd(dr)) sr = zv_neg(sr);
2393 4011 : for (i = 1; i < l; i++)
2394 2394 : if (sr[i] != t[i]) { t[i] = sr[i], vr1[i]++; }
2395 1617 : if (!dr) break;
2396 182 : u = v; v = r;
2397 : }
2398 1435 : if (single) return gc_stoi(av,vr1[1]);
2399 1428 : return gerepileupto(av, zv_to_ZV(vr1));
2400 : }
2401 :
2402 : /* True nf; return the vector of signs of x; the matrix of such if x is a vector
2403 : * of nf elements */
2404 : GEN
2405 43911 : nfsign(GEN nf, GEN x)
2406 : {
2407 : long i, l;
2408 : GEN archp, S;
2409 :
2410 43911 : archp = identity_perm( nf_get_r1(nf) );
2411 43911 : if (typ(x) != t_VEC) return nfsign_arch(nf, x, archp);
2412 35938 : l = lg(x); S = cgetg(l, t_MAT);
2413 148063 : for (i=1; i<l; i++) gel(S,i) = nfsign_arch(nf, gel(x,i), archp);
2414 35937 : return S;
2415 : }
2416 :
2417 : /* x integral elt, A integral ideal in HNF; reduce x mod A */
2418 : static GEN
2419 8234401 : zk_modHNF(GEN x, GEN A)
2420 8234401 : { return (typ(x) == t_COL)? ZC_hnfrem(x, A): modii(x, gcoeff(A,1,1)); }
2421 :
2422 : /* given an element x in Z_K and an integral ideal y in HNF, coprime with x,
2423 : outputs an element inverse of x modulo y */
2424 : GEN
2425 175 : nfinvmodideal(GEN nf, GEN x, GEN y)
2426 : {
2427 175 : pari_sp av = avma;
2428 175 : GEN a, yZ = gcoeff(y,1,1);
2429 :
2430 175 : if (equali1(yZ)) return gen_0;
2431 175 : x = nf_to_scalar_or_basis(nf, x);
2432 175 : if (typ(x) == t_INT) return gerepileupto(av, Fp_inv(x, yZ));
2433 :
2434 119 : a = hnfmerge_get_1(idealhnf_principal(nf,x), y);
2435 119 : if (!a) pari_err_INV("nfinvmodideal", x);
2436 119 : return gerepileupto(av, zk_modHNF(nfdiv(nf,a,x), y));
2437 : }
2438 :
2439 : static GEN
2440 3053066 : nfsqrmodideal(GEN nf, GEN x, GEN id)
2441 3053066 : { return zk_modHNF(nfsqri(nf,x), id); }
2442 : static GEN
2443 7557187 : nfmulmodideal(GEN nf, GEN x, GEN y, GEN id)
2444 7557187 : { return x? zk_modHNF(nfmuli(nf,x,y), id): y; }
2445 : /* assume x integral, k integer, A in HNF */
2446 : GEN
2447 5736206 : nfpowmodideal(GEN nf,GEN x,GEN k,GEN A)
2448 : {
2449 5736206 : long s = signe(k);
2450 : pari_sp av;
2451 : GEN y;
2452 :
2453 5736206 : if (!s) return gen_1;
2454 5736206 : av = avma;
2455 5736206 : x = nf_to_scalar_or_basis(nf, x);
2456 5736498 : if (typ(x) != t_COL) return Fp_pow(x, k, gcoeff(A,1,1));
2457 2560595 : if (s < 0) { k = negi(k); x = nfinvmodideal(nf, x,A); }
2458 2560595 : if (equali1(k)) return gerepileupto(av, s > 0? zk_modHNF(x, A): x);
2459 1209460 : for(y = NULL;;)
2460 : {
2461 4262687 : if (mpodd(k)) y = nfmulmodideal(nf,y,x,A);
2462 4262682 : k = shifti(k,-1); if (!signe(k)) break;
2463 3052624 : x = nfsqrmodideal(nf,x,A);
2464 : }
2465 1209470 : return gerepileupto(av, y);
2466 : }
2467 :
2468 : /* a * g^n mod id */
2469 : static GEN
2470 4604264 : nfmulpowmodideal(GEN nf, GEN a, GEN g, GEN n, GEN id)
2471 : {
2472 4604264 : return nfmulmodideal(nf, a, nfpowmodideal(nf,g,n,id), id);
2473 : }
2474 :
2475 : /* assume (num(g[i]), id) = 1 for all i. Return prod g[i]^e[i] mod id.
2476 : * EX = multiple of exponent of (O_K/id)^* */
2477 : GEN
2478 2613810 : famat_to_nf_modideal_coprime(GEN nf, GEN g, GEN e, GEN id, GEN EX)
2479 : {
2480 2613810 : GEN EXo2, plus = NULL, minus = NULL, idZ = gcoeff(id,1,1);
2481 2613810 : long i, lx = lg(g);
2482 :
2483 2613810 : if (equali1(idZ)) return gen_1; /* id = Z_K */
2484 2613327 : EXo2 = (expi(EX) > 10)? shifti(EX,-1): NULL;
2485 8071803 : for (i = 1; i < lx; i++)
2486 : {
2487 5458613 : GEN h, n = centermodii(gel(e,i), EX, EXo2);
2488 5457946 : long sn = signe(n);
2489 5457946 : if (!sn) continue;
2490 :
2491 3952798 : h = nf_to_scalar_or_basis(nf, gel(g,i));
2492 3953360 : switch(typ(h))
2493 : {
2494 2355418 : case t_INT: break;
2495 0 : case t_FRAC:
2496 0 : h = Fp_div(gel(h,1), gel(h,2), idZ); break;
2497 1597942 : default:
2498 : {
2499 : GEN dh;
2500 1597942 : h = Q_remove_denom(h, &dh);
2501 1598073 : if (dh) h = FpC_Fp_mul(h, Fp_inv(dh,idZ), idZ);
2502 : }
2503 : }
2504 3953424 : if (sn > 0)
2505 3951520 : plus = nfmulpowmodideal(nf, plus, h, n, id);
2506 : else /* sn < 0 */
2507 1904 : minus = nfmulpowmodideal(nf, minus, h, negi(n), id);
2508 : }
2509 2613190 : if (minus) plus = nfmulmodideal(nf, plus, nfinvmodideal(nf,minus,id), id);
2510 2613268 : return plus? plus: gen_1;
2511 : }
2512 :
2513 : /* given 2 integral ideals x, y in HNF s.t x | y | x^2, compute (1+x)/(1+y) in
2514 : * the form [[cyc],[gen], U], where U := ux^-1 as a pair [ZM, denom(U)] */
2515 : static GEN
2516 236584 : zidealij(GEN x, GEN y)
2517 : {
2518 236584 : GEN U, G, cyc, xp = gcoeff(x,1,1), xi = hnf_invscale(x, xp);
2519 : long j, N;
2520 :
2521 : /* x^(-1) y = relations between the 1 + x_i (HNF) */
2522 236570 : cyc = ZM_snf_group(ZM_Z_divexact(ZM_mul(xi, y), xp), &U, &G);
2523 236570 : N = lg(cyc); G = ZM_mul(x,G); settyp(G, t_VEC); /* new generators */
2524 573067 : for (j=1; j<N; j++)
2525 : {
2526 336504 : GEN c = gel(G,j);
2527 336504 : gel(c,1) = addiu(gel(c,1), 1); /* 1 + g_j */
2528 336469 : if (ZV_isscalar(c)) gel(G,j) = gel(c,1);
2529 : }
2530 236563 : return mkvec4(cyc, G, ZM_mul(U,xi), xp);
2531 : }
2532 :
2533 : /* lg(x) > 1, x + 1; shallow */
2534 : static GEN
2535 169573 : ZC_add1(GEN x)
2536 : {
2537 169573 : long i, l = lg(x);
2538 169573 : GEN y = cgetg(l, t_COL);
2539 395842 : for (i = 2; i < l; i++) gel(y,i) = gel(x,i);
2540 169575 : gel(y,1) = addiu(gel(x,1), 1); return y;
2541 : }
2542 : /* lg(x) > 1, x - 1; shallow */
2543 : static GEN
2544 70399 : ZC_sub1(GEN x)
2545 : {
2546 70399 : long i, l = lg(x);
2547 70399 : GEN y = cgetg(l, t_COL);
2548 176615 : for (i = 2; i < l; i++) gel(y,i) = gel(x,i);
2549 70398 : gel(y,1) = subiu(gel(x,1), 1); return y;
2550 : }
2551 :
2552 : /* x,y are t_INT or ZC */
2553 : static GEN
2554 0 : zkadd(GEN x, GEN y)
2555 : {
2556 0 : long tx = typ(x);
2557 0 : if (tx == typ(y))
2558 0 : return tx == t_INT? addii(x,y): ZC_add(x,y);
2559 : else
2560 0 : return tx == t_INT? ZC_Z_add(y,x): ZC_Z_add(x,y);
2561 : }
2562 : /* x a t_INT or ZC, x+1; shallow */
2563 : static GEN
2564 255190 : zkadd1(GEN x)
2565 : {
2566 255190 : long tx = typ(x);
2567 255190 : return tx == t_INT? addiu(x,1): ZC_add1(x);
2568 : }
2569 : /* x a t_INT or ZC, x-1; shallow */
2570 : static GEN
2571 255267 : zksub1(GEN x)
2572 : {
2573 255267 : long tx = typ(x);
2574 255267 : return tx == t_INT? subiu(x,1): ZC_sub1(x);
2575 : }
2576 : /* x,y are t_INT or ZC; x - y */
2577 : static GEN
2578 0 : zksub(GEN x, GEN y)
2579 : {
2580 0 : long tx = typ(x), ty = typ(y);
2581 0 : if (tx == ty)
2582 0 : return tx == t_INT? subii(x,y): ZC_sub(x,y);
2583 : else
2584 0 : return tx == t_INT? Z_ZC_sub(x,y): ZC_Z_sub(x,y);
2585 : }
2586 : /* x is t_INT or ZM (mult. map), y is t_INT or ZC; x * y */
2587 : static GEN
2588 255208 : zkmul(GEN x, GEN y)
2589 : {
2590 255208 : long tx = typ(x), ty = typ(y);
2591 255208 : if (ty == t_INT)
2592 184827 : return tx == t_INT? mulii(x,y): ZC_Z_mul(gel(x,1),y);
2593 : else
2594 70381 : return tx == t_INT? ZC_Z_mul(y,x): ZM_ZC_mul(x,y);
2595 : }
2596 :
2597 : /* (U,V) = 1 coprime ideals. Want z = x mod U, = y mod V; namely
2598 : * z =vx + uy = v(x-y) + y, where u + v = 1, u in U, v in V.
2599 : * zkc = [v, UV], v a t_INT or ZM (mult. by v map), UV a ZM (ideal in HNF);
2600 : * shallow */
2601 : GEN
2602 0 : zkchinese(GEN zkc, GEN x, GEN y)
2603 : {
2604 0 : GEN v = gel(zkc,1), UV = gel(zkc,2), z = zkadd(zkmul(v, zksub(x,y)), y);
2605 0 : return zk_modHNF(z, UV);
2606 : }
2607 : /* special case z = x mod U, = 1 mod V; shallow */
2608 : GEN
2609 255262 : zkchinese1(GEN zkc, GEN x)
2610 : {
2611 255262 : GEN v = gel(zkc,1), UV = gel(zkc,2), z = zkadd1(zkmul(v, zksub1(x)));
2612 255219 : return (typ(z) == t_INT)? z: ZC_hnfrem(z, UV);
2613 : }
2614 : static GEN
2615 237302 : zkVchinese1(GEN zkc, GEN v)
2616 : {
2617 : long i, ly;
2618 237302 : GEN y = cgetg_copy(v, &ly);
2619 492507 : for (i=1; i<ly; i++) gel(y,i) = zkchinese1(zkc, gel(v,i));
2620 237246 : return y;
2621 : }
2622 :
2623 : /* prepare to solve z = x (mod A), z = y mod (B) [zkchinese or zkchinese1] */
2624 : GEN
2625 237046 : zkchineseinit(GEN nf, GEN A, GEN B, GEN AB)
2626 : {
2627 237046 : GEN v = idealaddtoone_raw(nf, A, B);
2628 : long e;
2629 237040 : if ((e = gexpo(v)) > 5)
2630 : {
2631 83230 : GEN b = (typ(v) == t_COL)? v: scalarcol_shallow(v, nf_get_degree(nf));
2632 83230 : b= ZC_reducemodlll(b, AB);
2633 83231 : if (gexpo(b) < e) v = b;
2634 : }
2635 237038 : return mkvec2(zk_scalar_or_multable(nf,v), AB);
2636 : }
2637 : /* prepare to solve z = x (mod A), z = 1 mod (B)
2638 : * and then z = 1 (mod A), z = y mod (B) [zkchinese1 twice] */
2639 : static GEN
2640 259 : zkchinese1init2(GEN nf, GEN A, GEN B, GEN AB)
2641 : {
2642 259 : GEN zkc = zkchineseinit(nf, A, B, AB);
2643 259 : GEN mv = gel(zkc,1), mu;
2644 259 : if (typ(mv) == t_INT) return mkvec2(zkc, mkvec2(subui(1,mv),AB));
2645 35 : mu = RgM_Rg_add_shallow(ZM_neg(mv), gen_1);
2646 35 : return mkvec2(mkvec2(mv,AB), mkvec2(mu,AB));
2647 : }
2648 :
2649 : static GEN
2650 2144269 : apply_U(GEN L, GEN a)
2651 : {
2652 2144269 : GEN e, U = gel(L,3), dU = gel(L,4);
2653 2144269 : if (typ(a) == t_INT)
2654 665618 : e = ZC_Z_mul(gel(U,1), subiu(a, 1));
2655 : else
2656 : { /* t_COL */
2657 1478651 : GEN t = shallowcopy(a);
2658 1478705 : gel(t,1) = subiu(gel(t,1), 1); /* t = a - 1 */
2659 1478583 : e = ZM_ZC_mul(U, t);
2660 : }
2661 2144238 : return gdiv(e, dU);
2662 : }
2663 :
2664 : /* true nf; vectors of [[cyc],[g],U.X^-1]. Assume k > 1. */
2665 : static GEN
2666 168944 : principal_units(GEN nf, GEN pr, long k, GEN prk)
2667 : {
2668 : GEN list, prb;
2669 168944 : ulong mask = quadratic_prec_mask(k);
2670 168944 : long a = 1;
2671 :
2672 168944 : prb = pr_hnf(nf,pr);
2673 168949 : list = vectrunc_init(k);
2674 405533 : while (mask > 1)
2675 : {
2676 236587 : GEN pra = prb;
2677 236587 : long b = a << 1;
2678 :
2679 236587 : if (mask & 1) b--;
2680 236587 : mask >>= 1;
2681 : /* compute 1 + pr^a / 1 + pr^b, 2a <= b */
2682 236587 : prb = (b >= k)? prk: idealpows(nf,pr,b);
2683 236586 : vectrunc_append(list, zidealij(pra, prb));
2684 236586 : a = b;
2685 : }
2686 168946 : return list;
2687 : }
2688 : /* a = 1 mod (pr) return log(a) on local-gens of 1+pr/1+pr^k */
2689 : static GEN
2690 1325910 : log_prk1(GEN nf, GEN a, long nh, GEN L2, GEN prk)
2691 : {
2692 1325910 : GEN y = cgetg(nh+1, t_COL);
2693 1325910 : long j, iy, c = lg(L2)-1;
2694 3470135 : for (j = iy = 1; j <= c; j++)
2695 : {
2696 2144252 : GEN L = gel(L2,j), cyc = gel(L,1), gen = gel(L,2), E = apply_U(L,a);
2697 2144179 : long i, nc = lg(cyc)-1;
2698 2144179 : int last = (j == c);
2699 5797659 : for (i = 1; i <= nc; i++, iy++)
2700 : {
2701 3653434 : GEN t, e = gel(E,i);
2702 3653434 : if (typ(e) != t_INT) pari_err_COPRIME("zlog_prk1", a, prk);
2703 3653427 : t = Fp_neg(e, gel(cyc,i));
2704 3653362 : gel(y,iy) = negi(t);
2705 3653489 : if (!last && signe(t)) a = nfmulpowmodideal(nf, a, gel(gen,i), t, prk);
2706 : }
2707 : }
2708 1325883 : return y;
2709 : }
2710 : /* true nf */
2711 : static GEN
2712 56560 : principal_units_relations(GEN nf, GEN L2, GEN prk, long nh)
2713 : {
2714 56560 : GEN h = cgetg(nh+1,t_MAT);
2715 56560 : long ih, j, c = lg(L2)-1;
2716 180761 : for (j = ih = 1; j <= c; j++)
2717 : {
2718 124199 : GEN L = gel(L2,j), F = gel(L,1), G = gel(L,2);
2719 124199 : long k, lG = lg(G);
2720 303693 : for (k = 1; k < lG; k++,ih++)
2721 : { /* log(g^f) mod pr^e */
2722 179492 : GEN a = nfpowmodideal(nf,gel(G,k),gel(F,k),prk);
2723 179490 : gel(h,ih) = ZC_neg(log_prk1(nf, a, nh, L2, prk));
2724 179494 : gcoeff(h,ih,ih) = gel(F,k);
2725 : }
2726 : }
2727 56562 : return h;
2728 : }
2729 : /* true nf; k > 1; multiplicative group (1 + pr) / (1 + pr^k) */
2730 : static GEN
2731 168946 : idealprincipalunits_i(GEN nf, GEN pr, long k, GEN *pU)
2732 : {
2733 168946 : GEN cyc, gen, L2, prk = idealpows(nf, pr, k);
2734 :
2735 168945 : L2 = principal_units(nf, pr, k, prk);
2736 168949 : if (k == 2)
2737 : {
2738 112389 : GEN L = gel(L2,1);
2739 112389 : cyc = gel(L,1);
2740 112389 : gen = gel(L,2);
2741 112389 : if (pU) *pU = matid(lg(gen)-1);
2742 : }
2743 : else
2744 : {
2745 56560 : long c = lg(L2), j;
2746 56560 : GEN EX, h, Ui, vg = cgetg(c, t_VEC);
2747 180761 : for (j = 1; j < c; j++) gel(vg, j) = gmael(L2,j,2);
2748 56560 : vg = shallowconcat1(vg);
2749 56560 : h = principal_units_relations(nf, L2, prk, lg(vg)-1);
2750 56560 : h = ZM_hnfall_i(h, NULL, 0);
2751 56559 : cyc = ZM_snf_group(h, pU, &Ui);
2752 56560 : c = lg(Ui); gen = cgetg(c, t_VEC); EX = cyc_get_expo(cyc);
2753 187960 : for (j = 1; j < c; j++)
2754 131401 : gel(gen,j) = famat_to_nf_modideal_coprime(nf, vg, gel(Ui,j), prk, EX);
2755 : }
2756 168948 : return mkvec4(cyc, gen, prk, L2);
2757 : }
2758 : GEN
2759 140 : idealprincipalunits(GEN nf, GEN pr, long k)
2760 : {
2761 : pari_sp av;
2762 : GEN v;
2763 140 : nf = checknf(nf);
2764 140 : if (k == 1) { checkprid(pr); retmkvec3(gen_1,cgetg(1,t_VEC),cgetg(1,t_VEC)); }
2765 133 : av = avma; v = idealprincipalunits_i(nf, pr, k, NULL);
2766 133 : return gerepilecopy(av, mkvec3(powiu(pr_norm(pr), k-1), gel(v,1), gel(v,2)));
2767 : }
2768 :
2769 : /* true nf; given an ideal pr^k dividing an integral ideal x (in HNF form)
2770 : * compute an 'sprk', the structure of G = (Z_K/pr^k)^* [ x = NULL for x=pr^k ]
2771 : * Return a vector with at least 4 components [cyc],[gen],[HNF pr^k,pr,k],ff,
2772 : * where
2773 : * cyc : type of G as abelian group (SNF)
2774 : * gen : generators of G, coprime to x
2775 : * pr^k: in HNF
2776 : * ff : data for log_g in (Z_K/pr)^*
2777 : * Two extra components are present iff k > 1: L2, U
2778 : * L2 : list of data structures to compute local DL in (Z_K/pr)^*,
2779 : * and 1 + pr^a/ 1 + pr^b for various a < b <= min(2a, k)
2780 : * U : base change matrices to convert a vector of local DL to DL wrt gen
2781 : * If MOD is not NULL, initialize G / G^MOD instead */
2782 : static GEN
2783 425536 : sprkinit(GEN nf, GEN pr, long k, GEN x, GEN MOD)
2784 : {
2785 425536 : GEN T, p, Ld, modpr, cyc, gen, g, g0, A, prk, U, L2, ord0 = NULL;
2786 425536 : long f = pr_get_f(pr);
2787 :
2788 425532 : if(DEBUGLEVEL>3) err_printf("treating pr^%ld, pr = %Ps\n",k,pr);
2789 425532 : modpr = nf_to_Fq_init(nf, &pr,&T,&p);
2790 425550 : if (MOD)
2791 : {
2792 378036 : GEN A = subiu(powiu(p,f), 1), d = gcdii(A, MOD), fa = Z_factor(d);
2793 378017 : ord0 = mkvec2(A, fa); /* true order, factorization of order in G/G^MOD */
2794 378016 : Ld = gel(fa,1);
2795 378016 : if (lg(Ld) > 1 && equaliu(gel(Ld,1),2)) Ld = vecslice(Ld,2,lg(Ld)-1);
2796 : }
2797 : /* (Z_K / pr)^* */
2798 425540 : if (f == 1)
2799 : {
2800 336508 : g0 = g = MOD? pgener_Fp_local(p, Ld): pgener_Fp(p);
2801 336522 : if (!ord0) ord0 = get_arith_ZZM(subiu(p,1));
2802 : }
2803 : else
2804 : {
2805 89032 : g0 = g = MOD? gener_FpXQ_local(T, p, Ld): gener_FpXQ(T,p, &ord0);
2806 89032 : g = Fq_to_nf(g, modpr);
2807 89032 : if (typ(g) == t_POL) g = poltobasis(nf, g);
2808 : }
2809 425558 : A = gel(ord0, 1); /* Norm(pr)-1 */
2810 : /* If MOD != NULL, d = gcd(A, MOD): g^(A/d) has order d */
2811 425558 : if (k == 1)
2812 : {
2813 256747 : cyc = mkvec(A);
2814 256741 : gen = mkvec(g);
2815 256737 : prk = pr_hnf(nf,pr);
2816 256746 : L2 = U = NULL;
2817 : }
2818 : else
2819 : { /* local-gens of (1 + pr)/(1 + pr^k) = SNF-gens * U */
2820 : GEN AB, B, u, v, w;
2821 : long j, l;
2822 168811 : w = idealprincipalunits_i(nf, pr, k, &U);
2823 : /* incorporate (Z_K/pr)^*, order A coprime to B = expo(1+pr/1+pr^k)*/
2824 168810 : cyc = leafcopy(gel(w,1)); B = cyc_get_expo(cyc); AB = mulii(A,B);
2825 168796 : gen = leafcopy(gel(w,2));
2826 168807 : prk = gel(w,3);
2827 168807 : g = nfpowmodideal(nf, g, B, prk);
2828 168815 : g0 = Fq_pow(g0, modii(B,A), T, p); /* update primitive root */
2829 168808 : L2 = mkvec3(A, g, gel(w,4));
2830 168814 : gel(cyc,1) = AB;
2831 168814 : gel(gen,1) = nfmulmodideal(nf, gel(gen,1), g, prk);
2832 168804 : u = mulii(Fp_inv(A,B), A);
2833 168799 : v = subui(1, u); l = lg(U);
2834 504846 : for (j = 1; j < l; j++) gcoeff(U,1,j) = Fp_mul(u, gcoeff(U,1,j), AB);
2835 168805 : U = mkvec2(Rg_col_ei(v, lg(gen)-1, 1), U);
2836 : }
2837 : /* local-gens of (Z_K/pr^k)^* = SNF-gens * U */
2838 425560 : if (x)
2839 : {
2840 236801 : GEN uv = zkchineseinit(nf, idealmulpowprime(nf,x,pr,utoineg(k)), prk, x);
2841 236784 : gen = zkVchinese1(uv, gen);
2842 : }
2843 425500 : return mkvecn(U? 6: 4, cyc, gen, prk, mkvec3(modpr,g0,ord0), L2, U);
2844 : }
2845 : GEN
2846 3971108 : sprk_get_cyc(GEN s) { return gel(s,1); }
2847 : GEN
2848 1962460 : sprk_get_expo(GEN s) { return cyc_get_expo(sprk_get_cyc(s)); }
2849 : GEN
2850 335558 : sprk_get_gen(GEN s) { return gel(s,2); }
2851 : GEN
2852 4898777 : sprk_get_prk(GEN s) { return gel(s,3); }
2853 : GEN
2854 2535323 : sprk_get_ff(GEN s) { return gel(s,4); }
2855 : GEN
2856 2596355 : sprk_get_pr(GEN s) { GEN ff = gel(s,4); return modpr_get_pr(gel(ff,1)); }
2857 : /* L2 to 1 + pr / 1 + pr^k */
2858 : static GEN
2859 1208589 : sprk_get_L2(GEN s) { return gmael(s,5,3); }
2860 : /* lift to nf of primitive root of k(pr) */
2861 : static GEN
2862 318953 : sprk_get_gnf(GEN s) { return gmael(s,5,2); }
2863 : /* A = Npr-1, <g> = (Z_K/pr)^*, L2 to 1 + pr / 1 + pr^k */
2864 : void
2865 0 : sprk_get_AgL2(GEN s, GEN *A, GEN *g, GEN *L2)
2866 0 : { GEN v = gel(s,5); *A = gel(v,1); *g = gel(v,2); *L2 = gel(v,3); }
2867 : void
2868 1200015 : sprk_get_U2(GEN s, GEN *U1, GEN *U2)
2869 1200015 : { GEN v = gel(s,6); *U1 = gel(v,1); *U2 = gel(v,2); }
2870 : static int
2871 2535340 : sprk_is_prime(GEN s) { return lg(s) == 5; }
2872 :
2873 : GEN
2874 1962266 : famat_zlog_pr(GEN nf, GEN g, GEN e, GEN sprk, GEN mod)
2875 : {
2876 1962266 : GEN x, expo = sprk_get_expo(sprk);
2877 1962262 : if (mod) expo = gcdii(expo,mod);
2878 1962239 : x = famat_makecoprime(nf, g, e, sprk_get_pr(sprk), sprk_get_prk(sprk), expo);
2879 1962264 : return log_prk(nf, x, sprk, mod);
2880 : }
2881 : /* famat_zlog_pr assuming (g,sprk.pr) = 1 */
2882 : static GEN
2883 196 : famat_zlog_pr_coprime(GEN nf, GEN g, GEN e, GEN sprk, GEN MOD)
2884 : {
2885 196 : GEN x = famat_to_nf_modideal_coprime(nf, g, e, sprk_get_prk(sprk),
2886 : sprk_get_expo(sprk));
2887 196 : return log_prk(nf, x, sprk, MOD);
2888 : }
2889 :
2890 : /* o t_INT, O = [ord,fa] format for multiple of o (for Fq_log);
2891 : * return o in [ord,fa] format */
2892 : static GEN
2893 542518 : order_update(GEN o, GEN O)
2894 : {
2895 542518 : GEN p = gmael(O,2,1), z = o, P, E;
2896 542518 : long i, j, l = lg(p);
2897 542518 : P = cgetg(l, t_COL);
2898 542513 : E = cgetg(l, t_COL);
2899 593529 : for (i = j = 1; i < l; i++)
2900 : {
2901 593529 : long v = Z_pvalrem(z, gel(p,i), &z);
2902 593474 : if (v)
2903 : {
2904 578585 : gel(P,j) = gel(p,i);
2905 578585 : gel(E,j) = utoipos(v); j++;
2906 578611 : if (is_pm1(z)) break;
2907 : }
2908 : }
2909 542482 : setlg(P, j);
2910 542479 : setlg(E, j); return mkvec2(o, mkmat2(P,E));
2911 : }
2912 :
2913 : /* a in Z_K (t_COL or t_INT), pr prime ideal, sprk = sprkinit(nf,pr,k,x),
2914 : * mod positive t_INT or NULL (meaning mod=0).
2915 : * return log(a) modulo mod on SNF-generators of (Z_K/pr^k)^* */
2916 : GEN
2917 2609102 : log_prk(GEN nf, GEN a, GEN sprk, GEN mod)
2918 : {
2919 : GEN e, prk, g, U1, U2, y, ff, O, o, oN, gN, N, T, p, modpr, pr, cyc;
2920 :
2921 2609102 : if (typ(a) == t_MAT) return famat_zlog_pr(nf, gel(a,1), gel(a,2), sprk, mod);
2922 2535312 : N = NULL;
2923 2535312 : ff = sprk_get_ff(sprk);
2924 2535322 : pr = gel(ff,1); /* modpr */
2925 2535322 : g = gN = gel(ff,2);
2926 2535322 : O = gel(ff,3); /* order of g = |Fq^*|, in [ord, fa] format */
2927 2535322 : o = oN = gel(O,1); /* order as a t_INT */
2928 2535322 : prk = sprk_get_prk(sprk);
2929 2535342 : modpr = nf_to_Fq_init(nf, &pr, &T, &p);
2930 2535314 : if (mod)
2931 : {
2932 2019025 : GEN d = gcdii(o,mod);
2933 2018767 : if (!equalii(o, d))
2934 : {
2935 732801 : N = diviiexact(o,d); /* > 1, coprime to p */
2936 732764 : a = nfpowmodideal(nf, a, N, prk);
2937 732938 : oN = d; /* order of g^N mod pr */
2938 : }
2939 : }
2940 2535188 : if (equali1(oN))
2941 396407 : e = gen_0;
2942 : else
2943 : {
2944 2138868 : if (N) { O = order_update(oN, O); gN = Fq_pow(g, N, T, p); }
2945 2138871 : e = Fq_log(nf_to_Fq(nf,a,modpr), gN, O, T, p);
2946 : }
2947 : /* 0 <= e < oN is correct modulo oN */
2948 2535343 : if (sprk_is_prime(sprk)) return mkcol(e); /* k = 1 */
2949 :
2950 799028 : sprk_get_U2(sprk, &U1,&U2);
2951 799100 : cyc = sprk_get_cyc(sprk);
2952 799099 : if (mod)
2953 : {
2954 377805 : cyc = ZV_snf_gcd(cyc, mod);
2955 377771 : if (signe(remii(mod,p))) return ZV_ZV_mod(ZC_Z_mul(U1,e), cyc);
2956 : }
2957 745488 : if (signe(e))
2958 : {
2959 318953 : GEN E = N? mulii(e, N): e;
2960 318953 : a = nfmulpowmodideal(nf, a, sprk_get_gnf(sprk), Fp_neg(E, o), prk);
2961 : }
2962 : /* a = 1 mod pr */
2963 745488 : y = log_prk1(nf, a, lg(U2)-1, sprk_get_L2(sprk), prk);
2964 745514 : if (N)
2965 : { /* from DL(a^N) to DL(a) */
2966 135330 : GEN E = gel(sprk_get_cyc(sprk), 1), q = powiu(p, Z_pval(E, p));
2967 135329 : y = ZC_Z_mul(y, Fp_inv(N, q));
2968 : }
2969 745513 : y = ZC_lincomb(gen_1, e, ZM_ZC_mul(U2,y), U1);
2970 745520 : return ZV_ZV_mod(y, cyc);
2971 : }
2972 : /* true nf */
2973 : GEN
2974 90059 : log_prk_init(GEN nf, GEN pr, long k, GEN MOD)
2975 90059 : { return sprkinit(nf,pr,k,NULL,MOD);}
2976 : GEN
2977 497 : veclog_prk(GEN nf, GEN v, GEN sprk)
2978 : {
2979 497 : long l = lg(v), i;
2980 497 : GEN w = cgetg(l, t_MAT);
2981 1232 : for (i = 1; i < l; i++) gel(w,i) = log_prk(nf, gel(v,i), sprk, NULL);
2982 497 : return w;
2983 : }
2984 :
2985 : static GEN
2986 1371109 : famat_zlog(GEN nf, GEN fa, GEN sgn, zlog_S *S)
2987 : {
2988 1371109 : long i, l0, l = lg(S->U);
2989 1371109 : GEN g = gel(fa,1), e = gel(fa,2), y = cgetg(l, t_COL);
2990 1371108 : l0 = lg(S->sprk); /* = l (trivial arch. part), or l-1 */
2991 2842161 : for (i=1; i < l0; i++) gel(y,i) = famat_zlog_pr(nf, g, e, gel(S->sprk,i), S->mod);
2992 1371102 : if (l0 != l)
2993 : {
2994 187592 : if (!sgn) sgn = nfsign_arch(nf, fa, S->archp);
2995 187592 : gel(y,l0) = Flc_to_ZC(sgn);
2996 : }
2997 1371102 : return y;
2998 : }
2999 :
3000 : /* assume that cyclic factors are normalized, in particular != [1] */
3001 : static GEN
3002 257146 : split_U(GEN U, GEN Sprk)
3003 : {
3004 257146 : long t = 0, k, n, l = lg(Sprk);
3005 257146 : GEN vU = cgetg(l+1, t_VEC);
3006 591913 : for (k = 1; k < l; k++)
3007 : {
3008 334759 : n = lg(sprk_get_cyc(gel(Sprk,k))) - 1; /* > 0 */
3009 334758 : gel(vU,k) = vecslice(U, t+1, t+n);
3010 334769 : t += n;
3011 : }
3012 : /* t+1 .. lg(U)-1 */
3013 257154 : n = lg(U) - t - 1; /* can be 0 */
3014 257154 : if (!n) setlg(vU,l); else gel(vU,l) = vecslice(U, t+1, t+n);
3015 257156 : return vU;
3016 : }
3017 :
3018 : static void
3019 1986321 : init_zlog_mod(zlog_S *S, GEN bid, GEN mod)
3020 : {
3021 1986321 : GEN fa2 = bid_get_fact2(bid);
3022 1986315 : S->U = bid_get_U(bid);
3023 1986314 : S->hU = lg(bid_get_cyc(bid))-1;
3024 1986307 : S->archp = bid_get_archp(bid);
3025 1986305 : S->sprk = bid_get_sprk(bid);
3026 1986300 : S->bid = bid;
3027 1986300 : S->mod = mod;
3028 1986300 : S->P = gel(fa2,1);
3029 1986300 : S->k = gel(fa2,2);
3030 1986300 : S->no2 = lg(S->P) == lg(gel(bid_get_fact(bid),1));
3031 1986302 : }
3032 : void
3033 379611 : init_zlog(zlog_S *S, GEN bid)
3034 : {
3035 379611 : return init_zlog_mod(S, bid, NULL);
3036 : }
3037 :
3038 : /* a a t_FRAC/t_INT, reduce mod bid */
3039 : static GEN
3040 14 : Q_mod_bid(GEN bid, GEN a)
3041 : {
3042 14 : GEN xZ = gcoeff(bid_get_ideal(bid),1,1);
3043 14 : GEN b = Rg_to_Fp(a, xZ);
3044 14 : if (gsigne(a) < 0) b = subii(b, xZ);
3045 14 : return signe(b)? b: xZ;
3046 : }
3047 : /* Return decomposition of a on the CRT generators blocks attached to the
3048 : * S->sprk and sarch; sgn = sign(a, S->arch), NULL if unknown */
3049 : static GEN
3050 380548 : zlog(GEN nf, GEN a, GEN sgn, zlog_S *S)
3051 : {
3052 : long k, l;
3053 : GEN y;
3054 380548 : a = nf_to_scalar_or_basis(nf, a);
3055 380542 : switch(typ(a))
3056 : {
3057 162138 : case t_INT: break;
3058 14 : case t_FRAC: a = Q_mod_bid(S->bid, a); break;
3059 218390 : default: /* case t_COL: */
3060 : {
3061 : GEN den;
3062 218390 : check_nfelt(a, &den);
3063 218403 : if (den)
3064 : {
3065 105 : a = Q_muli_to_int(a, den);
3066 105 : a = mkmat2(mkcol2(a, den), mkcol2(gen_1, gen_m1));
3067 105 : return famat_zlog(nf, a, sgn, S);
3068 : }
3069 : }
3070 : }
3071 380438 : if (sgn)
3072 373858 : sgn = (lg(sgn) == 1)? NULL: leafcopy(sgn);
3073 : else
3074 6580 : sgn = (lg(S->archp) == 1)? NULL: nfsign_arch(nf, a, S->archp);
3075 380442 : l = lg(S->sprk);
3076 380442 : y = cgetg(sgn? l+1: l, t_COL);
3077 920784 : for (k = 1; k < l; k++)
3078 : {
3079 540379 : GEN sprk = gel(S->sprk,k);
3080 540379 : gel(y,k) = log_prk(nf, a, sprk, S->mod);
3081 : }
3082 380405 : if (sgn) gel(y,l) = Flc_to_ZC(sgn);
3083 380418 : return y;
3084 : }
3085 :
3086 : /* true nf */
3087 : GEN
3088 43596 : pr_basis_perm(GEN nf, GEN pr)
3089 : {
3090 43596 : long f = pr_get_f(pr);
3091 : GEN perm;
3092 43596 : if (f == nf_get_degree(nf)) return identity_perm(f);
3093 38031 : perm = cgetg(f+1, t_VECSMALL);
3094 38031 : perm[1] = 1;
3095 38031 : if (f > 1)
3096 : {
3097 2912 : GEN H = pr_hnf(nf,pr);
3098 : long i, k;
3099 10815 : for (i = k = 2; k <= f; i++)
3100 7903 : if (!equali1(gcoeff(H,i,i))) perm[k++] = i;
3101 : }
3102 38031 : return perm;
3103 : }
3104 :
3105 : /* \sum U[i]*y[i], U[i] ZM, y[i] ZC. We allow lg(y) > lg(U). */
3106 : static GEN
3107 1751613 : ZMV_ZCV_mul(GEN U, GEN y)
3108 : {
3109 1751613 : long i, l = lg(U);
3110 1751613 : GEN z = NULL;
3111 1751613 : if (l == 1) return cgetg(1,t_COL);
3112 4124522 : for (i = 1; i < l; i++)
3113 : {
3114 2372972 : GEN u = ZM_ZC_mul(gel(U,i), gel(y,i));
3115 2372941 : z = z? ZC_add(z, u): u;
3116 : }
3117 1751550 : return z;
3118 : }
3119 : /* A * (U[1], ..., U[d] */
3120 : static GEN
3121 518 : ZM_ZMV_mul(GEN A, GEN U)
3122 : {
3123 : long i, l;
3124 518 : GEN V = cgetg_copy(U,&l);
3125 1057 : for (i = 1; i < l; i++) gel(V,i) = ZM_mul(A,gel(U,i));
3126 518 : return V;
3127 : }
3128 :
3129 : /* a = 1 mod pr, sprk mod pr^e, e >= 1 */
3130 : static GEN
3131 400931 : sprk_log_prk1_2(GEN nf, GEN a, GEN sprk)
3132 : {
3133 400931 : GEN U1, U2, y, L2 = sprk_get_L2(sprk);
3134 400931 : sprk_get_U2(sprk, &U1,&U2);
3135 400931 : y = ZM_ZC_mul(U2, log_prk1(nf, a, lg(U2)-1, L2, sprk_get_prk(sprk)));
3136 400921 : return ZV_ZV_mod(y, sprk_get_cyc(sprk));
3137 : }
3138 : /* true nf; assume e >= 2 */
3139 : GEN
3140 105511 : sprk_log_gen_pr2(GEN nf, GEN sprk, long e)
3141 : {
3142 105511 : GEN M, G, pr = sprk_get_pr(sprk);
3143 : long i, l;
3144 105511 : if (e == 2)
3145 : {
3146 62167 : GEN L2 = sprk_get_L2(sprk), L = gel(L2,1);
3147 62167 : G = gel(L,2); l = lg(G);
3148 : }
3149 : else
3150 : {
3151 43344 : GEN perm = pr_basis_perm(nf,pr), PI = nfpow_u(nf, pr_get_gen(pr), e-1);
3152 43344 : l = lg(perm);
3153 43344 : G = cgetg(l, t_VEC);
3154 43344 : if (typ(PI) == t_INT)
3155 : { /* zk_ei_mul doesn't allow t_INT */
3156 5558 : long N = nf_get_degree(nf);
3157 5558 : gel(G,1) = addiu(PI,1);
3158 8533 : for (i = 2; i < l; i++)
3159 : {
3160 2975 : GEN z = col_ei(N, 1);
3161 2975 : gel(G,i) = z; gel(z, perm[i]) = PI;
3162 : }
3163 : }
3164 : else
3165 : {
3166 37786 : gel(G,1) = nfadd(nf, gen_1, PI);
3167 44569 : for (i = 2; i < l; i++)
3168 6783 : gel(G,i) = nfadd(nf, gen_1, zk_ei_mul(nf, PI, perm[i]));
3169 : }
3170 : }
3171 105511 : M = cgetg(l, t_MAT);
3172 233632 : for (i = 1; i < l; i++) gel(M,i) = sprk_log_prk1_2(nf, gel(G,i), sprk);
3173 105498 : return M;
3174 : }
3175 : /* Log on bid.gen of generators of P_{1,I pr^{e-1}} / P_{1,I pr^e} (I,pr) = 1,
3176 : * defined implicitly via CRT. 'ind' is the index of pr in modulus
3177 : * factorization; true nf */
3178 : GEN
3179 413113 : log_gen_pr(zlog_S *S, long ind, GEN nf, long e)
3180 : {
3181 413113 : GEN Uind = gel(S->U, ind);
3182 413113 : if (e == 1) retmkmat( gel(Uind,1) );
3183 102808 : return ZM_mul(Uind, sprk_log_gen_pr2(nf, gel(S->sprk,ind), e));
3184 : }
3185 : /* true nf */
3186 : GEN
3187 2037 : sprk_log_gen_pr(GEN nf, GEN sprk, long e)
3188 : {
3189 2037 : if (e == 1)
3190 : {
3191 0 : long n = lg(sprk_get_cyc(sprk))-1;
3192 0 : retmkmat(col_ei(n, 1));
3193 : }
3194 2037 : return sprk_log_gen_pr2(nf, sprk, e);
3195 : }
3196 : /* a = 1 mod pr */
3197 : GEN
3198 272797 : sprk_log_prk1(GEN nf, GEN a, GEN sprk)
3199 : {
3200 272797 : if (lg(sprk) == 5) return mkcol(gen_0); /* mod pr */
3201 272797 : return sprk_log_prk1_2(nf, a, sprk);
3202 : }
3203 : /* Log on bid.gen of generator of P_{1,f} / P_{1,f v[index]}
3204 : * v = vector of r1 real places */
3205 : GEN
3206 85870 : log_gen_arch(zlog_S *S, long index)
3207 : {
3208 85870 : GEN U = gel(S->U, lg(S->U)-1);
3209 85870 : return gel(U, index);
3210 : }
3211 :
3212 : /* compute bid.clgp: [h,cyc] or [h,cyc,gen] */
3213 : static GEN
3214 258189 : bid_grp(GEN nf, GEN U, GEN cyc, GEN g, GEN F, GEN sarch)
3215 : {
3216 258189 : GEN G, h = ZV_prod(cyc);
3217 : long c;
3218 258197 : if (!U) return mkvec2(h,cyc);
3219 257854 : c = lg(U);
3220 257854 : G = cgetg(c,t_VEC);
3221 257861 : if (c > 1)
3222 : {
3223 227803 : GEN U0, Uoo, EX = cyc_get_expo(cyc); /* exponent of bid */
3224 227802 : long i, hU = nbrows(U), nba = lg(sarch_get_cyc(sarch))-1; /* #f_oo */
3225 227817 : if (!nba) { U0 = U; Uoo = NULL; }
3226 80292 : else if (nba == hU) { U0 = NULL; Uoo = U; }
3227 : else
3228 : {
3229 71199 : U0 = rowslice(U, 1, hU-nba);
3230 71204 : Uoo = rowslice(U, hU-nba+1, hU);
3231 : }
3232 694742 : for (i = 1; i < c; i++)
3233 : {
3234 466940 : GEN t = gen_1;
3235 466940 : if (U0) t = famat_to_nf_modideal_coprime(nf, g, gel(U0,i), F, EX);
3236 466915 : if (Uoo) t = set_sign_mod_divisor(nf, ZV_to_Flv(gel(Uoo,i),2), t, sarch);
3237 466927 : gel(G,i) = t;
3238 : }
3239 : }
3240 257860 : return mkvec3(h, cyc, G);
3241 : }
3242 :
3243 : /* remove prime ideals of norm 2 with exponent 1 from factorization */
3244 : static GEN
3245 258525 : famat_strip2(GEN fa)
3246 : {
3247 258525 : GEN P = gel(fa,1), E = gel(fa,2), Q, F;
3248 258525 : long l = lg(P), i, j;
3249 258525 : Q = cgetg(l, t_COL);
3250 258520 : F = cgetg(l, t_COL);
3251 633287 : for (i = j = 1; i < l; i++)
3252 : {
3253 374750 : GEN pr = gel(P,i), e = gel(E,i);
3254 374750 : if (!absequaliu(pr_get_p(pr), 2) || itou(e) != 1 || pr_get_f(pr) != 1)
3255 : {
3256 336162 : gel(Q,j) = pr;
3257 336162 : gel(F,j) = e; j++;
3258 : }
3259 : }
3260 258537 : setlg(Q,j);
3261 258532 : setlg(F,j); return mkmat2(Q,F);
3262 : }
3263 : static int
3264 134027 : checkarchp(GEN v, long r1)
3265 : {
3266 134027 : long i, l = lg(v);
3267 134027 : pari_sp av = avma;
3268 : GEN p;
3269 134027 : if (l == 1) return 1;
3270 47098 : if (l == 2) return v[1] > 0 && v[1] <= r1;
3271 21990 : p = zero_zv(r1);
3272 66086 : for (i = 1; i < l; i++)
3273 : {
3274 44121 : long j = v[i];
3275 44121 : if (j <= 0 || j > r1 || p[j]) return gc_long(av, 0);
3276 44086 : p[j] = 1;
3277 : }
3278 21965 : return gc_long(av, 1);
3279 : }
3280 :
3281 : /* True nf. Put ideal to form [[ideal,arch]] and set fa and fa2 to its
3282 : * factorization, archp to the indices of arch places */
3283 : GEN
3284 258497 : check_mod_factored(GEN nf, GEN ideal, GEN *fa_, GEN *fa2_, GEN *archp_, GEN MOD)
3285 : {
3286 : GEN arch, x, fa, fa2, archp;
3287 : long R1;
3288 :
3289 258497 : R1 = nf_get_r1(nf);
3290 258501 : if (typ(ideal) == t_VEC && lg(ideal) == 3)
3291 : {
3292 178476 : arch = gel(ideal,2);
3293 178476 : ideal= gel(ideal,1);
3294 178476 : switch(typ(arch))
3295 : {
3296 44449 : case t_VEC:
3297 44449 : if (lg(arch) != R1+1)
3298 7 : pari_err_TYPE("Idealstar [incorrect archimedean component]",arch);
3299 44442 : archp = vec01_to_indices(arch);
3300 44442 : break;
3301 134027 : case t_VECSMALL:
3302 134027 : if (!checkarchp(arch, R1))
3303 35 : pari_err_TYPE("Idealstar [incorrect archimedean component]",arch);
3304 134002 : archp = arch;
3305 134002 : arch = indices_to_vec01(archp, R1);
3306 133996 : break;
3307 0 : default:
3308 0 : pari_err_TYPE("Idealstar [incorrect archimedean component]",arch);
3309 : return NULL;/*LCOV_EXCL_LINE*/
3310 : }
3311 : }
3312 : else
3313 : {
3314 80025 : arch = zerovec(R1);
3315 80014 : archp = cgetg(1, t_VECSMALL);
3316 : }
3317 258453 : if (MOD)
3318 : {
3319 213912 : if (typ(MOD) != t_INT) pari_err_TYPE("bnrinit [incorrect cycmod]", MOD);
3320 213912 : if (mpodd(MOD) && lg(archp) != 1)
3321 224 : MOD = shifti(MOD, 1); /* ensure elements of G^MOD are >> 0 */
3322 : }
3323 258464 : if (is_nf_factor(ideal))
3324 : {
3325 40081 : fa = ideal;
3326 40081 : x = factorbackprime(nf, gel(fa,1), gel(fa,2));
3327 : }
3328 : else
3329 : {
3330 218373 : fa = idealfactor(nf, ideal);
3331 218399 : x = ideal;
3332 : }
3333 258480 : if (typ(x) != t_MAT) x = idealhnf_shallow(nf, x);
3334 258470 : if (lg(x) == 1) pari_err_DOMAIN("Idealstar", "ideal","=",gen_0,x);
3335 258470 : if (typ(gcoeff(x,1,1)) != t_INT)
3336 7 : pari_err_DOMAIN("Idealstar","denominator(ideal)", "!=",gen_1,x);
3337 :
3338 258463 : fa2 = famat_strip2(fa);
3339 258458 : if (fa_ != NULL) *fa_ = fa;
3340 258458 : if (fa2_ != NULL) *fa2_ = fa2;
3341 258458 : if (fa2_ != NULL) *archp_ = archp;
3342 258458 : return mkvec2(x, arch);
3343 : }
3344 :
3345 : /* True nf. Compute [[ideal,arch], [h,[cyc],[gen]], idealfact, [liste], U]
3346 : flag may include nf_GEN | nf_INIT */
3347 : static GEN
3348 257859 : Idealstarmod_i(GEN nf, GEN ideal, long flag, GEN MOD)
3349 : {
3350 : long i, nbp;
3351 257859 : GEN y, cyc, U, u1 = NULL, fa, fa2, sprk, x_arch, x, arch, archp, E, P, sarch, gen;
3352 :
3353 257859 : x_arch = check_mod_factored(nf, ideal, &fa, &fa2, &archp, MOD);
3354 257834 : x = gel(x_arch, 1);
3355 257834 : arch = gel(x_arch, 2);
3356 :
3357 257834 : sarch = nfarchstar(nf, x, archp);
3358 257851 : P = gel(fa2,1);
3359 257851 : E = gel(fa2,2);
3360 257851 : nbp = lg(P)-1;
3361 257851 : sprk = cgetg(nbp+1,t_VEC);
3362 257859 : if (nbp)
3363 : {
3364 218666 : GEN t = (lg(gel(fa,1))==2)? NULL: x; /* beware fa != fa2 */
3365 218666 : cyc = cgetg(nbp+2,t_VEC);
3366 218658 : gen = cgetg(nbp+1,t_VEC);
3367 554157 : for (i = 1; i <= nbp; i++)
3368 : {
3369 335488 : GEN L = sprkinit(nf, gel(P,i), itou(gel(E,i)), t, MOD);
3370 335499 : gel(sprk,i) = L;
3371 335499 : gel(cyc,i) = sprk_get_cyc(L);
3372 : /* true gens are congruent to those mod x AND positive at archp */
3373 335496 : gel(gen,i) = sprk_get_gen(L);
3374 : }
3375 218669 : gel(cyc,i) = sarch_get_cyc(sarch);
3376 218669 : cyc = shallowconcat1(cyc);
3377 218670 : gen = shallowconcat1(gen);
3378 218676 : cyc = ZV_snf_group(cyc, &U, (flag & nf_GEN)? &u1: NULL);
3379 : }
3380 : else
3381 : {
3382 39193 : cyc = sarch_get_cyc(sarch);
3383 39193 : gen = cgetg(1,t_VEC);
3384 39193 : U = matid(lg(cyc)-1);
3385 39193 : if (flag & nf_GEN) u1 = U;
3386 : }
3387 257831 : y = bid_grp(nf, u1, cyc, gen, x, sarch);
3388 257852 : if (!(flag & nf_INIT)) return y;
3389 257040 : U = split_U(U, sprk);
3390 257049 : return mkvec5(mkvec2(x, arch), y, mkvec2(fa,fa2), mkvec2(sprk, sarch), U);
3391 : }
3392 :
3393 : static long
3394 63 : idealHNF_norm_pval(GEN x, GEN p)
3395 : {
3396 63 : long i, v = 0, l = lg(x);
3397 175 : for (i = 1; i < l; i++) v += Z_pval(gcoeff(x,i,i), p);
3398 63 : return v;
3399 : }
3400 : static long
3401 63 : sprk_get_k(GEN sprk)
3402 : {
3403 : GEN pr, prk;
3404 63 : if (sprk_is_prime(sprk)) return 1;
3405 63 : pr = sprk_get_pr(sprk);
3406 63 : prk = sprk_get_prk(sprk);
3407 63 : return idealHNF_norm_pval(prk, pr_get_p(pr)) / pr_get_f(pr);
3408 : }
3409 : /* true nf, L a sprk */
3410 : GEN
3411 63 : sprk_to_bid(GEN nf, GEN L, long flag)
3412 : {
3413 63 : GEN y, cyc, U, u1 = NULL, fa, fa2, arch, sarch, gen, sprk;
3414 :
3415 63 : arch = zerovec(nf_get_r1(nf));
3416 63 : fa = to_famat_shallow(sprk_get_pr(L), utoipos(sprk_get_k(L)));
3417 63 : sarch = nfarchstar(nf, NULL, cgetg(1, t_VECSMALL));
3418 63 : fa2 = famat_strip2(fa);
3419 63 : sprk = mkvec(L);
3420 63 : cyc = shallowconcat(sprk_get_cyc(L), sarch_get_cyc(sarch));
3421 63 : gen = sprk_get_gen(L);
3422 63 : cyc = ZV_snf_group(cyc, &U, (flag & nf_GEN)? &u1: NULL);
3423 63 : y = bid_grp(nf, u1, cyc, gen, NULL, sarch);
3424 63 : if (!(flag & nf_INIT)) return y;
3425 63 : return mkvec5(mkvec2(sprk_get_prk(L), arch), y, mkvec2(fa,fa2),
3426 : mkvec2(sprk, sarch), split_U(U, sprk));
3427 : }
3428 : GEN
3429 257595 : Idealstarmod(GEN nf, GEN ideal, long flag, GEN MOD)
3430 : {
3431 257595 : pari_sp av = avma;
3432 257595 : nf = nf? checknf(nf): nfinit(pol_x(0), DEFAULTPREC);
3433 257595 : return gerepilecopy(av, Idealstarmod_i(nf, ideal, flag, MOD));
3434 : }
3435 : GEN
3436 952 : Idealstar(GEN nf, GEN ideal, long flag) { return Idealstarmod(nf, ideal, flag, NULL); }
3437 : GEN
3438 273 : Idealstarprk(GEN nf, GEN pr, long k, long flag)
3439 : {
3440 273 : pari_sp av = avma;
3441 273 : GEN z = Idealstarmod_i(nf, mkmat2(mkcol(pr),mkcols(k)), flag, NULL);
3442 273 : return gerepilecopy(av, z);
3443 : }
3444 :
3445 : /* FIXME: obsolete */
3446 : GEN
3447 0 : zidealstarinitgen(GEN nf, GEN ideal)
3448 0 : { return Idealstar(nf,ideal, nf_INIT|nf_GEN); }
3449 : GEN
3450 0 : zidealstarinit(GEN nf, GEN ideal)
3451 0 : { return Idealstar(nf,ideal, nf_INIT); }
3452 : GEN
3453 0 : zidealstar(GEN nf, GEN ideal)
3454 0 : { return Idealstar(nf,ideal, nf_GEN); }
3455 :
3456 : GEN
3457 98 : idealstarmod(GEN nf, GEN ideal, long flag, GEN MOD)
3458 : {
3459 98 : switch(flag)
3460 : {
3461 0 : case 0: return Idealstarmod(nf,ideal, nf_GEN, MOD);
3462 84 : case 1: return Idealstarmod(nf,ideal, nf_INIT, MOD);
3463 14 : case 2: return Idealstarmod(nf,ideal, nf_INIT|nf_GEN, MOD);
3464 0 : default: pari_err_FLAG("idealstar");
3465 : }
3466 : return NULL; /* LCOV_EXCL_LINE */
3467 : }
3468 : GEN
3469 0 : idealstar0(GEN nf, GEN ideal,long flag) { return idealstarmod(nf, ideal, flag, NULL); }
3470 :
3471 : void
3472 218405 : check_nfelt(GEN x, GEN *den)
3473 : {
3474 218405 : long l = lg(x), i;
3475 218405 : GEN t, d = NULL;
3476 218405 : if (typ(x) != t_COL) pari_err_TYPE("check_nfelt", x);
3477 806594 : for (i=1; i<l; i++)
3478 : {
3479 588189 : t = gel(x,i);
3480 588189 : switch (typ(t))
3481 : {
3482 587958 : case t_INT: break;
3483 231 : case t_FRAC:
3484 231 : if (!d) d = gel(t,2); else d = lcmii(d, gel(t,2));
3485 231 : break;
3486 0 : default: pari_err_TYPE("check_nfelt", x);
3487 : }
3488 : }
3489 218405 : *den = d;
3490 218405 : }
3491 :
3492 : GEN
3493 2141297 : ZV_snf_gcd(GEN x, GEN mod)
3494 4938162 : { pari_APPLY_same(gcdii(gel(x,i), mod)); }
3495 :
3496 : /* assume a true bnf and bid */
3497 : GEN
3498 226763 : ideallog_units0(GEN bnf, GEN bid, GEN MOD)
3499 : {
3500 226763 : GEN nf = bnf_get_nf(bnf), D, y, C, cyc;
3501 226764 : long j, lU = lg(bnf_get_logfu(bnf)); /* r1+r2 */
3502 : zlog_S S;
3503 226763 : init_zlog_mod(&S, bid, MOD);
3504 226768 : if (!S.hU) return zeromat(0,lU);
3505 226768 : cyc = bid_get_cyc(bid);
3506 226771 : if (MOD) cyc = ZV_snf_gcd(cyc, MOD);
3507 226708 : D = nfsign_fu(bnf, bid_get_archp(bid));
3508 226774 : y = cgetg(lU, t_MAT);
3509 226772 : if ((C = bnf_build_cheapfu(bnf)))
3510 373823 : { for (j = 1; j < lU; j++) gel(y,j) = zlog(nf, gel(C,j), gel(D,j), &S); }
3511 : else
3512 : {
3513 49 : long i, l = lg(S.U), l0 = lg(S.sprk);
3514 : GEN X, U;
3515 49 : if (!(C = bnf_compactfu_mat(bnf))) bnf_build_units(bnf); /* error */
3516 49 : X = gel(C,1); U = gel(C,2);
3517 147 : for (j = 1; j < lU; j++) gel(y,j) = cgetg(l, t_COL);
3518 126 : for (i = 1; i < l0; i++)
3519 : {
3520 77 : GEN sprk = gel(S.sprk, i);
3521 77 : GEN Xi = sunits_makecoprime(X, sprk_get_pr(sprk), sprk_get_prk(sprk));
3522 231 : for (j = 1; j < lU; j++)
3523 154 : gcoeff(y,i,j) = famat_zlog_pr_coprime(nf, Xi, gel(U,j), sprk, MOD);
3524 : }
3525 49 : if (l0 != l)
3526 56 : for (j = 1; j < lU; j++) gcoeff(y,l0,j) = Flc_to_ZC(gel(D,j));
3527 : }
3528 226763 : y = vec_prepend(y, zlog(nf, bnf_get_tuU(bnf), nfsign_tu(bnf, S.archp), &S));
3529 600696 : for (j = 1; j <= lU; j++)
3530 373947 : gel(y,j) = ZV_ZV_mod(ZMV_ZCV_mul(S.U, gel(y,j)), cyc);
3531 226749 : return y;
3532 : }
3533 : GEN
3534 84 : ideallog_units(GEN bnf, GEN bid)
3535 84 : { return ideallog_units0(bnf, bid, NULL); }
3536 : GEN
3537 518 : log_prk_units(GEN nf, GEN D, GEN sprk)
3538 : {
3539 518 : GEN L, Ltu = log_prk(nf, gel(D,1), sprk, NULL);
3540 518 : D = gel(D,2);
3541 518 : if (lg(D) != 3 || typ(gel(D,2)) != t_MAT) L = veclog_prk(nf, D, sprk);
3542 : else
3543 : {
3544 21 : GEN X = gel(D,1), U = gel(D,2);
3545 21 : long j, lU = lg(U);
3546 21 : X = sunits_makecoprime(X, sprk_get_pr(sprk), sprk_get_prk(sprk));
3547 21 : L = cgetg(lU, t_MAT);
3548 63 : for (j = 1; j < lU; j++)
3549 42 : gel(L,j) = famat_zlog_pr_coprime(nf, X, gel(U,j), sprk, NULL);
3550 : }
3551 518 : return vec_prepend(L, Ltu);
3552 : }
3553 :
3554 : static GEN
3555 1379963 : ideallog_i(GEN nf, GEN x, zlog_S *S)
3556 : {
3557 1379963 : pari_sp av = avma;
3558 : GEN y;
3559 1379963 : if (!S->hU) return cgetg(1, t_COL);
3560 1377689 : if (typ(x) == t_MAT)
3561 1371004 : y = famat_zlog(nf, x, NULL, S);
3562 : else
3563 6685 : y = zlog(nf, x, NULL, S);
3564 1377676 : y = ZMV_ZCV_mul(S->U, y);
3565 1377678 : return gerepileupto(av, ZV_ZV_mod(y, bid_get_cyc(S->bid)));
3566 : }
3567 : GEN
3568 1386640 : ideallogmod(GEN nf, GEN x, GEN bid, GEN mod)
3569 : {
3570 : zlog_S S;
3571 1386640 : if (!nf)
3572 : {
3573 6671 : if (mod) pari_err_IMPL("Zideallogmod");
3574 6671 : return Zideallog(bid, x);
3575 : }
3576 1379969 : checkbid(bid); init_zlog_mod(&S, bid, mod);
3577 1379963 : return ideallog_i(checknf(nf), x, &S);
3578 : }
3579 : GEN
3580 13755 : ideallog(GEN nf, GEN x, GEN bid) { return ideallogmod(nf, x, bid, NULL); }
3581 :
3582 : /*************************************************************************/
3583 : /** **/
3584 : /** JOIN BID STRUCTURES, IDEAL LISTS **/
3585 : /** **/
3586 : /*************************************************************************/
3587 : /* bid1, bid2: for coprime modules m1 and m2 (without arch. part).
3588 : * Output: bid for m1 m2 */
3589 : static GEN
3590 469 : join_bid(GEN nf, GEN bid1, GEN bid2)
3591 : {
3592 469 : pari_sp av = avma;
3593 : long nbgen, l1,l2;
3594 : GEN I1,I2, G1,G2, sprk1,sprk2, cyc1,cyc2, sarch;
3595 469 : GEN sprk, fa,fa2, U, cyc, y, u1 = NULL, x, gen;
3596 :
3597 469 : I1 = bid_get_ideal(bid1);
3598 469 : I2 = bid_get_ideal(bid2);
3599 469 : if (gequal1(gcoeff(I1,1,1))) return bid2; /* frequent trivial case */
3600 259 : G1 = bid_get_grp(bid1);
3601 259 : G2 = bid_get_grp(bid2);
3602 259 : x = idealmul(nf, I1,I2);
3603 259 : fa = famat_mul_shallow(bid_get_fact(bid1), bid_get_fact(bid2));
3604 259 : fa2= famat_mul_shallow(bid_get_fact2(bid1), bid_get_fact2(bid2));
3605 259 : sprk1 = bid_get_sprk(bid1);
3606 259 : sprk2 = bid_get_sprk(bid2);
3607 259 : sprk = shallowconcat(sprk1, sprk2);
3608 :
3609 259 : cyc1 = abgrp_get_cyc(G1); l1 = lg(cyc1);
3610 259 : cyc2 = abgrp_get_cyc(G2); l2 = lg(cyc2);
3611 259 : gen = (lg(G1)>3 && lg(G2)>3)? gen_1: NULL;
3612 259 : nbgen = l1+l2-2;
3613 259 : cyc = ZV_snf_group(shallowconcat(cyc1,cyc2), &U, gen? &u1: NULL);
3614 259 : if (nbgen)
3615 : {
3616 259 : GEN U1 = bid_get_U(bid1), U2 = bid_get_U(bid2);
3617 0 : U1 = l1==1? const_vec(lg(sprk1), cgetg(1,t_MAT))
3618 259 : : ZM_ZMV_mul(vecslice(U, 1, l1-1), U1);
3619 0 : U2 = l2==1? const_vec(lg(sprk2), cgetg(1,t_MAT))
3620 259 : : ZM_ZMV_mul(vecslice(U, l1, nbgen), U2);
3621 259 : U = shallowconcat(U1, U2);
3622 : }
3623 : else
3624 0 : U = const_vec(lg(sprk), cgetg(1,t_MAT));
3625 :
3626 259 : if (gen)
3627 : {
3628 259 : GEN uv = zkchinese1init2(nf, I2, I1, x);
3629 518 : gen = shallowconcat(zkVchinese1(gel(uv,1), abgrp_get_gen(G1)),
3630 259 : zkVchinese1(gel(uv,2), abgrp_get_gen(G2)));
3631 : }
3632 259 : sarch = bid_get_sarch(bid1); /* trivial */
3633 259 : y = bid_grp(nf, u1, cyc, gen, x, sarch);
3634 259 : x = mkvec2(x, bid_get_arch(bid1));
3635 259 : y = mkvec5(x, y, mkvec2(fa, fa2), mkvec2(sprk, sarch), U);
3636 259 : return gerepilecopy(av,y);
3637 : }
3638 :
3639 : typedef struct _ideal_data {
3640 : GEN nf, emb, L, pr, prL, sgnU, archp;
3641 : } ideal_data;
3642 :
3643 : /* z <- ( z | f(v[i])_{i=1..#v} ) */
3644 : static void
3645 759078 : concat_join(GEN *pz, GEN v, GEN (*f)(ideal_data*,GEN), ideal_data *data)
3646 : {
3647 759078 : long i, nz, lv = lg(v);
3648 : GEN z, Z;
3649 759078 : if (lv == 1) return;
3650 222967 : z = *pz; nz = lg(z)-1;
3651 222967 : *pz = Z = cgetg(lv + nz, typ(z));
3652 371700 : for (i = 1; i <=nz; i++) gel(Z,i) = gel(z,i);
3653 223350 : Z += nz;
3654 492055 : for (i = 1; i < lv; i++) gel(Z,i) = f(data, gel(v,i));
3655 : }
3656 : static GEN
3657 469 : join_idealinit(ideal_data *D, GEN x)
3658 469 : { return join_bid(D->nf, x, D->prL); }
3659 : static GEN
3660 268524 : join_ideal(ideal_data *D, GEN x)
3661 268524 : { return idealmulpowprime(D->nf, x, D->pr, D->L); }
3662 : static GEN
3663 448 : join_unit(ideal_data *D, GEN x)
3664 : {
3665 448 : GEN bid = join_idealinit(D, gel(x,1)), u = gel(x,2), v = mkvec(D->emb);
3666 448 : if (lg(u) != 1) v = shallowconcat(u, v);
3667 448 : return mkvec2(bid, v);
3668 : }
3669 :
3670 : GEN
3671 49 : log_prk_units_init(GEN bnf)
3672 : {
3673 49 : GEN U = bnf_has_fu(bnf);
3674 49 : if (U) U = matalgtobasis(bnf_get_nf(bnf), U);
3675 21 : else if (!(U = bnf_compactfu_mat(bnf))) (void)bnf_build_units(bnf);
3676 49 : return mkvec2(bnf_get_tuU(bnf), U);
3677 : }
3678 : /* flag & nf_GEN : generators, otherwise no
3679 : * flag &2 : units, otherwise no
3680 : * flag &4 : ideals in HNF, otherwise bid
3681 : * flag &8 : omit ideals which cannot be conductors (pr^1 with Npr=2) */
3682 : static GEN
3683 11333 : Ideallist(GEN bnf, ulong bound, long flag)
3684 : {
3685 11333 : const long do_units = flag & 2, big_id = !(flag & 4), cond = flag & 8;
3686 11333 : const long istar_flag = (flag & nf_GEN) | nf_INIT;
3687 : pari_sp av;
3688 : long i, j;
3689 11333 : GEN nf, z, p, fa, id, BOUND, U, empty = cgetg(1,t_VEC);
3690 : forprime_t S;
3691 : ideal_data ID;
3692 : GEN (*join_z)(ideal_data*, GEN);
3693 :
3694 11333 : if (do_units)
3695 : {
3696 21 : bnf = checkbnf(bnf);
3697 21 : nf = bnf_get_nf(bnf);
3698 21 : join_z = &join_unit;
3699 : }
3700 : else
3701 : {
3702 11312 : nf = checknf(bnf);
3703 11312 : join_z = big_id? &join_idealinit: &join_ideal;
3704 : }
3705 11333 : if ((long)bound <= 0) return empty;
3706 11333 : id = matid(nf_get_degree(nf));
3707 11333 : if (big_id) id = Idealstar(nf,id, istar_flag);
3708 :
3709 : /* z[i] will contain all "objects" of norm i. Depending on flag, this means
3710 : * an ideal, a bid, or a couple [bid, log(units)]. Such objects are stored
3711 : * in vectors, computed one primary component at a time; join_z
3712 : * reconstructs the global object */
3713 11333 : BOUND = utoipos(bound);
3714 11333 : z = const_vec(bound, empty);
3715 11333 : U = do_units? log_prk_units_init(bnf): NULL;
3716 11333 : gel(z,1) = mkvec(U? mkvec2(id, empty): id);
3717 11333 : ID.nf = nf;
3718 :
3719 11333 : p = cgetipos(3);
3720 11333 : u_forprime_init(&S, 2, bound);
3721 11333 : av = avma;
3722 92664 : while ((p[2] = u_forprime_next(&S)))
3723 : {
3724 81607 : if (DEBUGLEVEL>1) err_printf("%ld ",p[2]);
3725 81607 : fa = idealprimedec_limit_norm(nf, p, BOUND);
3726 162838 : for (j = 1; j < lg(fa); j++)
3727 : {
3728 81507 : GEN pr = gel(fa,j), z2 = leafcopy(z);
3729 81514 : ulong Q, q = upr_norm(pr);
3730 : long l;
3731 81513 : ID.pr = ID.prL = pr;
3732 81513 : if (cond && q == 2) { l = 2; Q = 4; } else { l = 1; Q = q; }
3733 184267 : for (; Q <= bound; l++, Q *= q) /* add pr^l */
3734 : {
3735 : ulong iQ;
3736 103043 : ID.L = utoipos(l);
3737 103040 : if (big_id) {
3738 211 : ID.prL = Idealstarprk(nf, pr, l, istar_flag);
3739 210 : if (U)
3740 189 : ID.emb = Q == 2? empty
3741 189 : : log_prk_units(nf, U, gel(bid_get_sprk(ID.prL),1));
3742 : }
3743 861673 : for (iQ = Q,i = 1; iQ <= bound; iQ += Q,i++)
3744 758919 : concat_join(&gel(z,iQ), gel(z2,i), join_z, &ID);
3745 : }
3746 : }
3747 81331 : if (gc_needed(av,1))
3748 : {
3749 18 : if(DEBUGMEM>1) pari_warn(warnmem,"Ideallist");
3750 18 : z = gerepilecopy(av, z);
3751 : }
3752 : }
3753 11333 : return z;
3754 : }
3755 : GEN
3756 63 : gideallist(GEN bnf, GEN B, long flag)
3757 : {
3758 63 : pari_sp av = avma;
3759 63 : if (typ(B) != t_INT)
3760 : {
3761 0 : B = gfloor(B);
3762 0 : if (typ(B) != t_INT) pari_err_TYPE("ideallist", B);
3763 0 : if (signe(B) < 0) B = gen_0;
3764 : }
3765 63 : if (signe(B) < 0)
3766 : {
3767 28 : if (flag != 4) pari_err_IMPL("ideallist with bid for single norm");
3768 28 : return gerepilecopy(av, ideals_by_norm(checknf(bnf), absi(B)));
3769 : }
3770 35 : if (flag < 0 || flag > 15) pari_err_FLAG("ideallist");
3771 35 : return gerepilecopy(av, Ideallist(bnf, itou(B), flag));
3772 : }
3773 : GEN
3774 11298 : ideallist0(GEN bnf, long bound, long flag)
3775 : {
3776 11298 : pari_sp av = avma;
3777 11298 : if (flag < 0 || flag > 15) pari_err_FLAG("ideallist");
3778 11298 : return gerepilecopy(av, Ideallist(bnf, bound, flag));
3779 : }
3780 : GEN
3781 10563 : ideallist(GEN bnf,long bound) { return ideallist0(bnf,bound,4); }
3782 :
3783 : /* bid = for module m (without arch. part), arch = archimedean part.
3784 : * Output: bid for [m,arch] */
3785 : static GEN
3786 42 : join_bid_arch(GEN nf, GEN bid, GEN archp)
3787 : {
3788 42 : pari_sp av = avma;
3789 : GEN G, U;
3790 42 : GEN sprk, cyc, y, u1 = NULL, x, sarch, gen;
3791 :
3792 42 : checkbid(bid);
3793 42 : G = bid_get_grp(bid);
3794 42 : x = bid_get_ideal(bid);
3795 42 : sarch = nfarchstar(nf, bid_get_ideal(bid), archp);
3796 42 : sprk = bid_get_sprk(bid);
3797 :
3798 42 : gen = (lg(G)>3)? gel(G,3): NULL;
3799 42 : cyc = diagonal_shallow(shallowconcat(gel(G,2), sarch_get_cyc(sarch)));
3800 42 : cyc = ZM_snf_group(cyc, &U, gen? &u1: NULL);
3801 42 : y = bid_grp(nf, u1, cyc, gen, x, sarch);
3802 42 : U = split_U(U, sprk);
3803 42 : y = mkvec5(mkvec2(x, archp), y, gel(bid,3), mkvec2(sprk, sarch), U);
3804 42 : return gerepilecopy(av,y);
3805 : }
3806 : static GEN
3807 42 : join_arch(ideal_data *D, GEN x) {
3808 42 : return join_bid_arch(D->nf, x, D->archp);
3809 : }
3810 : static GEN
3811 14 : join_archunit(ideal_data *D, GEN x) {
3812 14 : GEN bid = join_arch(D, gel(x,1)), u = gel(x,2), v = mkvec(D->emb);
3813 14 : if (lg(u) != 1) v = shallowconcat(u, v);
3814 14 : return mkvec2(bid, v);
3815 : }
3816 :
3817 : /* L from ideallist, add archimedean part */
3818 : GEN
3819 14 : ideallistarch(GEN bnf, GEN L, GEN arch)
3820 : {
3821 : pari_sp av;
3822 14 : long i, j, l = lg(L), lz;
3823 : GEN v, z, V, nf;
3824 : ideal_data ID;
3825 : GEN (*join_z)(ideal_data*, GEN);
3826 :
3827 14 : if (typ(L) != t_VEC) pari_err_TYPE("ideallistarch",L);
3828 14 : if (l == 1) return cgetg(1,t_VEC);
3829 14 : z = gel(L,1);
3830 14 : if (typ(z) != t_VEC) pari_err_TYPE("ideallistarch",z);
3831 14 : z = gel(z,1); /* either a bid or [bid,U] */
3832 14 : ID.archp = vec01_to_indices(arch);
3833 14 : if (lg(z) == 3)
3834 : { /* [bid,U]: do units */
3835 7 : bnf = checkbnf(bnf); nf = bnf_get_nf(bnf);
3836 7 : if (typ(z) != t_VEC) pari_err_TYPE("ideallistarch",z);
3837 7 : ID.emb = zm_to_ZM( rowpermute(nfsign_units(bnf,NULL,1), ID.archp) );
3838 7 : join_z = &join_archunit;
3839 : }
3840 : else
3841 : {
3842 7 : join_z = &join_arch;
3843 7 : nf = checknf(bnf);
3844 : }
3845 14 : ID.nf = nf;
3846 14 : av = avma; V = cgetg(l, t_VEC);
3847 63 : for (i = 1; i < l; i++)
3848 : {
3849 49 : z = gel(L,i); lz = lg(z);
3850 49 : gel(V,i) = v = cgetg(lz,t_VEC);
3851 91 : for (j=1; j<lz; j++) gel(v,j) = join_z(&ID, gel(z,j));
3852 : }
3853 14 : return gerepilecopy(av,V);
3854 : }
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