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 39167731 : get_tab(GEN nf, long *N)
33 : {
34 39167731 : GEN tab = (typ(nf) == t_MAT)? nf: gel(nf,9);
35 39167731 : *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 1074417530 : _mulii(GEN x, GEN y) {
41 1735353781 : return is_pm1(x)? (signe(x) < 0)? negi(y): y
42 1735215261 : : mulii(x, y);
43 : }
44 :
45 : GEN
46 21902 : tablemul_ei_ej(GEN M, long i, long j)
47 : {
48 : long N;
49 21902 : GEN tab = get_tab(M, &N);
50 21902 : 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 11473 : tablemul_ei(GEN M, GEN x, long i)
58 : {
59 : long j, k, N;
60 : GEN v, tab;
61 :
62 11473 : if (i==1) return gcopy(x);
63 11473 : tab = get_tab(M, &N);
64 11473 : if (typ(x) != t_COL) { v = zerocol(N); gel(v,i) = gcopy(x); return v; }
65 11473 : tab += (i-1)*N; v = cgetg(N+1,t_COL);
66 : /* wi . x = [ sum_j tab[k,j] x[j] ]_k */
67 78155 : for (k=1; k<=N; k++)
68 : {
69 66682 : pari_sp av = avma;
70 66682 : GEN s = gen_0;
71 472206 : for (j=1; j<=N; j++)
72 : {
73 405524 : GEN c = gcoeff(tab,k,j);
74 405524 : if (!gequal0(c)) s = gadd(s, gmul(c, gel(x,j)));
75 : }
76 66682 : gel(v,k) = gerepileupto(av,s);
77 : }
78 11473 : return v;
79 : }
80 : /* as tablemul_ei, assume x a ZV of correct length */
81 : GEN
82 23727743 : zk_ei_mul(GEN nf, GEN x, long i)
83 : {
84 : long j, k, N;
85 : GEN v, tab;
86 :
87 23727743 : if (i==1) return ZC_copy(x);
88 23727743 : tab = get_tab(nf, &N); tab += (i-1)*N;
89 23728750 : v = cgetg(N+1,t_COL);
90 168908150 : for (k=1; k<=N; k++)
91 : {
92 145182649 : pari_sp av = avma;
93 145182649 : GEN s = gen_0;
94 2138887143 : for (j=1; j<=N; j++)
95 : {
96 1993856878 : GEN c = gcoeff(tab,k,j);
97 1993856878 : if (signe(c)) s = addii(s, _mulii(c, gel(x,j)));
98 : }
99 145030265 : gel(v,k) = gerepileuptoint(av, s);
100 : }
101 23725501 : return v;
102 : }
103 :
104 : /* table of multiplication by wi in R[w1,..., wN] */
105 : GEN
106 39081 : ei_multable(GEN TAB, long i)
107 : {
108 : long k,N;
109 39081 : GEN m, tab = get_tab(TAB, &N);
110 39081 : tab += (i-1)*N;
111 39081 : m = cgetg(N+1,t_MAT);
112 153293 : for (k=1; k<=N; k++) gel(m,k) = gel(tab,k);
113 39081 : return m;
114 : }
115 :
116 : GEN
117 10655046 : zk_multable(GEN nf, GEN x)
118 : {
119 10655046 : long i, l = lg(x);
120 10655046 : GEN mul = cgetg(l,t_MAT);
121 10654860 : gel(mul,1) = x; /* assume w_1 = 1 */
122 34023021 : for (i=2; i<l; i++) gel(mul,i) = zk_ei_mul(nf,x,i);
123 10651941 : return mul;
124 : }
125 : GEN
126 2604 : multable(GEN M, GEN x)
127 : {
128 : long i, N;
129 : GEN mul;
130 2604 : 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 4841776 : zk_scalar_or_multable(GEN nf, GEN x)
143 : {
144 4841776 : long tx = typ(x);
145 4841776 : if (tx == t_MAT || tx == t_INT) return x;
146 4680808 : x = nf_to_scalar_or_basis(nf, x);
147 4680657 : return (typ(x) == t_COL)? zk_multable(nf, x): x;
148 : }
149 :
150 : GEN
151 21305 : nftrace(GEN nf, GEN x)
152 : {
153 21305 : pari_sp av = avma;
154 21305 : nf = checknf(nf);
155 21305 : x = nf_to_scalar_or_basis(nf, x);
156 21285 : x = (typ(x) == t_COL)? RgV_dotproduct(x, gel(nf_get_Tr(nf),1))
157 21306 : : gmulgu(x, nf_get_degree(nf));
158 21307 : return gerepileupto(av, x);
159 : }
160 : GEN
161 1015 : rnfelttrace(GEN rnf, GEN x)
162 : {
163 1015 : pari_sp av = avma;
164 1015 : checkrnf(rnf);
165 : /* avoid rnfabstorel special t_POL case misinterpretation */
166 1008 : if (typ(x) == t_POL && varn(x) == rnf_get_varn(rnf))
167 63 : x = gmodulo(x, rnf_get_pol(rnf));
168 1008 : x = rnfeltabstorel(rnf, x);
169 693 : x = (typ(x) == t_POLMOD)? rnfeltdown(rnf, gtrace(x))
170 798 : : gmulgu(x, rnf_get_degree(rnf));
171 798 : return gerepileupto(av, x);
172 : }
173 :
174 : static GEN
175 35 : famatQ_to_famatZ(GEN fa)
176 : {
177 35 : GEN E, F, Q, P = gel(fa,1);
178 35 : long i, j, l = lg(P);
179 35 : if (l == 1 || RgV_is_ZV(P)) return fa;
180 7 : Q = cgetg(2*l, t_COL);
181 7 : F = cgetg(2*l, t_COL); E = gel(fa, 2);
182 35 : for (i = j = 1; i < l; i++)
183 : {
184 28 : GEN p = gel(P,i);
185 28 : if (typ(p) == t_INT)
186 14 : { gel(Q, j) = p; gel(F, j) = gel(E, i); j++; }
187 : else
188 : {
189 14 : gel(Q, j) = gel(p,1); gel(F, j) = gel(E, i); j++;
190 14 : gel(Q, j) = gel(p,2); gel(F, j) = negi(gel(E, i)); j++;
191 : }
192 : }
193 7 : setlg(Q, j); setlg(F, j); return mkmat2(Q, F);
194 : }
195 : static GEN
196 35 : famat_cba(GEN fa)
197 : {
198 35 : GEN Q, F, P = gel(fa, 1), E = gel(fa, 2);
199 35 : long i, j, lQ, l = lg(P);
200 35 : if (l == 1) return fa;
201 28 : Q = ZV_cba(P); lQ = lg(Q);
202 28 : F = cgetg(lQ, t_COL);
203 77 : for (j = 1; j < lQ; j++)
204 : {
205 49 : GEN v = gen_0, q = gel(Q,j);
206 49 : if (!equali1(q))
207 203 : for (i = 1; i < l; i++)
208 : {
209 161 : long e = Z_pval(gel(P,i), q);
210 161 : if (e) v = addii(v, muliu(gel(E,i), e));
211 : }
212 49 : gel(F, j) = v;
213 : }
214 28 : return mkmat2(Q, F);
215 : }
216 : static long
217 35 : famat_sign(GEN fa)
218 : {
219 35 : GEN P = gel(fa,1), E = gel(fa,2);
220 35 : long i, l = lg(P), s = 1;
221 126 : for (i = 1; i < l; i++)
222 91 : if (signe(gel(P,i)) < 0 && mpodd(gel(E,i))) s = -s;
223 35 : return s;
224 : }
225 : static GEN
226 35 : famat_abs(GEN fa)
227 : {
228 35 : GEN Q, P = gel(fa,1);
229 : long i, l;
230 35 : Q = cgetg_copy(P, &l);
231 126 : for (i = 1; i < l; i++) gel(Q,i) = absi_shallow(gel(P,i));
232 35 : return mkmat2(Q, gel(fa,2));
233 : }
234 :
235 : /* assume nf is a genuine nf, fa a famat */
236 : static GEN
237 35 : famat_norm(GEN nf, GEN fa)
238 : {
239 35 : pari_sp av = avma;
240 35 : GEN G, g = gel(fa,1);
241 : long i, l, s;
242 :
243 35 : G = cgetg_copy(g, &l);
244 112 : for (i = 1; i < l; i++) gel(G,i) = nfnorm(nf, gel(g,i));
245 35 : fa = mkmat2(G, gel(fa,2));
246 35 : fa = famatQ_to_famatZ(fa);
247 35 : s = famat_sign(fa);
248 35 : fa = famat_reduce(famat_abs(fa));
249 35 : fa = famat_cba(fa);
250 35 : g = factorback(fa);
251 35 : return gerepileupto(av, s < 0? gneg(g): g);
252 : }
253 : GEN
254 222728 : nfnorm(GEN nf, GEN x)
255 : {
256 222728 : pari_sp av = avma;
257 : GEN c, den;
258 : long n;
259 222728 : nf = checknf(nf);
260 222728 : n = nf_get_degree(nf);
261 222728 : if (typ(x) == t_MAT) return famat_norm(nf, x);
262 222693 : x = nf_to_scalar_or_basis(nf, x);
263 222693 : if (typ(x)!=t_COL)
264 126819 : return gerepileupto(av, gpowgs(x, n));
265 95874 : x = nf_to_scalar_or_alg(nf, Q_primitive_part(x, &c));
266 95874 : x = Q_remove_denom(x, &den);
267 95873 : x = ZX_resultant_all(nf_get_pol(nf), x, den, 0);
268 95873 : return gerepileupto(av, c ? gmul(x, gpowgs(c, n)): x);
269 : }
270 :
271 : static GEN
272 119 : to_RgX(GEN P, long vx)
273 : {
274 119 : return varn(P) == vx ? P: scalarpol_shallow(P, vx);
275 : }
276 :
277 : GEN
278 462 : rnfeltnorm(GEN rnf, GEN x)
279 : {
280 462 : pari_sp av = avma;
281 : GEN nf, pol;
282 : long v;
283 462 : checkrnf(rnf);
284 455 : v = rnf_get_varn(rnf);
285 : /* avoid rnfabstorel special t_POL case misinterpretation */
286 455 : if (typ(x) == t_POL && varn(x) == v) x = gmodulo(x, rnf_get_pol(rnf));
287 455 : x = liftpol_shallow(rnfeltabstorel(rnf, x));
288 245 : nf = rnf_get_nf(rnf); pol = rnf_get_pol(rnf);
289 490 : x = (typ(x) == t_POL)
290 119 : ? rnfeltdown(rnf, nfX_resultant(nf,pol,to_RgX(x,v)))
291 245 : : gpowgs(x, rnf_get_degree(rnf));
292 245 : return gerepileupto(av, x);
293 : }
294 :
295 : /* x + y in nf */
296 : GEN
297 23432667 : nfadd(GEN nf, GEN x, GEN y)
298 : {
299 23432667 : pari_sp av = avma;
300 : GEN z;
301 :
302 23432667 : nf = checknf(nf);
303 23432667 : x = nf_to_scalar_or_basis(nf, x);
304 23432667 : y = nf_to_scalar_or_basis(nf, y);
305 23432667 : if (typ(x) != t_COL)
306 17669273 : { z = (typ(y) == t_COL)? RgC_Rg_add(y, x): gadd(x,y); }
307 : else
308 5763394 : { z = (typ(y) == t_COL)? RgC_add(x, y): RgC_Rg_add(x, y); }
309 23432667 : return gerepileupto(av, z);
310 : }
311 : /* x - y in nf */
312 : GEN
313 1810800 : nfsub(GEN nf, GEN x, GEN y)
314 : {
315 1810800 : pari_sp av = avma;
316 : GEN z;
317 :
318 1810800 : nf = checknf(nf);
319 1810800 : x = nf_to_scalar_or_basis(nf, x);
320 1810800 : y = nf_to_scalar_or_basis(nf, y);
321 1810800 : if (typ(x) != t_COL)
322 1279089 : { z = (typ(y) == t_COL)? Rg_RgC_sub(x,y): gsub(x,y); }
323 : else
324 531711 : { z = (typ(y) == t_COL)? RgC_sub(x,y): RgC_Rg_sub(x,y); }
325 1810800 : return gerepileupto(av, z);
326 : }
327 :
328 : /* product of ZC x,y in (true) nf; ( sum_i x_i sum_j y_j m^{i,j}_k )_k */
329 : static GEN
330 8483310 : nfmuli_ZC(GEN nf, GEN x, GEN y)
331 : {
332 : long i, j, k, N;
333 8483310 : GEN TAB = get_tab(nf, &N), v = cgetg(N+1,t_COL);
334 :
335 41618642 : for (k = 1; k <= N; k++)
336 : {
337 33135397 : pari_sp av = avma;
338 33135397 : GEN s, TABi = TAB;
339 33135397 : if (k == 1)
340 8483303 : s = mulii(gel(x,1),gel(y,1));
341 : else
342 24652018 : s = addii(mulii(gel(x,1),gel(y,k)),
343 24652094 : mulii(gel(x,k),gel(y,1)));
344 217190892 : for (i=2; i<=N; i++)
345 : {
346 184059470 : GEN t, xi = gel(x,i);
347 184059470 : TABi += N;
348 184059470 : if (!signe(xi)) continue;
349 :
350 93094443 : t = NULL;
351 1061041355 : for (j=2; j<=N; j++)
352 : {
353 967949879 : GEN p1, c = gcoeff(TABi, k, j); /* m^{i,j}_k */
354 967949879 : if (!signe(c)) continue;
355 280654446 : p1 = _mulii(c, gel(y,j));
356 280659904 : t = t? addii(t, p1): p1;
357 : }
358 93091476 : if (t) s = addii(s, mulii(xi, t));
359 : }
360 33131422 : gel(v,k) = gerepileuptoint(av,s);
361 : }
362 8483245 : return v;
363 : }
364 : static int
365 74497082 : is_famat(GEN x) { return typ(x) == t_MAT && lg(x) == 3; }
366 : /* product of x and y in nf */
367 : GEN
368 36295180 : nfmul(GEN nf, GEN x, GEN y)
369 : {
370 : GEN z;
371 36295180 : pari_sp av = avma;
372 :
373 36295180 : if (x == y) return nfsqr(nf,x);
374 :
375 32204188 : nf = checknf(nf);
376 32204187 : if (is_famat(x) || is_famat(y)) return famat_mul(x, y);
377 32203878 : x = nf_to_scalar_or_basis(nf, x);
378 32203877 : y = nf_to_scalar_or_basis(nf, y);
379 32203877 : if (typ(x) != t_COL)
380 : {
381 21793590 : if (isintzero(x)) return gen_0;
382 15735049 : z = (typ(y) == t_COL)? RgC_Rg_mul(y, x): gmul(x,y); }
383 : else
384 : {
385 10410287 : if (typ(y) != t_COL)
386 : {
387 4538549 : if (isintzero(y)) return gen_0;
388 1608981 : z = RgC_Rg_mul(x, y);
389 : }
390 : else
391 : {
392 : GEN dx, dy;
393 5871738 : x = Q_remove_denom(x, &dx);
394 5871738 : y = Q_remove_denom(y, &dy);
395 5871738 : z = nfmuli_ZC(nf,x,y);
396 5871737 : dx = mul_denom(dx,dy);
397 5871736 : if (dx) z = ZC_Z_div(z, dx);
398 : }
399 : }
400 23215764 : return gerepileupto(av, z);
401 : }
402 : /* square of ZC x in nf */
403 : static GEN
404 6885529 : nfsqri_ZC(GEN nf, GEN x)
405 : {
406 : long i, j, k, N;
407 6885529 : GEN TAB = get_tab(nf, &N), v = cgetg(N+1,t_COL);
408 :
409 38242404 : for (k = 1; k <= N; k++)
410 : {
411 31356988 : pari_sp av = avma;
412 31356988 : GEN s, TABi = TAB;
413 31356988 : if (k == 1)
414 6885634 : s = sqri(gel(x,1));
415 : else
416 24471354 : s = shifti(mulii(gel(x,1),gel(x,k)), 1);
417 252468353 : for (i=2; i<=N; i++)
418 : {
419 221130036 : GEN p1, c, t, xi = gel(x,i);
420 221130036 : TABi += N;
421 221130036 : if (!signe(xi)) continue;
422 :
423 79308852 : c = gcoeff(TABi, k, i);
424 79308852 : t = signe(c)? _mulii(c,xi): NULL;
425 674550947 : for (j=i+1; j<=N; j++)
426 : {
427 595241752 : c = gcoeff(TABi, k, j);
428 595241752 : if (!signe(c)) continue;
429 231752161 : p1 = _mulii(c, shifti(gel(x,j),1));
430 231758826 : t = t? addii(t, p1): p1;
431 : }
432 79309195 : if (t) s = addii(s, mulii(xi, t));
433 : }
434 31338317 : gel(v,k) = gerepileuptoint(av,s);
435 : }
436 6885416 : return v;
437 : }
438 : /* square of x in nf */
439 : GEN
440 8906049 : nfsqr(GEN nf, GEN x)
441 : {
442 8906049 : pari_sp av = avma;
443 : GEN z;
444 :
445 8906049 : nf = checknf(nf);
446 8906053 : if (is_famat(x)) return famat_sqr(x);
447 8906057 : x = nf_to_scalar_or_basis(nf, x);
448 8906057 : if (typ(x) != t_COL) z = gsqr(x);
449 : else
450 : {
451 : GEN dx;
452 2628520 : x = Q_remove_denom(x, &dx);
453 2628518 : z = nfsqri_ZC(nf,x);
454 2628524 : if (dx) z = RgC_Rg_div(z, sqri(dx));
455 : }
456 8906061 : return gerepileupto(av, z);
457 : }
458 :
459 : /* x a ZC, v a t_COL of ZC/Z */
460 : GEN
461 205386 : zkC_multable_mul(GEN v, GEN x)
462 : {
463 205386 : long i, l = lg(v);
464 205386 : GEN y = cgetg(l, t_COL);
465 799264 : for (i = 1; i < l; i++)
466 : {
467 593877 : GEN c = gel(v,i);
468 593877 : if (typ(c)!=t_COL) {
469 0 : if (!isintzero(c)) c = ZC_Z_mul(gel(x,1), c);
470 : } else {
471 593877 : c = ZM_ZC_mul(x,c);
472 593878 : if (ZV_isscalar(c)) c = gel(c,1);
473 : }
474 593878 : gel(y,i) = c;
475 : }
476 205387 : return y;
477 : }
478 :
479 : GEN
480 56891 : nfC_multable_mul(GEN v, GEN x)
481 : {
482 56891 : long i, l = lg(v);
483 56891 : GEN y = cgetg(l, t_COL);
484 384047 : for (i = 1; i < l; i++)
485 : {
486 327156 : GEN c = gel(v,i);
487 327156 : if (typ(c)!=t_COL) {
488 272602 : if (!isintzero(c)) c = RgC_Rg_mul(gel(x,1), c);
489 : } else {
490 54554 : c = RgM_RgC_mul(x,c);
491 54554 : if (QV_isscalar(c)) c = gel(c,1);
492 : }
493 327156 : gel(y,i) = c;
494 : }
495 56891 : return y;
496 : }
497 :
498 : GEN
499 197964 : nfC_nf_mul(GEN nf, GEN v, GEN x)
500 : {
501 : long tx;
502 : GEN y;
503 :
504 197964 : x = nf_to_scalar_or_basis(nf, x);
505 197964 : tx = typ(x);
506 197964 : if (tx != t_COL)
507 : {
508 : long l, i;
509 149577 : if (tx == t_INT)
510 : {
511 140470 : long s = signe(x);
512 140470 : if (!s) return zerocol(lg(v)-1);
513 133108 : if (is_pm1(x)) return s > 0? leafcopy(v): RgC_neg(v);
514 : }
515 48503 : l = lg(v); y = cgetg(l, t_COL);
516 347452 : for (i=1; i < l; i++)
517 : {
518 298949 : GEN c = gel(v,i);
519 298949 : if (typ(c) != t_COL) c = gmul(c, x); else c = RgC_Rg_mul(c, x);
520 298949 : gel(y,i) = c;
521 : }
522 48503 : return y;
523 : }
524 : else
525 : {
526 : GEN dx;
527 48387 : x = zk_multable(nf, Q_remove_denom(x,&dx));
528 48387 : y = nfC_multable_mul(v, x);
529 48387 : return dx? RgC_Rg_div(y, dx): y;
530 : }
531 : }
532 : static GEN
533 10919 : mulbytab(GEN M, GEN c)
534 10919 : { return typ(c) == t_COL? RgM_RgC_mul(M,c): RgC_Rg_mul(gel(M,1), c); }
535 : GEN
536 2604 : tablemulvec(GEN M, GEN x, GEN v)
537 : {
538 : long l, i;
539 : GEN y;
540 :
541 2604 : if (typ(x) == t_COL && RgV_isscalar(x))
542 : {
543 0 : x = gel(x,1);
544 0 : return typ(v) == t_POL? RgX_Rg_mul(v,x): RgV_Rg_mul(v,x);
545 : }
546 2604 : x = multable(M, x); /* multiplication table by x */
547 2604 : y = cgetg_copy(v, &l);
548 2604 : if (typ(v) == t_POL)
549 : {
550 2604 : y[1] = v[1];
551 13523 : for (i=2; i < l; i++) gel(y,i) = mulbytab(x, gel(v,i));
552 2604 : y = normalizepol(y);
553 : }
554 : else
555 : {
556 0 : for (i=1; i < l; i++) gel(y,i) = mulbytab(x, gel(v,i));
557 : }
558 2604 : return y;
559 : }
560 :
561 : GEN
562 1260415 : zkmultable_capZ(GEN mx) { return Q_denom(zkmultable_inv(mx)); }
563 : GEN
564 1575652 : zkmultable_inv(GEN mx) { return ZM_gauss(mx, col_ei(lg(mx)-1,1)); }
565 : /* nf a true nf, x a ZC */
566 : GEN
567 315243 : zk_inv(GEN nf, GEN x) { return zkmultable_inv(zk_multable(nf,x)); }
568 :
569 : /* inverse of x in nf */
570 : GEN
571 239169 : nfinv(GEN nf, GEN x)
572 : {
573 239169 : pari_sp av = avma;
574 : GEN z;
575 :
576 239169 : nf = checknf(nf);
577 239169 : if (is_famat(x)) return famat_inv(x);
578 239169 : x = nf_to_scalar_or_basis(nf, x);
579 239169 : if (typ(x) == t_COL)
580 : {
581 : GEN d;
582 190536 : x = Q_remove_denom(x, &d);
583 190536 : z = zk_inv(nf, x);
584 190536 : if (d) z = RgC_Rg_mul(z, d);
585 : }
586 : else
587 48633 : z = ginv(x);
588 239169 : return gerepileupto(av, z);
589 : }
590 :
591 : /* quotient of x and y in nf */
592 : GEN
593 37019 : nfdiv(GEN nf, GEN x, GEN y)
594 : {
595 37019 : pari_sp av = avma;
596 : GEN z;
597 :
598 37019 : nf = checknf(nf);
599 37019 : if (is_famat(x) || is_famat(y)) return famat_div(x,y);
600 36928 : y = nf_to_scalar_or_basis(nf, y);
601 36928 : if (typ(y) != t_COL)
602 : {
603 22099 : x = nf_to_scalar_or_basis(nf, x);
604 22099 : z = (typ(x) == t_COL)? RgC_Rg_div(x, y): gdiv(x,y);
605 : }
606 : else
607 : {
608 : GEN d;
609 14829 : y = Q_remove_denom(y, &d);
610 14829 : z = nfmul(nf, x, zk_inv(nf,y));
611 14829 : if (d) z = typ(z) == t_COL? RgC_Rg_mul(z, d): gmul(z, d);
612 : }
613 36928 : return gerepileupto(av, z);
614 : }
615 :
616 : /* product of INTEGERS (t_INT or ZC) x and y in (true) nf */
617 : GEN
618 3968599 : nfmuli(GEN nf, GEN x, GEN y)
619 : {
620 3968599 : if (typ(x) == t_INT) return (typ(y) == t_COL)? ZC_Z_mul(y, x): mulii(x,y);
621 2842251 : if (typ(y) == t_INT) return ZC_Z_mul(x, y);
622 2611543 : return nfmuli_ZC(nf, x, y);
623 : }
624 : GEN
625 4257040 : nfsqri(GEN nf, GEN x)
626 4257040 : { return (typ(x) == t_INT)? sqri(x): nfsqri_ZC(nf, x); }
627 :
628 : /* both x and y are RgV */
629 : GEN
630 0 : tablemul(GEN TAB, GEN x, GEN y)
631 : {
632 : long i, j, k, N;
633 : GEN s, v;
634 0 : if (typ(x) != t_COL) return gmul(x, y);
635 0 : if (typ(y) != t_COL) return gmul(y, x);
636 0 : N = lg(x)-1;
637 0 : v = cgetg(N+1,t_COL);
638 0 : for (k=1; k<=N; k++)
639 : {
640 0 : pari_sp av = avma;
641 0 : GEN TABi = TAB;
642 0 : if (k == 1)
643 0 : s = gmul(gel(x,1),gel(y,1));
644 : else
645 0 : s = gadd(gmul(gel(x,1),gel(y,k)),
646 0 : gmul(gel(x,k),gel(y,1)));
647 0 : for (i=2; i<=N; i++)
648 : {
649 0 : GEN t, xi = gel(x,i);
650 0 : TABi += N;
651 0 : if (gequal0(xi)) continue;
652 :
653 0 : t = NULL;
654 0 : for (j=2; j<=N; j++)
655 : {
656 0 : GEN p1, c = gcoeff(TABi, k, j); /* m^{i,j}_k */
657 0 : if (gequal0(c)) continue;
658 0 : p1 = gmul(c, gel(y,j));
659 0 : t = t? gadd(t, p1): p1;
660 : }
661 0 : if (t) s = gadd(s, gmul(xi, t));
662 : }
663 0 : gel(v,k) = gerepileupto(av,s);
664 : }
665 0 : return v;
666 : }
667 : GEN
668 49370 : tablesqr(GEN TAB, GEN x)
669 : {
670 : long i, j, k, N;
671 : GEN s, v;
672 :
673 49370 : if (typ(x) != t_COL) return gsqr(x);
674 49370 : N = lg(x)-1;
675 49370 : v = cgetg(N+1,t_COL);
676 :
677 351166 : for (k=1; k<=N; k++)
678 : {
679 301796 : pari_sp av = avma;
680 301796 : GEN TABi = TAB;
681 301796 : if (k == 1)
682 49370 : s = gsqr(gel(x,1));
683 : else
684 252426 : s = gmul2n(gmul(gel(x,1),gel(x,k)), 1);
685 1917310 : for (i=2; i<=N; i++)
686 : {
687 1615514 : GEN p1, c, t, xi = gel(x,i);
688 1615514 : TABi += N;
689 1615514 : if (gequal0(xi)) continue;
690 :
691 416836 : c = gcoeff(TABi, k, i);
692 416836 : t = !gequal0(c)? gmul(c,xi): NULL;
693 1668464 : for (j=i+1; j<=N; j++)
694 : {
695 1251628 : c = gcoeff(TABi, k, j);
696 1251628 : if (gequal0(c)) continue;
697 642908 : p1 = gmul(gmul2n(c,1), gel(x,j));
698 642908 : t = t? gadd(t, p1): p1;
699 : }
700 416836 : if (t) s = gadd(s, gmul(xi, t));
701 : }
702 301796 : gel(v,k) = gerepileupto(av,s);
703 : }
704 49370 : return v;
705 : }
706 :
707 : static GEN
708 245009 : _mul(void *data, GEN x, GEN y) { return nfmuli((GEN)data,x,y); }
709 : static GEN
710 757571 : _sqr(void *data, GEN x) { return nfsqri((GEN)data,x); }
711 :
712 : /* Compute z^n in nf, left-shift binary powering */
713 : GEN
714 854775 : nfpow(GEN nf, GEN z, GEN n)
715 : {
716 854775 : pari_sp av = avma;
717 : long s;
718 : GEN x, cx;
719 :
720 854775 : if (typ(n)!=t_INT) pari_err_TYPE("nfpow",n);
721 854775 : nf = checknf(nf);
722 854772 : s = signe(n); if (!s) return gen_1;
723 854772 : if (is_famat(z)) return famat_pow(z, n);
724 794131 : x = nf_to_scalar_or_basis(nf, z);
725 794136 : if (typ(x) != t_COL) return powgi(x,n);
726 676394 : if (s < 0)
727 : { /* simplified nfinv */
728 : GEN d;
729 41535 : x = Q_remove_denom(x, &d);
730 41535 : x = zk_inv(nf, x);
731 41535 : x = primitive_part(x, &cx);
732 41535 : cx = mul_content(cx, d);
733 41535 : n = negi(n);
734 : }
735 : else
736 634859 : x = primitive_part(x, &cx);
737 676386 : x = gen_pow_i(x, n, (void*)nf, _sqr, _mul);
738 676394 : if (cx)
739 38068 : x = gerepileupto(av, gmul(x, powgi(cx, n)));
740 : else
741 638326 : x = gerepilecopy(av, x);
742 676401 : return x;
743 : }
744 : /* Compute z^n in nf, left-shift binary powering */
745 : GEN
746 342208 : nfpow_u(GEN nf, GEN z, ulong n)
747 : {
748 342208 : pari_sp av = avma;
749 : GEN x, cx;
750 :
751 342208 : if (!n) return gen_1;
752 342208 : x = nf_to_scalar_or_basis(nf, z);
753 342208 : if (typ(x) != t_COL) return gpowgs(x,n);
754 306002 : x = primitive_part(x, &cx);
755 306003 : x = gen_powu_i(x, n, (void*)nf, _sqr, _mul);
756 306004 : if (cx)
757 : {
758 111311 : x = gmul(x, powgi(cx, utoipos(n)));
759 111311 : return gerepileupto(av,x);
760 : }
761 194693 : return gerepilecopy(av, x);
762 : }
763 :
764 : long
765 588 : nfissquare(GEN nf, GEN z, GEN *px)
766 : {
767 588 : pari_sp av = avma;
768 588 : long v = fetch_var_higher();
769 : GEN R;
770 588 : nf = checknf(nf);
771 588 : if (nf_get_degree(nf) == 1)
772 : {
773 21 : z = algtobasis(nf, z);
774 21 : if (!issquareall(gel(z,1), px)) return gc_long(av, 0);
775 14 : if (px) *px = gerepileupto(av, *px); else set_avma(av);
776 14 : return 1;
777 : }
778 567 : z = nf_to_scalar_or_alg(nf, z);
779 567 : R = nfroots(nf, deg2pol_shallow(gen_m1, gen_0, z, v));
780 567 : delete_var(); if (lg(R) == 1) return gc_long(av, 0);
781 546 : if (px) *px = gerepilecopy(av, nf_to_scalar_or_basis(nf, gel(R,1)));
782 14 : else set_avma(av);
783 546 : return 1;
784 : }
785 :
786 : long
787 7746 : nfispower(GEN nf, GEN z, long n, GEN *px)
788 : {
789 7746 : pari_sp av = avma;
790 7746 : long v = fetch_var_higher();
791 : GEN R;
792 7746 : nf = checknf(nf);
793 7746 : if (nf_get_degree(nf) == 1)
794 : {
795 329 : z = algtobasis(nf, z);
796 329 : if (!ispower(gel(z,1), stoi(n), px)) return gc_long(av, 0);
797 147 : if (px) *px = gerepileupto(av, *px); else set_avma(av);
798 147 : return 1;
799 : }
800 7417 : if (n <= 0)
801 0 : pari_err_DOMAIN("nfeltispower","exponent","<=",gen_0,stoi(n));
802 7417 : z = nf_to_scalar_or_alg(nf, z);
803 7417 : if (n==1)
804 : {
805 0 : if (px) *px = gerepilecopy(av, z);
806 0 : return 1;
807 : }
808 7417 : R = nfroots(nf, gsub(pol_xn(n, v), z));
809 7417 : delete_var(); if (lg(R) == 1) return gc_long(av, 0);
810 3157 : if (px) *px = gerepilecopy(av, nf_to_scalar_or_basis(nf, gel(R,1)));
811 3143 : else set_avma(av);
812 3157 : return 1;
813 : }
814 :
815 : static GEN
816 56 : idmulred(void *nf, GEN x, GEN y) { return idealmulred((GEN) nf, x, y); }
817 : static GEN
818 413 : idpowred(void *nf, GEN x, GEN n) { return idealpowred((GEN) nf, x, n); }
819 : static GEN
820 70361 : idmul(void *nf, GEN x, GEN y) { return idealmul((GEN) nf, x, y); }
821 : static GEN
822 86123 : idpow(void *nf, GEN x, GEN n) { return idealpow((GEN) nf, x, n); }
823 : GEN
824 85078 : idealfactorback(GEN nf, GEN L, GEN e, int red)
825 : {
826 85078 : nf = checknf(nf);
827 85078 : if (red) return gen_factorback(L, e, (void*)nf, &idmulred, &idpowred, NULL);
828 84721 : if (!e && typ(L) == t_MAT && lg(L) == 3) { e = gel(L,2); L = gel(L,1); }
829 84721 : if (is_vec_t(typ(L)) && RgV_is_prV(L))
830 : { /* don't use gen_factorback since *= pr^v can be done more efficiently */
831 64304 : pari_sp av = avma;
832 64304 : long i, l = lg(L);
833 : GEN a;
834 64304 : if (!e) e = const_vec(l-1, gen_1);
835 61448 : else switch(typ(e))
836 : {
837 7250 : case t_VECSMALL: e = zv_to_ZV(e); break;
838 54198 : case t_VEC: case t_COL:
839 54198 : if (!RgV_is_ZV(e))
840 0 : pari_err_TYPE("factorback [not an exponent vector]", e);
841 54198 : break;
842 0 : default: pari_err_TYPE("idealfactorback", e);
843 : }
844 64304 : if (l != lg(e))
845 0 : pari_err_TYPE("factorback [not an exponent vector]", e);
846 64304 : if (l == 1 || ZV_equal0(e)) return gc_const(av, gen_1);
847 20392 : a = idealpow(nf, gel(L,1), gel(e,1));
848 234853 : for (i = 2; i < l; i++)
849 214461 : if (signe(gel(e,i))) a = idealmulpowprime(nf, a, gel(L,i), gel(e,i));
850 20392 : return gerepileupto(av, a);
851 : }
852 20417 : return gen_factorback(L, e, (void*)nf, &idmul, &idpow, NULL);
853 : }
854 : static GEN
855 321203 : eltmul(void *nf, GEN x, GEN y) { return nfmul((GEN) nf, x, y); }
856 : static GEN
857 458806 : eltpow(void *nf, GEN x, GEN n) { return nfpow((GEN) nf, x, n); }
858 : GEN
859 265666 : nffactorback(GEN nf, GEN L, GEN e)
860 265666 : { return gen_factorback(L, e, (void*)checknf(nf), &eltmul, &eltpow, NULL); }
861 :
862 : static GEN
863 3080880 : _nf_red(void *E, GEN x) { (void)E; return gcopy(x); }
864 :
865 : static GEN
866 12632962 : _nf_add(void *E, GEN x, GEN y) { return nfadd((GEN)E,x,y); }
867 :
868 : static GEN
869 747819 : _nf_neg(void *E, GEN x) { (void)E; return gneg(x); }
870 :
871 : static GEN
872 15164880 : _nf_mul(void *E, GEN x, GEN y) { return nfmul((GEN)E,x,y); }
873 :
874 : static GEN
875 53161 : _nf_inv(void *E, GEN x) { return nfinv((GEN)E,x); }
876 :
877 : static GEN
878 10841 : _nf_s(void *E, long x) { (void)E; return stoi(x); }
879 :
880 : static const struct bb_field nf_field={_nf_red,_nf_add,_nf_mul,_nf_neg,
881 : _nf_inv,&gequal0,_nf_s };
882 :
883 225825 : const struct bb_field *get_nf_field(void **E, GEN nf)
884 225825 : { *E = (void*)nf; return &nf_field; }
885 :
886 : GEN
887 14 : nfM_det(GEN nf, GEN M)
888 : {
889 : void *E;
890 14 : const struct bb_field *S = get_nf_field(&E, nf);
891 14 : return gen_det(M, E, S);
892 : }
893 : GEN
894 10827 : nfM_inv(GEN nf, GEN M)
895 : {
896 : void *E;
897 10827 : const struct bb_field *S = get_nf_field(&E, nf);
898 10827 : return gen_Gauss(M, matid(lg(M)-1), E, S);
899 : }
900 :
901 : GEN
902 0 : nfM_ker(GEN nf, GEN M)
903 : {
904 : void *E;
905 0 : const struct bb_field *S = get_nf_field(&E, nf);
906 0 : return gen_ker(M, 0, E, S);
907 : }
908 :
909 : GEN
910 10358 : nfM_mul(GEN nf, GEN A, GEN B)
911 : {
912 : void *E;
913 10358 : const struct bb_field *S = get_nf_field(&E, nf);
914 10358 : return gen_matmul(A, B, E, S);
915 : }
916 : GEN
917 204626 : nfM_nfC_mul(GEN nf, GEN A, GEN B)
918 : {
919 : void *E;
920 204626 : const struct bb_field *S = get_nf_field(&E, nf);
921 204626 : return gen_matcolmul(A, B, E, S);
922 : }
923 :
924 : /* valuation of integral x (ZV), with resp. to prime ideal pr */
925 : long
926 24141700 : ZC_nfvalrem(GEN x, GEN pr, GEN *newx)
927 : {
928 24141700 : pari_sp av = avma;
929 : long i, v, l;
930 24141700 : GEN r, y, p = pr_get_p(pr), mul = pr_get_tau(pr);
931 :
932 : /* p inert */
933 24141707 : if (typ(mul) == t_INT) return newx? ZV_pvalrem(x, p, newx):ZV_pval(x, p);
934 23153279 : y = cgetg_copy(x, &l); /* will hold the new x */
935 23153663 : x = leafcopy(x);
936 36996394 : for(v=0;; v++)
937 : {
938 142229172 : for (i=1; i<l; i++)
939 : { /* is (x.b)[i] divisible by p ? */
940 128381337 : gel(y,i) = dvmdii(ZMrow_ZC_mul(mul,x,i),p,&r);
941 128384596 : if (r != gen_0) { if (newx) *newx = x; return v; }
942 : }
943 13847835 : swap(x, y);
944 13847835 : if (!newx && (v & 0xf) == 0xf) v += pr_get_e(pr) * ZV_pvalrem(x, p, &x);
945 13847835 : if (gc_needed(av,1))
946 : {
947 0 : if(DEBUGMEM>1) pari_warn(warnmem,"ZC_nfvalrem, v >= %ld", v);
948 0 : gerepileall(av, 2, &x, &y);
949 : }
950 : }
951 : }
952 : long
953 19891056 : ZC_nfval(GEN x, GEN P)
954 19891056 : { return ZC_nfvalrem(x, P, NULL); }
955 :
956 : /* v_P(x) != 0, x a ZV. Simpler version of ZC_nfvalrem */
957 : int
958 1246439 : ZC_prdvd(GEN x, GEN P)
959 : {
960 1246439 : pari_sp av = avma;
961 : long i, l;
962 1246439 : GEN p = pr_get_p(P), mul = pr_get_tau(P);
963 1246455 : if (typ(mul) == t_INT) return ZV_Z_dvd(x, p);
964 1245916 : l = lg(x);
965 5050076 : for (i=1; i<l; i++)
966 4534503 : if (!dvdii(ZMrow_ZC_mul(mul,x,i), p)) return gc_bool(av,0);
967 515573 : return gc_bool(av,1);
968 : }
969 :
970 : int
971 357 : pr_equal(GEN P, GEN Q)
972 : {
973 357 : GEN gQ, p = pr_get_p(P);
974 357 : long e = pr_get_e(P), f = pr_get_f(P), n;
975 357 : if (!equalii(p, pr_get_p(Q)) || e != pr_get_e(Q) || f != pr_get_f(Q))
976 336 : return 0;
977 21 : gQ = pr_get_gen(Q); n = lg(gQ)-1;
978 21 : if (2*e*f > n) return 1; /* room for only one such pr */
979 14 : return ZV_equal(pr_get_gen(P), gQ) || ZC_prdvd(gQ, P);
980 : }
981 :
982 : GEN
983 420721 : famat_nfvalrem(GEN nf, GEN x, GEN pr, GEN *py)
984 : {
985 420721 : pari_sp av = avma;
986 420721 : GEN P = gel(x,1), E = gel(x,2), V = gen_0, y = NULL;
987 420721 : long l = lg(P), simplify = 0, i;
988 420721 : if (py) { *py = gen_1; y = cgetg(l, t_COL); }
989 :
990 2258510 : for (i = 1; i < l; i++)
991 : {
992 1837789 : GEN e = gel(E,i);
993 : long v;
994 1837789 : if (!signe(e))
995 : {
996 7 : if (py) gel(y,i) = gen_1;
997 7 : simplify = 1; continue;
998 : }
999 1837782 : v = nfvalrem(nf, gel(P,i), pr, py? &gel(y,i): NULL);
1000 1837782 : if (v == LONG_MAX) { set_avma(av); if (py) *py = gen_0; return mkoo(); }
1001 1837782 : V = addmulii(V, stoi(v), e);
1002 : }
1003 420721 : if (!py) V = gerepileuptoint(av, V);
1004 : else
1005 : {
1006 42 : y = mkmat2(y, gel(x,2));
1007 42 : if (simplify) y = famat_remove_trivial(y);
1008 42 : gerepileall(av, 2, &V, &y); *py = y;
1009 : }
1010 420721 : return V;
1011 : }
1012 : long
1013 5621955 : nfval(GEN nf, GEN x, GEN pr)
1014 : {
1015 5621955 : pari_sp av = avma;
1016 : long w, e;
1017 : GEN cx, p;
1018 :
1019 5621955 : if (gequal0(x)) return LONG_MAX;
1020 5608619 : nf = checknf(nf);
1021 5608617 : checkprid(pr);
1022 5608609 : p = pr_get_p(pr);
1023 5608607 : e = pr_get_e(pr);
1024 5608601 : x = nf_to_scalar_or_basis(nf, x);
1025 5608521 : if (typ(x) != t_COL) return e*Q_pval(x,p);
1026 2376613 : x = Q_primitive_part(x, &cx);
1027 2376644 : w = ZC_nfval(x,pr);
1028 2376588 : if (cx) w += e*Q_pval(cx,p);
1029 2376588 : return gc_long(av,w);
1030 : }
1031 :
1032 : /* want to write p^v = uniformizer^(e*v) * z^v, z coprime to pr */
1033 : /* z := tau^e / p^(e-1), algebraic integer coprime to pr; return z^v */
1034 : static GEN
1035 951048 : powp(GEN nf, GEN pr, long v)
1036 : {
1037 : GEN b, z;
1038 : long e;
1039 951048 : if (!v) return gen_1;
1040 424473 : b = pr_get_tau(pr);
1041 424473 : if (typ(b) == t_INT) return gen_1;
1042 121898 : e = pr_get_e(pr);
1043 121898 : z = gel(b,1);
1044 121898 : if (e != 1) z = gdiv(nfpow_u(nf, z, e), powiu(pr_get_p(pr),e-1));
1045 121898 : if (v < 0) { v = -v; z = nfinv(nf, z); }
1046 121898 : if (v != 1) z = nfpow_u(nf, z, v);
1047 121898 : return z;
1048 : }
1049 : long
1050 3639109 : nfvalrem(GEN nf, GEN x, GEN pr, GEN *py)
1051 : {
1052 3639109 : pari_sp av = avma;
1053 : long w, e;
1054 : GEN cx, p, t;
1055 :
1056 3639109 : if (!py) return nfval(nf,x,pr);
1057 1787936 : if (gequal0(x)) { *py = gen_0; return LONG_MAX; }
1058 1787881 : nf = checknf(nf);
1059 1787881 : checkprid(pr);
1060 1787881 : p = pr_get_p(pr);
1061 1787881 : e = pr_get_e(pr);
1062 1787881 : x = nf_to_scalar_or_basis(nf, x);
1063 1787881 : if (typ(x) != t_COL) {
1064 538531 : w = Q_pvalrem(x,p, py);
1065 538531 : if (!w) { *py = gerepilecopy(av, x); return 0; }
1066 330239 : *py = gerepileupto(av, gmul(powp(nf, pr, w), *py));
1067 330239 : return e*w;
1068 : }
1069 1249350 : x = Q_primitive_part(x, &cx);
1070 1249351 : w = ZC_nfvalrem(x,pr, py);
1071 1249333 : if (cx)
1072 : {
1073 620809 : long v = Q_pvalrem(cx,p, &t);
1074 620809 : *py = nfmul(nf, *py, gmul(powp(nf,pr,v), t));
1075 620809 : *py = gerepileupto(av, *py);
1076 620809 : w += e*v;
1077 : }
1078 : else
1079 628524 : *py = gerepilecopy(av, *py);
1080 1249353 : return w;
1081 : }
1082 : GEN
1083 15015 : gpnfvalrem(GEN nf, GEN x, GEN pr, GEN *py)
1084 : {
1085 : long v;
1086 15015 : if (is_famat(x)) return famat_nfvalrem(nf, x, pr, py);
1087 15008 : v = nfvalrem(nf,x,pr,py);
1088 15008 : return v == LONG_MAX? mkoo(): stoi(v);
1089 : }
1090 :
1091 : /* true nf */
1092 : GEN
1093 305620 : coltoalg(GEN nf, GEN x)
1094 : {
1095 305620 : return mkpolmod( nf_to_scalar_or_alg(nf, x), nf_get_pol(nf) );
1096 : }
1097 :
1098 : GEN
1099 358712 : basistoalg(GEN nf, GEN x)
1100 : {
1101 : GEN T;
1102 :
1103 358712 : nf = checknf(nf);
1104 358712 : switch(typ(x))
1105 : {
1106 299355 : case t_COL: {
1107 299355 : pari_sp av = avma;
1108 299355 : return gerepilecopy(av, coltoalg(nf, x));
1109 : }
1110 33390 : case t_POLMOD:
1111 33390 : T = nf_get_pol(nf);
1112 33390 : if (!RgX_equal_var(T,gel(x,1)))
1113 0 : pari_err_MODULUS("basistoalg", T,gel(x,1));
1114 33390 : return gcopy(x);
1115 2359 : case t_POL:
1116 2359 : T = nf_get_pol(nf);
1117 2359 : if (varn(T) != varn(x)) pari_err_VAR("basistoalg",x,T);
1118 2359 : retmkpolmod(RgX_rem(x, T), ZX_copy(T));
1119 23608 : case t_INT:
1120 : case t_FRAC:
1121 23608 : T = nf_get_pol(nf);
1122 23608 : retmkpolmod(gcopy(x), ZX_copy(T));
1123 0 : default:
1124 0 : pari_err_TYPE("basistoalg",x);
1125 : return NULL; /* LCOV_EXCL_LINE */
1126 : }
1127 : }
1128 :
1129 : /* true nf, x a t_POL */
1130 : static GEN
1131 4556118 : pol_to_scalar_or_basis(GEN nf, GEN x)
1132 : {
1133 4556118 : GEN T = nf_get_pol(nf);
1134 4556114 : long l = lg(x);
1135 4556114 : if (varn(x) != varn(T)) pari_err_VAR("nf_to_scalar_or_basis", x,T);
1136 4556010 : if (l >= lg(T)) { x = RgX_rem(x, T); l = lg(x); }
1137 4556010 : if (l == 2) return gen_0;
1138 3542809 : if (l == 3)
1139 : {
1140 818440 : x = gel(x,2);
1141 818440 : if (!is_rational_t(typ(x))) pari_err_TYPE("nf_to_scalar_or_basis",x);
1142 818433 : return x;
1143 : }
1144 2724369 : return poltobasis(nf,x);
1145 : }
1146 : /* Assume nf is a genuine nf. */
1147 : GEN
1148 161130144 : nf_to_scalar_or_basis(GEN nf, GEN x)
1149 : {
1150 161130144 : switch(typ(x))
1151 : {
1152 96960187 : case t_INT: case t_FRAC:
1153 96960187 : return x;
1154 556260 : case t_POLMOD:
1155 556260 : x = checknfelt_mod(nf,x,"nf_to_scalar_or_basis");
1156 556126 : switch(typ(x))
1157 : {
1158 85427 : case t_INT: case t_FRAC: return x;
1159 470699 : case t_POL: return pol_to_scalar_or_basis(nf,x);
1160 : }
1161 0 : break;
1162 4085422 : case t_POL: return pol_to_scalar_or_basis(nf,x);
1163 59531637 : case t_COL:
1164 59531637 : if (lg(x)-1 != nf_get_degree(nf)) break;
1165 59531318 : return QV_isscalar(x)? gel(x,1): x;
1166 : }
1167 96 : pari_err_TYPE("nf_to_scalar_or_basis",x);
1168 : return NULL; /* LCOV_EXCL_LINE */
1169 : }
1170 : /* Let x be a polynomial with coefficients in Q or nf. Return the same
1171 : * polynomial with coefficients expressed as vectors (on the integral basis).
1172 : * No consistency checks, not memory-clean. */
1173 : GEN
1174 28767 : RgX_to_nfX(GEN nf, GEN x)
1175 : {
1176 : long i, l;
1177 28767 : GEN y = cgetg_copy(x, &l); y[1] = x[1];
1178 235475 : for (i=2; i<l; i++) gel(y,i) = nf_to_scalar_or_basis(nf, gel(x,i));
1179 28767 : return y;
1180 : }
1181 :
1182 : /* Assume nf is a genuine nf. */
1183 : GEN
1184 3877465 : nf_to_scalar_or_alg(GEN nf, GEN x)
1185 : {
1186 3877465 : switch(typ(x))
1187 : {
1188 84809 : case t_INT: case t_FRAC:
1189 84809 : return x;
1190 420 : case t_POLMOD:
1191 420 : x = checknfelt_mod(nf,x,"nf_to_scalar_or_alg");
1192 420 : if (typ(x) != t_POL) return x;
1193 : /* fall through */
1194 : case t_POL:
1195 : {
1196 5124 : GEN T = nf_get_pol(nf);
1197 5124 : long l = lg(x);
1198 5124 : if (varn(x) != varn(T)) pari_err_VAR("nf_to_scalar_or_alg", x,T);
1199 5124 : if (l >= lg(T)) { x = RgX_rem(x, T); l = lg(x); }
1200 5124 : if (l == 2) return gen_0;
1201 5124 : if (l == 3) return gel(x,2);
1202 3612 : return x;
1203 : }
1204 3787486 : case t_COL:
1205 : {
1206 : GEN dx;
1207 3787486 : if (lg(x)-1 != nf_get_degree(nf)) break;
1208 7498066 : if (QV_isscalar(x)) return gel(x,1);
1209 3710472 : x = Q_remove_denom(x, &dx);
1210 3710507 : x = RgV_RgC_mul(nf_get_zkprimpart(nf), x);
1211 3710630 : dx = mul_denom(dx, nf_get_zkden(nf));
1212 3710621 : return gdiv(x,dx);
1213 : }
1214 : }
1215 48 : pari_err_TYPE("nf_to_scalar_or_alg",x);
1216 : return NULL; /* LCOV_EXCL_LINE */
1217 : }
1218 :
1219 : /* gmul(A, RgX_to_RgC(x)), A t_MAT of compatible dimensions */
1220 : GEN
1221 1365 : RgM_RgX_mul(GEN A, GEN x)
1222 : {
1223 1365 : long i, l = lg(x)-1;
1224 : GEN z;
1225 1365 : if (l == 1) return zerocol(nbrows(A));
1226 1351 : z = gmul(gel(x,2), gel(A,1));
1227 2555 : for (i = 2; i < l; i++)
1228 1204 : if (!gequal0(gel(x,i+1))) z = gadd(z, gmul(gel(x,i+1), gel(A,i)));
1229 1351 : return z;
1230 : }
1231 : GEN
1232 10306848 : ZM_ZX_mul(GEN A, GEN x)
1233 : {
1234 10306848 : long i, l = lg(x)-1;
1235 : GEN z;
1236 10306848 : if (l == 1) return zerocol(nbrows(A));
1237 10305714 : z = ZC_Z_mul(gel(A,1), gel(x,2));
1238 32185865 : for (i = 2; i < l ; i++)
1239 21882785 : if (signe(gel(x,i+1))) z = ZC_add(z, ZC_Z_mul(gel(A,i), gel(x,i+1)));
1240 10303080 : return z;
1241 : }
1242 : /* x a t_POL, nf a genuine nf. No garbage collecting. No check. */
1243 : GEN
1244 9707503 : poltobasis(GEN nf, GEN x)
1245 : {
1246 9707503 : GEN d, T = nf_get_pol(nf);
1247 9707491 : if (varn(x) != varn(T)) pari_err_VAR( "poltobasis", x,T);
1248 9707358 : if (degpol(x) >= degpol(T)) x = RgX_rem(x,T);
1249 9707310 : x = Q_remove_denom(x, &d);
1250 9707567 : if (!RgX_is_ZX(x)) pari_err_TYPE("poltobasis",x);
1251 9707501 : x = ZM_ZX_mul(nf_get_invzk(nf), x);
1252 9705350 : if (d) x = RgC_Rg_div(x, d);
1253 9705365 : return x;
1254 : }
1255 :
1256 : GEN
1257 921325 : algtobasis(GEN nf, GEN x)
1258 : {
1259 : pari_sp av;
1260 :
1261 921325 : nf = checknf(nf);
1262 921326 : switch(typ(x))
1263 : {
1264 113254 : case t_POLMOD:
1265 113254 : if (!RgX_equal_var(nf_get_pol(nf),gel(x,1)))
1266 7 : pari_err_MODULUS("algtobasis", nf_get_pol(nf),gel(x,1));
1267 113247 : x = gel(x,2);
1268 113247 : switch(typ(x))
1269 : {
1270 8197 : case t_INT:
1271 8197 : case t_FRAC: return scalarcol(x, nf_get_degree(nf));
1272 105050 : case t_POL:
1273 105050 : av = avma;
1274 105050 : return gerepileupto(av,poltobasis(nf,x));
1275 : }
1276 0 : break;
1277 :
1278 249672 : case t_POL:
1279 249672 : av = avma;
1280 249672 : return gerepileupto(av,poltobasis(nf,x));
1281 :
1282 83152 : case t_COL:
1283 83152 : if (!RgV_is_QV(x)) pari_err_TYPE("nfalgtobasis",x);
1284 83146 : if (lg(x)-1 != nf_get_degree(nf)) pari_err_DIM("nfalgtobasis");
1285 83146 : return gcopy(x);
1286 :
1287 475249 : case t_INT:
1288 475249 : case t_FRAC: return scalarcol(x, nf_get_degree(nf));
1289 : }
1290 0 : pari_err_TYPE("algtobasis",x);
1291 : return NULL; /* LCOV_EXCL_LINE */
1292 : }
1293 :
1294 : GEN
1295 47488 : rnfbasistoalg(GEN rnf,GEN x)
1296 : {
1297 47488 : const char *f = "rnfbasistoalg";
1298 : long lx, i;
1299 47488 : pari_sp av = avma;
1300 : GEN z, nf, R, T;
1301 :
1302 47488 : checkrnf(rnf);
1303 47488 : nf = rnf_get_nf(rnf);
1304 47488 : T = nf_get_pol(nf);
1305 47488 : R = QXQX_to_mod_shallow(rnf_get_pol(rnf), T);
1306 47488 : switch(typ(x))
1307 : {
1308 875 : case t_COL:
1309 875 : z = cgetg_copy(x, &lx);
1310 2597 : for (i=1; i<lx; i++)
1311 : {
1312 1778 : GEN c = nf_to_scalar_or_alg(nf, gel(x,i));
1313 1722 : if (typ(c) == t_POL) c = mkpolmod(c,T);
1314 1722 : gel(z,i) = c;
1315 : }
1316 819 : z = RgV_RgC_mul(gel(rnf_get_zk(rnf),1), z);
1317 735 : return gerepileupto(av, gmodulo(z,R));
1318 :
1319 31227 : case t_POLMOD:
1320 31227 : x = polmod_nffix(f, rnf, x, 0);
1321 30954 : if (typ(x) != t_POL) break;
1322 14261 : retmkpolmod(RgX_copy(x), RgX_copy(R));
1323 1274 : case t_POL:
1324 1274 : if (varn(x) == varn(T)) { RgX_check_QX(x,f); x = gmodulo(x,T); break; }
1325 1029 : if (varn(x) == varn(R))
1326 : {
1327 973 : x = RgX_nffix(f,nf_get_pol(nf),x,0);
1328 973 : return gmodulo(x, R);
1329 : }
1330 56 : pari_err_VAR(f, x,R);
1331 : }
1332 30994 : retmkpolmod(scalarpol(x, varn(R)), RgX_copy(R));
1333 : }
1334 :
1335 : GEN
1336 2275 : matbasistoalg(GEN nf,GEN x)
1337 : {
1338 : long i, j, li, lx;
1339 2275 : GEN z = cgetg_copy(x, &lx);
1340 :
1341 2275 : if (lx == 1) return z;
1342 2268 : switch(typ(x))
1343 : {
1344 77 : case t_VEC: case t_COL:
1345 273 : for (i=1; i<lx; i++) gel(z,i) = basistoalg(nf, gel(x,i));
1346 77 : return z;
1347 2191 : case t_MAT: break;
1348 0 : default: pari_err_TYPE("matbasistoalg",x);
1349 : }
1350 2191 : li = lgcols(x);
1351 8183 : for (j=1; j<lx; j++)
1352 : {
1353 5992 : GEN c = cgetg(li,t_COL), xj = gel(x,j);
1354 5992 : gel(z,j) = c;
1355 28077 : for (i=1; i<li; i++) gel(c,i) = basistoalg(nf, gel(xj,i));
1356 : }
1357 2191 : return z;
1358 : }
1359 :
1360 : GEN
1361 30722 : matalgtobasis(GEN nf,GEN x)
1362 : {
1363 : long i, j, li, lx;
1364 30722 : GEN z = cgetg_copy(x, &lx);
1365 :
1366 30722 : if (lx == 1) return z;
1367 30260 : switch(typ(x))
1368 : {
1369 30253 : case t_VEC: case t_COL:
1370 79557 : for (i=1; i<lx; i++) gel(z,i) = algtobasis(nf, gel(x,i));
1371 30253 : return z;
1372 7 : case t_MAT: break;
1373 0 : default: pari_err_TYPE("matalgtobasis",x);
1374 : }
1375 7 : li = lgcols(x);
1376 14 : for (j=1; j<lx; j++)
1377 : {
1378 7 : GEN c = cgetg(li,t_COL), xj = gel(x,j);
1379 7 : gel(z,j) = c;
1380 21 : for (i=1; i<li; i++) gel(c,i) = algtobasis(nf, gel(xj,i));
1381 : }
1382 7 : return z;
1383 : }
1384 : GEN
1385 10953 : RgM_to_nfM(GEN nf,GEN x)
1386 : {
1387 : long i, j, li, lx;
1388 10953 : GEN z = cgetg_copy(x, &lx);
1389 :
1390 10953 : if (lx == 1) return z;
1391 10953 : li = lgcols(x);
1392 81711 : for (j=1; j<lx; j++)
1393 : {
1394 70758 : GEN c = cgetg(li,t_COL), xj = gel(x,j);
1395 70758 : gel(z,j) = c;
1396 462951 : for (i=1; i<li; i++) gel(c,i) = nf_to_scalar_or_basis(nf, gel(xj,i));
1397 : }
1398 10953 : return z;
1399 : }
1400 : GEN
1401 148531 : RgC_to_nfC(GEN nf, GEN x)
1402 908819 : { pari_APPLY_type(t_COL, nf_to_scalar_or_basis(nf, gel(x,i))) }
1403 :
1404 : /* x a t_POLMOD, supposedly in rnf = K[z]/(T), K = Q[y]/(Tnf) */
1405 : GEN
1406 141436 : polmod_nffix(const char *f, GEN rnf, GEN x, int lift)
1407 141436 : { return polmod_nffix2(f, rnf_get_nfpol(rnf), rnf_get_pol(rnf), x,lift); }
1408 : GEN
1409 141527 : polmod_nffix2(const char *f, GEN T, GEN R, GEN x, int lift)
1410 : {
1411 141527 : if (RgX_equal_var(gel(x,1), R))
1412 : {
1413 129087 : x = gel(x,2);
1414 129087 : if (typ(x) == t_POL && varn(x) == varn(R))
1415 : {
1416 98349 : x = RgX_nffix(f, T, x, lift);
1417 98349 : switch(lg(x))
1418 : {
1419 5817 : case 2: return gen_0;
1420 12197 : case 3: return gel(x,2);
1421 : }
1422 80335 : return x;
1423 : }
1424 : }
1425 43178 : return Rg_nffix(f, T, x, lift);
1426 : }
1427 : GEN
1428 1428 : rnfalgtobasis(GEN rnf,GEN x)
1429 : {
1430 1428 : const char *f = "rnfalgtobasis";
1431 1428 : pari_sp av = avma;
1432 : GEN T, R;
1433 :
1434 1428 : checkrnf(rnf);
1435 1428 : R = rnf_get_pol(rnf);
1436 1428 : T = rnf_get_nfpol(rnf);
1437 1428 : switch(typ(x))
1438 : {
1439 98 : case t_COL:
1440 98 : if (lg(x)-1 != rnf_get_degree(rnf)) pari_err_DIM(f);
1441 49 : x = RgV_nffix(f, T, x, 0);
1442 42 : return gerepilecopy(av, x);
1443 :
1444 1162 : case t_POLMOD:
1445 1162 : x = polmod_nffix(f, rnf, x, 0);
1446 1057 : if (typ(x) != t_POL) break;
1447 714 : return gerepileupto(av, RgM_RgX_mul(rnf_get_invzk(rnf), x));
1448 112 : case t_POL:
1449 112 : if (varn(x) == varn(T))
1450 : {
1451 42 : RgX_check_QX(x,f);
1452 28 : if (degpol(x) >= degpol(T)) x = RgX_rem(x,T);
1453 28 : x = mkpolmod(x,T); break;
1454 : }
1455 70 : x = RgX_nffix(f, T, x, 0);
1456 56 : if (degpol(x) >= degpol(R)) x = RgX_rem(x, R);
1457 56 : return gerepileupto(av, RgM_RgX_mul(rnf_get_invzk(rnf), x));
1458 : }
1459 427 : return gerepileupto(av, scalarcol(x, rnf_get_degree(rnf)));
1460 : }
1461 :
1462 : /* Given a and b in nf, gives an algebraic integer y in nf such that a-b.y
1463 : * is "small" */
1464 : GEN
1465 259 : nfdiveuc(GEN nf, GEN a, GEN b)
1466 : {
1467 259 : pari_sp av = avma;
1468 259 : a = nfdiv(nf,a,b);
1469 259 : return gerepileupto(av, ground(a));
1470 : }
1471 :
1472 : /* Given a and b in nf, gives a "small" algebraic integer r in nf
1473 : * of the form a-b.y */
1474 : GEN
1475 259 : nfmod(GEN nf, GEN a, GEN b)
1476 : {
1477 259 : pari_sp av = avma;
1478 259 : GEN p1 = gneg_i(nfmul(nf,b,ground(nfdiv(nf,a,b))));
1479 259 : return gerepileupto(av, nfadd(nf,a,p1));
1480 : }
1481 :
1482 : /* Given a and b in nf, gives a two-component vector [y,r] in nf such
1483 : * that r=a-b.y is "small". */
1484 : GEN
1485 259 : nfdivrem(GEN nf, GEN a, GEN b)
1486 : {
1487 259 : pari_sp av = avma;
1488 259 : GEN p1,z, y = ground(nfdiv(nf,a,b));
1489 :
1490 259 : p1 = gneg_i(nfmul(nf,b,y));
1491 259 : z = cgetg(3,t_VEC);
1492 259 : gel(z,1) = gcopy(y);
1493 259 : gel(z,2) = nfadd(nf,a,p1); return gerepileupto(av, z);
1494 : }
1495 :
1496 : /*************************************************************************/
1497 : /** **/
1498 : /** LOGARITHMIC EMBEDDINGS **/
1499 : /** **/
1500 : /*************************************************************************/
1501 :
1502 : static int
1503 4611593 : low_prec(GEN x)
1504 : {
1505 4611593 : switch(typ(x))
1506 : {
1507 0 : case t_INT: return !signe(x);
1508 4611593 : case t_REAL: return !signe(x) || realprec(x) <= DEFAULTPREC;
1509 0 : default: return 0;
1510 : }
1511 : }
1512 :
1513 : static GEN
1514 23163 : cxlog_1(GEN nf) { return zerocol(lg(nf_get_roots(nf))-1); }
1515 : static GEN
1516 545 : cxlog_m1(GEN nf, long prec)
1517 : {
1518 545 : long i, l = lg(nf_get_roots(nf)), r1 = nf_get_r1(nf);
1519 545 : GEN v = cgetg(l, t_COL), p, P;
1520 545 : p = mppi(prec); P = mkcomplex(gen_0, p);
1521 1224 : for (i = 1; i <= r1; i++) gel(v,i) = P; /* IPi*/
1522 545 : if (i < l) P = gmul2n(P,1);
1523 1160 : for ( ; i < l; i++) gel(v,i) = P; /* 2IPi */
1524 545 : return v;
1525 : }
1526 : static GEN
1527 1714931 : ZC_cxlog(GEN nf, GEN x, long prec)
1528 : {
1529 : long i, l, r1;
1530 : GEN v;
1531 1714931 : x = RgM_RgC_mul(nf_get_M(nf), Q_primpart(x));
1532 1714934 : l = lg(x); r1 = nf_get_r1(nf);
1533 4330666 : for (i = 1; i <= r1; i++)
1534 2615732 : if (low_prec(gel(x,i))) return NULL;
1535 3513991 : for ( ; i < l; i++)
1536 1799058 : if (low_prec(gnorm(gel(x,i)))) return NULL;
1537 1714933 : v = cgetg(l,t_COL);
1538 4330664 : for (i = 1; i <= r1; i++) gel(v,i) = glog(gel(x,i),prec);
1539 3513989 : for ( ; i < l; i++) gel(v,i) = gmul2n(glog(gel(x,i),prec),1);
1540 1714932 : return v;
1541 : }
1542 : static GEN
1543 223252 : famat_cxlog(GEN nf, GEN fa, long prec)
1544 : {
1545 223252 : GEN G, E, y = NULL;
1546 : long i, l;
1547 :
1548 223252 : if (typ(fa) != t_MAT) pari_err_TYPE("famat_cxlog",fa);
1549 223252 : if (lg(fa) == 1) return cxlog_1(nf);
1550 223252 : G = gel(fa,1);
1551 223252 : E = gel(fa,2); l = lg(E);
1552 1119146 : for (i = 1; i < l; i++)
1553 : {
1554 895894 : GEN t, e = gel(E,i), x = nf_to_scalar_or_basis(nf, gel(G,i));
1555 : /* multiplicative arch would be better (save logs), but exponents overflow
1556 : * [ could keep track of expo separately, but not worth it ] */
1557 895894 : switch(typ(x))
1558 : { /* ignore positive rationals */
1559 16412 : case t_FRAC: x = gel(x,1); /* fall through */
1560 266204 : case t_INT: if (signe(x) > 0) continue;
1561 97 : if (!mpodd(e)) continue;
1562 41 : t = cxlog_m1(nf, prec); /* we probably should not reach this line */
1563 41 : break;
1564 629690 : default: /* t_COL */
1565 629690 : t = ZC_cxlog(nf,x,prec); if (!t) return NULL;
1566 629690 : t = RgC_Rg_mul(t, e);
1567 : }
1568 629731 : y = y? RgV_add(y,t): t;
1569 : }
1570 223252 : return y ? y: cxlog_1(nf);
1571 : }
1572 : /* Archimedean components: [e_i Log( sigma_i(X) )], where X = primpart(x),
1573 : * and e_i = 1 (resp 2.) for i <= R1 (resp. > R1) */
1574 : GEN
1575 1309642 : nf_cxlog(GEN nf, GEN x, long prec)
1576 : {
1577 1309642 : if (typ(x) == t_MAT) return famat_cxlog(nf,x,prec);
1578 1086390 : x = nf_to_scalar_or_basis(nf,x);
1579 1086390 : switch(typ(x))
1580 : {
1581 0 : case t_FRAC: x = gel(x,1); /* fall through */
1582 1148 : case t_INT:
1583 1148 : return signe(x) > 0? cxlog_1(nf): cxlog_m1(nf, prec);
1584 1085242 : default:
1585 1085242 : return ZC_cxlog(nf, x, prec);
1586 : }
1587 : }
1588 : GEN
1589 97 : nfV_cxlog(GEN nf, GEN x, long prec)
1590 : {
1591 : long i, l;
1592 97 : GEN v = cgetg_copy(x, &l);
1593 167 : for (i = 1; i < l; i++)
1594 70 : if (!(gel(v,i) = nf_cxlog(nf, gel(x,i), prec))) return NULL;
1595 97 : return v;
1596 : }
1597 :
1598 : static GEN
1599 15232 : scalar_logembed(GEN nf, GEN u, GEN *emb)
1600 : {
1601 : GEN v, logu;
1602 15232 : long i, s = signe(u), RU = lg(nf_get_roots(nf))-1, R1 = nf_get_r1(nf);
1603 :
1604 15232 : if (!s) pari_err_DOMAIN("nflogembed","argument","=",gen_0,u);
1605 15232 : v = cgetg(RU+1, t_COL); logu = logr_abs(u);
1606 17213 : for (i = 1; i <= R1; i++) gel(v,i) = logu;
1607 15232 : if (i <= RU)
1608 : {
1609 14350 : GEN logu2 = shiftr(logu,1);
1610 55839 : for ( ; i <= RU; i++) gel(v,i) = logu2;
1611 : }
1612 15232 : if (emb) *emb = const_col(RU, u);
1613 15232 : return v;
1614 : }
1615 :
1616 : static GEN
1617 1309 : famat_logembed(GEN nf,GEN x,GEN *emb,long prec)
1618 : {
1619 1309 : GEN A, M, T, a, t, g = gel(x,1), e = gel(x,2);
1620 1309 : long i, l = lg(e);
1621 :
1622 1309 : if (l == 1) return scalar_logembed(nf, real_1(prec), emb);
1623 1309 : A = NULL; T = emb? cgetg(l, t_COL): NULL;
1624 1309 : if (emb) *emb = M = mkmat2(T, e);
1625 62132 : for (i = 1; i < l; i++)
1626 : {
1627 60823 : a = nflogembed(nf, gel(g,i), &t, prec);
1628 60823 : if (!a) return NULL;
1629 60823 : a = RgC_Rg_mul(a, gel(e,i));
1630 60823 : A = A? RgC_add(A, a): a;
1631 60823 : if (emb) gel(T,i) = t;
1632 : }
1633 1309 : return A;
1634 : }
1635 :
1636 : /* Get archimedean components: [e_i log( | sigma_i(x) | )], with e_i = 1
1637 : * (resp 2.) for i <= R1 (resp. > R1) and set emb to the embeddings of x.
1638 : * Return NULL if precision problem */
1639 : GEN
1640 98651 : nflogembed(GEN nf, GEN x, GEN *emb, long prec)
1641 : {
1642 : long i, l, r1;
1643 : GEN v, t;
1644 :
1645 98651 : if (typ(x) == t_MAT) return famat_logembed(nf,x,emb,prec);
1646 97342 : x = nf_to_scalar_or_basis(nf,x);
1647 97342 : if (typ(x) != t_COL) return scalar_logembed(nf, gtofp(x,prec), emb);
1648 82110 : x = RgM_RgC_mul(nf_get_M(nf), x);
1649 82110 : l = lg(x); r1 = nf_get_r1(nf); v = cgetg(l,t_COL);
1650 109039 : for (i = 1; i <= r1; i++)
1651 : {
1652 26929 : t = gabs(gel(x,i),prec); if (low_prec(t)) return NULL;
1653 26929 : gel(v,i) = glog(t,prec);
1654 : }
1655 251986 : for ( ; i < l; i++)
1656 : {
1657 169876 : t = gnorm(gel(x,i)); if (low_prec(t)) return NULL;
1658 169876 : gel(v,i) = glog(t,prec);
1659 : }
1660 82110 : if (emb) *emb = x;
1661 82110 : return v;
1662 : }
1663 :
1664 : /*************************************************************************/
1665 : /** **/
1666 : /** REAL EMBEDDINGS **/
1667 : /** **/
1668 : /*************************************************************************/
1669 : static GEN
1670 486224 : sarch_get_cyc(GEN sarch) { return gel(sarch,1); }
1671 : static GEN
1672 665698 : sarch_get_archp(GEN sarch) { return gel(sarch,2); }
1673 : static GEN
1674 163621 : sarch_get_MI(GEN sarch) { return gel(sarch,3); }
1675 : static GEN
1676 163620 : sarch_get_lambda(GEN sarch) { return gel(sarch,4); }
1677 : static GEN
1678 163620 : sarch_get_F(GEN sarch) { return gel(sarch,5); }
1679 :
1680 : /* x not a scalar, true nf, return number of positive roots of char_x */
1681 : static long
1682 1262 : num_positive(GEN nf, GEN x)
1683 : {
1684 1262 : GEN T = nf_get_pol(nf), B, charx;
1685 : long dnf, vnf, N;
1686 1262 : x = nf_to_scalar_or_alg(nf, x); /* not a scalar */
1687 1262 : charx = ZXQ_charpoly(x, T, 0);
1688 1262 : charx = ZX_radical(charx);
1689 1262 : N = degpol(T) / degpol(charx);
1690 : /* real places are unramified ? */
1691 1262 : if (N == 1 || ZX_sturm(charx) * N == nf_get_r1(nf))
1692 1255 : return ZX_sturmpart(charx, mkvec2(gen_0,mkoo())) * N;
1693 : /* painful case, multiply by random square until primitive */
1694 7 : dnf = nf_get_degree(nf);
1695 7 : vnf = varn(T);
1696 7 : B = int2n(10);
1697 : for(;;)
1698 0 : {
1699 7 : GEN y = RgXQ_sqr(random_FpX(dnf, vnf, B), T);
1700 7 : y = RgXQ_mul(x, y, T);
1701 7 : charx = ZXQ_charpoly(y, T, 0);
1702 7 : if (ZX_is_squarefree(charx))
1703 7 : return ZX_sturmpart(charx, mkvec2(gen_0,mkoo())) * N;
1704 : }
1705 : }
1706 :
1707 : /* x a QC: return sigma_k(x) where 1 <= k <= r1+r2; correct but inefficient
1708 : * if x in Q. M = nf_get_M(nf) */
1709 : static GEN
1710 2134 : nfembed_i(GEN M, GEN x, long k)
1711 : {
1712 2134 : long i, l = lg(M);
1713 2134 : GEN z = gel(x,1);
1714 24356 : for (i = 2; i < l; i++) z = gadd(z, gmul(gcoeff(M,k,i), gel(x,i)));
1715 2134 : return z;
1716 : }
1717 : GEN
1718 0 : nfembed(GEN nf, GEN x, long k)
1719 : {
1720 0 : pari_sp av = avma;
1721 0 : nf = checknf(nf);
1722 0 : x = nf_to_scalar_or_basis(nf,x);
1723 0 : if (typ(x) != t_COL) return gerepilecopy(av, x);
1724 0 : return gerepileupto(av, nfembed_i(nf_get_M(nf),x,k));
1725 : }
1726 :
1727 : /* x a ZC */
1728 : static GEN
1729 894149 : zk_embed(GEN M, GEN x, long k)
1730 : {
1731 894149 : long i, l = lg(x);
1732 894149 : GEN z = gel(x,1); /* times M[k,1], which is 1 */
1733 2984250 : for (i = 2; i < l; i++) z = mpadd(z, mpmul(gcoeff(M,k,i), gel(x,i)));
1734 894152 : return z;
1735 : }
1736 :
1737 : /* Given floating point approximation z of sigma_k(x), decide its sign
1738 : * [0/+, 1/- and -1 for FAIL] */
1739 : static long
1740 875877 : eval_sign_embed(GEN z)
1741 : {
1742 875877 : if (typ(z) == t_REAL)
1743 : {
1744 875877 : long l = realprec(z);
1745 875877 : if (l <= LOWDEFAULTPREC
1746 875877 : || (l == LOWDEFAULTPREC + 1 && !z[l-1])) return -1; /* dubious, fail */
1747 875078 : if (expo(z) < 16 - l) return -1; /* same */
1748 : }
1749 875035 : return (signe(z) < 1)? 1: 0;
1750 : }
1751 : /* return v such that (-1)^v = sign(sigma_k(x)), x primitive ZC */
1752 : static long
1753 779285 : eval_sign(GEN M, GEN x, long k)
1754 779285 : { return eval_sign_embed( zk_embed(M, x, k) ); }
1755 :
1756 : /* check that signs[i..#signs] == s; signs = NULL encodes "totally positive" */
1757 : static int
1758 0 : oksigns(long l, GEN signs, long i, long s)
1759 : {
1760 0 : if (!signs) return s == 0;
1761 0 : for (; i < l; i++)
1762 0 : if (signs[i] != s) return 0;
1763 0 : return 1;
1764 : }
1765 : /* check that signs[i] = s and signs[i+1..#signs] = 1-s */
1766 : static int
1767 0 : oksigns2(long l, GEN signs, long i, long s)
1768 : {
1769 0 : if (!signs) return s == 0 && i == l-1;
1770 0 : return signs[i] == s && oksigns(l, signs, i+1, 1-s);
1771 : }
1772 :
1773 : /* true nf, x a ZC (primitive for efficiency) which is not a scalar; embx its
1774 : * embeddings or NULL */
1775 : static int
1776 80245 : nfchecksigns_i(GEN nf, GEN x, GEN embx, GEN signs, GEN archp)
1777 : {
1778 80245 : long l = lg(archp), i;
1779 80245 : GEN M = nf_get_M(nf), sarch = NULL;
1780 80245 : long np = -1;
1781 126449 : for (i = 1; i < l; i++)
1782 : {
1783 : long s;
1784 97957 : if (embx)
1785 96593 : s = eval_sign_embed(gel(embx,i));
1786 : else
1787 1364 : s = eval_sign(M, x, archp[i]);
1788 : /* 0 / + or 1 / -; -1 for FAIL */
1789 97957 : if (s < 0) /* failure */
1790 : {
1791 0 : long ni, r1 = nf_get_r1(nf);
1792 : GEN xi;
1793 0 : if (np < 0)
1794 : {
1795 0 : np = num_positive(nf, x);
1796 0 : if (np == 0) return oksigns(l, signs, i, 1);
1797 0 : if (np == r1) return oksigns(l, signs, i, 0);
1798 0 : sarch = nfarchstar(nf, NULL, identity_perm(r1));
1799 : }
1800 0 : xi = set_sign_mod_divisor(nf, vecsmall_ei(r1, archp[i]), gen_1, sarch);
1801 0 : xi = Q_primpart(xi);
1802 0 : ni = num_positive(nf, nfmuli(nf,x,xi));
1803 0 : if (ni == 0) return oksigns2(l, signs, i, 0);
1804 0 : if (ni == r1) return oksigns2(l, signs, i, 1);
1805 0 : s = ni < np? 0: 1;
1806 : }
1807 97957 : if (s != (signs? signs[i]: 0)) return 0;
1808 : }
1809 28492 : return 1;
1810 : }
1811 : static void
1812 775 : pl_convert(GEN pl, GEN *psigns, GEN *parchp)
1813 : {
1814 775 : long i, j, l = lg(pl);
1815 775 : GEN signs = cgetg(l, t_VECSMALL);
1816 775 : GEN archp = cgetg(l, t_VECSMALL);
1817 2576 : for (i = j = 1; i < l; i++)
1818 : {
1819 1801 : if (!pl[i]) continue;
1820 1403 : archp[j] = i;
1821 1403 : signs[j] = (pl[i] < 0)? 1: 0;
1822 1403 : j++;
1823 : }
1824 775 : setlg(archp, j); *parchp = archp;
1825 775 : setlg(signs, j); *psigns = signs;
1826 775 : }
1827 : /* pl : requested signs for real embeddings, 0 = no sign constraint */
1828 : int
1829 14719 : nfchecksigns(GEN nf, GEN x, GEN pl)
1830 : {
1831 14719 : pari_sp av = avma;
1832 : GEN signs, archp;
1833 14719 : nf = checknf(nf);
1834 14719 : x = nf_to_scalar_or_basis(nf,x);
1835 14719 : if (typ(x) != t_COL)
1836 : {
1837 13944 : long i, l = lg(pl), s = gsigne(x);
1838 27853 : for (i = 1; i < l; i++)
1839 13909 : if (pl[i] && pl[i] != s) return gc_bool(av,0);
1840 13944 : return gc_bool(av,1);
1841 : }
1842 775 : pl_convert(pl, &signs, &archp);
1843 775 : return gc_bool(av, nfchecksigns_i(nf, x, NULL, signs, archp));
1844 : }
1845 :
1846 : /* signs = NULL: totally positive, else sign[i] = 0 (+) or 1 (-) */
1847 : static GEN
1848 163621 : get_C(GEN lambda, long l, GEN signs)
1849 : {
1850 : long i;
1851 : GEN C, mlambda;
1852 163621 : if (!signs) return const_vec(l-1, lambda);
1853 133878 : C = cgetg(l, t_COL); mlambda = gneg(lambda);
1854 342869 : for (i = 1; i < l; i++) gel(C,i) = signs[i]? mlambda: lambda;
1855 133879 : return C;
1856 : }
1857 : /* signs = NULL: totally positive at archp.
1858 : * Assume that a t_COL x is not a scalar */
1859 : static GEN
1860 277343 : nfsetsigns(GEN nf, GEN signs, GEN x, GEN sarch)
1861 : {
1862 277343 : long i, l = lg(sarch_get_archp(sarch));
1863 : GEN ex;
1864 : /* Is signature already correct ? */
1865 277344 : if (typ(x) != t_COL)
1866 : {
1867 197879 : long s = gsigne(x);
1868 197879 : if (!s) i = 1;
1869 197858 : else if (!signs)
1870 7427 : i = (s < 0)? 1: l;
1871 : else
1872 : {
1873 190431 : s = s < 0? 1: 0;
1874 324131 : for (i = 1; i < l; i++)
1875 245438 : if (signs[i] != s) break;
1876 : }
1877 197879 : ex = (i < l)? const_col(l-1, x): NULL;
1878 : }
1879 : else
1880 : { /* inefficient if x scalar, wrong if x = 0 */
1881 79465 : pari_sp av = avma;
1882 79465 : GEN cex, M = nf_get_M(nf), archp = sarch_get_archp(sarch);
1883 79470 : GEN xp = Q_primitive_part(x,&cex);
1884 79469 : ex = cgetg(l,t_COL);
1885 194337 : for (i = 1; i < l; i++) gel(ex,i) = zk_embed(M,xp,archp[i]);
1886 79470 : if (nfchecksigns_i(nf, xp, ex, signs, archp)) { ex = NULL; set_avma(av); }
1887 51715 : else if (cex) ex = RgC_Rg_mul(ex, cex); /* put back content */
1888 : }
1889 277352 : if (ex)
1890 : { /* If no, fix it */
1891 163620 : GEN MI = sarch_get_MI(sarch), F = sarch_get_F(sarch);
1892 163620 : GEN lambda = sarch_get_lambda(sarch);
1893 163620 : GEN t = RgC_sub(get_C(lambda, l, signs), ex);
1894 163616 : t = grndtoi(RgM_RgC_mul(MI,t), NULL);
1895 163610 : if (lg(F) != 1) t = ZM_ZC_mul(F, t);
1896 163616 : x = typ(x) == t_COL? RgC_add(t, x): RgC_Rg_add(t, x);
1897 : }
1898 277346 : return x;
1899 : }
1900 : /* - true nf
1901 : * - sarch = nfarchstar(nf, F);
1902 : * - x encodes a vector of signs at arch.archp: either a t_VECSMALL
1903 : * (vector of signs as {0,1}-vector), NULL (totally positive at archp),
1904 : * or a nonzero number field element (replaced by its signature at archp);
1905 : * - y is a nonzero number field element
1906 : * Return z = y (mod F) with signs(y, archp) = signs(x) (a {0,1}-vector).
1907 : * Not stack-clean */
1908 : GEN
1909 308887 : set_sign_mod_divisor(GEN nf, GEN x, GEN y, GEN sarch)
1910 : {
1911 308887 : GEN archp = sarch_get_archp(sarch);
1912 308887 : if (lg(archp) == 1) return y;
1913 275581 : if (x && typ(x) != t_VECSMALL) x = nfsign_arch(nf, x, archp);
1914 275581 : return nfsetsigns(nf, x, nf_to_scalar_or_basis(nf,y), sarch);
1915 : }
1916 :
1917 : static GEN
1918 83458 : setsigns_init(GEN nf, GEN archp, GEN F, GEN DATA)
1919 : {
1920 83458 : GEN lambda, Mr = rowpermute(nf_get_M(nf), archp), MI = F? RgM_mul(Mr,F): Mr;
1921 83459 : lambda = gmul2n(matrixnorm(MI,DEFAULTPREC), -1);
1922 83455 : if (typ(lambda) != t_REAL) lambda = gmul(lambda, uutoQ(1001,1000));
1923 83455 : if (lg(archp) < lg(MI))
1924 : {
1925 58922 : GEN perm = gel(indexrank(MI), 2);
1926 58924 : if (!F) F = matid(nf_get_degree(nf));
1927 58924 : MI = vecpermute(MI, perm);
1928 58925 : F = vecpermute(F, perm);
1929 : }
1930 83458 : if (!F) F = cgetg(1,t_MAT);
1931 83458 : MI = RgM_inv(MI);
1932 83456 : return mkvec5(DATA, archp, MI, lambda, F);
1933 : }
1934 : /* F nonzero integral ideal in HNF (or NULL: Z_K), compute elements in 1+F
1935 : * whose sign matrix at archp is identity; archp in 'indices' format */
1936 : GEN
1937 259456 : nfarchstar(GEN nf, GEN F, GEN archp)
1938 : {
1939 259456 : long nba = lg(archp) - 1;
1940 259456 : if (!nba) return mkvec2(cgetg(1,t_VEC), archp);
1941 81696 : if (F && equali1(gcoeff(F,1,1))) F = NULL;
1942 81696 : if (F) F = idealpseudored(F, nf_get_roundG(nf));
1943 81690 : return setsigns_init(nf, archp, F, const_vec(nba, gen_2));
1944 : }
1945 :
1946 : /*************************************************************************/
1947 : /** **/
1948 : /** IDEALCHINESE **/
1949 : /** **/
1950 : /*************************************************************************/
1951 : static int
1952 4206 : isprfact(GEN x)
1953 : {
1954 : long i, l;
1955 : GEN L, E;
1956 4206 : if (typ(x) != t_MAT || lg(x) != 3) return 0;
1957 4206 : L = gel(x,1); l = lg(L);
1958 4206 : E = gel(x,2);
1959 13993 : for(i=1; i<l; i++)
1960 : {
1961 9787 : checkprid(gel(L,i));
1962 9787 : if (typ(gel(E,i)) != t_INT) return 0;
1963 : }
1964 4206 : return 1;
1965 : }
1966 :
1967 : /* initialize projectors mod pr[i]^e[i] for idealchinese */
1968 : static GEN
1969 4206 : pr_init(GEN nf, GEN fa, GEN w, GEN dw)
1970 : {
1971 4206 : GEN U, E, F, FZ, L = gel(fa,1), E0 = gel(fa,2);
1972 4206 : long i, r = lg(L);
1973 :
1974 4206 : if (w && lg(w) != r) pari_err_TYPE("idealchinese", w);
1975 4206 : if (r == 1 && !dw) return cgetg(1,t_VEC);
1976 4192 : E = leafcopy(E0); /* do not destroy fa[2] */
1977 13979 : for (i = 1; i < r; i++)
1978 9787 : if (signe(gel(E,i)) < 0) gel(E,i) = gen_0;
1979 4192 : F = factorbackprime(nf, L, E);
1980 4192 : if (dw)
1981 : {
1982 693 : F = ZM_Z_mul(F, dw);
1983 1596 : for (i = 1; i < r; i++)
1984 : {
1985 903 : GEN pr = gel(L,i);
1986 903 : long e = itos(gel(E0,i)), v = idealval(nf, dw, pr);
1987 903 : if (e >= 0)
1988 896 : gel(E,i) = addiu(gel(E,i), v);
1989 7 : else if (v + e <= 0)
1990 0 : F = idealmulpowprime(nf, F, pr, stoi(-v)); /* coprime to pr */
1991 : else
1992 : {
1993 7 : F = idealmulpowprime(nf, F, pr, stoi(e));
1994 7 : gel(E,i) = stoi(v + e);
1995 : }
1996 : }
1997 : }
1998 4192 : U = cgetg(r, t_VEC);
1999 13979 : for (i = 1; i < r; i++)
2000 : {
2001 : GEN u;
2002 9787 : if (w && gequal0(gel(w,i))) u = gen_0; /* unused */
2003 : else
2004 : {
2005 9710 : GEN pr = gel(L,i), e = gel(E,i), t;
2006 9710 : t = idealdivpowprime(nf,F, pr, e);
2007 9710 : u = hnfmerge_get_1(t, idealpow(nf, pr, e));
2008 9710 : if (!u) pari_err_COPRIME("idealchinese", t,pr);
2009 : }
2010 9787 : gel(U,i) = u;
2011 : }
2012 4192 : FZ = gcoeff(F, 1, 1);
2013 4192 : F = idealpseudored(F, nf_get_roundG(nf));
2014 4192 : return mkvec2(mkvec2(F, FZ), U);
2015 : }
2016 :
2017 : static GEN
2018 2261 : pl_normalize(GEN nf, GEN pl)
2019 : {
2020 2261 : const char *fun = "idealchinese";
2021 2261 : if (lg(pl)-1 != nf_get_r1(nf)) pari_err_TYPE(fun,pl);
2022 2261 : switch(typ(pl))
2023 : {
2024 707 : case t_VEC: RgV_check_ZV(pl,fun); pl = ZV_to_zv(pl);
2025 : /* fall through */
2026 2261 : case t_VECSMALL: break;
2027 0 : default: pari_err_TYPE(fun,pl);
2028 : }
2029 2261 : return pl;
2030 : }
2031 :
2032 : static int
2033 9443 : is_chineseinit(GEN x)
2034 : {
2035 : GEN fa, pl;
2036 : long l;
2037 9443 : if (typ(x) != t_VEC || lg(x)!=3) return 0;
2038 7602 : fa = gel(x,1);
2039 7602 : pl = gel(x,2);
2040 7602 : if (typ(fa) != t_VEC || typ(pl) != t_VEC) return 0;
2041 4207 : l = lg(fa);
2042 4207 : if (l != 1)
2043 : {
2044 : GEN z;
2045 4165 : if (l != 3) return 0;
2046 4165 : z = gel(fa, 1);
2047 4165 : if (typ(z) != t_VEC || lg(z) != 3 || typ(gel(z,1)) != t_MAT
2048 4158 : || typ(gel(z,2)) != t_INT
2049 4158 : || typ(gel(fa,2)) != t_VEC)
2050 7 : return 0;
2051 : }
2052 4200 : l = lg(pl);
2053 4200 : if (l != 1)
2054 : {
2055 665 : if (l != 6 || typ(gel(pl,3)) != t_MAT || typ(gel(pl,1)) != t_VECSMALL
2056 665 : || typ(gel(pl,2)) != t_VECSMALL)
2057 0 : return 0;
2058 : }
2059 4200 : return 1;
2060 : }
2061 :
2062 : /* nf a true 'nf' */
2063 : static GEN
2064 4661 : chineseinit_i(GEN nf, GEN fa, GEN w, GEN dw)
2065 : {
2066 4661 : const char *fun = "idealchineseinit";
2067 4661 : GEN archp = NULL, pl = NULL;
2068 4661 : switch(typ(fa))
2069 : {
2070 2261 : case t_VEC:
2071 2261 : if (is_chineseinit(fa))
2072 : {
2073 0 : if (dw) pari_err_DOMAIN(fun, "denom(y)", "!=", gen_1, w);
2074 0 : return fa;
2075 : }
2076 2261 : if (lg(fa) != 3) pari_err_TYPE(fun, fa);
2077 : /* of the form [x,s] */
2078 2261 : pl = pl_normalize(nf, gel(fa,2));
2079 2261 : fa = gel(fa,1);
2080 2261 : archp = vecsmall01_to_indices(pl);
2081 : /* keep pr_init, reset pl */
2082 2261 : if (is_chineseinit(fa)) { fa = gel(fa,1); break; }
2083 : /* fall through */
2084 : case t_MAT: /* factorization? */
2085 4206 : if (isprfact(fa)) { fa = pr_init(nf, fa, w, dw); break; }
2086 0 : default: pari_err_TYPE(fun,fa);
2087 : }
2088 :
2089 4661 : if (!pl) pl = cgetg(1,t_VEC);
2090 : else
2091 : {
2092 2261 : long r = lg(archp);
2093 2261 : if (r == 1) pl = cgetg(1, t_VEC);
2094 : else
2095 : {
2096 1757 : GEN F = (lg(fa) == 1)? NULL: gmael(fa,1,1), signs = cgetg(r, t_VECSMALL);
2097 : long i;
2098 5082 : for (i = 1; i < r; i++) signs[i] = (pl[archp[i]] < 0)? 1: 0;
2099 1757 : pl = setsigns_init(nf, archp, F, signs);
2100 : }
2101 : }
2102 4661 : return mkvec2(fa, pl);
2103 : }
2104 :
2105 : /* Given a prime ideal factorization x, possibly with 0 or negative exponents,
2106 : * and a vector w of elements of nf, gives b such that
2107 : * v_p(b-w_p)>=v_p(x) for all prime ideals p in the ideal factorization
2108 : * and v_p(b)>=0 for all other p, using the standard proof given in GTM 138. */
2109 : GEN
2110 8406 : idealchinese(GEN nf, GEN x0, GEN w)
2111 : {
2112 8406 : const char *fun = "idealchinese";
2113 8406 : pari_sp av = avma;
2114 8406 : GEN x = x0, x1, x2, s, dw, F;
2115 :
2116 8406 : nf = checknf(nf);
2117 8406 : if (!w) return gerepilecopy(av, chineseinit_i(nf,x,NULL,NULL));
2118 :
2119 4921 : if (typ(w) != t_VEC) pari_err_TYPE(fun,w);
2120 4921 : w = Q_remove_denom(matalgtobasis(nf,w), &dw);
2121 4921 : if (!is_chineseinit(x)) x = chineseinit_i(nf,x,w,dw);
2122 : /* x is a 'chineseinit' */
2123 4921 : x1 = gel(x,1); s = NULL;
2124 4921 : x2 = gel(x,2);
2125 4921 : if (lg(x1) == 1) { F = NULL; dw = NULL; }
2126 : else
2127 : {
2128 4879 : GEN U = gel(x1,2), FZ;
2129 4879 : long i, r = lg(w);
2130 4879 : F = gmael(x1,1,1); FZ = gmael(x1,1,2);
2131 17624 : for (i=1; i<r; i++)
2132 12745 : if (!ZV_equal0(gel(w,i)))
2133 : {
2134 9640 : GEN t = nfmuli(nf, gel(U,i), gel(w,i));
2135 9640 : s = s? ZC_add(s,t): t;
2136 : }
2137 4879 : if (s)
2138 : {
2139 4858 : s = ZC_reducemodmatrix(s, F);
2140 4858 : if (dw && x == x0) /* input was a chineseinit */
2141 : {
2142 7 : dw = modii(dw, FZ);
2143 7 : s = FpC_Fp_mul(s, Fp_inv(dw, FZ), FZ);
2144 7 : dw = NULL;
2145 : }
2146 4858 : if (ZV_isscalar(s)) s = icopy(gel(s,1));
2147 : }
2148 : }
2149 4921 : if (lg(x2) != 1)
2150 : {
2151 1764 : s = nfsetsigns(nf, gel(x2,1), s? s: gen_0, x2);
2152 1764 : if (typ(s) == t_COL && QV_isscalar(s))
2153 : {
2154 294 : s = gel(s,1); if (!dw) s = gcopy(s);
2155 : }
2156 : }
2157 3157 : else if (!s) return gc_const(av, gen_0);
2158 4872 : return gerepileupto(av, dw? gdiv(s, dw): s);
2159 : }
2160 :
2161 : /*************************************************************************/
2162 : /** **/
2163 : /** (Z_K/I)^* **/
2164 : /** **/
2165 : /*************************************************************************/
2166 : GEN
2167 2261 : vecsmall01_to_indices(GEN v)
2168 : {
2169 2261 : long i, k, l = lg(v);
2170 2261 : GEN p = new_chunk(l) + l;
2171 6636 : for (k=1, i=l-1; i; i--)
2172 4375 : if (v[i]) { *--p = i; k++; }
2173 2261 : *--p = _evallg(k) | evaltyp(t_VECSMALL);
2174 2261 : set_avma((pari_sp)p); return p;
2175 : }
2176 : GEN
2177 1087041 : vec01_to_indices(GEN v)
2178 : {
2179 : long i, k, l;
2180 : GEN p;
2181 :
2182 1087041 : switch (typ(v))
2183 : {
2184 1040309 : case t_VECSMALL: return v;
2185 46732 : case t_VEC: break;
2186 0 : default: pari_err_TYPE("vec01_to_indices",v);
2187 : }
2188 46732 : l = lg(v);
2189 46732 : p = new_chunk(l) + l;
2190 140553 : for (k=1, i=l-1; i; i--)
2191 93821 : if (signe(gel(v,i))) { *--p = i; k++; }
2192 46732 : *--p = _evallg(k) | evaltyp(t_VECSMALL);
2193 46732 : set_avma((pari_sp)p); return p;
2194 : }
2195 : GEN
2196 136890 : indices_to_vec01(GEN p, long r)
2197 : {
2198 136890 : long i, l = lg(p);
2199 136890 : GEN v = zerovec(r);
2200 206630 : for (i = 1; i < l; i++) gel(v, p[i]) = gen_1;
2201 136889 : return v;
2202 : }
2203 :
2204 : /* return (column) vector of R1 signatures of x (0 or 1) */
2205 : GEN
2206 1040308 : nfsign_arch(GEN nf, GEN x, GEN arch)
2207 : {
2208 1040308 : GEN sarch, M, V, archp = vec01_to_indices(arch);
2209 1040309 : long i, s, np, n = lg(archp)-1;
2210 : pari_sp av;
2211 :
2212 1040309 : if (!n) return cgetg(1,t_VECSMALL);
2213 838710 : if (typ(x) == t_MAT)
2214 : { /* factorisation */
2215 276147 : GEN g = gel(x,1), e = gel(x,2);
2216 276147 : long l = lg(g);
2217 276147 : V = zero_zv(n);
2218 800482 : for (i = 1; i < l; i++)
2219 524336 : if (mpodd(gel(e,i)))
2220 430391 : Flv_add_inplace(V, nfsign_arch(nf,gel(g,i),archp), 2);
2221 276146 : set_avma((pari_sp)V); return V;
2222 : }
2223 562563 : av = avma; V = cgetg(n+1,t_VECSMALL);
2224 562563 : x = nf_to_scalar_or_basis(nf, x);
2225 562564 : switch(typ(x))
2226 : {
2227 179965 : case t_INT:
2228 179965 : s = signe(x);
2229 179965 : if (!s) pari_err_DOMAIN("nfsign_arch","element","=",gen_0,x);
2230 179965 : set_avma(av); return const_vecsmall(n, (s < 0)? 1: 0);
2231 644 : case t_FRAC:
2232 644 : s = signe(gel(x,1));
2233 644 : set_avma(av); return const_vecsmall(n, (s < 0)? 1: 0);
2234 : }
2235 381955 : x = Q_primpart(x); M = nf_get_M(nf); sarch = NULL; np = -1;
2236 1159068 : for (i = 1; i <= n; i++)
2237 : {
2238 777921 : long s = eval_sign(M, x, archp[i]);
2239 777920 : if (s < 0) /* failure */
2240 : {
2241 842 : long ni, r1 = nf_get_r1(nf);
2242 : GEN xi;
2243 842 : if (np < 0)
2244 : {
2245 842 : np = num_positive(nf, x);
2246 842 : if (np == 0) { set_avma(av); return const_vecsmall(n, 1); }
2247 799 : if (np == r1){ set_avma(av); return const_vecsmall(n, 0); }
2248 420 : sarch = nfarchstar(nf, NULL, identity_perm(r1));
2249 : }
2250 420 : xi = set_sign_mod_divisor(nf, vecsmall_ei(r1, archp[i]), gen_1, sarch);
2251 420 : xi = Q_primpart(xi);
2252 420 : ni = num_positive(nf, nfmuli(nf,x,xi));
2253 420 : if (ni == 0) { set_avma(av); V = const_vecsmall(n, 1); V[i] = 0; return V; }
2254 413 : if (ni == r1){ set_avma(av); V = const_vecsmall(n, 0); V[i] = 1; return V; }
2255 35 : s = ni < np? 0: 1;
2256 : }
2257 777113 : V[i] = s;
2258 : }
2259 381147 : set_avma((pari_sp)V); return V;
2260 : }
2261 : static void
2262 35483 : chk_ind(const char *s, long i, long r1)
2263 : {
2264 35483 : if (i <= 0) pari_err_DOMAIN(s, "index", "<=", gen_0, stoi(i));
2265 35469 : if (i > r1) pari_err_DOMAIN(s, "index", ">", utoi(r1), utoi(i));
2266 35434 : }
2267 : static GEN
2268 126378 : parse_embed(GEN ind, long r, const char *f)
2269 : {
2270 : long l, i;
2271 126378 : if (!ind) return identity_perm(r);
2272 33418 : switch(typ(ind))
2273 : {
2274 70 : case t_INT: ind = mkvecsmall(itos(ind)); break;
2275 84 : case t_VEC: case t_COL: ind = vec_to_vecsmall(ind); break;
2276 33264 : case t_VECSMALL: break;
2277 0 : default: pari_err_TYPE(f, ind);
2278 : }
2279 33418 : l = lg(ind);
2280 68852 : for (i = 1; i < l; i++) chk_ind(f, ind[i], r);
2281 33369 : return ind;
2282 : }
2283 : GEN
2284 124061 : nfeltsign(GEN nf, GEN x, GEN ind0)
2285 : {
2286 124061 : pari_sp av = avma;
2287 : long i, l;
2288 : GEN v, ind;
2289 124061 : nf = checknf(nf);
2290 124061 : ind = parse_embed(ind0, nf_get_r1(nf), "nfeltsign");
2291 124040 : l = lg(ind);
2292 124040 : if (is_rational_t(typ(x)))
2293 : { /* nfsign_arch would test this, but avoid converting t_VECSMALL -> t_VEC */
2294 : GEN s;
2295 30975 : switch(gsigne(x))
2296 : {
2297 16366 : case -1:s = gen_m1; break;
2298 14602 : case 1: s = gen_1; break;
2299 7 : default: s = gen_0; break;
2300 : }
2301 30975 : set_avma(av);
2302 30975 : return (ind0 && typ(ind0) == t_INT)? s: const_vec(l-1, s);
2303 : }
2304 93065 : v = nfsign_arch(nf, x, ind);
2305 93065 : if (ind0 && typ(ind0) == t_INT) { set_avma(av); return v[1]? gen_m1: gen_1; }
2306 93051 : settyp(v, t_VEC);
2307 262311 : for (i = 1; i < l; i++) gel(v,i) = v[i]? gen_m1: gen_1;
2308 93051 : return gerepileupto(av, v);
2309 : }
2310 :
2311 : /* true nf */
2312 : GEN
2313 728 : nfeltembed_i(GEN *pnf, GEN x, GEN ind0, long prec0)
2314 : {
2315 : long i, e, l, r1, r2, prec, prec1;
2316 728 : GEN v, ind, cx, nf = *pnf;
2317 728 : nf_get_sign(nf,&r1,&r2);
2318 728 : x = nf_to_scalar_or_basis(nf, x);
2319 721 : ind = parse_embed(ind0, r1+r2, "nfeltembed");
2320 714 : l = lg(ind);
2321 714 : if (typ(x) != t_COL)
2322 : {
2323 224 : if (!(ind0 && typ(ind0) == t_INT)) x = const_vec(l-1, x);
2324 224 : return x;
2325 : }
2326 490 : x = Q_primitive_part(x, &cx);
2327 490 : prec1 = prec0; e = gexpo(x);
2328 490 : if (e > 8) prec1 += nbits2extraprec(e);
2329 490 : prec = prec1;
2330 490 : if (nf_get_prec(nf) < prec) nf = nfnewprec_shallow(nf, prec);
2331 490 : v = cgetg(l, t_VEC);
2332 : for(;;)
2333 132 : {
2334 622 : GEN M = nf_get_M(nf);
2335 2624 : for (i = 1; i < l; i++)
2336 : {
2337 2134 : GEN t = nfembed_i(M, x, ind[i]);
2338 2134 : long e = gexpo(t);
2339 2134 : if (gequal0(t) || precision(t) < prec0
2340 2134 : || (e < 0 && prec < prec1 + nbits2extraprec(-e)) ) break;
2341 2002 : if (cx) t = gmul(t, cx);
2342 2002 : gel(v,i) = t;
2343 : }
2344 622 : if (i == l) break;
2345 132 : prec = precdbl(prec);
2346 132 : if (DEBUGLEVEL>1) pari_warn(warnprec,"eltnfembed", prec);
2347 132 : *pnf = nf = nfnewprec_shallow(nf, prec);
2348 : }
2349 490 : if (ind0 && typ(ind0) == t_INT) v = gel(v,1);
2350 490 : return v;
2351 : }
2352 : GEN
2353 728 : nfeltembed(GEN nf, GEN x, GEN ind0, long prec0)
2354 : {
2355 728 : pari_sp av = avma; nf = checknf(nf);
2356 728 : return gerepilecopy(av, nfeltembed_i(&nf, x, ind0, prec0));
2357 : }
2358 :
2359 : /* number of distinct roots of sigma(f) */
2360 : GEN
2361 1596 : nfpolsturm(GEN nf, GEN f, GEN ind0)
2362 : {
2363 1596 : pari_sp av = avma;
2364 : long d, l, r1, single;
2365 : GEN ind, u, v, vr1, T, s, t;
2366 :
2367 1596 : nf = checknf(nf); T = nf_get_pol(nf); r1 = nf_get_r1(nf);
2368 1596 : ind = parse_embed(ind0, r1, "nfpolsturm");
2369 1575 : single = ind0 && typ(ind0) == t_INT;
2370 1575 : l = lg(ind);
2371 :
2372 1575 : if (gequal0(f)) pari_err_ROOTS0("nfpolsturm");
2373 1568 : if (typ(f) == t_POL && varn(f) != varn(T))
2374 : {
2375 1547 : f = RgX_nffix("nfpolsturm", T, f,1);
2376 1547 : if (lg(f) == 3) f = NULL;
2377 : }
2378 : else
2379 : {
2380 21 : (void)Rg_nffix("nfpolsturm", T, f, 0);
2381 21 : f = NULL;
2382 : }
2383 1568 : if (!f) { set_avma(av); return single? gen_0: zerovec(l-1); }
2384 1547 : d = degpol(f);
2385 1547 : if (d == 1) { set_avma(av); return single? gen_1: const_vec(l-1,gen_1); }
2386 :
2387 1505 : vr1 = const_vecsmall(l-1, 1);
2388 1505 : u = Q_primpart(f); s = ZV_to_zv(nfeltsign(nf, gel(u,d+2), ind));
2389 1505 : v = RgX_deriv(u); t = odd(d)? leafcopy(s): zv_neg(s);
2390 : for(;;)
2391 182 : {
2392 1687 : GEN r = RgX_neg( Q_primpart(RgX_pseudorem(u, v)) ), sr;
2393 1687 : long i, dr = degpol(r);
2394 1687 : if (dr < 0) break;
2395 1687 : sr = ZV_to_zv(nfeltsign(nf, gel(r,dr+2), ind));
2396 4144 : for (i = 1; i < l; i++)
2397 2457 : if (sr[i] != s[i]) { s[i] = sr[i], vr1[i]--; }
2398 1687 : if (odd(dr)) sr = zv_neg(sr);
2399 4144 : for (i = 1; i < l; i++)
2400 2457 : if (sr[i] != t[i]) { t[i] = sr[i], vr1[i]++; }
2401 1687 : if (!dr) break;
2402 182 : u = v; v = r;
2403 : }
2404 1505 : if (single) return gc_stoi(av,vr1[1]);
2405 1498 : return gerepileupto(av, zv_to_ZV(vr1));
2406 : }
2407 :
2408 : /* True nf; return the vector of signs of x; the matrix of such if x is a vector
2409 : * of nf elements */
2410 : GEN
2411 43958 : nfsign(GEN nf, GEN x)
2412 : {
2413 : long i, l;
2414 : GEN archp, S;
2415 :
2416 43958 : archp = identity_perm( nf_get_r1(nf) );
2417 43958 : if (typ(x) != t_VEC) return nfsign_arch(nf, x, archp);
2418 35936 : l = lg(x); S = cgetg(l, t_MAT);
2419 148053 : for (i=1; i<l; i++) gel(S,i) = nfsign_arch(nf, gel(x,i), archp);
2420 35937 : return S;
2421 : }
2422 :
2423 : /* x integral elt, A integral ideal in HNF; reduce x mod A */
2424 : static GEN
2425 7784519 : zk_modHNF(GEN x, GEN A)
2426 7784519 : { return (typ(x) == t_COL)? ZC_hnfrem(x, A): modii(x, gcoeff(A,1,1)); }
2427 :
2428 : /* given an element x in Z_K and an integral ideal y in HNF, coprime with x,
2429 : outputs an element inverse of x modulo y */
2430 : GEN
2431 189 : nfinvmodideal(GEN nf, GEN x, GEN y)
2432 : {
2433 189 : pari_sp av = avma;
2434 189 : GEN a, yZ = gcoeff(y,1,1);
2435 :
2436 189 : if (equali1(yZ)) return gen_0;
2437 189 : x = nf_to_scalar_or_basis(nf, x);
2438 189 : if (typ(x) == t_INT) return gerepileupto(av, Fp_inv(x, yZ));
2439 :
2440 79 : a = hnfmerge_get_1(idealhnf_principal(nf,x), y);
2441 79 : if (!a) pari_err_INV("nfinvmodideal", x);
2442 79 : return gerepileupto(av, zk_modHNF(nfdiv(nf,a,x), y));
2443 : }
2444 :
2445 : static GEN
2446 2685267 : nfsqrmodideal(GEN nf, GEN x, GEN id)
2447 2685267 : { return zk_modHNF(nfsqri(nf,x), id); }
2448 : static GEN
2449 7266748 : nfmulmodideal(GEN nf, GEN x, GEN y, GEN id)
2450 7266748 : { return x? zk_modHNF(nfmuli(nf,x,y), id): y; }
2451 : /* assume x integral, k integer, A in HNF */
2452 : GEN
2453 5823179 : nfpowmodideal(GEN nf,GEN x,GEN k,GEN A)
2454 : {
2455 5823179 : long s = signe(k);
2456 : pari_sp av;
2457 : GEN y;
2458 :
2459 5823179 : if (!s) return gen_1;
2460 5823179 : av = avma;
2461 5823179 : x = nf_to_scalar_or_basis(nf, x);
2462 5823399 : if (typ(x) != t_COL) return Fp_pow(x, k, gcoeff(A,1,1));
2463 2619345 : if (s < 0) { k = negi(k); x = nfinvmodideal(nf, x,A); }
2464 2619345 : if (equali1(k)) return gerepileupto(av, s > 0? zk_modHNF(x, A): x);
2465 1148701 : for(y = NULL;;)
2466 : {
2467 3833957 : if (mpodd(k)) y = nfmulmodideal(nf,y,x,A);
2468 3833944 : k = shifti(k,-1); if (!signe(k)) break;
2469 2684940 : x = nfsqrmodideal(nf,x,A);
2470 : }
2471 1148694 : return gerepileupto(av, y);
2472 : }
2473 :
2474 : /* a * g^n mod id */
2475 : static GEN
2476 4672821 : nfmulpowmodideal(GEN nf, GEN a, GEN g, GEN n, GEN id)
2477 : {
2478 4672821 : return nfmulmodideal(nf, a, nfpowmodideal(nf,g,n,id), id);
2479 : }
2480 :
2481 : /* assume (num(g[i]), id) = 1 for all i. Return prod g[i]^e[i] mod id.
2482 : * EX = multiple of exponent of (O_K/id)^* */
2483 : GEN
2484 2621567 : famat_to_nf_modideal_coprime(GEN nf, GEN g, GEN e, GEN id, GEN EX)
2485 : {
2486 2621567 : GEN EXo2, plus = NULL, minus = NULL, idZ = gcoeff(id,1,1);
2487 2621567 : long i, lx = lg(g);
2488 :
2489 2621567 : if (equali1(idZ)) return gen_1; /* id = Z_K */
2490 2621079 : EXo2 = (expi(EX) > 10)? shifti(EX,-1): NULL;
2491 8283468 : for (i = 1; i < lx; i++)
2492 : {
2493 5662451 : GEN h, n = centermodii(gel(e,i), EX, EXo2);
2494 5662021 : long sn = signe(n);
2495 5662021 : if (!sn) continue;
2496 :
2497 4021345 : h = nf_to_scalar_or_basis(nf, gel(g,i));
2498 4021698 : switch(typ(h))
2499 : {
2500 2371174 : case t_INT: break;
2501 0 : case t_FRAC:
2502 0 : h = Fp_div(gel(h,1), gel(h,2), idZ); break;
2503 1650524 : default:
2504 : {
2505 : GEN dh;
2506 1650524 : h = Q_remove_denom(h, &dh);
2507 1650672 : if (dh) h = FpC_Fp_mul(h, Fp_inv(dh,idZ), idZ);
2508 : }
2509 : }
2510 4021774 : if (sn > 0)
2511 4019883 : plus = nfmulpowmodideal(nf, plus, h, n, id);
2512 : else /* sn < 0 */
2513 1891 : minus = nfmulpowmodideal(nf, minus, h, negi(n), id);
2514 : }
2515 2621017 : if (minus) plus = nfmulmodideal(nf, plus, nfinvmodideal(nf,minus,id), id);
2516 2621100 : return plus? plus: gen_1;
2517 : }
2518 :
2519 : /* given 2 integral ideals x, y in HNF s.t x | y | x^2, compute (1+x)/(1+y) in
2520 : * the form [[cyc],[gen], U], where U := ux^-1 as a pair [ZM, denom(U)] */
2521 : static GEN
2522 236892 : zidealij(GEN x, GEN y)
2523 : {
2524 236892 : GEN U, G, cyc, xp = gcoeff(x,1,1), xi = hnf_invscale(x, xp);
2525 : long j, N;
2526 :
2527 : /* x^(-1) y = relations between the 1 + x_i (HNF) */
2528 236884 : cyc = ZM_snf_group(ZM_Z_divexact(ZM_mul(xi, y), xp), &U, &G);
2529 236885 : N = lg(cyc); G = ZM_mul(x,G); settyp(G, t_VEC); /* new generators */
2530 573765 : for (j=1; j<N; j++)
2531 : {
2532 336896 : GEN c = gel(G,j);
2533 336896 : gel(c,1) = addiu(gel(c,1), 1); /* 1 + g_j */
2534 336889 : if (ZV_isscalar(c)) gel(G,j) = gel(c,1);
2535 : }
2536 236869 : return mkvec4(cyc, G, ZM_mul(U,xi), xp);
2537 : }
2538 :
2539 : /* lg(x) > 1, x + 1; shallow */
2540 : static GEN
2541 169708 : ZC_add1(GEN x)
2542 : {
2543 169708 : long i, l = lg(x);
2544 169708 : GEN y = cgetg(l, t_COL);
2545 396246 : for (i = 2; i < l; i++) gel(y,i) = gel(x,i);
2546 169714 : gel(y,1) = addiu(gel(x,1), 1); return y;
2547 : }
2548 : /* lg(x) > 1, x - 1; shallow */
2549 : static GEN
2550 70496 : ZC_sub1(GEN x)
2551 : {
2552 70496 : long i, l = lg(x);
2553 70496 : GEN y = cgetg(l, t_COL);
2554 176937 : for (i = 2; i < l; i++) gel(y,i) = gel(x,i);
2555 70496 : gel(y,1) = subiu(gel(x,1), 1); return y;
2556 : }
2557 :
2558 : /* x,y are t_INT or ZC */
2559 : static GEN
2560 0 : zkadd(GEN x, GEN y)
2561 : {
2562 0 : long tx = typ(x);
2563 0 : if (tx == typ(y))
2564 0 : return tx == t_INT? addii(x,y): ZC_add(x,y);
2565 : else
2566 0 : return tx == t_INT? ZC_Z_add(y,x): ZC_Z_add(x,y);
2567 : }
2568 : /* x a t_INT or ZC, x+1; shallow */
2569 : static GEN
2570 255396 : zkadd1(GEN x)
2571 : {
2572 255396 : long tx = typ(x);
2573 255396 : return tx == t_INT? addiu(x,1): ZC_add1(x);
2574 : }
2575 : /* x a t_INT or ZC, x-1; shallow */
2576 : static GEN
2577 255430 : zksub1(GEN x)
2578 : {
2579 255430 : long tx = typ(x);
2580 255430 : return tx == t_INT? subiu(x,1): ZC_sub1(x);
2581 : }
2582 : /* x,y are t_INT or ZC; x - y */
2583 : static GEN
2584 0 : zksub(GEN x, GEN y)
2585 : {
2586 0 : long tx = typ(x), ty = typ(y);
2587 0 : if (tx == ty)
2588 0 : return tx == t_INT? subii(x,y): ZC_sub(x,y);
2589 : else
2590 0 : return tx == t_INT? Z_ZC_sub(x,y): ZC_Z_sub(x,y);
2591 : }
2592 : /* x is t_INT or ZM (mult. map), y is t_INT or ZC; x * y */
2593 : static GEN
2594 255396 : zkmul(GEN x, GEN y)
2595 : {
2596 255396 : long tx = typ(x), ty = typ(y);
2597 255396 : if (ty == t_INT)
2598 184917 : return tx == t_INT? mulii(x,y): ZC_Z_mul(gel(x,1),y);
2599 : else
2600 70479 : return tx == t_INT? ZC_Z_mul(y,x): ZM_ZC_mul(x,y);
2601 : }
2602 :
2603 : /* (U,V) = 1 coprime ideals. Want z = x mod U, = y mod V; namely
2604 : * z =vx + uy = v(x-y) + y, where u + v = 1, u in U, v in V.
2605 : * zkc = [v, UV], v a t_INT or ZM (mult. by v map), UV a ZM (ideal in HNF);
2606 : * shallow */
2607 : GEN
2608 0 : zkchinese(GEN zkc, GEN x, GEN y)
2609 : {
2610 0 : GEN v = gel(zkc,1), UV = gel(zkc,2), z = zkadd(zkmul(v, zksub(x,y)), y);
2611 0 : return zk_modHNF(z, UV);
2612 : }
2613 : /* special case z = x mod U, = 1 mod V; shallow */
2614 : GEN
2615 255430 : zkchinese1(GEN zkc, GEN x)
2616 : {
2617 255430 : GEN v = gel(zkc,1), UV = gel(zkc,2), z = zkadd1(zkmul(v, zksub1(x)));
2618 255419 : return (typ(z) == t_INT)? z: ZC_hnfrem(z, UV);
2619 : }
2620 : static GEN
2621 237403 : zkVchinese1(GEN zkc, GEN v)
2622 : {
2623 : long i, ly;
2624 237403 : GEN y = cgetg_copy(v, &ly);
2625 492801 : for (i=1; i<ly; i++) gel(y,i) = zkchinese1(zkc, gel(v,i));
2626 237373 : return y;
2627 : }
2628 :
2629 : /* prepare to solve z = x (mod A), z = y mod (B) [zkchinese or zkchinese1] */
2630 : GEN
2631 237152 : zkchineseinit(GEN nf, GEN A, GEN B, GEN AB)
2632 : {
2633 237152 : GEN v = idealaddtoone_raw(nf, A, B);
2634 : long e;
2635 237152 : if ((e = gexpo(v)) > 5)
2636 : {
2637 83283 : GEN b = (typ(v) == t_COL)? v: scalarcol_shallow(v, nf_get_degree(nf));
2638 83283 : b= ZC_reducemodlll(b, AB);
2639 83285 : if (gexpo(b) < e) v = b;
2640 : }
2641 237153 : return mkvec2(zk_scalar_or_multable(nf,v), AB);
2642 : }
2643 : /* prepare to solve z = x (mod A), z = 1 mod (B)
2644 : * and then z = 1 (mod A), z = y mod (B) [zkchinese1 twice] */
2645 : static GEN
2646 259 : zkchinese1init2(GEN nf, GEN A, GEN B, GEN AB)
2647 : {
2648 259 : GEN zkc = zkchineseinit(nf, A, B, AB);
2649 259 : GEN mv = gel(zkc,1), mu;
2650 259 : if (typ(mv) == t_INT) return mkvec2(zkc, mkvec2(subui(1,mv),AB));
2651 35 : mu = RgM_Rg_add_shallow(ZM_neg(mv), gen_1);
2652 35 : return mkvec2(mkvec2(mv,AB), mkvec2(mu,AB));
2653 : }
2654 :
2655 : static GEN
2656 2151434 : apply_U(GEN L, GEN a)
2657 : {
2658 2151434 : GEN e, U = gel(L,3), dU = gel(L,4);
2659 2151434 : if (typ(a) == t_INT)
2660 672135 : e = ZC_Z_mul(gel(U,1), subiu(a, 1));
2661 : else
2662 : { /* t_COL */
2663 1479299 : GEN t = shallowcopy(a);
2664 1479330 : gel(t,1) = subiu(gel(t,1), 1); /* t = a - 1 */
2665 1479225 : e = ZM_ZC_mul(U, t);
2666 : }
2667 2151346 : return gdiv(e, dU);
2668 : }
2669 :
2670 : /* true nf; vectors of [[cyc],[g],U.X^-1]. Assume k > 1. */
2671 : static GEN
2672 169103 : principal_units(GEN nf, GEN pr, long k, GEN prk)
2673 : {
2674 : GEN list, prb;
2675 169103 : ulong mask = quadratic_prec_mask(k);
2676 169103 : long a = 1;
2677 :
2678 169103 : prb = pr_hnf(nf,pr);
2679 169105 : list = vectrunc_init(k);
2680 405995 : while (mask > 1)
2681 : {
2682 236897 : GEN pra = prb;
2683 236897 : long b = a << 1;
2684 :
2685 236897 : if (mask & 1) b--;
2686 236897 : mask >>= 1;
2687 : /* compute 1 + pr^a / 1 + pr^b, 2a <= b */
2688 236897 : prb = (b >= k)? prk: idealpows(nf,pr,b);
2689 236894 : vectrunc_append(list, zidealij(pra, prb));
2690 236892 : a = b;
2691 : }
2692 169098 : return list;
2693 : }
2694 : /* a = 1 mod (pr) return log(a) on local-gens of 1+pr/1+pr^k */
2695 : static GEN
2696 1329275 : log_prk1(GEN nf, GEN a, long nh, GEN L2, GEN prk)
2697 : {
2698 1329275 : GEN y = cgetg(nh+1, t_COL);
2699 1329278 : long j, iy, c = lg(L2)-1;
2700 3480659 : for (j = iy = 1; j <= c; j++)
2701 : {
2702 2151428 : GEN L = gel(L2,j), cyc = gel(L,1), gen = gel(L,2), E = apply_U(L,a);
2703 2151255 : long i, nc = lg(cyc)-1;
2704 2151255 : int last = (j == c);
2705 5814021 : for (i = 1; i <= nc; i++, iy++)
2706 : {
2707 3662640 : GEN t, e = gel(E,i);
2708 3662640 : if (typ(e) != t_INT) pari_err_COPRIME("zlog_prk1", a, prk);
2709 3662633 : t = Fp_neg(e, gel(cyc,i));
2710 3662658 : gel(y,iy) = negi(t);
2711 3662759 : if (!last && signe(t)) a = nfmulpowmodideal(nf, a, gel(gen,i), t, prk);
2712 : }
2713 : }
2714 1329231 : return y;
2715 : }
2716 : /* true nf */
2717 : static GEN
2718 56642 : principal_units_relations(GEN nf, GEN L2, GEN prk, long nh)
2719 : {
2720 56642 : GEN h = cgetg(nh+1,t_MAT);
2721 56642 : long ih, j, c = lg(L2)-1;
2722 181079 : for (j = ih = 1; j <= c; j++)
2723 : {
2724 124437 : GEN L = gel(L2,j), F = gel(L,1), G = gel(L,2);
2725 124437 : long k, lG = lg(G);
2726 304237 : for (k = 1; k < lG; k++,ih++)
2727 : { /* log(g^f) mod pr^e */
2728 179800 : GEN a = nfpowmodideal(nf,gel(G,k),gel(F,k),prk);
2729 179800 : gel(h,ih) = ZC_neg(log_prk1(nf, a, nh, L2, prk));
2730 179800 : gcoeff(h,ih,ih) = gel(F,k);
2731 : }
2732 : }
2733 56642 : return h;
2734 : }
2735 : /* true nf; k > 1; multiplicative group (1 + pr) / (1 + pr^k) */
2736 : static GEN
2737 169097 : idealprincipalunits_i(GEN nf, GEN pr, long k, GEN *pU)
2738 : {
2739 169097 : GEN cyc, gen, L2, prk = idealpows(nf, pr, k);
2740 :
2741 169103 : L2 = principal_units(nf, pr, k, prk);
2742 169103 : if (k == 2)
2743 : {
2744 112461 : GEN L = gel(L2,1);
2745 112461 : cyc = gel(L,1);
2746 112461 : gen = gel(L,2);
2747 112461 : if (pU) *pU = matid(lg(gen)-1);
2748 : }
2749 : else
2750 : {
2751 56642 : long c = lg(L2), j;
2752 56642 : GEN EX, h, Ui, vg = cgetg(c, t_VEC);
2753 181077 : for (j = 1; j < c; j++) gel(vg, j) = gmael(L2,j,2);
2754 56642 : vg = shallowconcat1(vg);
2755 56642 : h = principal_units_relations(nf, L2, prk, lg(vg)-1);
2756 56644 : h = ZM_hnfall_i(h, NULL, 0);
2757 56644 : cyc = ZM_snf_group(h, pU, &Ui);
2758 56644 : c = lg(Ui); gen = cgetg(c, t_VEC); EX = cyc_get_expo(cyc);
2759 188254 : for (j = 1; j < c; j++)
2760 131610 : gel(gen,j) = famat_to_nf_modideal_coprime(nf, vg, gel(Ui,j), prk, EX);
2761 : }
2762 169106 : return mkvec4(cyc, gen, prk, L2);
2763 : }
2764 : GEN
2765 154 : idealprincipalunits(GEN nf, GEN pr, long k)
2766 : {
2767 : pari_sp av;
2768 : GEN v;
2769 154 : nf = checknf(nf);
2770 154 : if (k == 1) { checkprid(pr); retmkvec3(gen_1,cgetg(1,t_VEC),cgetg(1,t_VEC)); }
2771 147 : av = avma; v = idealprincipalunits_i(nf, pr, k, NULL);
2772 147 : return gerepilecopy(av, mkvec3(powiu(pr_norm(pr), k-1), gel(v,1), gel(v,2)));
2773 : }
2774 :
2775 : /* true nf; given an ideal pr^k dividing an integral ideal x (in HNF form)
2776 : * compute an 'sprk', the structure of G = (Z_K/pr^k)^* [ x = NULL for x=pr^k ]
2777 : * Return a vector with at least 4 components [cyc],[gen],[HNF pr^k,pr,k],ff,
2778 : * where
2779 : * cyc : type of G as abelian group (SNF)
2780 : * gen : generators of G, coprime to x
2781 : * pr^k: in HNF
2782 : * ff : data for log_g in (Z_K/pr)^*
2783 : * Two extra components are present iff k > 1: L2, U
2784 : * L2 : list of data structures to compute local DL in (Z_K/pr)^*,
2785 : * and 1 + pr^a/ 1 + pr^b for various a < b <= min(2a, k)
2786 : * U : base change matrices to convert a vector of local DL to DL wrt gen
2787 : * If MOD is not NULL, initialize G / G^MOD instead */
2788 : static GEN
2789 425826 : sprkinit(GEN nf, GEN pr, long k, GEN x, GEN MOD)
2790 : {
2791 425826 : GEN T, p, Ld, modpr, cyc, gen, g, g0, A, prk, U, L2, ord0 = NULL;
2792 425826 : long f = pr_get_f(pr);
2793 :
2794 425824 : if(DEBUGLEVEL>3) err_printf("treating pr^%ld, pr = %Ps\n",k,pr);
2795 425824 : modpr = nf_to_Fq_init(nf, &pr,&T,&p);
2796 425868 : if (MOD)
2797 : {
2798 378324 : GEN o = subiu(powiu(p,f), 1), d = gcdii(o, MOD), fa = Z_factor(d);
2799 378285 : ord0 = mkvec2(o, fa); /* true order, factorization of order in G/G^MOD */
2800 378286 : Ld = gel(fa,1);
2801 378286 : if (lg(Ld) > 1 && equaliu(gel(Ld,1),2)) Ld = vecslice(Ld,2,lg(Ld)-1);
2802 : }
2803 : /* (Z_K / pr)^* */
2804 425838 : if (f == 1)
2805 : {
2806 336707 : g0 = g = MOD? pgener_Fp_local(p, Ld): pgener_Fp(p);
2807 336722 : if (!ord0) ord0 = get_arith_ZZM(subiu(p,1));
2808 : }
2809 : else
2810 : {
2811 89131 : g0 = g = MOD? gener_FpXQ_local(T, p, Ld): gener_FpXQ(T,p, &ord0);
2812 89130 : g = Fq_to_nf(g, modpr);
2813 89130 : if (typ(g) == t_POL) g = poltobasis(nf, g);
2814 : }
2815 425851 : A = gel(ord0, 1); /* Norm(pr)-1 */
2816 : /* If MOD != NULL, d = gcd(A, MOD): g^(A/d) has order d */
2817 425851 : if (k == 1)
2818 : {
2819 256901 : cyc = mkvec(A);
2820 256900 : gen = mkvec(g);
2821 256895 : prk = pr_hnf(nf,pr);
2822 256912 : L2 = U = NULL;
2823 : }
2824 : else
2825 : { /* local-gens of (1 + pr)/(1 + pr^k) = SNF-gens * U */
2826 : GEN AB, B, u, v, w;
2827 : long j, l;
2828 168950 : w = idealprincipalunits_i(nf, pr, k, &U);
2829 : /* incorporate (Z_K/pr)^*, order A coprime to B = expo(1+pr/1+pr^k)*/
2830 168958 : cyc = leafcopy(gel(w,1)); B = cyc_get_expo(cyc); AB = mulii(A,B);
2831 168947 : gen = leafcopy(gel(w,2));
2832 168945 : prk = gel(w,3);
2833 168945 : g = nfpowmodideal(nf, g, B, prk);
2834 168958 : g0 = Fq_pow(g0, modii(B,A), T, p); /* update primitive root */
2835 168948 : L2 = mkvec3(A, g, gel(w,4));
2836 168950 : gel(cyc,1) = AB;
2837 168950 : gel(gen,1) = nfmulmodideal(nf, gel(gen,1), g, prk);
2838 168947 : u = mulii(Fp_inv(A,B), A);
2839 168943 : v = subui(1, u); l = lg(U);
2840 505388 : for (j = 1; j < l; j++) gcoeff(U,1,j) = Fp_mul(u, gcoeff(U,1,j), AB);
2841 168937 : U = mkvec2(Rg_col_ei(v, lg(gen)-1, 1), U);
2842 : }
2843 : /* local-gens of (Z_K/pr^k)^* = SNF-gens * U */
2844 425862 : if (x)
2845 : {
2846 236904 : GEN uv = zkchineseinit(nf, idealmulpowprime(nf,x,pr,utoineg(k)), prk, x);
2847 236886 : gen = zkVchinese1(uv, gen);
2848 : }
2849 425813 : return mkvecn(U? 6: 4, cyc, gen, prk, mkvec3(modpr,g0,ord0), L2, U);
2850 : }
2851 : GEN
2852 3981640 : sprk_get_cyc(GEN s) { return gel(s,1); }
2853 : GEN
2854 1969596 : sprk_get_expo(GEN s) { return cyc_get_expo(sprk_get_cyc(s)); }
2855 : GEN
2856 335784 : sprk_get_gen(GEN s) { return gel(s,2); }
2857 : GEN
2858 4915569 : sprk_get_prk(GEN s) { return gel(s,3); }
2859 : GEN
2860 2543221 : sprk_get_ff(GEN s) { return gel(s,4); }
2861 : GEN
2862 2603902 : sprk_get_pr(GEN s) { GEN ff = gel(s,4); return modpr_get_pr(gel(ff,1)); }
2863 : /* L2 to 1 + pr / 1 + pr^k */
2864 : static GEN
2865 1211730 : sprk_get_L2(GEN s) { return gmael(s,5,3); }
2866 : /* lift to nf of primitive root of k(pr) */
2867 : static GEN
2868 318598 : sprk_get_gnf(GEN s) { return gmael(s,5,2); }
2869 : /* A = Npr-1, <g> = (Z_K/pr)^*, L2 to 1 + pr / 1 + pr^k */
2870 : void
2871 0 : sprk_get_AgL2(GEN s, GEN *A, GEN *g, GEN *L2)
2872 0 : { GEN v = gel(s,5); *A = gel(v,1); *g = gel(v,2); *L2 = gel(v,3); }
2873 : void
2874 1203070 : sprk_get_U2(GEN s, GEN *U1, GEN *U2)
2875 1203070 : { GEN v = gel(s,6); *U1 = gel(v,1); *U2 = gel(v,2); }
2876 : static int
2877 2543218 : sprk_is_prime(GEN s) { return lg(s) == 5; }
2878 :
2879 : GEN
2880 1969400 : famat_zlog_pr(GEN nf, GEN g, GEN e, GEN sprk, GEN mod)
2881 : {
2882 1969400 : GEN x, expo = sprk_get_expo(sprk);
2883 1969399 : if (mod) expo = gcdii(expo,mod);
2884 1969390 : x = famat_makecoprime(nf, g, e, sprk_get_pr(sprk), sprk_get_prk(sprk), expo);
2885 1969402 : return log_prk(nf, x, sprk, mod);
2886 : }
2887 : /* famat_zlog_pr assuming (g,sprk.pr) = 1 */
2888 : static GEN
2889 196 : famat_zlog_pr_coprime(GEN nf, GEN g, GEN e, GEN sprk, GEN MOD)
2890 : {
2891 196 : GEN x = famat_to_nf_modideal_coprime(nf, g, e, sprk_get_prk(sprk),
2892 : sprk_get_expo(sprk));
2893 196 : return log_prk(nf, x, sprk, MOD);
2894 : }
2895 :
2896 : /* o t_INT, O = [ord,fa] format for multiple of o (for Fq_log);
2897 : * return o in [ord,fa] format */
2898 : static GEN
2899 560572 : order_update(GEN o, GEN O)
2900 : {
2901 560572 : GEN p = gmael(O,2,1), z = o, P, E;
2902 560572 : long i, j, l = lg(p);
2903 560572 : P = cgetg(l, t_COL);
2904 560573 : E = cgetg(l, t_COL);
2905 619619 : for (i = j = 1; i < l; i++)
2906 : {
2907 619619 : long v = Z_pvalrem(z, gel(p,i), &z);
2908 619571 : if (v)
2909 : {
2910 604630 : gel(P,j) = gel(p,i);
2911 604630 : gel(E,j) = utoipos(v); j++;
2912 604640 : if (is_pm1(z)) break;
2913 : }
2914 : }
2915 560527 : setlg(P, j);
2916 560525 : setlg(E, j); return mkvec2(o, mkmat2(P,E));
2917 : }
2918 :
2919 : /* a in Z_K (t_COL or t_INT), pr prime ideal, sprk = sprkinit(nf,pr,k,x),
2920 : * mod positive t_INT or NULL (meaning mod=0).
2921 : * return log(a) modulo mod on SNF-generators of (Z_K/pr^k)^* */
2922 : GEN
2923 2617007 : log_prk(GEN nf, GEN a, GEN sprk, GEN mod)
2924 : {
2925 : GEN e, prk, g, U1, U2, y, ff, O, o, oN, gN, N, T, p, modpr, pr, cyc;
2926 :
2927 2617007 : if (typ(a) == t_MAT) return famat_zlog_pr(nf, gel(a,1), gel(a,2), sprk, mod);
2928 2543214 : N = NULL;
2929 2543214 : ff = sprk_get_ff(sprk);
2930 2543218 : pr = gel(ff,1); /* modpr */
2931 2543218 : g = gN = gel(ff,2);
2932 2543218 : O = gel(ff,3); /* order of g = |Fq^*|, in [ord, fa] format */
2933 2543218 : o = oN = gel(O,1); /* order as a t_INT */
2934 2543218 : prk = sprk_get_prk(sprk);
2935 2543226 : modpr = nf_to_Fq_init(nf, &pr, &T, &p);
2936 2543245 : if (mod)
2937 : {
2938 2026890 : GEN d = gcdii(o,mod);
2939 2026631 : if (!equalii(o, d))
2940 : {
2941 750751 : N = diviiexact(o,d); /* > 1, coprime to p */
2942 750694 : a = nfpowmodideal(nf, a, N, prk);
2943 750898 : oN = d; /* order of g^N mod pr */
2944 : }
2945 : }
2946 2543090 : if (equali1(oN))
2947 397516 : e = gen_0;
2948 : else
2949 : {
2950 2145649 : if (N) { O = order_update(oN, O); gN = Fq_pow(g, N, T, p); }
2951 2145642 : e = Fq_log(nf_to_Fq(nf,a,modpr), gN, O, T, p);
2952 : }
2953 : /* 0 <= e < oN is correct modulo oN */
2954 2543231 : if (sprk_is_prime(sprk)) return mkcol(e); /* k = 1 */
2955 :
2956 800294 : sprk_get_U2(sprk, &U1,&U2);
2957 800390 : cyc = sprk_get_cyc(sprk);
2958 800391 : if (mod)
2959 : {
2960 379036 : cyc = ZV_snf_gcd(cyc, mod);
2961 379015 : if (signe(remii(mod,p))) return ZV_ZV_mod(ZC_Z_mul(U1,e), cyc);
2962 : }
2963 746762 : if (signe(e))
2964 : {
2965 318598 : GEN E = N? mulii(e, N): e;
2966 318598 : a = nfmulpowmodideal(nf, a, sprk_get_gnf(sprk), Fp_neg(E, o), prk);
2967 : }
2968 : /* a = 1 mod pr */
2969 746762 : y = log_prk1(nf, a, lg(U2)-1, sprk_get_L2(sprk), prk);
2970 746800 : if (N)
2971 : { /* from DL(a^N) to DL(a) */
2972 135236 : GEN E = gel(sprk_get_cyc(sprk), 1), q = powiu(p, Z_pval(E, p));
2973 135234 : y = ZC_Z_mul(y, Fp_inv(N, q));
2974 : }
2975 746799 : y = ZC_lincomb(gen_1, e, ZM_ZC_mul(U2,y), U1);
2976 746792 : return ZV_ZV_mod(y, cyc);
2977 : }
2978 : /* true nf */
2979 : GEN
2980 90118 : log_prk_init(GEN nf, GEN pr, long k, GEN MOD)
2981 90118 : { return sprkinit(nf,pr,k,NULL,MOD);}
2982 : GEN
2983 497 : veclog_prk(GEN nf, GEN v, GEN sprk)
2984 : {
2985 497 : long l = lg(v), i;
2986 497 : GEN w = cgetg(l, t_MAT);
2987 1232 : for (i = 1; i < l; i++) gel(w,i) = log_prk(nf, gel(v,i), sprk, NULL);
2988 497 : return w;
2989 : }
2990 :
2991 : static GEN
2992 1374215 : famat_zlog(GEN nf, GEN fa, GEN sgn, zlog_S *S)
2993 : {
2994 1374215 : long i, l0, l = lg(S->U);
2995 1374215 : GEN g = gel(fa,1), e = gel(fa,2), y = cgetg(l, t_COL);
2996 1374216 : l0 = lg(S->sprk); /* = l (trivial arch. part), or l-1 */
2997 2852344 : for (i=1; i < l0; i++) gel(y,i) = famat_zlog_pr(nf, g, e, gel(S->sprk,i), S->mod);
2998 1374216 : if (l0 != l)
2999 : {
3000 190755 : if (!sgn) sgn = nfsign_arch(nf, fa, S->archp);
3001 190755 : gel(y,l0) = Flc_to_ZC(sgn);
3002 : }
3003 1374216 : return y;
3004 : }
3005 :
3006 : /* assume that cyclic factors are normalized, in particular != [1] */
3007 : static GEN
3008 257418 : split_U(GEN U, GEN Sprk)
3009 : {
3010 257418 : long t = 0, k, n, l = lg(Sprk);
3011 257418 : GEN vU = cgetg(l+1, t_VEC);
3012 592441 : for (k = 1; k < l; k++)
3013 : {
3014 335023 : n = lg(sprk_get_cyc(gel(Sprk,k))) - 1; /* > 0 */
3015 335024 : gel(vU,k) = vecslice(U, t+1, t+n);
3016 335028 : t += n;
3017 : }
3018 : /* t+1 .. lg(U)-1 */
3019 257418 : n = lg(U) - t - 1; /* can be 0 */
3020 257418 : if (!n) setlg(vU,l); else gel(vU,l) = vecslice(U, t+1, t+n);
3021 257418 : return vU;
3022 : }
3023 :
3024 : static void
3025 1990534 : init_zlog_mod(zlog_S *S, GEN bid, GEN mod)
3026 : {
3027 1990534 : GEN fa2 = bid_get_fact2(bid), MOD = bid_get_MOD(bid);
3028 1990525 : S->U = bid_get_U(bid);
3029 1990525 : S->hU = lg(bid_get_cyc(bid))-1;
3030 1990515 : S->archp = bid_get_archp(bid);
3031 1990514 : S->sprk = bid_get_sprk(bid);
3032 1990514 : S->bid = bid;
3033 1990514 : if (MOD) mod = mod? gcdii(mod, MOD): MOD;
3034 1990428 : S->mod = mod;
3035 1990428 : S->P = gel(fa2,1);
3036 1990428 : S->k = gel(fa2,2);
3037 1990428 : S->no2 = lg(S->P) == lg(gel(bid_get_fact(bid),1));
3038 1990453 : }
3039 : void
3040 380059 : init_zlog(zlog_S *S, GEN bid)
3041 : {
3042 380059 : return init_zlog_mod(S, bid, NULL);
3043 : }
3044 :
3045 : /* a a t_FRAC/t_INT, reduce mod bid */
3046 : static GEN
3047 14 : Q_mod_bid(GEN bid, GEN a)
3048 : {
3049 14 : GEN xZ = gcoeff(bid_get_ideal(bid),1,1);
3050 14 : GEN b = Rg_to_Fp(a, xZ);
3051 14 : if (gsigne(a) < 0) b = subii(b, xZ);
3052 14 : return signe(b)? b: xZ;
3053 : }
3054 : /* Return decomposition of a on the CRT generators blocks attached to the
3055 : * S->sprk and sarch; sgn = sign(a, S->arch), NULL if unknown */
3056 : static GEN
3057 381282 : zlog(GEN nf, GEN a, GEN sgn, zlog_S *S)
3058 : {
3059 : long k, l;
3060 : GEN y;
3061 381282 : a = nf_to_scalar_or_basis(nf, a);
3062 381255 : switch(typ(a))
3063 : {
3064 162481 : case t_INT: break;
3065 14 : case t_FRAC: a = Q_mod_bid(S->bid, a); break;
3066 218760 : default: /* case t_COL: */
3067 : {
3068 : GEN den;
3069 218760 : check_nfelt(a, &den);
3070 218791 : if (den)
3071 : {
3072 107 : a = Q_muli_to_int(a, den);
3073 105 : a = mkmat2(mkcol2(a, den), mkcol2(gen_1, gen_m1));
3074 105 : return famat_zlog(nf, a, sgn, S);
3075 : }
3076 : }
3077 : }
3078 381158 : if (sgn)
3079 374200 : sgn = (lg(sgn) == 1)? NULL: leafcopy(sgn);
3080 : else
3081 6958 : sgn = (lg(S->archp) == 1)? NULL: nfsign_arch(nf, a, S->archp);
3082 381164 : l = lg(S->sprk);
3083 381164 : y = cgetg(sgn? l+1: l, t_COL);
3084 922256 : for (k = 1; k < l; k++)
3085 : {
3086 541138 : GEN sprk = gel(S->sprk,k);
3087 541138 : gel(y,k) = log_prk(nf, a, sprk, S->mod);
3088 : }
3089 381118 : if (sgn) gel(y,l) = Flc_to_ZC(sgn);
3090 381130 : return y;
3091 : }
3092 :
3093 : /* true nf */
3094 : GEN
3095 43813 : pr_basis_perm(GEN nf, GEN pr)
3096 : {
3097 43813 : long f = pr_get_f(pr);
3098 : GEN perm;
3099 43813 : if (f == nf_get_degree(nf)) return identity_perm(f);
3100 38164 : perm = cgetg(f+1, t_VECSMALL);
3101 38163 : perm[1] = 1;
3102 38163 : if (f > 1)
3103 : {
3104 2912 : GEN H = pr_hnf(nf,pr);
3105 : long i, k;
3106 10808 : for (i = k = 2; k <= f; i++)
3107 7896 : if (!equali1(gcoeff(H,i,i))) perm[k++] = i;
3108 : }
3109 38163 : return perm;
3110 : }
3111 :
3112 : /* \sum U[i]*y[i], U[i] ZM, y[i] ZC. We allow lg(y) > lg(U). */
3113 : static GEN
3114 1755443 : ZMV_ZCV_mul(GEN U, GEN y)
3115 : {
3116 1755443 : long i, l = lg(U);
3117 1755443 : GEN z = NULL;
3118 1755443 : if (l == 1) return cgetg(1,t_COL);
3119 4139617 : for (i = 1; i < l; i++)
3120 : {
3121 2384227 : GEN u = ZM_ZC_mul(gel(U,i), gel(y,i));
3122 2384207 : z = z? ZC_add(z, u): u;
3123 : }
3124 1755390 : return z;
3125 : }
3126 : /* A * (U[1], ..., U[d] */
3127 : static GEN
3128 518 : ZM_ZMV_mul(GEN A, GEN U)
3129 : {
3130 : long i, l;
3131 518 : GEN V = cgetg_copy(U,&l);
3132 1057 : for (i = 1; i < l; i++) gel(V,i) = ZM_mul(A,gel(U,i));
3133 518 : return V;
3134 : }
3135 :
3136 : /* a = 1 mod pr, sprk mod pr^e, e >= 1 */
3137 : static GEN
3138 402698 : sprk_log_prk1_2(GEN nf, GEN a, GEN sprk)
3139 : {
3140 402698 : GEN U1, U2, y, L2 = sprk_get_L2(sprk);
3141 402698 : sprk_get_U2(sprk, &U1,&U2);
3142 402698 : y = ZM_ZC_mul(U2, log_prk1(nf, a, lg(U2)-1, L2, sprk_get_prk(sprk)));
3143 402686 : return ZV_ZV_mod(y, sprk_get_cyc(sprk));
3144 : }
3145 : /* true nf; assume e >= 2 */
3146 : GEN
3147 105835 : sprk_log_gen_pr2(GEN nf, GEN sprk, long e)
3148 : {
3149 105835 : GEN M, G, pr = sprk_get_pr(sprk);
3150 : long i, l;
3151 105835 : if (e == 2)
3152 : {
3153 62277 : GEN L2 = sprk_get_L2(sprk), L = gel(L2,1);
3154 62277 : G = gel(L,2); l = lg(G);
3155 : }
3156 : else
3157 : {
3158 43558 : GEN perm = pr_basis_perm(nf,pr), PI = nfpow_u(nf, pr_get_gen(pr), e-1);
3159 43561 : l = lg(perm);
3160 43561 : G = cgetg(l, t_VEC);
3161 43561 : if (typ(PI) == t_INT)
3162 : { /* zk_ei_mul doesn't allow t_INT */
3163 5642 : long N = nf_get_degree(nf);
3164 5642 : gel(G,1) = addiu(PI,1);
3165 8645 : for (i = 2; i < l; i++)
3166 : {
3167 3003 : GEN z = col_ei(N, 1);
3168 3003 : gel(G,i) = z; gel(z, perm[i]) = PI;
3169 : }
3170 : }
3171 : else
3172 : {
3173 37919 : gel(G,1) = nfadd(nf, gen_1, PI);
3174 44702 : for (i = 2; i < l; i++)
3175 6783 : gel(G,i) = nfadd(nf, gen_1, zk_ei_mul(nf, PI, perm[i]));
3176 : }
3177 : }
3178 105838 : M = cgetg(l, t_MAT);
3179 234353 : for (i = 1; i < l; i++) gel(M,i) = sprk_log_prk1_2(nf, gel(G,i), sprk);
3180 105830 : return M;
3181 : }
3182 : /* Log on bid.gen of generators of P_{1,I pr^{e-1}} / P_{1,I pr^e} (I,pr) = 1,
3183 : * defined implicitly via CRT. 'ind' is the index of pr in modulus
3184 : * factorization; true nf */
3185 : GEN
3186 413801 : log_gen_pr(zlog_S *S, long ind, GEN nf, long e)
3187 : {
3188 413801 : GEN Uind = gel(S->U, ind);
3189 413801 : if (e == 1) retmkmat( gel(Uind,1) );
3190 103128 : return ZM_mul(Uind, sprk_log_gen_pr2(nf, gel(S->sprk,ind), e));
3191 : }
3192 : /* true nf */
3193 : GEN
3194 2037 : sprk_log_gen_pr(GEN nf, GEN sprk, long e)
3195 : {
3196 2037 : if (e == 1)
3197 : {
3198 0 : long n = lg(sprk_get_cyc(sprk))-1;
3199 0 : retmkmat(col_ei(n, 1));
3200 : }
3201 2037 : return sprk_log_gen_pr2(nf, sprk, e);
3202 : }
3203 : /* a = 1 mod pr */
3204 : GEN
3205 274175 : sprk_log_prk1(GEN nf, GEN a, GEN sprk)
3206 : {
3207 274175 : if (lg(sprk) == 5) return mkcol(gen_0); /* mod pr */
3208 274175 : return sprk_log_prk1_2(nf, a, sprk);
3209 : }
3210 : /* Log on bid.gen of generator of P_{1,f} / P_{1,f v[index]}
3211 : * v = vector of r1 real places */
3212 : GEN
3213 86253 : log_gen_arch(zlog_S *S, long index) { return gel(veclast(S->U), index); }
3214 :
3215 : /* compute bid.clgp: [h,cyc] or [h,cyc,gen] */
3216 : static GEN
3217 258440 : bid_grp(GEN nf, GEN U, GEN cyc, GEN g, GEN F, GEN sarch)
3218 : {
3219 258440 : GEN G, h = ZV_prod(cyc);
3220 : long c;
3221 258445 : if (!U) return mkvec2(h,cyc);
3222 258088 : c = lg(U);
3223 258088 : G = cgetg(c,t_VEC);
3224 258093 : if (c > 1)
3225 : {
3226 228018 : GEN U0, Uoo, EX = cyc_get_expo(cyc); /* exponent of bid */
3227 228017 : long i, hU = nbrows(U), nba = lg(sarch_get_cyc(sarch))-1; /* #f_oo */
3228 228028 : if (!nba) { U0 = U; Uoo = NULL; }
3229 80417 : else if (nba == hU) { U0 = NULL; Uoo = U; }
3230 : else
3231 : {
3232 71275 : U0 = rowslice(U, 1, hU-nba);
3233 71278 : Uoo = rowslice(U, hU-nba+1, hU);
3234 : }
3235 695425 : for (i = 1; i < c; i++)
3236 : {
3237 467395 : GEN t = gen_1;
3238 467395 : if (U0) t = famat_to_nf_modideal_coprime(nf, g, gel(U0,i), F, EX);
3239 467398 : if (Uoo) t = set_sign_mod_divisor(nf, ZV_to_Flv(gel(Uoo,i),2), t, sarch);
3240 467400 : gel(G,i) = t;
3241 : }
3242 : }
3243 258105 : return mkvec3(h, cyc, G);
3244 : }
3245 :
3246 : /* remove prime ideals of norm 2 with exponent 1 from factorization */
3247 : static GEN
3248 258801 : famat_strip2(GEN fa)
3249 : {
3250 258801 : GEN P = gel(fa,1), E = gel(fa,2), Q, F;
3251 258801 : long l = lg(P), i, j;
3252 258801 : Q = cgetg(l, t_COL);
3253 258795 : F = cgetg(l, t_COL);
3254 633855 : for (i = j = 1; i < l; i++)
3255 : {
3256 375059 : GEN pr = gel(P,i), e = gel(E,i);
3257 375059 : if (!absequaliu(pr_get_p(pr), 2) || itou(e) != 1 || pr_get_f(pr) != 1)
3258 : {
3259 336417 : gel(Q,j) = pr;
3260 336417 : gel(F,j) = e; j++;
3261 : }
3262 : }
3263 258796 : setlg(Q,j);
3264 258795 : setlg(F,j); return mkmat2(Q,F);
3265 : }
3266 : static int
3267 134091 : checkarchp(GEN v, long r1)
3268 : {
3269 134091 : long i, l = lg(v);
3270 134091 : pari_sp av = avma;
3271 : GEN p;
3272 134091 : if (l == 1) return 1;
3273 47158 : if (l == 2) return v[1] > 0 && v[1] <= r1;
3274 22021 : p = zero_zv(r1);
3275 66150 : for (i = 1; i < l; i++)
3276 : {
3277 44163 : long j = v[i];
3278 44163 : if (j <= 0 || j > r1 || p[j]) return gc_long(av, 0);
3279 44128 : p[j] = 1;
3280 : }
3281 21987 : return gc_long(av, 1);
3282 : }
3283 :
3284 : /* True nf. Put ideal to form [[ideal,arch]] and set fa and fa2 to its
3285 : * factorization, archp to the indices of arch places */
3286 : GEN
3287 258795 : check_mod_factored(GEN nf, GEN ideal, GEN *fa_, GEN *fa2_, GEN *archp_, GEN MOD)
3288 : {
3289 : GEN arch, x, fa, fa2, archp;
3290 : long R1;
3291 :
3292 258795 : R1 = nf_get_r1(nf);
3293 258794 : if (typ(ideal) == t_VEC && lg(ideal) == 3)
3294 : {
3295 178688 : arch = gel(ideal,2);
3296 178688 : ideal= gel(ideal,1);
3297 178688 : switch(typ(arch))
3298 : {
3299 44597 : case t_VEC:
3300 44597 : if (lg(arch) != R1+1)
3301 7 : pari_err_TYPE("Idealstar [incorrect archimedean component]",arch);
3302 44590 : archp = vec01_to_indices(arch);
3303 44590 : break;
3304 134091 : case t_VECSMALL:
3305 134091 : if (!checkarchp(arch, R1))
3306 35 : pari_err_TYPE("Idealstar [incorrect archimedean component]",arch);
3307 134055 : archp = arch;
3308 134055 : arch = indices_to_vec01(archp, R1);
3309 134053 : break;
3310 0 : default:
3311 0 : pari_err_TYPE("Idealstar [incorrect archimedean component]",arch);
3312 : return NULL;/*LCOV_EXCL_LINE*/
3313 : }
3314 : }
3315 : else
3316 : {
3317 80106 : arch = zerovec(R1);
3318 80104 : archp = cgetg(1, t_VECSMALL);
3319 : }
3320 258743 : if (MOD)
3321 : {
3322 214174 : if (typ(MOD) != t_INT) pari_err_TYPE("bnrinit [incorrect cycmod]", MOD);
3323 214174 : if (mpodd(MOD) && lg(archp) != 1)
3324 231 : MOD = shifti(MOD, 1); /* ensure elements of G^MOD are >> 0 */
3325 : }
3326 258739 : if (is_nf_factor(ideal))
3327 : {
3328 40262 : fa = ideal;
3329 40262 : x = factorbackprime(nf, gel(fa,1), gel(fa,2));
3330 : }
3331 : else
3332 : {
3333 218476 : fa = idealfactor(nf, ideal);
3334 218492 : x = ideal;
3335 : }
3336 258756 : if (typ(x) != t_MAT) x = idealhnf_shallow(nf, x);
3337 258744 : if (lg(x) == 1) pari_err_DOMAIN("Idealstar", "ideal","=",gen_0,x);
3338 258744 : if (typ(gcoeff(x,1,1)) != t_INT)
3339 7 : pari_err_DOMAIN("Idealstar","denominator(ideal)", "!=",gen_1,x);
3340 :
3341 258737 : fa2 = famat_strip2(fa);
3342 258729 : if (fa_ != NULL) *fa_ = fa;
3343 258729 : if (fa2_ != NULL) *fa2_ = fa2;
3344 258729 : if (fa2_ != NULL) *archp_ = archp;
3345 258729 : return mkvec2(x, arch);
3346 : }
3347 :
3348 : /* True nf. Compute [[ideal,arch], [h,[cyc],[gen]], idealfact, [liste], U]
3349 : flag may include nf_GEN | nf_INIT */
3350 : static GEN
3351 258158 : Idealstarmod_i(GEN nf, GEN ideal, long flag, GEN MOD)
3352 : {
3353 : long i, nbp;
3354 258158 : GEN y, cyc, U, u1 = NULL, fa, fa2, sprk, x_arch, x, arch, archp, E, P, sarch, gen;
3355 :
3356 258158 : x_arch = check_mod_factored(nf, ideal, &fa, &fa2, &archp, MOD);
3357 258104 : x = gel(x_arch, 1);
3358 258104 : arch = gel(x_arch, 2);
3359 :
3360 258104 : sarch = nfarchstar(nf, x, archp);
3361 258102 : P = gel(fa2,1);
3362 258102 : E = gel(fa2,2);
3363 258102 : nbp = lg(P)-1;
3364 258102 : sprk = cgetg(nbp+1,t_VEC);
3365 258108 : if (nbp)
3366 : {
3367 218846 : GEN t = (lg(gel(fa,1))==2)? NULL: x; /* beware fa != fa2 */
3368 218846 : cyc = cgetg(nbp+2,t_VEC);
3369 218828 : gen = cgetg(nbp+1,t_VEC);
3370 554563 : for (i = 1; i <= nbp; i++)
3371 : {
3372 335711 : GEN L = sprkinit(nf, gel(P,i), itou(gel(E,i)), t, MOD);
3373 335723 : gel(sprk,i) = L;
3374 335723 : gel(cyc,i) = sprk_get_cyc(L);
3375 : /* true gens are congruent to those mod x AND positive at archp */
3376 335722 : gel(gen,i) = sprk_get_gen(L);
3377 : }
3378 218852 : gel(cyc,i) = sarch_get_cyc(sarch);
3379 218852 : cyc = shallowconcat1(cyc);
3380 218847 : gen = shallowconcat1(gen);
3381 218854 : cyc = ZV_snf_group(cyc, &U, (flag & nf_GEN)? &u1: NULL);
3382 : }
3383 : else
3384 : {
3385 39262 : cyc = sarch_get_cyc(sarch);
3386 39262 : gen = cgetg(1,t_VEC);
3387 39262 : U = matid(lg(cyc)-1);
3388 39262 : if (flag & nf_GEN) u1 = U;
3389 : }
3390 258093 : if (MOD) cyc = ZV_snf_gcd(cyc, MOD);
3391 258082 : y = bid_grp(nf, u1, cyc, gen, x, sarch);
3392 258111 : if (!(flag & nf_INIT)) return y;
3393 257313 : U = split_U(U, sprk);
3394 514625 : return mkvec5(mkvec2(x, arch), y, mkvec2(fa,fa2),
3395 257312 : MOD? mkvec3(sprk, sarch, MOD): mkvec2(sprk, sarch),
3396 : U);
3397 : }
3398 :
3399 : static long
3400 63 : idealHNF_norm_pval(GEN x, GEN p)
3401 : {
3402 63 : long i, v = 0, l = lg(x);
3403 175 : for (i = 1; i < l; i++) v += Z_pval(gcoeff(x,i,i), p);
3404 63 : return v;
3405 : }
3406 : static long
3407 63 : sprk_get_k(GEN sprk)
3408 : {
3409 : GEN pr, prk;
3410 63 : if (sprk_is_prime(sprk)) return 1;
3411 63 : pr = sprk_get_pr(sprk);
3412 63 : prk = sprk_get_prk(sprk);
3413 63 : return idealHNF_norm_pval(prk, pr_get_p(pr)) / pr_get_f(pr);
3414 : }
3415 : /* true nf, L a sprk */
3416 : GEN
3417 63 : sprk_to_bid(GEN nf, GEN L, long flag)
3418 : {
3419 63 : GEN y, cyc, U, u1 = NULL, fa, fa2, arch, sarch, gen, sprk;
3420 :
3421 63 : arch = zerovec(nf_get_r1(nf));
3422 63 : fa = to_famat_shallow(sprk_get_pr(L), utoipos(sprk_get_k(L)));
3423 63 : sarch = nfarchstar(nf, NULL, cgetg(1, t_VECSMALL));
3424 63 : fa2 = famat_strip2(fa);
3425 63 : sprk = mkvec(L);
3426 63 : cyc = shallowconcat(sprk_get_cyc(L), sarch_get_cyc(sarch));
3427 63 : gen = sprk_get_gen(L);
3428 63 : cyc = ZV_snf_group(cyc, &U, (flag & nf_GEN)? &u1: NULL);
3429 63 : y = bid_grp(nf, u1, cyc, gen, NULL, sarch);
3430 63 : if (!(flag & nf_INIT)) return y;
3431 63 : return mkvec5(mkvec2(sprk_get_prk(L), arch), y, mkvec2(fa,fa2),
3432 : mkvec2(sprk, sarch), split_U(U, sprk));
3433 : }
3434 : GEN
3435 257885 : Idealstarmod(GEN nf, GEN ideal, long flag, GEN MOD)
3436 : {
3437 257885 : pari_sp av = avma;
3438 257885 : nf = nf? checknf(nf): nfinit(pol_x(0), DEFAULTPREC);
3439 257886 : return gerepilecopy(av, Idealstarmod_i(nf, ideal, flag, MOD));
3440 : }
3441 : GEN
3442 938 : Idealstar(GEN nf, GEN ideal, long flag) { return Idealstarmod(nf, ideal, flag, NULL); }
3443 : GEN
3444 273 : Idealstarprk(GEN nf, GEN pr, long k, long flag)
3445 : {
3446 273 : pari_sp av = avma;
3447 273 : GEN z = Idealstarmod_i(nf, mkmat2(mkcol(pr),mkcols(k)), flag, NULL);
3448 273 : return gerepilecopy(av, z);
3449 : }
3450 :
3451 : /* FIXME: obsolete */
3452 : GEN
3453 0 : zidealstarinitgen(GEN nf, GEN ideal)
3454 0 : { return Idealstar(nf,ideal, nf_INIT|nf_GEN); }
3455 : GEN
3456 0 : zidealstarinit(GEN nf, GEN ideal)
3457 0 : { return Idealstar(nf,ideal, nf_INIT); }
3458 : GEN
3459 0 : zidealstar(GEN nf, GEN ideal)
3460 0 : { return Idealstar(nf,ideal, nf_GEN); }
3461 :
3462 : GEN
3463 112 : idealstarmod(GEN nf, GEN ideal, long flag, GEN MOD)
3464 : {
3465 112 : switch(flag)
3466 : {
3467 0 : case 0: return Idealstarmod(nf,ideal, nf_GEN, MOD);
3468 98 : case 1: return Idealstarmod(nf,ideal, nf_INIT, MOD);
3469 14 : case 2: return Idealstarmod(nf,ideal, nf_INIT|nf_GEN, MOD);
3470 0 : default: pari_err_FLAG("idealstar");
3471 : }
3472 : return NULL; /* LCOV_EXCL_LINE */
3473 : }
3474 : GEN
3475 0 : idealstar0(GEN nf, GEN ideal,long flag) { return idealstarmod(nf, ideal, flag, NULL); }
3476 :
3477 : void
3478 218782 : check_nfelt(GEN x, GEN *den)
3479 : {
3480 218782 : long l = lg(x), i;
3481 218782 : GEN t, d = NULL;
3482 218782 : if (typ(x) != t_COL) pari_err_TYPE("check_nfelt", x);
3483 808324 : for (i=1; i<l; i++)
3484 : {
3485 589539 : t = gel(x,i);
3486 589539 : switch (typ(t))
3487 : {
3488 589308 : case t_INT: break;
3489 231 : case t_FRAC:
3490 231 : if (!d) d = gel(t,2); else d = lcmii(d, gel(t,2));
3491 231 : break;
3492 0 : default: pari_err_TYPE("check_nfelt", x);
3493 : }
3494 : }
3495 218785 : *den = d;
3496 218785 : }
3497 :
3498 : GEN
3499 1952563 : ZV_snf_gcd(GEN x, GEN mod)
3500 4357094 : { pari_APPLY_same(gcdii(gel(x,i), mod)); }
3501 :
3502 : /* assume a true bnf and bid */
3503 : GEN
3504 227018 : ideallog_units0(GEN bnf, GEN bid, GEN MOD)
3505 : {
3506 227018 : GEN nf = bnf_get_nf(bnf), D, y, C, cyc;
3507 227017 : long j, lU = lg(bnf_get_logfu(bnf)); /* r1+r2 */
3508 : zlog_S S;
3509 227017 : init_zlog_mod(&S, bid, MOD);
3510 226998 : if (!S.hU) return zeromat(0,lU);
3511 226998 : cyc = bid_get_cyc(bid);
3512 226994 : D = nfsign_fu(bnf, bid_get_archp(bid));
3513 227001 : y = cgetg(lU, t_MAT);
3514 227000 : if ((C = bnf_build_cheapfu(bnf)))
3515 374172 : { for (j = 1; j < lU; j++) gel(y,j) = zlog(nf, gel(C,j), gel(D,j), &S); }
3516 : else
3517 : {
3518 49 : long i, l = lg(S.U), l0 = lg(S.sprk);
3519 : GEN X, U;
3520 49 : if (!(C = bnf_compactfu_mat(bnf))) bnf_build_units(bnf); /* error */
3521 49 : X = gel(C,1); U = gel(C,2);
3522 147 : for (j = 1; j < lU; j++) gel(y,j) = cgetg(l, t_COL);
3523 126 : for (i = 1; i < l0; i++)
3524 : {
3525 77 : GEN sprk = gel(S.sprk, i);
3526 77 : GEN Xi = sunits_makecoprime(X, sprk_get_pr(sprk), sprk_get_prk(sprk));
3527 231 : for (j = 1; j < lU; j++)
3528 154 : gcoeff(y,i,j) = famat_zlog_pr_coprime(nf, Xi, gel(U,j), sprk, MOD);
3529 : }
3530 49 : if (l0 != l)
3531 56 : for (j = 1; j < lU; j++) gcoeff(y,l0,j) = Flc_to_ZC(gel(D,j));
3532 : }
3533 226998 : y = vec_prepend(y, zlog(nf, bnf_get_tuU(bnf), nfsign_tu(bnf, S.archp), &S));
3534 601292 : for (j = 1; j <= lU; j++)
3535 374295 : gel(y,j) = ZV_ZV_mod(ZMV_ZCV_mul(S.U, gel(y,j)), cyc);
3536 226997 : return y;
3537 : }
3538 : GEN
3539 84 : ideallog_units(GEN bnf, GEN bid)
3540 84 : { return ideallog_units0(bnf, bid, NULL); }
3541 : GEN
3542 518 : log_prk_units(GEN nf, GEN D, GEN sprk)
3543 : {
3544 518 : GEN L, Ltu = log_prk(nf, gel(D,1), sprk, NULL);
3545 518 : D = gel(D,2);
3546 518 : if (lg(D) != 3 || typ(gel(D,2)) != t_MAT) L = veclog_prk(nf, D, sprk);
3547 : else
3548 : {
3549 21 : GEN X = gel(D,1), U = gel(D,2);
3550 21 : long j, lU = lg(U);
3551 21 : X = sunits_makecoprime(X, sprk_get_pr(sprk), sprk_get_prk(sprk));
3552 21 : L = cgetg(lU, t_MAT);
3553 63 : for (j = 1; j < lU; j++)
3554 42 : gel(L,j) = famat_zlog_pr_coprime(nf, X, gel(U,j), sprk, NULL);
3555 : }
3556 518 : return vec_prepend(L, Ltu);
3557 : }
3558 :
3559 : static GEN
3560 1383469 : ideallog_i(GEN nf, GEN x, zlog_S *S)
3561 : {
3562 1383469 : pari_sp av = avma;
3563 : GEN y;
3564 1383469 : if (!S->hU) return cgetg(1, t_COL);
3565 1381173 : if (typ(x) == t_MAT)
3566 1374110 : y = famat_zlog(nf, x, NULL, S);
3567 : else
3568 7063 : y = zlog(nf, x, NULL, S);
3569 1381171 : y = ZMV_ZCV_mul(S->U, y);
3570 1381169 : return gerepileupto(av, ZV_ZV_mod(y, bid_get_cyc(S->bid)));
3571 : }
3572 : GEN
3573 1390151 : ideallogmod(GEN nf, GEN x, GEN bid, GEN mod)
3574 : {
3575 : zlog_S S;
3576 1390151 : if (!nf)
3577 : {
3578 6671 : if (mod) pari_err_IMPL("Zideallogmod");
3579 6671 : return Zideallog(bid, x);
3580 : }
3581 1383480 : checkbid(bid); init_zlog_mod(&S, bid, mod);
3582 1383469 : return ideallog_i(checknf(nf), x, &S);
3583 : }
3584 : GEN
3585 13769 : ideallog(GEN nf, GEN x, GEN bid) { return ideallogmod(nf, x, bid, NULL); }
3586 :
3587 : /*************************************************************************/
3588 : /** **/
3589 : /** JOIN BID STRUCTURES, IDEAL LISTS **/
3590 : /** **/
3591 : /*************************************************************************/
3592 : /* bid1, bid2: for coprime modules m1 and m2 (without arch. part).
3593 : * Output: bid for m1 m2 */
3594 : static GEN
3595 469 : join_bid(GEN nf, GEN bid1, GEN bid2)
3596 : {
3597 469 : pari_sp av = avma;
3598 : long nbgen, l1,l2;
3599 : GEN I1,I2, G1,G2, sprk1,sprk2, cyc1,cyc2, sarch;
3600 469 : GEN sprk, fa,fa2, U, cyc, y, u1 = NULL, x, gen;
3601 :
3602 469 : I1 = bid_get_ideal(bid1);
3603 469 : I2 = bid_get_ideal(bid2);
3604 469 : if (gequal1(gcoeff(I1,1,1))) return bid2; /* frequent trivial case */
3605 259 : G1 = bid_get_grp(bid1);
3606 259 : G2 = bid_get_grp(bid2);
3607 259 : x = idealmul(nf, I1,I2);
3608 259 : fa = famat_mul_shallow(bid_get_fact(bid1), bid_get_fact(bid2));
3609 259 : fa2= famat_mul_shallow(bid_get_fact2(bid1), bid_get_fact2(bid2));
3610 259 : sprk1 = bid_get_sprk(bid1);
3611 259 : sprk2 = bid_get_sprk(bid2);
3612 259 : sprk = shallowconcat(sprk1, sprk2);
3613 :
3614 259 : cyc1 = abgrp_get_cyc(G1); l1 = lg(cyc1);
3615 259 : cyc2 = abgrp_get_cyc(G2); l2 = lg(cyc2);
3616 259 : gen = (lg(G1)>3 && lg(G2)>3)? gen_1: NULL;
3617 259 : nbgen = l1+l2-2;
3618 259 : cyc = ZV_snf_group(shallowconcat(cyc1,cyc2), &U, gen? &u1: NULL);
3619 259 : if (nbgen)
3620 : {
3621 259 : GEN U1 = bid_get_U(bid1), U2 = bid_get_U(bid2);
3622 0 : U1 = l1==1? const_vec(lg(sprk1), cgetg(1,t_MAT))
3623 259 : : ZM_ZMV_mul(vecslice(U, 1, l1-1), U1);
3624 0 : U2 = l2==1? const_vec(lg(sprk2), cgetg(1,t_MAT))
3625 259 : : ZM_ZMV_mul(vecslice(U, l1, nbgen), U2);
3626 259 : U = shallowconcat(U1, U2);
3627 : }
3628 : else
3629 0 : U = const_vec(lg(sprk), cgetg(1,t_MAT));
3630 :
3631 259 : if (gen)
3632 : {
3633 259 : GEN uv = zkchinese1init2(nf, I2, I1, x);
3634 518 : gen = shallowconcat(zkVchinese1(gel(uv,1), abgrp_get_gen(G1)),
3635 259 : zkVchinese1(gel(uv,2), abgrp_get_gen(G2)));
3636 : }
3637 259 : sarch = bid_get_sarch(bid1); /* trivial */
3638 259 : y = bid_grp(nf, u1, cyc, gen, x, sarch);
3639 259 : x = mkvec2(x, bid_get_arch(bid1));
3640 259 : y = mkvec5(x, y, mkvec2(fa, fa2), mkvec2(sprk, sarch), U);
3641 259 : return gerepilecopy(av,y);
3642 : }
3643 :
3644 : typedef struct _ideal_data {
3645 : GEN nf, emb, L, pr, prL, sgnU, archp;
3646 : } ideal_data;
3647 :
3648 : /* z <- ( z | f(v[i])_{i=1..#v} ) */
3649 : static void
3650 758599 : concat_join(GEN *pz, GEN v, GEN (*f)(ideal_data*,GEN), ideal_data *data)
3651 : {
3652 758599 : long i, nz, lv = lg(v);
3653 : GEN z, Z;
3654 758599 : if (lv == 1) return;
3655 222876 : z = *pz; nz = lg(z)-1;
3656 222876 : *pz = Z = cgetg(lv + nz, typ(z));
3657 371620 : for (i = 1; i <=nz; i++) gel(Z,i) = gel(z,i);
3658 223300 : Z += nz;
3659 492115 : for (i = 1; i < lv; i++) gel(Z,i) = f(data, gel(v,i));
3660 : }
3661 : static GEN
3662 469 : join_idealinit(ideal_data *D, GEN x)
3663 469 : { return join_bid(D->nf, x, D->prL); }
3664 : static GEN
3665 268476 : join_ideal(ideal_data *D, GEN x)
3666 268476 : { return idealmulpowprime(D->nf, x, D->pr, D->L); }
3667 : static GEN
3668 448 : join_unit(ideal_data *D, GEN x)
3669 : {
3670 448 : GEN bid = join_idealinit(D, gel(x,1)), u = gel(x,2), v = mkvec(D->emb);
3671 448 : if (lg(u) != 1) v = shallowconcat(u, v);
3672 448 : return mkvec2(bid, v);
3673 : }
3674 :
3675 : GEN
3676 49 : log_prk_units_init(GEN bnf)
3677 : {
3678 49 : GEN U = bnf_has_fu(bnf);
3679 49 : if (U) U = matalgtobasis(bnf_get_nf(bnf), U);
3680 21 : else if (!(U = bnf_compactfu_mat(bnf))) (void)bnf_build_units(bnf);
3681 49 : return mkvec2(bnf_get_tuU(bnf), U);
3682 : }
3683 : /* flag & nf_GEN : generators, otherwise no
3684 : * flag &2 : units, otherwise no
3685 : * flag &4 : ideals in HNF, otherwise bid
3686 : * flag &8 : omit ideals which cannot be conductors (pr^1 with Npr=2) */
3687 : static GEN
3688 11333 : Ideallist(GEN bnf, ulong bound, long flag)
3689 : {
3690 11333 : const long do_units = flag & 2, big_id = !(flag & 4), cond = flag & 8;
3691 11333 : const long istar_flag = (flag & nf_GEN) | nf_INIT;
3692 : pari_sp av;
3693 : long i, j;
3694 11333 : GEN nf, z, p, fa, id, BOUND, U, empty = cgetg(1,t_VEC);
3695 : forprime_t S;
3696 : ideal_data ID;
3697 : GEN (*join_z)(ideal_data*, GEN);
3698 :
3699 11333 : if (do_units)
3700 : {
3701 21 : bnf = checkbnf(bnf);
3702 21 : nf = bnf_get_nf(bnf);
3703 21 : join_z = &join_unit;
3704 : }
3705 : else
3706 : {
3707 11312 : nf = checknf(bnf);
3708 11312 : join_z = big_id? &join_idealinit: &join_ideal;
3709 : }
3710 11333 : if ((long)bound <= 0) return empty;
3711 11333 : id = matid(nf_get_degree(nf));
3712 11333 : if (big_id) id = Idealstar(nf,id, istar_flag);
3713 :
3714 : /* z[i] will contain all "objects" of norm i. Depending on flag, this means
3715 : * an ideal, a bid, or a couple [bid, log(units)]. Such objects are stored
3716 : * in vectors, computed one primary component at a time; join_z
3717 : * reconstructs the global object */
3718 11333 : BOUND = utoipos(bound);
3719 11333 : z = const_vec(bound, empty);
3720 11333 : U = do_units? log_prk_units_init(bnf): NULL;
3721 11333 : gel(z,1) = mkvec(U? mkvec2(id, empty): id);
3722 11333 : ID.nf = nf;
3723 :
3724 11333 : p = cgetipos(3);
3725 11333 : u_forprime_init(&S, 2, bound);
3726 11333 : av = avma;
3727 92866 : while ((p[2] = u_forprime_next(&S)))
3728 : {
3729 81611 : if (DEBUGLEVEL>1) err_printf("%ld ",p[2]);
3730 81611 : fa = idealprimedec_limit_norm(nf, p, BOUND);
3731 163043 : for (j = 1; j < lg(fa); j++)
3732 : {
3733 81510 : GEN pr = gel(fa,j), z2 = leafcopy(z);
3734 81514 : ulong Q, q = upr_norm(pr);
3735 : long l;
3736 81513 : ID.pr = ID.prL = pr;
3737 81513 : if (cond && q == 2) { l = 2; Q = 4; } else { l = 1; Q = q; }
3738 184469 : for (; Q <= bound; l++, Q *= q) /* add pr^l */
3739 : {
3740 : ulong iQ;
3741 103042 : ID.L = utoipos(l);
3742 103039 : if (big_id) {
3743 211 : ID.prL = Idealstarprk(nf, pr, l, istar_flag);
3744 210 : if (U)
3745 189 : ID.emb = Q == 2? empty
3746 189 : : log_prk_units(nf, U, gel(bid_get_sprk(ID.prL),1));
3747 : }
3748 861551 : for (iQ = Q,i = 1; iQ <= bound; iQ += Q,i++)
3749 758595 : concat_join(&gel(z,iQ), gel(z2,i), join_z, &ID);
3750 : }
3751 : }
3752 81533 : if (gc_needed(av,1))
3753 : {
3754 18 : if(DEBUGMEM>1) pari_warn(warnmem,"Ideallist");
3755 18 : z = gerepilecopy(av, z);
3756 : }
3757 : }
3758 11333 : return z;
3759 : }
3760 : GEN
3761 63 : gideallist(GEN bnf, GEN B, long flag)
3762 : {
3763 63 : pari_sp av = avma;
3764 63 : if (typ(B) != t_INT)
3765 : {
3766 0 : B = gfloor(B);
3767 0 : if (typ(B) != t_INT) pari_err_TYPE("ideallist", B);
3768 0 : if (signe(B) < 0) B = gen_0;
3769 : }
3770 63 : if (signe(B) < 0)
3771 : {
3772 28 : if (flag != 4) pari_err_IMPL("ideallist with bid for single norm");
3773 28 : return gerepilecopy(av, ideals_by_norm(checknf(bnf), absi(B)));
3774 : }
3775 35 : if (flag < 0 || flag > 15) pari_err_FLAG("ideallist");
3776 35 : return gerepilecopy(av, Ideallist(bnf, itou(B), flag));
3777 : }
3778 : GEN
3779 11298 : ideallist0(GEN bnf, long bound, long flag)
3780 : {
3781 11298 : pari_sp av = avma;
3782 11298 : if (flag < 0 || flag > 15) pari_err_FLAG("ideallist");
3783 11298 : return gerepilecopy(av, Ideallist(bnf, bound, flag));
3784 : }
3785 : GEN
3786 10563 : ideallist(GEN bnf,long bound) { return ideallist0(bnf,bound,4); }
3787 :
3788 : /* bid = for module m (without arch. part), arch = archimedean part.
3789 : * Output: bid for [m,arch] */
3790 : static GEN
3791 42 : join_bid_arch(GEN nf, GEN bid, GEN archp)
3792 : {
3793 42 : pari_sp av = avma;
3794 : GEN G, U;
3795 42 : GEN sprk, cyc, y, u1 = NULL, x, sarch, gen;
3796 :
3797 42 : checkbid(bid);
3798 42 : G = bid_get_grp(bid);
3799 42 : x = bid_get_ideal(bid);
3800 42 : sarch = nfarchstar(nf, bid_get_ideal(bid), archp);
3801 42 : sprk = bid_get_sprk(bid);
3802 :
3803 42 : gen = (lg(G)>3)? gel(G,3): NULL;
3804 42 : cyc = diagonal_shallow(shallowconcat(gel(G,2), sarch_get_cyc(sarch)));
3805 42 : cyc = ZM_snf_group(cyc, &U, gen? &u1: NULL);
3806 42 : y = bid_grp(nf, u1, cyc, gen, x, sarch);
3807 42 : U = split_U(U, sprk);
3808 42 : y = mkvec5(mkvec2(x, archp), y, gel(bid,3), mkvec2(sprk, sarch), U);
3809 42 : return gerepilecopy(av,y);
3810 : }
3811 : static GEN
3812 42 : join_arch(ideal_data *D, GEN x) {
3813 42 : return join_bid_arch(D->nf, x, D->archp);
3814 : }
3815 : static GEN
3816 14 : join_archunit(ideal_data *D, GEN x) {
3817 14 : GEN bid = join_arch(D, gel(x,1)), u = gel(x,2), v = mkvec(D->emb);
3818 14 : if (lg(u) != 1) v = shallowconcat(u, v);
3819 14 : return mkvec2(bid, v);
3820 : }
3821 :
3822 : /* L from ideallist, add archimedean part */
3823 : GEN
3824 14 : ideallistarch(GEN bnf, GEN L, GEN arch)
3825 : {
3826 : pari_sp av;
3827 14 : long i, j, l = lg(L), lz;
3828 : GEN v, z, V, nf;
3829 : ideal_data ID;
3830 : GEN (*join_z)(ideal_data*, GEN);
3831 :
3832 14 : if (typ(L) != t_VEC) pari_err_TYPE("ideallistarch",L);
3833 14 : if (l == 1) return cgetg(1,t_VEC);
3834 14 : z = gel(L,1);
3835 14 : if (typ(z) != t_VEC) pari_err_TYPE("ideallistarch",z);
3836 14 : z = gel(z,1); /* either a bid or [bid,U] */
3837 14 : ID.archp = vec01_to_indices(arch);
3838 14 : if (lg(z) == 3)
3839 : { /* [bid,U]: do units */
3840 7 : bnf = checkbnf(bnf); nf = bnf_get_nf(bnf);
3841 7 : if (typ(z) != t_VEC) pari_err_TYPE("ideallistarch",z);
3842 7 : ID.emb = zm_to_ZM( rowpermute(nfsign_units(bnf,NULL,1), ID.archp) );
3843 7 : join_z = &join_archunit;
3844 : }
3845 : else
3846 : {
3847 7 : join_z = &join_arch;
3848 7 : nf = checknf(bnf);
3849 : }
3850 14 : ID.nf = nf;
3851 14 : av = avma; V = cgetg(l, t_VEC);
3852 63 : for (i = 1; i < l; i++)
3853 : {
3854 49 : z = gel(L,i); lz = lg(z);
3855 49 : gel(V,i) = v = cgetg(lz,t_VEC);
3856 91 : for (j=1; j<lz; j++) gel(v,j) = join_z(&ID, gel(z,j));
3857 : }
3858 14 : return gerepilecopy(av,V);
3859 : }
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