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 : /* (continued) */
19 : /* */
20 : /*******************************************************************/
21 : #include "pari.h"
22 : #include "paripriv.h"
23 :
24 : #define DEBUGLEVEL DEBUGLEVEL_nf
25 :
26 : /*******************************************************************/
27 : /* */
28 : /* IDEAL OPERATIONS */
29 : /* */
30 : /*******************************************************************/
31 :
32 : /* A valid ideal is either principal (valid nf_element), or prime, or a matrix
33 : * on the integer basis in HNF.
34 : * A prime ideal is of the form [p,a,e,f,b], where the ideal is p.Z_K+a.Z_K,
35 : * p is a rational prime, a belongs to Z_K, e=e(P/p), f=f(P/p), and b
36 : * is Lenstra's constant, such that p.P^(-1)= p Z_K + b Z_K.
37 : *
38 : * An extended ideal is a couple [I,F] where I is an ideal and F is either an
39 : * algebraic number, or a factorization matrix attached to an algebraic number.
40 : * All routines work with either extended ideals or ideals (an omitted F is
41 : * assumed to be factor(1)). All ideals are output in HNF form. */
42 :
43 : /* types and conversions */
44 :
45 : long
46 8705355 : idealtyp(GEN *ideal, GEN *arch)
47 : {
48 8705355 : GEN x = *ideal;
49 8705355 : long t,lx,tx = typ(x);
50 :
51 8705355 : if (tx!=t_VEC || lg(x)!=3) *arch = NULL;
52 : else
53 : {
54 212524 : GEN a = gel(x,2);
55 212524 : if (typ(a) == t_MAT && lg(a) != 3)
56 : { /* allow [;] */
57 14 : if (lg(a) != 1) pari_err_TYPE("idealtyp [extended ideal]",x);
58 7 : a = trivial_fact();
59 : }
60 212517 : *arch = a;
61 212517 : x = gel(x,1); tx = typ(x);
62 : }
63 8705348 : switch(tx)
64 : {
65 3456439 : case t_MAT: lx = lg(x);
66 3456439 : if (lx == 1) { t = id_PRINCIPAL; x = gen_0; break; }
67 3456278 : if (lx != lgcols(x)) pari_err_TYPE("idealtyp [nonsquare t_MAT]",x);
68 3456268 : t = id_MAT;
69 3456268 : break;
70 :
71 4616239 : case t_VEC: if (lg(x)!=6) pari_err_TYPE("idealtyp",x);
72 4616224 : t = id_PRIME; break;
73 :
74 632818 : case t_POL: case t_POLMOD: case t_COL:
75 : case t_INT: case t_FRAC:
76 632818 : t = id_PRINCIPAL; break;
77 0 : default:
78 0 : pari_err_TYPE("idealtyp",x);
79 : return 0; /*LCOV_EXCL_LINE*/
80 : }
81 8705471 : *ideal = x; return t;
82 : }
83 :
84 : /* true nf; v = [a,x,...], a in Z. Return (a,x) */
85 : GEN
86 668536 : idealhnf_two(GEN nf, GEN v)
87 : {
88 668536 : GEN p = gel(v,1), pi = gel(v,2), m = zk_scalar_or_multable(nf, pi);
89 668599 : if (typ(m) == t_INT) return scalarmat(gcdii(m,p), nf_get_degree(nf));
90 599451 : return ZM_hnfmodid(m, p);
91 : }
92 : /* true nf */
93 : GEN
94 4004790 : pr_hnf(GEN nf, GEN pr)
95 : {
96 4004790 : GEN p = pr_get_p(pr), m;
97 4004753 : if (pr_is_inert(pr)) return scalarmat(p, nf_get_degree(nf));
98 3579724 : m = zk_scalar_or_multable(nf, pr_get_gen(pr));
99 3579449 : return ZM_hnfmodprime(m, p);
100 : }
101 :
102 : GEN
103 479577 : idealhnf_principal(GEN nf, GEN x)
104 : {
105 : GEN cx;
106 479577 : x = nf_to_scalar_or_basis(nf, x);
107 479577 : switch(typ(x))
108 : {
109 297849 : case t_COL: break;
110 152986 : case t_INT: if (!signe(x)) return cgetg(1,t_MAT);
111 152223 : return scalarmat(absi_shallow(x), nf_get_degree(nf));
112 28741 : case t_FRAC:
113 28741 : return scalarmat(Q_abs_shallow(x), nf_get_degree(nf));
114 1 : default: pari_err_TYPE("idealhnf",x);
115 : }
116 297849 : x = Q_primitive_part(x, &cx);
117 297847 : RgV_check_ZV(x, "idealhnf");
118 297847 : x = zk_multable(nf, x);
119 297849 : x = ZM_hnfmodid(x, zkmultable_capZ(x));
120 297846 : return cx? ZM_Q_mul(x,cx): x;
121 : }
122 :
123 : /* x integral ideal in t_MAT form, nx columns */
124 : static GEN
125 7 : vec_mulid(GEN nf, GEN x, long nx, long N)
126 : {
127 7 : GEN m = cgetg(nx*N + 1, t_MAT);
128 : long i, j, k;
129 21 : for (i=k=1; i<=nx; i++)
130 56 : for (j=1; j<=N; j++) gel(m, k++) = zk_ei_mul(nf, gel(x,i),j);
131 7 : return m;
132 : }
133 : /* true nf */
134 : GEN
135 792626 : idealhnf_shallow(GEN nf, GEN x)
136 : {
137 792626 : long tx = typ(x), lx = lg(x), N;
138 :
139 : /* cannot use idealtyp because here we allow nonsquare matrices */
140 792626 : if (tx == t_VEC && lx == 3) { x = gel(x,1); tx = typ(x); lx = lg(x); }
141 792626 : if (tx == t_VEC && lx == 6) return pr_hnf(nf,x); /* PRIME */
142 429575 : switch(tx)
143 : {
144 87723 : case t_MAT:
145 : {
146 : GEN cx;
147 87723 : long nx = lx-1;
148 87723 : N = nf_get_degree(nf);
149 87724 : if (nx == 0) return cgetg(1, t_MAT);
150 87703 : if (nbrows(x) != N) pari_err_TYPE("idealhnf [wrong dimension]",x);
151 87696 : if (nx == 1) return idealhnf_principal(nf, gel(x,1));
152 :
153 80717 : if (nx == N && RgM_is_ZM(x) && ZM_ishnf(x)) return x;
154 47208 : x = Q_primitive_part(x, &cx);
155 47208 : if (nx < N) x = vec_mulid(nf, x, nx, N);
156 47208 : x = ZM_hnfmod(x, ZM_detmult(x));
157 47208 : return cx? ZM_Q_mul(x,cx): x;
158 : }
159 14 : case t_QFB:
160 : {
161 14 : pari_sp av = avma;
162 14 : GEN u, D = nf_get_disc(nf), T = nf_get_pol(nf), f = nf_get_index(nf);
163 14 : GEN A = gel(x,1), B = gel(x,2);
164 14 : N = nf_get_degree(nf);
165 14 : if (N != 2)
166 0 : pari_err_TYPE("idealhnf [Qfb for nonquadratic fields]", x);
167 14 : if (!equalii(qfb_disc(x), D))
168 7 : pari_err_DOMAIN("idealhnf [Qfb]", "disc(q)", "!=", D, x);
169 : /* x -> A Z + (-B + sqrt(D)) / 2 Z
170 : K = Q[t]/T(t), t^2 + ut + v = 0, u^2 - 4v = Df^2
171 : => t = (-u + sqrt(D) f)/2
172 : => sqrt(D)/2 = (t + u/2)/f */
173 7 : u = gel(T,3);
174 7 : B = deg1pol_shallow(ginv(f),
175 : gsub(gdiv(u, shifti(f,1)), gdiv(B,gen_2)),
176 7 : varn(T));
177 7 : return gerepileupto(av, idealhnf_two(nf, mkvec2(A,B)));
178 : }
179 341838 : default: return idealhnf_principal(nf, x); /* PRINCIPAL */
180 : }
181 : }
182 : /* true nf */
183 : GEN
184 238 : idealhnf(GEN nf, GEN x)
185 : {
186 238 : pari_sp av = avma;
187 238 : GEN y = idealhnf_shallow(nf, x);
188 231 : return (avma == av)? gcopy(y): gerepileupto(av, y);
189 : }
190 :
191 : /* GP functions */
192 :
193 : GEN
194 63 : idealtwoelt0(GEN nf, GEN x, GEN a)
195 : {
196 63 : if (!a) return idealtwoelt(nf,x);
197 42 : return idealtwoelt2(nf,x,a);
198 : }
199 :
200 : GEN
201 56 : idealpow0(GEN nf, GEN x, GEN n, long flag)
202 : {
203 56 : if (flag) return idealpowred(nf,x,n);
204 49 : return idealpow(nf,x,n);
205 : }
206 :
207 : GEN
208 56 : idealmul0(GEN nf, GEN x, GEN y, long flag)
209 : {
210 56 : if (flag) return idealmulred(nf,x,y);
211 49 : return idealmul(nf,x,y);
212 : }
213 :
214 : GEN
215 56 : idealdiv0(GEN nf, GEN x, GEN y, long flag)
216 : {
217 56 : switch(flag)
218 : {
219 28 : case 0: return idealdiv(nf,x,y);
220 28 : case 1: return idealdivexact(nf,x,y);
221 0 : default: pari_err_FLAG("idealdiv");
222 : }
223 : return NULL; /* LCOV_EXCL_LINE */
224 : }
225 :
226 : GEN
227 70 : idealaddtoone0(GEN nf, GEN arg1, GEN arg2)
228 : {
229 70 : if (!arg2) return idealaddmultoone(nf,arg1);
230 35 : return idealaddtoone(nf,arg1,arg2);
231 : }
232 :
233 : /* b not a scalar */
234 : static GEN
235 28 : hnf_Z_ZC(GEN nf, GEN a, GEN b) { return hnfmodid(zk_multable(nf,b), a); }
236 : /* b not a scalar */
237 : static GEN
238 21 : hnf_Z_QC(GEN nf, GEN a, GEN b)
239 : {
240 : GEN db;
241 21 : b = Q_remove_denom(b, &db);
242 21 : if (db) a = mulii(a, db);
243 21 : b = hnf_Z_ZC(nf,a,b);
244 21 : return db? RgM_Rg_div(b, db): b;
245 : }
246 : /* b not a scalar (not point in trying to optimize for this case) */
247 : static GEN
248 28 : hnf_Q_QC(GEN nf, GEN a, GEN b)
249 : {
250 : GEN da, db;
251 28 : if (typ(a) == t_INT) return hnf_Z_QC(nf, a, b);
252 7 : da = gel(a,2);
253 7 : a = gel(a,1);
254 7 : b = Q_remove_denom(b, &db);
255 : /* write da = d*A, db = d*B, gcd(A,B) = 1
256 : * gcd(a/(d A), b/(d B)) = gcd(a B, A b) / A B d = gcd(a B, b) / A B d */
257 7 : if (db)
258 : {
259 7 : GEN d = gcdii(da,db);
260 7 : if (!is_pm1(d)) db = diviiexact(db,d); /* B */
261 7 : if (!is_pm1(db))
262 : {
263 7 : a = mulii(a, db); /* a B */
264 7 : da = mulii(da, db); /* A B d = lcm(denom(a),denom(b)) */
265 : }
266 : }
267 7 : return RgM_Rg_div(hnf_Z_ZC(nf,a,b), da);
268 : }
269 : static GEN
270 7 : hnf_QC_QC(GEN nf, GEN a, GEN b)
271 : {
272 : GEN da, db, d, x;
273 7 : a = Q_remove_denom(a, &da);
274 7 : b = Q_remove_denom(b, &db);
275 7 : if (da) b = ZC_Z_mul(b, da);
276 7 : if (db) a = ZC_Z_mul(a, db);
277 7 : d = mul_denom(da, db);
278 7 : a = zk_multable(nf,a); da = zkmultable_capZ(a);
279 7 : b = zk_multable(nf,b); db = zkmultable_capZ(b);
280 7 : x = ZM_hnfmodid(shallowconcat(a,b), gcdii(da,db));
281 7 : return d? RgM_Rg_div(x, d): x;
282 : }
283 : static GEN
284 21 : hnf_Q_Q(GEN nf, GEN a, GEN b) {return scalarmat(Q_gcd(a,b), nf_get_degree(nf));}
285 : GEN
286 147 : idealhnf0(GEN nf, GEN a, GEN b)
287 : {
288 : long ta, tb;
289 : pari_sp av;
290 : GEN x;
291 147 : nf = checknf(nf);
292 147 : if (!b) return idealhnf(nf,a);
293 :
294 : /* HNF of aZ_K+bZ_K */
295 63 : av = avma;
296 63 : a = nf_to_scalar_or_basis(nf,a); ta = typ(a);
297 63 : b = nf_to_scalar_or_basis(nf,b); tb = typ(b);
298 56 : if (ta == t_COL)
299 14 : x = (tb==t_COL)? hnf_QC_QC(nf, a,b): hnf_Q_QC(nf, b,a);
300 : else
301 42 : x = (tb==t_COL)? hnf_Q_QC(nf, a,b): hnf_Q_Q(nf, a,b);
302 56 : return gerepileupto(av, x);
303 : }
304 :
305 : /*******************************************************************/
306 : /* */
307 : /* TWO-ELEMENT FORM */
308 : /* */
309 : /*******************************************************************/
310 : static GEN idealapprfact_i(GEN nf, GEN x, int nored);
311 :
312 : static int
313 431027 : ok_elt(GEN x, GEN xZ, GEN y)
314 : {
315 431027 : pari_sp av = avma;
316 431027 : return gc_bool(av, ZM_equal(x, ZM_hnfmodid(y, xZ)));
317 : }
318 :
319 : static GEN
320 135704 : addmul_col(GEN a, long s, GEN b)
321 : {
322 : long i,l;
323 135704 : if (!s) return a? leafcopy(a): a;
324 135409 : if (!a) return gmulsg(s,b);
325 125019 : l = lg(a);
326 602924 : for (i=1; i<l; i++)
327 477907 : if (signe(gel(b,i))) gel(a,i) = addii(gel(a,i), mulsi(s, gel(b,i)));
328 125017 : return a;
329 : }
330 :
331 : /* a <-- a + s * b, all coeffs integers */
332 : static GEN
333 68433 : addmul_mat(GEN a, long s, GEN b)
334 : {
335 : long j,l;
336 : /* copy otherwise next call corrupts a */
337 68433 : if (!s) return a? RgM_shallowcopy(a): a;
338 63995 : if (!a) return gmulsg(s,b);
339 32431 : l = lg(a);
340 144889 : for (j=1; j<l; j++)
341 112458 : (void)addmul_col(gel(a,j), s, gel(b,j));
342 32431 : return a;
343 : }
344 :
345 : static GEN
346 260249 : get_random_a(GEN nf, GEN x, GEN xZ)
347 : {
348 : pari_sp av;
349 260249 : long i, lm, l = lg(x);
350 : GEN a, z, beta, mul;
351 :
352 260249 : beta= cgetg(l, t_VEC);
353 260249 : mul = cgetg(l, t_VEC); lm = 1; /* = lg(mul) */
354 : /* look for a in x such that a O/xZ = x O/xZ */
355 468055 : for (i = 2; i < l; i++)
356 : {
357 457665 : GEN xi = gel(x,i);
358 457665 : GEN t = FpM_red(zk_multable(nf,xi), xZ); /* ZM, cannot be a scalar */
359 457662 : if (gequal0(t)) continue;
360 399463 : if (ok_elt(x,xZ, t)) return xi;
361 149604 : gel(beta,lm) = xi;
362 : /* mul[i] = { canonical generators for x[i] O/xZ as Z-module } */
363 149604 : gel(mul,lm) = t; lm++;
364 : }
365 10390 : setlg(mul, lm);
366 10390 : setlg(beta,lm);
367 10390 : z = cgetg(lm, t_VECSMALL);
368 31668 : for(av = avma;; set_avma(av))
369 : {
370 100101 : for (a=NULL,i=1; i<lm; i++)
371 : {
372 68433 : long t = random_bits(4) - 7; /* in [-7,8] */
373 68433 : z[i] = t;
374 68433 : a = addmul_mat(a, t, gel(mul,i));
375 : }
376 : /* a = matrix (NOT HNF) of ideal generated by beta.z in O/xZ */
377 31668 : if (a && ok_elt(x,xZ, a)) break;
378 : }
379 33636 : for (a=NULL,i=1; i<lm; i++)
380 23246 : a = addmul_col(a, z[i], gel(beta,i));
381 10390 : return a;
382 : }
383 :
384 : /* x square matrix, assume it is HNF */
385 : static GEN
386 393920 : mat_ideal_two_elt(GEN nf, GEN x)
387 : {
388 : GEN y, a, cx, xZ;
389 393920 : long N = nf_get_degree(nf);
390 : pari_sp av, tetpil;
391 :
392 393920 : if (lg(x)-1 != N) pari_err_DIM("idealtwoelt");
393 393906 : if (N == 2) return mkvec2copy(gcoeff(x,1,1), gel(x,2));
394 :
395 279944 : y = cgetg(3,t_VEC); av = avma;
396 279944 : cx = Q_content(x);
397 279945 : xZ = gcoeff(x,1,1);
398 279945 : if (gequal(xZ, cx)) /* x = (cx) */
399 : {
400 4819 : gel(y,1) = cx;
401 4819 : gel(y,2) = gen_0; return y;
402 : }
403 275124 : if (equali1(cx)) cx = NULL;
404 : else
405 : {
406 1526 : x = Q_div_to_int(x, cx);
407 1526 : xZ = gcoeff(x,1,1);
408 : }
409 275124 : if (N < 6)
410 245403 : a = get_random_a(nf, x, xZ);
411 : else
412 : {
413 29721 : const long FB[] = { _evallg(15+1) | evaltyp(t_VECSMALL),
414 : 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47
415 : };
416 29721 : GEN P, E, a1 = Z_lsmoothen(xZ, (GEN)FB, &P, &E);
417 29721 : if (!a1) /* factors completely */
418 14876 : a = idealapprfact_i(nf, idealfactor(nf,x), 1);
419 14845 : else if (lg(P) == 1) /* no small factors */
420 10859 : a = get_random_a(nf, x, xZ);
421 : else /* general case */
422 : {
423 : GEN A0, A1, a0, u0, u1, v0, v1, pi0, pi1, t, u;
424 3986 : a0 = diviiexact(xZ, a1);
425 3986 : A0 = ZM_hnfmodid(x, a0); /* smooth part of x */
426 3986 : A1 = ZM_hnfmodid(x, a1); /* cofactor */
427 3986 : pi0 = idealapprfact_i(nf, idealfactor(nf,A0), 1);
428 3986 : pi1 = get_random_a(nf, A1, a1);
429 3986 : (void)bezout(a0, a1, &v0,&v1);
430 3986 : u0 = mulii(a0, v0);
431 3986 : u1 = mulii(a1, v1);
432 3986 : if (typ(pi0) != t_COL) t = addmulii(u0, pi0, u1);
433 : else
434 3986 : { t = ZC_Z_mul(pi0, u1); gel(t,1) = addii(gel(t,1), u0); }
435 3986 : u = ZC_Z_mul(pi1, u0); gel(u,1) = addii(gel(u,1), u1);
436 3986 : a = nfmuli(nf, centermod(u, xZ), centermod(t, xZ));
437 : }
438 : }
439 275126 : if (cx)
440 : {
441 1526 : a = centermod(a, xZ);
442 1526 : tetpil = avma;
443 1526 : if (typ(cx) == t_INT)
444 : {
445 70 : gel(y,1) = mulii(xZ, cx);
446 70 : gel(y,2) = ZC_Z_mul(a, cx);
447 : }
448 : else
449 : {
450 1456 : gel(y,1) = gmul(xZ, cx);
451 1456 : gel(y,2) = RgC_Rg_mul(a, cx);
452 : }
453 : }
454 : else
455 : {
456 273600 : tetpil = avma;
457 273600 : gel(y,1) = icopy(xZ);
458 273600 : gel(y,2) = centermod(a, xZ);
459 : }
460 275123 : gerepilecoeffssp(av,tetpil,y+1,2); return y;
461 : }
462 :
463 : /* Given an ideal x, returns [a,alpha] such that a is in Q,
464 : * x = a Z_K + alpha Z_K, alpha in K^*
465 : * a = 0 or alpha = 0 are possible, but do not try to determine whether
466 : * x is principal. */
467 : GEN
468 97874 : idealtwoelt(GEN nf, GEN x)
469 : {
470 : pari_sp av;
471 : GEN z;
472 97874 : long tx = idealtyp(&x,&z);
473 97868 : nf = checknf(nf);
474 97868 : if (tx == id_MAT) return mat_ideal_two_elt(nf,x);
475 1029 : if (tx == id_PRIME) return mkvec2copy(gel(x,1), gel(x,2));
476 : /* id_PRINCIPAL */
477 1022 : av = avma; x = nf_to_scalar_or_basis(nf, x);
478 1848 : return gerepilecopy(av, typ(x)==t_COL? mkvec2(gen_0,x):
479 917 : mkvec2(Q_abs_shallow(x),gen_0));
480 : }
481 :
482 : /*******************************************************************/
483 : /* */
484 : /* FACTORIZATION */
485 : /* */
486 : /*******************************************************************/
487 : /* x integral ideal in HNF, Zval = v_p(x \cap Z) > 0; return v_p(Nx) */
488 : static long
489 737310 : idealHNF_norm_pval(GEN x, GEN p, long Zval)
490 : {
491 737310 : long i, v = Zval, l = lg(x);
492 2350618 : for (i = 2; i < l; i++) v += Z_pval(gcoeff(x,i,i), p);
493 737310 : return v;
494 : }
495 :
496 : /* x integral in HNF, f0 = partial factorization of a multiple of
497 : * x[1,1] = x\cap Z */
498 : GEN
499 269997 : idealHNF_Z_factor_i(GEN x, GEN f0, GEN *pvN, GEN *pvZ)
500 : {
501 269997 : GEN P, E, vN, vZ, xZ = gcoeff(x,1,1), f = f0? f0: Z_factor(xZ);
502 : long i, l;
503 270013 : P = gel(f,1); l = lg(P);
504 270013 : E = gel(f,2);
505 270013 : *pvN = vN = cgetg(l, t_VECSMALL);
506 270019 : *pvZ = vZ = cgetg(l, t_VECSMALL);
507 703031 : for (i = 1; i < l; i++)
508 : {
509 433008 : GEN p = gel(P,i);
510 433008 : vZ[i] = f0? Z_pval(xZ, p): (long) itou(gel(E,i));
511 433009 : vN[i] = idealHNF_norm_pval(x,p, vZ[i]);
512 : }
513 270023 : return P;
514 : }
515 : /* return P, primes dividing Nx and xZ = x\cap Z, set v_p(Nx), v_p(xZ);
516 : * x integral in HNF */
517 : GEN
518 0 : idealHNF_Z_factor(GEN x, GEN *pvN, GEN *pvZ)
519 0 : { return idealHNF_Z_factor_i(x, NULL, pvN, pvZ); }
520 :
521 : /* v_P(A)*f(P) <= Nval [e.g. Nval = v_p(Norm A)], Zval = v_p(A \cap Z).
522 : * Return v_P(A) */
523 : static long
524 889859 : idealHNF_val(GEN A, GEN P, long Nval, long Zval)
525 : {
526 889859 : long f = pr_get_f(P), vmax, v, e, i, j, k, l;
527 : GEN mul, B, a, y, r, p, pk, cx, vals;
528 : pari_sp av;
529 :
530 889857 : if (Nval < f) return 0;
531 889318 : p = pr_get_p(P);
532 889321 : e = pr_get_e(P);
533 : /* v_P(A) <= max [ e * v_p(A \cap Z), floor[v_p(Nix) / f ] */
534 889321 : vmax = minss(Zval * e, Nval / f);
535 889323 : mul = pr_get_tau(P);
536 889320 : l = lg(mul);
537 889320 : B = cgetg(l,t_MAT);
538 : /* B[1] not needed: v_pr(A[1]) = v_pr(A \cap Z) is known already */
539 889695 : gel(B,1) = gen_0; /* dummy */
540 2694464 : for (j = 2; j < l; j++)
541 : {
542 2031230 : GEN x = gel(A,j);
543 2031230 : gel(B,j) = y = cgetg(l, t_COL);
544 17116153 : for (i = 1; i < l; i++)
545 : { /* compute a = (x.t0)_i, A in HNF ==> x[j+1..l-1] = 0 */
546 15311384 : a = mulii(gel(x,1), gcoeff(mul,i,1));
547 141302750 : for (k = 2; k <= j; k++) a = addii(a, mulii(gel(x,k), gcoeff(mul,i,k)));
548 : /* p | a ? */
549 15311036 : gel(y,i) = dvmdii(a,p,&r); if (signe(r)) return 0;
550 : }
551 : }
552 663234 : vals = cgetg(l, t_VECSMALL);
553 : /* vals[1] not needed */
554 2292031 : for (j = 2; j < l; j++)
555 : {
556 1628743 : gel(B,j) = Q_primitive_part(gel(B,j), &cx);
557 1628752 : vals[j] = cx? 1 + e * Q_pval(cx, p): 1;
558 : }
559 663288 : pk = powiu(p, ceildivuu(vmax, e));
560 663275 : av = avma; y = cgetg(l,t_COL);
561 : /* can compute mod p^ceil((vmax-v)/e) */
562 1342817 : for (v = 1; v < vmax; v++)
563 : { /* we know v_pr(Bj) >= v for all j */
564 704910 : if (e == 1 || (vmax - v) % e == 0) pk = diviiexact(pk, p);
565 3990785 : for (j = 2; j < l; j++)
566 : {
567 3311267 : GEN x = gel(B,j); if (v < vals[j]) continue;
568 20767913 : for (i = 1; i < l; i++)
569 : {
570 18817624 : pari_sp av2 = avma;
571 18817624 : a = mulii(gel(x,1), gcoeff(mul,i,1));
572 299764634 : for (k = 2; k < l; k++) a = addii(a, mulii(gel(x,k), gcoeff(mul,i,k)));
573 : /* a = (x.t_0)_i; p | a ? */
574 18816666 : a = dvmdii(a,p,&r); if (signe(r)) return v;
575 18791214 : if (lgefint(a) > lgefint(pk)) a = remii(a, pk);
576 18791211 : gel(y,i) = gerepileuptoint(av2, a);
577 : }
578 1950289 : gel(B,j) = y; y = x;
579 1950289 : if (gc_needed(av,3))
580 : {
581 0 : if(DEBUGMEM>1) pari_warn(warnmem,"idealval");
582 0 : gerepileall(av,3, &y,&B,&pk);
583 : }
584 : }
585 : }
586 637907 : return v;
587 : }
588 : /* true nf, x != 0 integral ideal in HNF, cx t_INT or NULL,
589 : * FA integer factorization matrix or NULL. Return partial factorization of
590 : * cx * x above primes in FA (complete factorization if !FA)*/
591 : static GEN
592 269994 : idealHNF_factor_i(GEN nf, GEN x, GEN cx, GEN FA)
593 : {
594 269994 : const long N = lg(x)-1;
595 : long i, j, k, l, v;
596 269994 : GEN vN, vZ, vP, vE, vp = idealHNF_Z_factor_i(x, FA, &vN,&vZ);
597 :
598 270022 : l = lg(vp);
599 270022 : i = cx? expi(cx)+1: 1;
600 270023 : vP = cgetg((l+i-2)*N+1, t_COL);
601 270021 : vE = cgetg((l+i-2)*N+1, t_COL);
602 703020 : for (i = k = 1; i < l; i++)
603 : {
604 433008 : GEN L, p = gel(vp,i);
605 433008 : long Nval = vN[i], Zval = vZ[i], vc = cx? Z_pvalrem(cx,p,&cx): 0;
606 433007 : if (vc)
607 : {
608 43009 : L = idealprimedec(nf,p);
609 43008 : if (is_pm1(cx)) cx = NULL;
610 : }
611 : else
612 389998 : L = idealprimedec_limit_f(nf,p,Nval);
613 1018550 : for (j = 1; Nval && j < lg(L); j++) /* !Nval => only cx contributes */
614 : {
615 585553 : GEN P = gel(L,j);
616 585553 : pari_sp av = avma;
617 585553 : v = idealHNF_val(x, P, Nval, Zval);
618 585528 : set_avma(av);
619 585529 : Nval -= v*pr_get_f(P);
620 585532 : v += vc * pr_get_e(P); if (!v) continue;
621 477923 : gel(vP,k) = P;
622 477923 : gel(vE,k) = utoipos(v); k++;
623 : }
624 476154 : if (vc) for (; j<lg(L); j++)
625 : {
626 43164 : GEN P = gel(L,j);
627 43164 : gel(vP,k) = P;
628 43164 : gel(vE,k) = utoipos(vc * pr_get_e(P)); k++;
629 : }
630 : }
631 270012 : if (cx && !FA)
632 : { /* complete factorization */
633 59717 : GEN f = Z_factor(cx), cP = gel(f,1), cE = gel(f,2);
634 59717 : long lc = lg(cP);
635 126761 : for (i=1; i<lc; i++)
636 : {
637 67046 : GEN p = gel(cP,i), L = idealprimedec(nf,p);
638 67044 : long vc = itos(gel(cE,i));
639 144000 : for (j=1; j<lg(L); j++)
640 : {
641 76956 : GEN P = gel(L,j);
642 76956 : gel(vP,k) = P;
643 76956 : gel(vE,k) = utoipos(vc * pr_get_e(P)); k++;
644 : }
645 : }
646 : }
647 270010 : setlg(vP, k);
648 270011 : setlg(vE, k); return mkmat2(vP, vE);
649 : }
650 : /* true nf, x integral ideal */
651 : static GEN
652 225091 : idealHNF_factor(GEN nf, GEN x, ulong lim)
653 : {
654 225091 : GEN cx, F = NULL;
655 225091 : if (lim)
656 : {
657 : GEN P, E;
658 : long i;
659 : /* strict useless because of prime table */
660 42 : F = absZ_factor_limit(gcoeff(x,1,1), lim);
661 42 : P = gel(F,1);
662 42 : E = gel(F,2);
663 : /* filter out entries > lim */
664 77 : for (i = lg(P)-1; i; i--)
665 77 : if (cmpiu(gel(P,i), lim) <= 0) break;
666 42 : setlg(P, i+1);
667 42 : setlg(E, i+1);
668 : }
669 225091 : x = Q_primitive_part(x, &cx);
670 225072 : return idealHNF_factor_i(nf, x, cx, F);
671 : }
672 : /* c * vector(#L,i,L[i].e), assume results fit in ulong */
673 : static GEN
674 32556 : prV_e_muls(GEN L, long c)
675 : {
676 32556 : long j, l = lg(L);
677 32556 : GEN z = cgetg(l, t_COL);
678 66438 : for (j = 1; j < l; j++) gel(z,j) = stoi(c * pr_get_e(gel(L,j)));
679 32575 : return z;
680 : }
681 : /* true nf, y in Q */
682 : static GEN
683 25257 : Q_nffactor(GEN nf, GEN y, ulong lim)
684 : {
685 : GEN f, P, E;
686 : long l, i;
687 25257 : if (typ(y) == t_INT)
688 : {
689 25233 : if (!signe(y)) pari_err_DOMAIN("idealfactor", "ideal", "=",gen_0,y);
690 25219 : if (is_pm1(y)) return trivial_fact();
691 : }
692 17054 : y = Q_abs_shallow(y);
693 17065 : if (!lim) f = Q_factor(y);
694 : else
695 : {
696 36 : f = Q_factor_limit(y, lim);
697 35 : P = gel(f,1);
698 35 : E = gel(f,2);
699 77 : for (i = lg(P)-1; i > 0; i--)
700 63 : if (abscmpiu(gel(P,i), lim) < 0) break;
701 35 : setlg(P,i+1); setlg(E,i+1);
702 : }
703 17051 : P = gel(f,1); l = lg(P); if (l == 1) return f;
704 17037 : E = gel(f,2);
705 49639 : for (i = 1; i < l; i++)
706 : {
707 32578 : gel(P,i) = idealprimedec(nf, gel(P,i));
708 32543 : gel(E,i) = prV_e_muls(gel(P,i), itos(gel(E,i)));
709 : }
710 17061 : P = shallowconcat1(P); gel(f,1) = P; settyp(P, t_COL);
711 17077 : E = shallowconcat1(E); gel(f,2) = E; return f;
712 : }
713 :
714 : GEN
715 3738 : idealfactor_partial(GEN nf, GEN x, GEN L)
716 : {
717 3738 : pari_sp av = avma;
718 : long i, j, l;
719 : GEN P, E;
720 3738 : if (!L) return idealfactor(nf, x);
721 2968 : if (typ(L) == t_INT) return idealfactor_limit(nf, x, itou(L));
722 2947 : l = lg(L); if (l == 1) return trivial_fact();
723 2163 : P = cgetg(l, t_VEC);
724 4466 : for (i = 1; i < l; i++)
725 : {
726 2303 : GEN p = gel(L,i);
727 2303 : gel(P,i) = typ(p) == t_INT? idealprimedec(nf, p): mkvec(p);
728 : }
729 2163 : P = shallowconcat1(P); settyp(P, t_COL);
730 2163 : P = gen_sort_uniq(P, (void*)&cmp_prime_ideal, &cmp_nodata);
731 2163 : E = cgetg_copy(P, &l);
732 12194 : for (i = j = 1; i < l; i++)
733 : {
734 10031 : long v = idealval(nf, x, gel(P,i));
735 10031 : if (v) { gel(P,j) = gel(P,i); gel(E,j) = stoi(v); j++; }
736 : }
737 2163 : setlg(P,j);
738 2163 : setlg(E,j); return gerepilecopy(av, mkmat2(P, E));
739 : }
740 : GEN
741 250392 : idealfactor_limit(GEN nf, GEN x, ulong lim)
742 : {
743 250392 : pari_sp av = avma;
744 : GEN fa, y;
745 250392 : long tx = idealtyp(&x,&y);
746 :
747 250398 : if (tx == id_PRIME)
748 : {
749 63 : if (lim && abscmpiu(pr_get_p(x), lim) >= 0) return trivial_fact();
750 56 : retmkmat2(mkcolcopy(x), mkcol(gen_1));
751 : }
752 250335 : nf = checknf(nf);
753 250331 : if (tx == id_PRINCIPAL)
754 : {
755 26633 : y = nf_to_scalar_or_basis(nf, x);
756 26637 : if (typ(y) != t_COL) return gerepilecopy(av, Q_nffactor(nf, y, lim));
757 : }
758 225068 : y = idealnumden(nf, x);
759 225080 : fa = idealHNF_factor(nf, gel(y,1), lim);
760 225077 : if (!isint1(gel(y,2)))
761 14 : fa = famat_div_shallow(fa, idealHNF_factor(nf, gel(y,2), lim));
762 225077 : fa = gerepilecopy(av, fa);
763 225085 : return sort_factor(fa, (void*)&cmp_prime_ideal, &cmp_nodata);
764 : }
765 : GEN
766 250212 : idealfactor(GEN nf, GEN x) { return idealfactor_limit(nf, x, 0); }
767 : GEN
768 154 : gpidealfactor(GEN nf, GEN x, GEN lim)
769 : {
770 154 : ulong L = 0;
771 154 : if (lim)
772 : {
773 70 : if (typ(lim) != t_INT || signe(lim) < 0) pari_err_FLAG("idealfactor");
774 70 : L = itou(lim);
775 : }
776 154 : return idealfactor_limit(nf, x, L);
777 : }
778 :
779 : static GEN
780 6664 : ramified_root(GEN nf, GEN R, GEN A, long n)
781 : {
782 6664 : GEN v, P = gel(idealfactor(nf, R), 1);
783 6664 : long i, l = lg(P);
784 6664 : v = cgetg(l, t_VECSMALL);
785 7140 : for (i = 1; i < l; i++)
786 : {
787 483 : long w = idealval(nf, A, gel(P,i));
788 483 : if (w % n) return NULL;
789 476 : v[i] = w / n;
790 : }
791 6657 : return idealfactorback(nf, P, v, 0);
792 : }
793 : static int
794 0 : ramified_root_simple(GEN nf, long n, GEN P, GEN v)
795 : {
796 0 : long i, l = lg(v);
797 0 : for (i = 1; i < l; i++) if (v[i])
798 : {
799 0 : GEN vpr = idealprimedec(nf, gel(P,i));
800 0 : long lpr = lg(vpr), j;
801 0 : for (j = 1; j < lpr; j++)
802 : {
803 0 : long e = pr_get_e(gel(vpr,j));
804 0 : if ((e * v[i]) % n) return 0;
805 : }
806 : }
807 0 : return 1;
808 : }
809 : /* true nf; A is assumed to be the n-th power of an integral ideal,
810 : * return its n-th root; n > 1 */
811 : static long
812 6664 : idealsqrtn_int(GEN nf, GEN A, long n, GEN *pB)
813 : {
814 : GEN C, root;
815 : long i, l;
816 :
817 6664 : if (typ(A) == t_INT) /* > 0 */
818 : {
819 3332 : GEN P = nf_get_ramified_primes(nf), v, q;
820 3332 : l = lg(P); v = cgetg(l, t_VECSMALL);
821 15638 : for (i = 1; i < l; i++) v[i] = Z_pvalrem(A, gel(P,i), &A);
822 3332 : C = gen_1;
823 3332 : if (!isint1(A) && !Z_ispowerall(A, n, pB? &C: NULL)) return 0;
824 3332 : if (!pB) return ramified_root_simple(nf, n, P, v);
825 3332 : q = factorback2(P, v);
826 3332 : root = ramified_root(nf, q, q, n);
827 3332 : if (!root) return 0;
828 3332 : if (!equali1(C)) root = isint1(root)? C: ZM_Z_mul(root, C);
829 3332 : *pB = root; return 1;
830 : }
831 : /* compute valuations at ramified primes */
832 3332 : root = ramified_root(nf, idealadd(nf, nf_get_diff(nf), A), A, n);
833 3332 : if (!root) return 0;
834 : /* remove ramified primes */
835 3325 : if (isint1(root))
836 2891 : root = matid(nf_get_degree(nf));
837 : else
838 434 : A = idealdivexact(nf, A, idealpows(nf,root,n));
839 3325 : A = Q_primitive_part(A, &C);
840 3325 : if (C)
841 : {
842 0 : if (!Z_ispowerall(C,n,&C)) return 0;
843 0 : if (pB) root = ZM_Z_mul(root, C);
844 : }
845 :
846 : /* compute final n-th root, at most degree(nf)-1 iterations */
847 3325 : for (i = 0;; i++)
848 2247 : {
849 5572 : GEN J, b, a = gcoeff(A,1,1); /* A \cap Z */
850 5572 : if (is_pm1(a)) break;
851 2268 : if (!Z_ispowerall(a,n,&b)) return 0;
852 2247 : J = idealadd(nf, b, A);
853 2247 : A = idealdivexact(nf, idealpows(nf,J,n), A);
854 : /* div and not divexact here */
855 2247 : if (pB) root = odd(i)? idealdiv(nf, root, J): idealmul(nf, root, J);
856 : }
857 3304 : if (pB) *pB = root;
858 3304 : return 1;
859 : }
860 :
861 : /* A is assumed to be the n-th power of an ideal in nf
862 : returns its n-th root. */
863 : long
864 3353 : idealispower(GEN nf, GEN A, long n, GEN *pB)
865 : {
866 3353 : pari_sp av = avma;
867 : GEN v, N, D;
868 3353 : nf = checknf(nf);
869 3353 : if (n <= 0) pari_err_DOMAIN("idealispower", "n", "<=", gen_0, stoi(n));
870 3353 : if (n == 1) { if (pB) *pB = idealhnf(nf,A); return 1; }
871 3346 : v = idealnumden(nf,A);
872 3346 : if (gequal0(gel(v,1))) { set_avma(av); if (pB) *pB = cgetg(1,t_MAT); return 1; }
873 3346 : if (!idealsqrtn_int(nf, gel(v,1), n, pB? &N: NULL)) return 0;
874 3318 : if (!idealsqrtn_int(nf, gel(v,2), n, pB? &D: NULL)) return 0;
875 3318 : if (pB) *pB = gerepileupto(av, idealdiv(nf,N,D)); else set_avma(av);
876 3318 : return 1;
877 : }
878 :
879 : /* x t_INT or integral nonzero ideal in HNF */
880 : static GEN
881 95576 : idealredmodpower_i(GEN nf, GEN x, ulong k, ulong B)
882 : {
883 : GEN cx, y, U, N, F, Q;
884 95576 : if (typ(x) == t_INT)
885 : {
886 50044 : if (!signe(x) || is_pm1(x)) return gen_1;
887 1792 : F = Z_factor_limit(x, B);
888 1792 : gel(F,2) = gdiventgs(gel(F,2), k);
889 1792 : return ginv(factorback(F));
890 : }
891 45532 : N = gcoeff(x,1,1); if (is_pm1(N)) return gen_1;
892 44923 : F = absZ_factor_limit_strict(N, B, &U);
893 44924 : if (U)
894 : {
895 182 : GEN M = powii(gel(U,1), gel(U,2));
896 182 : y = hnfmodid(x, M); /* coprime part to B! */
897 182 : if (!idealispower(nf, y, k, &U)) U = NULL;
898 182 : x = hnfmodid(x, diviiexact(N, M));
899 : }
900 : /* x = B-smooth part of initial x */
901 44924 : x = Q_primitive_part(x, &cx);
902 44923 : F = idealHNF_factor_i(nf, x, cx, F);
903 44923 : gel(F,2) = gdiventgs(gel(F,2), k);
904 44923 : Q = idealfactorback(nf, gel(F,1), gel(F,2), 0);
905 44923 : if (U) Q = idealmul(nf,Q,U);
906 44923 : if (typ(Q) == t_INT) return Q;
907 13844 : y = idealred_elt(nf, idealHNF_inv_Z(nf, Q));
908 13844 : return gdiv(y, gcoeff(Q,1,1));
909 : }
910 : GEN
911 47794 : idealredmodpower(GEN nf, GEN x, ulong n, ulong B)
912 : {
913 47794 : pari_sp av = avma;
914 : GEN a, b;
915 47794 : nf = checknf(nf);
916 47794 : if (!n) pari_err_DOMAIN("idealredmodpower","n", "=", gen_0, gen_0);
917 47794 : x = idealnumden(nf, x);
918 47796 : a = gel(x,1);
919 47796 : if (isintzero(a)) { set_avma(av); return gen_1; }
920 47788 : a = idealredmodpower_i(nf, gel(x,1), n, B);
921 47788 : b = idealredmodpower_i(nf, gel(x,2), n, B);
922 47788 : if (!isint1(b)) a = nf_to_scalar_or_basis(nf, nfdiv(nf, a, b));
923 47788 : return gerepilecopy(av, a);
924 : }
925 :
926 : /* P prime ideal in idealprimedec format. Return valuation(A) at P */
927 : long
928 872593 : idealval(GEN nf, GEN A, GEN P)
929 : {
930 872593 : pari_sp av = avma;
931 : GEN a, p, cA;
932 872593 : long vcA, v, Zval, tx = idealtyp(&A,&a);
933 :
934 872588 : if (tx == id_PRINCIPAL) return nfval(nf,A,P);
935 790524 : checkprid(P);
936 790526 : if (tx == id_PRIME) return pr_equal(P, A)? 1: 0;
937 : /* id_MAT */
938 790498 : nf = checknf(nf);
939 790498 : A = Q_primitive_part(A, &cA);
940 790495 : p = pr_get_p(P);
941 790496 : vcA = cA? Q_pval(cA,p): 0;
942 790496 : if (pr_is_inert(P)) return gc_long(av,vcA);
943 778995 : Zval = Z_pval(gcoeff(A,1,1), p);
944 778995 : if (!Zval) v = 0;
945 : else
946 : {
947 304306 : long Nval = idealHNF_norm_pval(A, p, Zval);
948 304307 : v = idealHNF_val(A, P, Nval, Zval);
949 : }
950 778999 : return gc_long(av, vcA? v + vcA*pr_get_e(P): v);
951 : }
952 : GEN
953 6615 : gpidealval(GEN nf, GEN ix, GEN P)
954 : {
955 6615 : long v = idealval(nf,ix,P);
956 6615 : return v == LONG_MAX? mkoo(): stoi(v);
957 : }
958 :
959 : /* gcd and generalized Bezout */
960 :
961 : GEN
962 70630 : idealadd(GEN nf, GEN x, GEN y)
963 : {
964 70630 : pari_sp av = avma;
965 : long tx, ty;
966 : GEN z, a, dx, dy, dz;
967 :
968 70630 : tx = idealtyp(&x,&z);
969 70630 : ty = idealtyp(&y,&z); nf = checknf(nf);
970 70630 : if (tx != id_MAT) x = idealhnf_shallow(nf,x);
971 70630 : if (ty != id_MAT) y = idealhnf_shallow(nf,y);
972 70630 : if (lg(x) == 1) return gerepilecopy(av,y);
973 70504 : if (lg(y) == 1) return gerepilecopy(av,x); /* check for 0 ideal */
974 70049 : dx = Q_denom(x);
975 70049 : dy = Q_denom(y); dz = lcmii(dx,dy);
976 70049 : if (is_pm1(dz)) dz = NULL; else {
977 13048 : x = Q_muli_to_int(x, dz);
978 13048 : y = Q_muli_to_int(y, dz);
979 : }
980 70049 : a = gcdii(gcoeff(x,1,1), gcoeff(y,1,1));
981 70049 : if (is_pm1(a))
982 : {
983 33521 : long N = lg(x)-1;
984 33521 : if (!dz) { set_avma(av); return matid(N); }
985 3773 : return gerepileupto(av, scalarmat(ginv(dz), N));
986 : }
987 36528 : z = ZM_hnfmodid(shallowconcat(x,y), a);
988 36528 : if (dz) z = RgM_Rg_div(z,dz);
989 36528 : return gerepileupto(av,z);
990 : }
991 :
992 : static GEN
993 28 : trivial_merge(GEN x)
994 28 : { return (lg(x) == 1 || !is_pm1(gcoeff(x,1,1)))? NULL: gen_1; }
995 : /* true nf */
996 : static GEN
997 607972 : _idealaddtoone(GEN nf, GEN x, GEN y, long red)
998 : {
999 : GEN a;
1000 607972 : long tx = idealtyp(&x, &a/*junk*/);
1001 607958 : long ty = idealtyp(&y, &a/*junk*/);
1002 : long ea;
1003 607954 : if (tx != id_MAT) x = idealhnf_shallow(nf, x);
1004 607989 : if (ty != id_MAT) y = idealhnf_shallow(nf, y);
1005 607989 : if (lg(x) == 1)
1006 14 : a = trivial_merge(y);
1007 607975 : else if (lg(y) == 1)
1008 14 : a = trivial_merge(x);
1009 : else
1010 607961 : a = hnfmerge_get_1(x, y);
1011 607979 : if (!a) pari_err_COPRIME("idealaddtoone",x,y);
1012 607959 : if (red && (ea = gexpo(a)) > 10)
1013 : {
1014 5572 : GEN b = (typ(a) == t_COL)? a: scalarcol_shallow(a, nf_get_degree(nf));
1015 5572 : b = ZC_reducemodlll(b, idealHNF_mul(nf,x,y));
1016 5572 : if (gexpo(b) < ea) a = b;
1017 : }
1018 607959 : return a;
1019 : }
1020 : /* true nf */
1021 : GEN
1022 18144 : idealaddtoone_i(GEN nf, GEN x, GEN y)
1023 18144 : { return _idealaddtoone(nf, x, y, 1); }
1024 : /* true nf */
1025 : GEN
1026 589826 : idealaddtoone_raw(GEN nf, GEN x, GEN y)
1027 589826 : { return _idealaddtoone(nf, x, y, 0); }
1028 :
1029 : GEN
1030 98 : idealaddtoone(GEN nf, GEN x, GEN y)
1031 : {
1032 98 : GEN z = cgetg(3,t_VEC), a;
1033 98 : pari_sp av = avma;
1034 98 : nf = checknf(nf);
1035 98 : a = gerepileupto(av, idealaddtoone_i(nf,x,y));
1036 84 : gel(z,1) = a;
1037 84 : gel(z,2) = typ(a) == t_COL? Z_ZC_sub(gen_1,a): subui(1,a);
1038 84 : return z;
1039 : }
1040 :
1041 : /* assume elements of list are integral ideals */
1042 : GEN
1043 35 : idealaddmultoone(GEN nf, GEN list)
1044 : {
1045 35 : pari_sp av = avma;
1046 35 : long N, i, l, nz, tx = typ(list);
1047 : GEN H, U, perm, L;
1048 :
1049 35 : nf = checknf(nf); N = nf_get_degree(nf);
1050 35 : if (!is_vec_t(tx)) pari_err_TYPE("idealaddmultoone",list);
1051 35 : l = lg(list);
1052 35 : L = cgetg(l, t_VEC);
1053 35 : if (l == 1)
1054 0 : pari_err_DOMAIN("idealaddmultoone", "sum(ideals)", "!=", gen_1, L);
1055 35 : nz = 0; /* number of nonzero ideals in L */
1056 98 : for (i=1; i<l; i++)
1057 : {
1058 70 : GEN I = gel(list,i);
1059 70 : if (typ(I) != t_MAT) I = idealhnf_shallow(nf,I);
1060 70 : if (lg(I) != 1)
1061 : {
1062 42 : nz++; RgM_check_ZM(I,"idealaddmultoone");
1063 35 : if (lgcols(I) != N+1) pari_err_TYPE("idealaddmultoone [not an ideal]", I);
1064 : }
1065 63 : gel(L,i) = I;
1066 : }
1067 28 : H = ZM_hnfperm(shallowconcat1(L), &U, &perm);
1068 28 : if (lg(H) == 1 || !equali1(gcoeff(H,1,1)))
1069 7 : pari_err_DOMAIN("idealaddmultoone", "sum(ideals)", "!=", gen_1, L);
1070 49 : for (i=1; i<=N; i++)
1071 49 : if (perm[i] == 1) break;
1072 21 : U = gel(U,(nz-1)*N + i); /* (L[1]|...|L[nz]) U = 1 */
1073 21 : nz = 0;
1074 63 : for (i=1; i<l; i++)
1075 : {
1076 42 : GEN c = gel(L,i);
1077 42 : if (lg(c) == 1)
1078 14 : c = gen_0;
1079 : else {
1080 28 : c = ZM_ZC_mul(c, vecslice(U, nz*N + 1, (nz+1)*N));
1081 28 : nz++;
1082 : }
1083 42 : gel(L,i) = c;
1084 : }
1085 21 : return gerepilecopy(av, L);
1086 : }
1087 :
1088 : /* multiplication */
1089 :
1090 : /* x integral ideal (without archimedean component) in HNF form
1091 : * y = [a,alpha] corresponds to the integral ideal aZ_K+alpha Z_K, a in Z,
1092 : * alpha a ZV or a ZM (multiplication table). Multiply them */
1093 : static GEN
1094 2909837 : idealHNF_mul_two(GEN nf, GEN x, GEN y)
1095 : {
1096 2909837 : GEN m, a = gel(y,1), alpha = gel(y,2);
1097 : long i, N;
1098 :
1099 2909837 : if (typ(alpha) != t_MAT)
1100 : {
1101 2595093 : alpha = zk_scalar_or_multable(nf, alpha);
1102 2595097 : if (typ(alpha) == t_INT) /* e.g. y inert ? 0 should not (but may) occur */
1103 18309 : return signe(a)? ZM_Z_mul(x, gcdii(a, alpha)): cgetg(1,t_MAT);
1104 : }
1105 2891532 : N = lg(x)-1; m = cgetg((N<<1)+1,t_MAT);
1106 10813820 : for (i=1; i<=N; i++) gel(m,i) = ZM_ZC_mul(alpha,gel(x,i));
1107 10812536 : for (i=1; i<=N; i++) gel(m,i+N) = ZC_Z_mul(gel(x,i), a);
1108 2890493 : return ZM_hnfmodid(m, mulii(a, gcoeff(x,1,1)));
1109 : }
1110 :
1111 : /* Assume x and y are integral in HNF form [NOT extended]. Not memory clean.
1112 : * HACK: ideal in y can be of the form [a,b], a in Z, b in Z_K */
1113 : GEN
1114 963930 : idealHNF_mul(GEN nf, GEN x, GEN y)
1115 : {
1116 : GEN z;
1117 963930 : if (typ(y) == t_VEC)
1118 677641 : z = idealHNF_mul_two(nf,x,y);
1119 : else
1120 : { /* reduce one ideal to two-elt form. The smallest */
1121 286289 : GEN xZ = gcoeff(x,1,1), yZ = gcoeff(y,1,1);
1122 286289 : if (cmpii(xZ, yZ) < 0)
1123 : {
1124 116576 : if (is_pm1(xZ)) return gcopy(y);
1125 101012 : z = idealHNF_mul_two(nf, y, mat_ideal_two_elt(nf,x));
1126 : }
1127 : else
1128 : {
1129 169714 : if (is_pm1(yZ)) return gcopy(x);
1130 110877 : z = idealHNF_mul_two(nf, x, mat_ideal_two_elt(nf,y));
1131 : }
1132 : }
1133 889529 : return z;
1134 : }
1135 :
1136 : /* operations on elements in factored form */
1137 :
1138 : GEN
1139 229866 : famat_mul_shallow(GEN f, GEN g)
1140 : {
1141 229866 : if (typ(f) != t_MAT) f = to_famat_shallow(f,gen_1);
1142 229866 : if (typ(g) != t_MAT) g = to_famat_shallow(g,gen_1);
1143 229866 : if (lgcols(f) == 1) return g;
1144 178473 : if (lgcols(g) == 1) return f;
1145 176845 : return mkmat2(shallowconcat(gel(f,1), gel(g,1)),
1146 176842 : shallowconcat(gel(f,2), gel(g,2)));
1147 : }
1148 : GEN
1149 84105 : famat_mulpow_shallow(GEN f, GEN g, GEN e)
1150 : {
1151 84105 : if (!signe(e)) return f;
1152 56555 : return famat_mul_shallow(f, famat_pow_shallow(g, e));
1153 : }
1154 :
1155 : GEN
1156 145948 : famat_mulpows_shallow(GEN f, GEN g, long e)
1157 : {
1158 145948 : if (e==0) return f;
1159 122536 : return famat_mul_shallow(f, famat_pows_shallow(g, e));
1160 : }
1161 :
1162 : GEN
1163 9772 : famat_div_shallow(GEN f, GEN g)
1164 9772 : { return famat_mul_shallow(f, famat_inv_shallow(g)); }
1165 :
1166 : GEN
1167 0 : to_famat(GEN x, GEN y) { retmkmat2(mkcolcopy(x), mkcolcopy(y)); }
1168 : GEN
1169 1281781 : to_famat_shallow(GEN x, GEN y) { return mkmat2(mkcol(x), mkcol(y)); }
1170 :
1171 : /* concat the single elt x; not gconcat since x may be a t_COL */
1172 : static GEN
1173 16315 : append(GEN v, GEN x)
1174 : {
1175 16315 : long i, l = lg(v);
1176 16315 : GEN w = cgetg(l+1, typ(v));
1177 119174 : for (i=1; i<l; i++) gel(w,i) = gcopy(gel(v,i));
1178 16315 : gel(w,i) = gcopy(x); return w;
1179 : }
1180 : /* add x^1 to famat f */
1181 : static GEN
1182 42178 : famat_add(GEN f, GEN x)
1183 : {
1184 42178 : GEN h = cgetg(3,t_MAT);
1185 42178 : if (lgcols(f) == 1)
1186 : {
1187 25947 : gel(h,1) = mkcolcopy(x);
1188 25947 : gel(h,2) = mkcol(gen_1);
1189 : }
1190 : else
1191 : {
1192 16231 : gel(h,1) = append(gel(f,1), x);
1193 16231 : gel(h,2) = gconcat(gel(f,2), gen_1);
1194 : }
1195 42178 : return h;
1196 : }
1197 : /* add x^-1 to famat f */
1198 : static GEN
1199 84 : famat_sub(GEN f, GEN x)
1200 : {
1201 84 : GEN h = cgetg(3,t_MAT);
1202 84 : if (lgcols(f) == 1)
1203 : {
1204 0 : gel(h,1) = mkcolcopy(x);
1205 0 : gel(h,2) = mkcol(gen_m1);
1206 : }
1207 : else
1208 : {
1209 84 : gel(h,1) = append(gel(f,1), x);
1210 84 : gel(h,2) = gconcat(gel(f,2), gen_m1);
1211 : }
1212 84 : return h;
1213 : }
1214 :
1215 : GEN
1216 72636 : famat_mul(GEN f, GEN g)
1217 : {
1218 : GEN h;
1219 72636 : if (typ(g) != t_MAT) {
1220 42136 : if (typ(f) == t_MAT) return famat_add(f, g);
1221 0 : h = cgetg(3, t_MAT);
1222 0 : gel(h,1) = mkcol2(gcopy(f), gcopy(g));
1223 0 : gel(h,2) = mkcol2(gen_1, gen_1);
1224 : }
1225 30500 : if (typ(f) != t_MAT) return famat_add(g, f);
1226 30458 : if (lgcols(f) == 1) return gcopy(g);
1227 10035 : if (lgcols(g) == 1) return gcopy(f);
1228 7129 : h = cgetg(3,t_MAT);
1229 7129 : gel(h,1) = gconcat(gel(f,1), gel(g,1));
1230 7129 : gel(h,2) = gconcat(gel(f,2), gel(g,2));
1231 7129 : return h;
1232 : }
1233 :
1234 : GEN
1235 91 : famat_div(GEN f, GEN g)
1236 : {
1237 : GEN h;
1238 91 : if (typ(g) != t_MAT) {
1239 42 : if (typ(f) == t_MAT) return famat_sub(f, g);
1240 0 : h = cgetg(3, t_MAT);
1241 0 : gel(h,1) = mkcol2(gcopy(f), gcopy(g));
1242 0 : gel(h,2) = mkcol2(gen_1, gen_m1);
1243 : }
1244 49 : if (typ(f) != t_MAT) return famat_sub(g, f);
1245 7 : if (lgcols(f) == 1) return famat_inv(g);
1246 7 : if (lgcols(g) == 1) return gcopy(f);
1247 7 : h = cgetg(3,t_MAT);
1248 7 : gel(h,1) = gconcat(gel(f,1), gel(g,1));
1249 7 : gel(h,2) = gconcat(gel(f,2), gneg(gel(g,2)));
1250 7 : return h;
1251 : }
1252 :
1253 : GEN
1254 14695 : famat_sqr(GEN f)
1255 : {
1256 : GEN h;
1257 14695 : if (typ(f) != t_MAT) return to_famat(f,gen_2);
1258 14695 : if (lgcols(f) == 1) return gcopy(f);
1259 8212 : h = cgetg(3,t_MAT);
1260 8212 : gel(h,1) = gcopy(gel(f,1));
1261 8212 : gel(h,2) = gmul2n(gel(f,2),1);
1262 8212 : return h;
1263 : }
1264 :
1265 : GEN
1266 26306 : famat_inv_shallow(GEN f)
1267 : {
1268 26306 : if (typ(f) != t_MAT) return to_famat_shallow(f,gen_m1);
1269 9919 : if (lgcols(f) == 1) return f;
1270 9919 : return mkmat2(gel(f,1), ZC_neg(gel(f,2)));
1271 : }
1272 : GEN
1273 18835 : famat_inv(GEN f)
1274 : {
1275 18835 : if (typ(f) != t_MAT) return to_famat(f,gen_m1);
1276 18835 : if (lgcols(f) == 1) return gcopy(f);
1277 1487 : retmkmat2(gcopy(gel(f,1)), ZC_neg(gel(f,2)));
1278 : }
1279 : GEN
1280 14 : famat_pow(GEN f, GEN n)
1281 : {
1282 14 : if (typ(f) != t_MAT) return to_famat(f,n);
1283 14 : if (lgcols(f) == 1) return gcopy(f);
1284 14 : retmkmat2(gcopy(gel(f,1)), ZC_Z_mul(gel(f,2),n));
1285 : }
1286 : GEN
1287 65597 : famat_pow_shallow(GEN f, GEN n)
1288 : {
1289 65597 : if (is_pm1(n)) return signe(n) > 0? f: famat_inv_shallow(f);
1290 35936 : if (typ(f) != t_MAT) return to_famat_shallow(f,n);
1291 9938 : if (lgcols(f) == 1) return f;
1292 6195 : return mkmat2(gel(f,1), ZC_Z_mul(gel(f,2),n));
1293 : }
1294 :
1295 : GEN
1296 149752 : famat_pows_shallow(GEN f, long n)
1297 : {
1298 149752 : if (n==1) return f;
1299 63266 : if (n==-1) return famat_inv_shallow(f);
1300 63252 : if (typ(f) != t_MAT) return to_famat_shallow(f, stoi(n));
1301 26659 : if (lgcols(f) == 1) return f;
1302 26661 : return mkmat2(gel(f,1), ZC_z_mul(gel(f,2),n));
1303 : }
1304 :
1305 : GEN
1306 0 : famat_Z_gcd(GEN M, GEN n)
1307 : {
1308 0 : pari_sp av=avma;
1309 0 : long i, j, l=lgcols(M);
1310 0 : GEN F=cgetg(3,t_MAT);
1311 0 : gel(F,1)=cgetg(l,t_COL);
1312 0 : gel(F,2)=cgetg(l,t_COL);
1313 0 : for (i=1, j=1; i<l; i++)
1314 : {
1315 0 : GEN p = gcoeff(M,i,1);
1316 0 : GEN e = gminsg(Z_pval(n,p),gcoeff(M,i,2));
1317 0 : if (signe(e))
1318 : {
1319 0 : gcoeff(F,j,1)=p;
1320 0 : gcoeff(F,j,2)=e;
1321 0 : j++;
1322 : }
1323 : }
1324 0 : setlg(gel(F,1),j); setlg(gel(F,2),j);
1325 0 : return gerepilecopy(av,F);
1326 : }
1327 :
1328 : /* x assumed to be a t_MATs (factorization matrix), or compatible with
1329 : * the element_* functions. */
1330 : static GEN
1331 25622 : ext_sqr(GEN nf, GEN x)
1332 25622 : { return (typ(x)==t_MAT)? famat_sqr(x): nfsqr(nf, x); }
1333 : static GEN
1334 84409 : ext_mul(GEN nf, GEN x, GEN y)
1335 84409 : { return (typ(x)==t_MAT)? famat_mul(x,y): nfmul(nf, x, y); }
1336 : static GEN
1337 18835 : ext_inv(GEN nf, GEN x)
1338 18835 : { return (typ(x)==t_MAT)? famat_inv(x): nfinv(nf, x); }
1339 : static GEN
1340 0 : ext_pow(GEN nf, GEN x, GEN n)
1341 0 : { return (typ(x)==t_MAT)? famat_pow(x,n): nfpow(nf, x, n); }
1342 :
1343 : GEN
1344 0 : famat_to_nf(GEN nf, GEN f)
1345 : {
1346 : GEN t, x, e;
1347 : long i;
1348 0 : if (lgcols(f) == 1) return gen_1;
1349 0 : x = gel(f,1);
1350 0 : e = gel(f,2);
1351 0 : t = nfpow(nf, gel(x,1), gel(e,1));
1352 0 : for (i=lg(x)-1; i>1; i--)
1353 0 : t = nfmul(nf, t, nfpow(nf, gel(x,i), gel(e,i)));
1354 0 : return t;
1355 : }
1356 :
1357 : GEN
1358 0 : famat_idealfactor(GEN nf, GEN x)
1359 : {
1360 : long i, l;
1361 0 : GEN g = gel(x,1), e = gel(x,2), h = cgetg_copy(g, &l);
1362 0 : for (i = 1; i < l; i++) gel(h,i) = idealfactor(nf, gel(g,i));
1363 0 : h = famat_reduce(famatV_factorback(h,e));
1364 0 : return sort_factor(h, (void*)&cmp_prime_ideal, &cmp_nodata);
1365 : }
1366 :
1367 : GEN
1368 135639 : famat_reduce(GEN fa)
1369 : {
1370 : GEN E, G, L, g, e;
1371 : long i, k, l;
1372 :
1373 135639 : if (typ(fa) != t_MAT || lgcols(fa) == 1) return fa;
1374 122721 : g = gel(fa,1); l = lg(g);
1375 122721 : e = gel(fa,2);
1376 122721 : L = gen_indexsort(g, (void*)&cmp_universal, &cmp_nodata);
1377 122719 : G = cgetg(l, t_COL);
1378 122717 : E = cgetg(l, t_COL);
1379 : /* merge */
1380 1764454 : for (k=i=1; i<l; i++,k++)
1381 : {
1382 1641730 : gel(G,k) = gel(g,L[i]);
1383 1641730 : gel(E,k) = gel(e,L[i]);
1384 1641730 : if (k > 1 && gidentical(gel(G,k), gel(G,k-1)))
1385 : {
1386 1002390 : gel(E,k-1) = addii(gel(E,k), gel(E,k-1));
1387 1002387 : k--;
1388 : }
1389 : }
1390 : /* kill 0 exponents */
1391 122724 : l = k;
1392 762079 : for (k=i=1; i<l; i++)
1393 639351 : if (!gequal0(gel(E,i)))
1394 : {
1395 629907 : gel(G,k) = gel(G,i);
1396 629907 : gel(E,k) = gel(E,i); k++;
1397 : }
1398 122728 : setlg(G, k);
1399 122722 : setlg(E, k); return mkmat2(G,E);
1400 : }
1401 : GEN
1402 63 : matreduce(GEN f)
1403 63 : { pari_sp av = avma;
1404 63 : switch(typ(f))
1405 : {
1406 21 : case t_VEC: case t_COL:
1407 : {
1408 21 : GEN e; f = vec_reduce(f, &e); settyp(f, t_COL);
1409 21 : return gerepilecopy(av, mkmat2(f, zc_to_ZC(e)));
1410 : }
1411 35 : case t_MAT:
1412 35 : if (lg(f) == 3) break;
1413 : default:
1414 14 : pari_err_TYPE("matreduce", f);
1415 : }
1416 28 : if (typ(gel(f,1)) == t_VECSMALL)
1417 0 : f = famatsmall_reduce(f);
1418 : else
1419 : {
1420 28 : if (!RgV_is_ZV(gel(f,2))) pari_err_TYPE("matreduce",f);
1421 21 : f = famat_reduce(f);
1422 : }
1423 21 : return gerepilecopy(av, f);
1424 : }
1425 :
1426 : GEN
1427 14673 : famatsmall_reduce(GEN fa)
1428 : {
1429 : GEN E, G, L, g, e;
1430 : long i, k, l;
1431 14673 : if (lgcols(fa) == 1) return fa;
1432 14673 : g = gel(fa,1); l = lg(g);
1433 14673 : e = gel(fa,2);
1434 14673 : L = vecsmall_indexsort(g);
1435 14673 : G = cgetg(l, t_VECSMALL);
1436 14673 : E = cgetg(l, t_VECSMALL);
1437 : /* merge */
1438 131197 : for (k=i=1; i<l; i++,k++)
1439 : {
1440 116524 : G[k] = g[L[i]];
1441 116524 : E[k] = e[L[i]];
1442 116524 : if (k > 1 && G[k] == G[k-1])
1443 : {
1444 7073 : E[k-1] += E[k];
1445 7073 : k--;
1446 : }
1447 : }
1448 : /* kill 0 exponents */
1449 14673 : l = k;
1450 124124 : for (k=i=1; i<l; i++)
1451 109451 : if (E[i])
1452 : {
1453 105703 : G[k] = G[i];
1454 105703 : E[k] = E[i]; k++;
1455 : }
1456 14673 : setlg(G, k);
1457 14673 : setlg(E, k); return mkmat2(G,E);
1458 : }
1459 :
1460 : GEN
1461 53985 : famat_remove_trivial(GEN fa)
1462 : {
1463 53985 : GEN P, E, p = gel(fa,1), e = gel(fa,2);
1464 53985 : long j, k, l = lg(p);
1465 53985 : P = cgetg(l, t_COL);
1466 53984 : E = cgetg(l, t_COL);
1467 1632984 : for (j = k = 1; j < l; j++)
1468 1579000 : if (signe(gel(e,j))) { gel(P,k) = gel(p,j); gel(E,k++) = gel(e,j); }
1469 53984 : setlg(P, k); setlg(E, k); return mkmat2(P,E);
1470 : }
1471 :
1472 : GEN
1473 15048 : famatV_factorback(GEN v, GEN e)
1474 : {
1475 15048 : long i, l = lg(e);
1476 : GEN V;
1477 15048 : if (l == 1) return trivial_fact();
1478 14740 : V = signe(gel(e,1))? famat_pow_shallow(gel(v,1), gel(e,1)): trivial_fact();
1479 55634 : for (i = 2; i < l; i++) V = famat_mulpow_shallow(V, gel(v,i), gel(e,i));
1480 14740 : return V;
1481 : }
1482 :
1483 : GEN
1484 46438 : famatV_zv_factorback(GEN v, GEN e)
1485 : {
1486 46438 : long i, l = lg(e);
1487 : GEN V;
1488 46438 : if (l == 1) return trivial_fact();
1489 44058 : V = uel(e,1)? famat_pows_shallow(gel(v,1), uel(e,1)): trivial_fact();
1490 138306 : for (i = 2; i < l; i++) V = famat_mulpows_shallow(V, gel(v,i), uel(e,i));
1491 44058 : return V;
1492 : }
1493 :
1494 : GEN
1495 45199 : ZM_famat_limit(GEN fa, GEN limit)
1496 : {
1497 : pari_sp av;
1498 : GEN E, G, g, e, r;
1499 : long i, k, l, n, lG;
1500 :
1501 45199 : if (lgcols(fa) == 1) return fa;
1502 45192 : g = gel(fa,1); l = lg(g);
1503 45192 : e = gel(fa,2);
1504 90776 : for(n=0, i=1; i<l; i++)
1505 45584 : if (cmpii(gel(g,i),limit)<=0) n++;
1506 45192 : lG = n<l-1 ? n+2 : n+1;
1507 45192 : G = cgetg(lG, t_COL);
1508 45192 : E = cgetg(lG, t_COL);
1509 45192 : av = avma;
1510 90776 : for (i=1, k=1, r = gen_1; i<l; i++)
1511 : {
1512 45584 : if (cmpii(gel(g,i),limit)<=0)
1513 : {
1514 45451 : gel(G,k) = gel(g,i);
1515 45451 : gel(E,k) = gel(e,i);
1516 45451 : k++;
1517 133 : } else r = mulii(r, powii(gel(g,i), gel(e,i)));
1518 : }
1519 45192 : if (k<i)
1520 : {
1521 133 : gel(G, k) = gerepileuptoint(av, r);
1522 133 : gel(E, k) = gen_1;
1523 : }
1524 45192 : return mkmat2(G,E);
1525 : }
1526 :
1527 : /* assume pr has degree 1 and coprime to Q_denom(x) */
1528 : static GEN
1529 119453 : to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1530 : {
1531 119453 : GEN d, r, p = modpr_get_p(modpr);
1532 119453 : x = nf_to_scalar_or_basis(nf,x);
1533 119453 : if (typ(x) != t_COL) return Rg_to_Fp(x,p);
1534 118886 : x = Q_remove_denom(x, &d);
1535 118886 : r = zk_to_Fq(x, modpr);
1536 118886 : if (d) r = Fp_div(r, d, p);
1537 118886 : return r;
1538 : }
1539 :
1540 : /* pr coprime to all denominators occurring in x */
1541 : static GEN
1542 784 : famat_to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1543 : {
1544 784 : GEN p = modpr_get_p(modpr);
1545 784 : GEN t = NULL, g = gel(x,1), e = gel(x,2), q = subiu(p,1);
1546 784 : long i, l = lg(g);
1547 3423 : for (i = 1; i < l; i++)
1548 : {
1549 2639 : GEN n = modii(gel(e,i), q);
1550 2639 : if (signe(n))
1551 : {
1552 2625 : GEN h = to_Fp_coprime(nf, gel(g,i), modpr);
1553 2625 : h = Fp_pow(h, n, p);
1554 2625 : t = t? Fp_mul(t, h, p): h;
1555 : }
1556 : }
1557 784 : return t? modii(t, p): gen_1;
1558 : }
1559 :
1560 : /* cf famat_to_nf_modideal_coprime, modpr attached to prime of degree 1 */
1561 : GEN
1562 117612 : nf_to_Fp_coprime(GEN nf, GEN x, GEN modpr)
1563 : {
1564 784 : return typ(x)==t_MAT? famat_to_Fp_coprime(nf, x, modpr)
1565 118396 : : to_Fp_coprime(nf, x, modpr);
1566 : }
1567 :
1568 : static long
1569 1185488 : zk_pvalrem(GEN x, GEN p, GEN *py)
1570 1185488 : { return (typ(x) == t_INT)? Z_pvalrem(x, p, py): ZV_pvalrem(x, p, py); }
1571 : /* x a QC or Q. Return a ZC or Z, whose content is coprime to Z. Set v, dx
1572 : * such that x = p^v (newx / dx); dx = NULL if 1 */
1573 : static GEN
1574 1281675 : nf_remove_denom_p(GEN nf, GEN x, GEN p, GEN *pdx, long *pv)
1575 : {
1576 : long vcx;
1577 : GEN dx;
1578 1281675 : x = nf_to_scalar_or_basis(nf, x);
1579 1281670 : x = Q_remove_denom(x, &dx);
1580 1281677 : if (dx)
1581 : {
1582 178014 : vcx = - Z_pvalrem(dx, p, &dx);
1583 178013 : if (!vcx) vcx = zk_pvalrem(x, p, &x);
1584 178013 : if (isint1(dx)) dx = NULL;
1585 : }
1586 : else
1587 : {
1588 1103663 : vcx = zk_pvalrem(x, p, &x);
1589 1103673 : dx = NULL;
1590 : }
1591 1281686 : *pv = vcx;
1592 1281686 : *pdx = dx; return x;
1593 : }
1594 : /* x = b^e/p^(e-1) in Z_K; x = 0 mod p/pr^e, (x,pr) = 1. Return NULL
1595 : * if p inert (instead of 1) */
1596 : static GEN
1597 68832 : p_makecoprime(GEN pr)
1598 : {
1599 68832 : GEN B = pr_get_tau(pr), b;
1600 : long i, e;
1601 :
1602 68832 : if (typ(B) == t_INT) return NULL;
1603 68692 : b = gel(B,1); /* B = multiplication table by b */
1604 68692 : e = pr_get_e(pr);
1605 68692 : if (e == 1) return b;
1606 : /* one could also divide (exactly) by p in each iteration */
1607 39943 : for (i = 1; i < e; i++) b = ZM_ZC_mul(B, b);
1608 19509 : return ZC_Z_divexact(b, powiu(pr_get_p(pr), e-1));
1609 : }
1610 :
1611 : /* Compute A = prod g[i]^e[i] mod pr^k, assuming (A, pr) = 1.
1612 : * Method: modify each g[i] so that it becomes coprime to pr,
1613 : * g[i] *= (b/p)^v_pr(g[i]), where b/p = pr^(-1) times something integral
1614 : * and prime to p; globally, we multiply by (b/p)^v_pr(A) = 1.
1615 : * Optimizations:
1616 : * 1) remove all powers of p from contents, and consider extra generator p^vp;
1617 : * modified as p * (b/p)^e = b^e / p^(e-1)
1618 : * 2) remove denominators, coprime to p, by multiplying by inverse mod prk\cap Z
1619 : *
1620 : * EX = multiple of exponent of (O_K / pr^k)^* used to reduce the product in
1621 : * case the e[i] are large */
1622 : GEN
1623 560894 : famat_makecoprime(GEN nf, GEN g, GEN e, GEN pr, GEN prk, GEN EX)
1624 : {
1625 560894 : GEN G, E, t, vp = NULL, p = pr_get_p(pr), prkZ = gcoeff(prk, 1,1);
1626 560894 : long i, l = lg(g);
1627 :
1628 560894 : G = cgetg(l+1, t_VEC);
1629 560900 : E = cgetg(l+1, t_VEC); /* l+1: room for "modified p" */
1630 1842563 : for (i=1; i < l; i++)
1631 : {
1632 : long vcx;
1633 1281671 : GEN dx, x = nf_remove_denom_p(nf, gel(g,i), p, &dx, &vcx);
1634 1281684 : if (vcx) /* = v_p(content(g[i])) */
1635 : {
1636 113421 : GEN a = mulsi(vcx, gel(e,i));
1637 113416 : vp = vp? addii(vp, a): a;
1638 : }
1639 : /* x integral, content coprime to p; dx coprime to p */
1640 1281681 : if (typ(x) == t_INT)
1641 : { /* x coprime to p, hence to pr */
1642 165090 : x = modii(x, prkZ);
1643 165090 : if (dx) x = Fp_div(x, dx, prkZ);
1644 : }
1645 : else
1646 : {
1647 1116591 : (void)ZC_nfvalrem(x, pr, &x); /* x *= (b/p)^v_pr(x) */
1648 1116565 : x = ZC_hnfrem(FpC_red(x,prkZ), prk);
1649 1116574 : if (dx) x = FpC_Fp_mul(x, Fp_inv(dx,prkZ), prkZ);
1650 : }
1651 1281658 : gel(G,i) = x;
1652 1281658 : gel(E,i) = gel(e,i);
1653 : }
1654 :
1655 560892 : t = vp? p_makecoprime(pr): NULL;
1656 560896 : if (!t)
1657 : { /* no need for extra generator */
1658 492225 : setlg(G,l);
1659 492225 : setlg(E,l);
1660 : }
1661 : else
1662 : {
1663 68671 : gel(G,i) = FpC_red(t, prkZ);
1664 68671 : gel(E,i) = vp;
1665 : }
1666 560898 : return famat_to_nf_modideal_coprime(nf, G, E, prk, EX);
1667 : }
1668 :
1669 : /* simplified version of famat_makecoprime for X = SUnits[1] */
1670 : GEN
1671 42 : sunits_makecoprime(GEN X, GEN pr, GEN prk)
1672 : {
1673 42 : GEN G, p = pr_get_p(pr), prkZ = gcoeff(prk,1,1);
1674 42 : long i, l = lg(X);
1675 :
1676 42 : G = cgetg(l, t_VEC);
1677 3710 : for (i = 1; i < l; i++)
1678 : {
1679 3668 : GEN x = gel(X,i);
1680 3668 : if (typ(x) == t_INT) /* a prime */
1681 938 : x = equalii(x,p)? p_makecoprime(pr): modii(x, prkZ);
1682 : else
1683 : {
1684 2730 : (void)ZC_nfvalrem(x, pr, &x); /* x *= (b/p)^v_pr(x) */
1685 2730 : x = ZC_hnfrem(FpC_red(x,prkZ), prk);
1686 : }
1687 3668 : gel(G,i) = x;
1688 : }
1689 42 : return G;
1690 : }
1691 :
1692 : /* prod g[i]^e[i] mod bid, assume (g[i], id) = 1 and 1 < lg(g) <= lg(e) */
1693 : GEN
1694 19880 : famat_to_nf_moddivisor(GEN nf, GEN g, GEN e, GEN bid)
1695 : {
1696 19880 : GEN t, cyc = bid_get_cyc(bid);
1697 19880 : if (lg(cyc) == 1)
1698 0 : t = gen_1;
1699 : else
1700 19880 : t = famat_to_nf_modideal_coprime(nf, g, e, bid_get_ideal(bid),
1701 : cyc_get_expo(cyc));
1702 19880 : return set_sign_mod_divisor(nf, mkmat2(g,e), t, bid_get_sarch(bid));
1703 : }
1704 :
1705 : GEN
1706 766998 : vecmul(GEN x, GEN y)
1707 : {
1708 766998 : if (is_scalar_t(typ(x))) return gmul(x,y);
1709 766997 : pari_APPLY_same(vecmul(gel(x,i), gel(y,i)))
1710 : }
1711 :
1712 : GEN
1713 0 : vecinv(GEN x)
1714 : {
1715 0 : if (is_scalar_t(typ(x))) return ginv(x);
1716 0 : pari_APPLY_same(vecinv(gel(x,i)))
1717 : }
1718 :
1719 : GEN
1720 0 : vecpow(GEN x, GEN n)
1721 : {
1722 0 : if (is_scalar_t(typ(x))) return powgi(x,n);
1723 0 : pari_APPLY_same(vecpow(gel(x,i), n))
1724 : }
1725 :
1726 : GEN
1727 903 : vecdiv(GEN x, GEN y)
1728 : {
1729 903 : if (is_scalar_t(typ(x))) return gdiv(x,y);
1730 903 : pari_APPLY_same(vecdiv(gel(x,i), gel(y,i)))
1731 : }
1732 :
1733 : /* A ideal as a square t_MAT */
1734 : static GEN
1735 225042 : idealmulelt(GEN nf, GEN x, GEN A)
1736 : {
1737 : long i, lx;
1738 : GEN dx, dA, D;
1739 225042 : if (lg(A) == 1) return cgetg(1, t_MAT);
1740 225042 : x = nf_to_scalar_or_basis(nf,x);
1741 225042 : if (typ(x) != t_COL)
1742 89263 : return isintzero(x)? cgetg(1,t_MAT): RgM_Rg_mul(A, Q_abs_shallow(x));
1743 135779 : x = Q_remove_denom(x, &dx);
1744 135779 : A = Q_remove_denom(A, &dA);
1745 135779 : x = zk_multable(nf, x);
1746 135779 : D = mulii(zkmultable_capZ(x), gcoeff(A,1,1));
1747 135779 : x = zkC_multable_mul(A, x);
1748 135779 : settyp(x, t_MAT); lx = lg(x);
1749 : /* x may contain scalars (at most 1 since the ideal is nonzero)*/
1750 466628 : for (i=1; i<lx; i++)
1751 341475 : if (typ(gel(x,i)) == t_INT)
1752 : {
1753 10626 : if (i > 1) swap(gel(x,1), gel(x,i)); /* help HNF */
1754 10626 : gel(x,1) = scalarcol_shallow(gel(x,1), lx-1);
1755 10626 : break;
1756 : }
1757 135779 : x = ZM_hnfmodid(x, D);
1758 135779 : dx = mul_denom(dx,dA);
1759 135779 : return dx? gdiv(x,dx): x;
1760 : }
1761 :
1762 : /* nf a true nf, tx <= ty */
1763 : static GEN
1764 2150049 : idealmul_aux(GEN nf, GEN x, GEN y, long tx, long ty)
1765 : {
1766 : GEN z, cx, cy;
1767 2150049 : switch(tx)
1768 : {
1769 276281 : case id_PRINCIPAL:
1770 276281 : switch(ty)
1771 : {
1772 50875 : case id_PRINCIPAL:
1773 50875 : return idealhnf_principal(nf, nfmul(nf,x,y));
1774 364 : case id_PRIME:
1775 : {
1776 364 : GEN p = pr_get_p(y), pi = pr_get_gen(y), cx;
1777 364 : if (pr_is_inert(y)) return RgM_Rg_mul(idealhnf_principal(nf,x),p);
1778 :
1779 210 : x = nf_to_scalar_or_basis(nf, x);
1780 210 : switch(typ(x))
1781 : {
1782 196 : case t_INT:
1783 196 : if (!signe(x)) return cgetg(1,t_MAT);
1784 196 : return ZM_Z_mul(pr_hnf(nf,y), absi_shallow(x));
1785 7 : case t_FRAC:
1786 7 : return RgM_Rg_mul(pr_hnf(nf,y), Q_abs_shallow(x));
1787 : }
1788 : /* t_COL */
1789 7 : x = Q_primitive_part(x, &cx);
1790 7 : x = zk_multable(nf, x);
1791 7 : z = shallowconcat(ZM_Z_mul(x,p), ZM_ZC_mul(x,pi));
1792 7 : z = ZM_hnfmodid(z, mulii(p, zkmultable_capZ(x)));
1793 7 : return cx? ZM_Q_mul(z, cx): z;
1794 : }
1795 225042 : default: /* id_MAT */
1796 225042 : return idealmulelt(nf, x,y);
1797 : }
1798 1596539 : case id_PRIME:
1799 1596539 : if (ty==id_PRIME)
1800 1592313 : { y = pr_hnf(nf,y); cy = NULL; }
1801 : else
1802 4226 : y = Q_primitive_part(y, &cy);
1803 1596539 : y = idealHNF_mul_two(nf,y,x);
1804 1596539 : return cy? ZM_Q_mul(y,cy): y;
1805 :
1806 277229 : default: /* id_MAT */
1807 : {
1808 277229 : long N = nf_get_degree(nf);
1809 277229 : if (lg(x)-1 != N || lg(y)-1 != N) pari_err_DIM("idealmul");
1810 277215 : x = Q_primitive_part(x, &cx);
1811 277215 : y = Q_primitive_part(y, &cy); cx = mul_content(cx,cy);
1812 277215 : y = idealHNF_mul(nf,x,y);
1813 277215 : return cx? ZM_Q_mul(y,cx): y;
1814 : }
1815 : }
1816 : }
1817 :
1818 : /* output the ideal product x.y */
1819 : GEN
1820 2150048 : idealmul(GEN nf, GEN x, GEN y)
1821 : {
1822 : pari_sp av;
1823 : GEN res, ax, ay, z;
1824 2150048 : long tx = idealtyp(&x,&ax);
1825 2150049 : long ty = idealtyp(&y,&ay), f;
1826 2150049 : if (tx>ty) { swap(ax,ay); swap(x,y); lswap(tx,ty); }
1827 2150049 : f = (ax||ay); res = f? cgetg(3,t_VEC): NULL; /*product is an extended ideal*/
1828 2150049 : av = avma;
1829 2150049 : z = gerepileupto(av, idealmul_aux(checknf(nf), x,y, tx,ty));
1830 2150035 : if (!f) return z;
1831 23523 : if (ax && ay)
1832 22312 : ax = ext_mul(nf, ax, ay);
1833 : else
1834 1211 : ax = gcopy(ax? ax: ay);
1835 23523 : gel(res,1) = z; gel(res,2) = ax; return res;
1836 : }
1837 :
1838 : /* Return x, integral in 2-elt form, such that pr^2 = c * x. cf idealpowprime
1839 : * nf = true nf */
1840 : static GEN
1841 213096 : idealsqrprime(GEN nf, GEN pr, GEN *pc)
1842 : {
1843 213096 : GEN p = pr_get_p(pr), q, gen;
1844 213096 : long e = pr_get_e(pr), f = pr_get_f(pr);
1845 :
1846 213097 : q = (e == 1)? sqri(p): p;
1847 213085 : if (e <= 2 && e * f == nf_get_degree(nf))
1848 : { /* pr^e = (p) */
1849 44026 : *pc = q;
1850 44026 : return mkvec2(gen_1,gen_0);
1851 : }
1852 169069 : gen = nfsqr(nf, pr_get_gen(pr));
1853 169076 : gen = FpC_red(gen, q);
1854 169061 : *pc = NULL;
1855 169061 : return mkvec2(q, gen);
1856 : }
1857 : /* cf idealpow_aux */
1858 : static GEN
1859 30193 : idealsqr_aux(GEN nf, GEN x, long tx)
1860 : {
1861 30193 : GEN T = nf_get_pol(nf), m, cx, a, alpha;
1862 30193 : long N = degpol(T);
1863 30193 : switch(tx)
1864 : {
1865 70 : case id_PRINCIPAL:
1866 70 : return idealhnf_principal(nf, nfsqr(nf,x));
1867 9691 : case id_PRIME:
1868 9691 : if (pr_is_inert(x)) return scalarmat(sqri(gel(x,1)), N);
1869 9523 : x = idealsqrprime(nf, x, &cx);
1870 9523 : x = idealhnf_two(nf,x);
1871 9523 : return cx? ZM_Z_mul(x, cx): x;
1872 20432 : default:
1873 20432 : x = Q_primitive_part(x, &cx);
1874 20432 : a = mat_ideal_two_elt(nf,x); alpha = gel(a,2); a = gel(a,1);
1875 20432 : alpha = nfsqr(nf,alpha);
1876 20432 : m = zk_scalar_or_multable(nf, alpha);
1877 20432 : if (typ(m) == t_INT) {
1878 1547 : x = gcdii(sqri(a), m);
1879 1547 : if (cx) x = gmul(x, gsqr(cx));
1880 1547 : x = scalarmat(x, N);
1881 : }
1882 : else
1883 : { /* could use gcdii(sqri(a), zkmultable_capZ(m)), but costly */
1884 18885 : x = ZM_hnfmodid(m, sqri(a));
1885 18885 : if (cx) cx = gsqr(cx);
1886 18885 : if (cx) x = ZM_Q_mul(x, cx);
1887 : }
1888 20432 : return x;
1889 : }
1890 : }
1891 : GEN
1892 30193 : idealsqr(GEN nf, GEN x)
1893 : {
1894 : pari_sp av;
1895 : GEN res, ax, z;
1896 30193 : long tx = idealtyp(&x,&ax);
1897 30193 : res = ax? cgetg(3,t_VEC): NULL; /*product is an extended ideal*/
1898 30193 : av = avma;
1899 30193 : z = gerepileupto(av, idealsqr_aux(checknf(nf), x, tx));
1900 30193 : if (!ax) return z;
1901 25622 : gel(res,1) = z;
1902 25622 : gel(res,2) = ext_sqr(nf, ax); return res;
1903 : }
1904 :
1905 : /* norm of an ideal */
1906 : GEN
1907 11137 : idealnorm(GEN nf, GEN x)
1908 : {
1909 : pari_sp av;
1910 : GEN y;
1911 : long tx;
1912 :
1913 11137 : switch(idealtyp(&x,&y))
1914 : {
1915 245 : case id_PRIME: return pr_norm(x);
1916 8764 : case id_MAT: return RgM_det_triangular(x);
1917 : }
1918 : /* id_PRINCIPAL */
1919 2128 : nf = checknf(nf); av = avma;
1920 2128 : x = nfnorm(nf, x);
1921 2128 : tx = typ(x);
1922 2128 : if (tx == t_INT) return gerepileuptoint(av, absi(x));
1923 406 : if (tx != t_FRAC) pari_err_TYPE("idealnorm",x);
1924 406 : return gerepileupto(av, Q_abs(x));
1925 : }
1926 :
1927 : /* x \cap Z */
1928 : GEN
1929 2975 : idealdown(GEN nf, GEN x)
1930 : {
1931 2975 : pari_sp av = avma;
1932 : GEN y, c;
1933 2975 : switch(idealtyp(&x,&y))
1934 : {
1935 7 : case id_PRIME: return icopy(pr_get_p(x));
1936 2100 : case id_MAT: return gcopy(gcoeff(x,1,1));
1937 : }
1938 : /* id_PRINCIPAL */
1939 868 : nf = checknf(nf); av = avma;
1940 868 : x = nf_to_scalar_or_basis(nf, x);
1941 868 : if (is_rational_t(typ(x))) return Q_abs(x);
1942 14 : x = Q_primitive_part(x, &c);
1943 14 : y = zkmultable_capZ(zk_multable(nf, x));
1944 14 : return gerepilecopy(av, mul_content(c, y));
1945 : }
1946 :
1947 : /* true nf */
1948 : static GEN
1949 28 : idealismaximal_int(GEN nf, GEN p)
1950 : {
1951 : GEN L;
1952 28 : if (!BPSW_psp(p)) return NULL;
1953 56 : if (!dvdii(nf_get_index(nf), p) &&
1954 42 : !FpX_is_irred(FpX_red(nf_get_pol(nf),p), p)) return NULL;
1955 14 : L = idealprimedec(nf, p);
1956 14 : return lg(L) == 2? gel(L,1): NULL;
1957 : }
1958 : /* true nf */
1959 : static GEN
1960 7 : idealismaximal_mat(GEN nf, GEN x)
1961 : {
1962 : GEN p, c, L;
1963 : long i, l, f;
1964 7 : x = Q_primitive_part(x, &c);
1965 7 : p = gcoeff(x,1,1);
1966 7 : if (c)
1967 : {
1968 0 : if (typ(c) == t_FRAC || !equali1(p)) return NULL;
1969 0 : return idealismaximal_int(nf, p);
1970 : }
1971 7 : if (!BPSW_psp(p)) return NULL;
1972 7 : l = lg(x); f = 1;
1973 21 : for (i = 2; i < l; i++)
1974 : {
1975 14 : c = gcoeff(x,i,i);
1976 14 : if (equalii(c, p)) f++; else if (!equali1(c)) return NULL;
1977 : }
1978 7 : L = idealprimedec_limit_f(nf, p, f);
1979 14 : for (i = lg(L)-1; i; i--)
1980 : {
1981 14 : GEN pr = gel(L,i);
1982 14 : if (pr_get_f(pr) != f) break;
1983 14 : if (idealval(nf, x, pr) == 1) return pr;
1984 : }
1985 0 : return NULL;
1986 : }
1987 : /* true nf */
1988 : static GEN
1989 42 : idealismaximal_i(GEN nf, GEN x)
1990 : {
1991 : GEN L, p, pr, c;
1992 : long i, l;
1993 42 : switch(idealtyp(&x,&c))
1994 : {
1995 7 : case id_PRIME: return x;
1996 7 : case id_MAT: return idealismaximal_mat(nf, x);
1997 : }
1998 : /* id_PRINCIPAL */
1999 28 : x = nf_to_scalar_or_basis(nf, x);
2000 28 : switch(typ(x))
2001 : {
2002 28 : case t_INT: return idealismaximal_int(nf, absi_shallow(x));
2003 0 : case t_FRAC: return NULL;
2004 : }
2005 0 : x = Q_primitive_part(x, &c);
2006 0 : if (c) return NULL;
2007 0 : p = zkmultable_capZ(zk_multable(nf, x));
2008 0 : L = idealprimedec(nf, p); l = lg(L); pr = NULL;
2009 0 : for (i = 1; i < l; i++)
2010 : {
2011 0 : long v = ZC_nfval(x, gel(L,i));
2012 0 : if (v > 1 || (v && pr)) return NULL;
2013 0 : pr = gel(L,i);
2014 : }
2015 0 : return pr;
2016 : }
2017 : GEN
2018 42 : idealismaximal(GEN nf, GEN x)
2019 : {
2020 42 : pari_sp av = avma;
2021 42 : x = idealismaximal_i(checknf(nf), x);
2022 42 : if (!x) { set_avma(av); return gen_0; }
2023 28 : return gerepilecopy(av, x);
2024 : }
2025 :
2026 : /* I^(-1) = { x \in K, Tr(x D^(-1) I) \in Z }, D different of K/Q
2027 : *
2028 : * nf[5][6] = pp( D^(-1) ) = pp( HNF( T^(-1) ) ), T = (Tr(wi wj))
2029 : * nf[5][7] = same in 2-elt form.
2030 : * Assume I integral. Return the integral ideal (I\cap Z) I^(-1) */
2031 : GEN
2032 193297 : idealHNF_inv_Z(GEN nf, GEN I)
2033 : {
2034 193297 : GEN J, dual, IZ = gcoeff(I,1,1); /* I \cap Z */
2035 193297 : if (isint1(IZ)) return matid(lg(I)-1);
2036 177750 : J = idealHNF_mul(nf,I, gmael(nf,5,7));
2037 : /* I in HNF, hence easily inverted; multiply by IZ to get integer coeffs
2038 : * missing content cancels while solving the linear equation */
2039 177749 : dual = shallowtrans( hnf_divscale(J, gmael(nf,5,6), IZ) );
2040 177750 : return ZM_hnfmodid(dual, IZ);
2041 : }
2042 : /* I HNF with rational coefficients (denominator d). */
2043 : GEN
2044 66131 : idealHNF_inv(GEN nf, GEN I)
2045 : {
2046 66131 : GEN J, IQ = gcoeff(I,1,1); /* I \cap Q; d IQ = dI \cap Z */
2047 66131 : J = idealHNF_inv_Z(nf, Q_remove_denom(I, NULL)); /* = (dI)^(-1) * (d IQ) */
2048 66131 : return equali1(IQ)? J: RgM_Rg_div(J, IQ);
2049 : }
2050 :
2051 : /* return p * P^(-1) [integral] */
2052 : GEN
2053 36109 : pr_inv_p(GEN pr)
2054 : {
2055 36109 : if (pr_is_inert(pr)) return matid(pr_get_f(pr));
2056 35451 : return ZM_hnfmodid(pr_get_tau(pr), pr_get_p(pr));
2057 : }
2058 : GEN
2059 17069 : pr_inv(GEN pr)
2060 : {
2061 17069 : GEN p = pr_get_p(pr);
2062 17069 : if (pr_is_inert(pr)) return scalarmat(ginv(p), pr_get_f(pr));
2063 16691 : return RgM_Rg_div(ZM_hnfmodid(pr_get_tau(pr),p), p);
2064 : }
2065 :
2066 : GEN
2067 130576 : idealinv(GEN nf, GEN x)
2068 : {
2069 : GEN res, ax;
2070 : pari_sp av;
2071 130576 : long tx = idealtyp(&x,&ax), N;
2072 :
2073 130576 : res = ax? cgetg(3,t_VEC): NULL;
2074 130576 : nf = checknf(nf); av = avma;
2075 130576 : N = nf_get_degree(nf);
2076 130576 : switch (tx)
2077 : {
2078 60846 : case id_MAT:
2079 60846 : if (lg(x)-1 != N) pari_err_DIM("idealinv");
2080 60846 : x = idealHNF_inv(nf,x); break;
2081 53676 : case id_PRINCIPAL:
2082 53676 : x = nf_to_scalar_or_basis(nf, x);
2083 53676 : if (typ(x) != t_COL)
2084 53627 : x = idealhnf_principal(nf,ginv(x));
2085 : else
2086 : { /* nfinv + idealhnf where we already know (x) \cap Z */
2087 : GEN c, d;
2088 49 : x = Q_remove_denom(x, &c);
2089 49 : x = zk_inv(nf, x);
2090 49 : x = Q_remove_denom(x, &d); /* true inverse is c/d * x */
2091 49 : if (!d) /* x and x^(-1) integral => x a unit */
2092 14 : x = c? scalarmat(c, N): matid(N);
2093 : else
2094 : {
2095 35 : c = c? gdiv(c,d): ginv(d);
2096 35 : x = zk_multable(nf, x);
2097 35 : x = ZM_Q_mul(ZM_hnfmodid(x,d), c);
2098 : }
2099 : }
2100 53676 : break;
2101 16054 : case id_PRIME:
2102 16054 : x = pr_inv(x); break;
2103 : }
2104 130574 : x = gerepileupto(av,x); if (!ax) return x;
2105 18835 : gel(res,1) = x;
2106 18835 : gel(res,2) = ext_inv(nf, ax); return res;
2107 : }
2108 :
2109 : /* write x = A/B, A,B coprime integral ideals */
2110 : GEN
2111 276432 : idealnumden(GEN nf, GEN x)
2112 : {
2113 276432 : pari_sp av = avma;
2114 : GEN x0, ax, c, d, A, B, J;
2115 276432 : long tx = idealtyp(&x,&ax);
2116 276430 : nf = checknf(nf);
2117 276433 : switch (tx)
2118 : {
2119 7 : case id_PRIME:
2120 7 : retmkvec2(idealhnf(nf, x), gen_1);
2121 50468 : case id_PRINCIPAL:
2122 : {
2123 : GEN xZ, mx;
2124 50468 : x = nf_to_scalar_or_basis(nf, x);
2125 50468 : switch(typ(x))
2126 : {
2127 2506 : case t_INT: return gerepilecopy(av, mkvec2(absi_shallow(x),gen_1));
2128 14 : case t_FRAC:return gerepilecopy(av, mkvec2(absi_shallow(gel(x,1)), gel(x,2)));
2129 : }
2130 : /* t_COL */
2131 47948 : x = Q_remove_denom(x, &d);
2132 47948 : if (!d) return gerepilecopy(av, mkvec2(idealhnf_shallow(nf, x), gen_1));
2133 180 : mx = zk_multable(nf, x);
2134 180 : xZ = zkmultable_capZ(mx);
2135 180 : x = ZM_hnfmodid(mx, xZ); /* principal ideal (x) */
2136 180 : x0 = mkvec2(xZ, mx); /* same, for fast multiplication */
2137 180 : break;
2138 : }
2139 225958 : default: /* id_MAT */
2140 : {
2141 225958 : long n = lg(x)-1;
2142 225958 : if (n == 0) return mkvec2(gen_0, gen_1);
2143 225958 : if (n != nf_get_degree(nf)) pari_err_DIM("idealnumden");
2144 225958 : x0 = x = Q_remove_denom(x, &d);
2145 225960 : if (!d) return gerepilecopy(av, mkvec2(x, gen_1));
2146 14 : break;
2147 : }
2148 : }
2149 194 : J = hnfmodid(x, d); /* = d/B */
2150 194 : c = gcoeff(J,1,1); /* (d/B) \cap Z, divides d */
2151 194 : B = idealHNF_inv_Z(nf, J); /* (d/B \cap Z) B/d */
2152 194 : if (!equalii(c,d)) B = ZM_Z_mul(B, diviiexact(d,c)); /* = B ! */
2153 194 : A = idealHNF_mul(nf, B, x0); /* d * (original x) * B = d A */
2154 194 : A = ZM_Z_divexact(A, d); /* = A ! */
2155 194 : return gerepilecopy(av, mkvec2(A, B));
2156 : }
2157 :
2158 : /* Return x, integral in 2-elt form, such that pr^n = c * x. Assume n != 0.
2159 : * nf = true nf */
2160 : static GEN
2161 1031111 : idealpowprime(GEN nf, GEN pr, GEN n, GEN *pc)
2162 : {
2163 1031111 : GEN p = pr_get_p(pr), q, gen;
2164 :
2165 1031096 : *pc = NULL;
2166 1031096 : if (is_pm1(n)) /* n = 1 special cased for efficiency */
2167 : {
2168 546077 : q = p;
2169 546077 : if (typ(pr_get_tau(pr)) == t_INT) /* inert */
2170 : {
2171 0 : *pc = (signe(n) >= 0)? p: ginv(p);
2172 0 : return mkvec2(gen_1,gen_0);
2173 : }
2174 546075 : if (signe(n) >= 0) gen = pr_get_gen(pr);
2175 : else
2176 : {
2177 150843 : gen = pr_get_tau(pr); /* possibly t_MAT */
2178 150859 : *pc = ginv(p);
2179 : }
2180 : }
2181 485065 : else if (equalis(n,2)) return idealsqrprime(nf, pr, pc);
2182 : else
2183 : {
2184 281490 : long e = pr_get_e(pr), f = pr_get_f(pr);
2185 281501 : GEN r, m = truedvmdis(n, e, &r);
2186 281491 : if (e * f == nf_get_degree(nf))
2187 : { /* pr^e = (p) */
2188 82328 : if (signe(m)) *pc = powii(p,m);
2189 82328 : if (!signe(r)) return mkvec2(gen_1,gen_0);
2190 40312 : q = p;
2191 40312 : gen = nfpow(nf, pr_get_gen(pr), r);
2192 : }
2193 : else
2194 : {
2195 199176 : m = absi_shallow(m);
2196 199176 : if (signe(r)) m = addiu(m,1);
2197 199176 : q = powii(p,m); /* m = ceil(|n|/e) */
2198 199176 : if (signe(n) >= 0) gen = nfpow(nf, pr_get_gen(pr), n);
2199 : else
2200 : {
2201 23757 : gen = pr_get_tau(pr);
2202 23758 : if (typ(gen) == t_MAT) gen = gel(gen,1);
2203 23758 : n = negi(n);
2204 23758 : gen = ZC_Z_divexact(nfpow(nf, gen, n), powii(p, subii(n,m)));
2205 23758 : *pc = ginv(q);
2206 : }
2207 : }
2208 239489 : gen = FpC_red(gen, q);
2209 : }
2210 785561 : return mkvec2(q, gen);
2211 : }
2212 :
2213 : /* True nf. x * pr^n. Assume x in HNF or scalar (possibly nonintegral) */
2214 : GEN
2215 647223 : idealmulpowprime(GEN nf, GEN x, GEN pr, GEN n)
2216 : {
2217 : GEN c, cx, y;
2218 647223 : long N = nf_get_degree(nf);
2219 :
2220 647252 : if (!signe(n)) return typ(x) == t_MAT? x: scalarmat_shallow(x, N);
2221 :
2222 : /* inert, special cased for efficiency */
2223 647245 : if (pr_is_inert(pr))
2224 : {
2225 64206 : GEN q = powii(pr_get_p(pr), n);
2226 62326 : return typ(x) == t_MAT? RgM_Rg_mul(x,q)
2227 126524 : : scalarmat_shallow(gmul(Q_abs(x),q), N);
2228 : }
2229 :
2230 583018 : y = idealpowprime(nf, pr, n, &c);
2231 582987 : if (typ(x) == t_MAT)
2232 580446 : { x = Q_primitive_part(x, &cx); if (is_pm1(gcoeff(x,1,1))) x = NULL; }
2233 : else
2234 2541 : { cx = x; x = NULL; }
2235 582950 : cx = mul_content(c,cx);
2236 582948 : if (x)
2237 423726 : x = idealHNF_mul_two(nf,x,y);
2238 : else
2239 159222 : x = idealhnf_two(nf,y);
2240 583125 : if (cx) x = ZM_Q_mul(x,cx);
2241 582792 : return x;
2242 : }
2243 : GEN
2244 4564 : idealdivpowprime(GEN nf, GEN x, GEN pr, GEN n)
2245 : {
2246 4564 : return idealmulpowprime(nf,x,pr, negi(n));
2247 : }
2248 :
2249 : /* nf = true nf */
2250 : static GEN
2251 692677 : idealpow_aux(GEN nf, GEN x, long tx, GEN n)
2252 : {
2253 692677 : GEN T = nf_get_pol(nf), m, cx, n1, a, alpha;
2254 692676 : long N = degpol(T), s = signe(n);
2255 692676 : if (!s) return matid(N);
2256 680839 : switch(tx)
2257 : {
2258 25963 : case id_PRINCIPAL:
2259 25963 : return idealhnf_principal(nf, nfpow(nf,x,n));
2260 533451 : case id_PRIME:
2261 533451 : if (pr_is_inert(x)) return scalarmat(powii(gel(x,1), n), N);
2262 448084 : x = idealpowprime(nf, x, n, &cx);
2263 448081 : x = idealhnf_two(nf,x);
2264 448093 : return cx? ZM_Q_mul(x, cx): x;
2265 121425 : default:
2266 121425 : if (is_pm1(n)) return (s < 0)? idealinv(nf, x): gcopy(x);
2267 64760 : n1 = (s < 0)? negi(n): n;
2268 :
2269 64760 : x = Q_primitive_part(x, &cx);
2270 64760 : a = mat_ideal_two_elt(nf,x); alpha = gel(a,2); a = gel(a,1);
2271 64760 : alpha = nfpow(nf,alpha,n1);
2272 64760 : m = zk_scalar_or_multable(nf, alpha);
2273 64760 : if (typ(m) == t_INT) {
2274 637 : x = gcdii(powii(a,n1), m);
2275 637 : if (s<0) x = ginv(x);
2276 637 : if (cx) x = gmul(x, powgi(cx,n));
2277 637 : x = scalarmat(x, N);
2278 : }
2279 : else
2280 : { /* could use gcdii(powii(a,n1), zkmultable_capZ(m)), but costly */
2281 64123 : x = ZM_hnfmodid(m, powii(a,n1));
2282 64123 : if (cx) cx = powgi(cx,n);
2283 64123 : if (s<0) {
2284 7 : GEN xZ = gcoeff(x,1,1);
2285 7 : cx = cx ? gdiv(cx, xZ): ginv(xZ);
2286 7 : x = idealHNF_inv_Z(nf,x);
2287 : }
2288 64123 : if (cx) x = ZM_Q_mul(x, cx);
2289 : }
2290 64760 : return x;
2291 : }
2292 : }
2293 :
2294 : /* raise the ideal x to the power n (in Z) */
2295 : GEN
2296 692681 : idealpow(GEN nf, GEN x, GEN n)
2297 : {
2298 : pari_sp av;
2299 : long tx;
2300 : GEN res, ax;
2301 :
2302 692681 : if (typ(n) != t_INT) pari_err_TYPE("idealpow",n);
2303 692681 : tx = idealtyp(&x,&ax);
2304 692681 : res = ax? cgetg(3,t_VEC): NULL;
2305 692681 : av = avma;
2306 692681 : x = gerepileupto(av, idealpow_aux(checknf(nf), x, tx, n));
2307 692670 : if (!ax) return x;
2308 0 : ax = ext_pow(nf, ax, n);
2309 0 : gel(res,1) = x;
2310 0 : gel(res,2) = ax;
2311 0 : return res;
2312 : }
2313 :
2314 : /* Return ideal^e in number field nf. e is a C integer. */
2315 : GEN
2316 246786 : idealpows(GEN nf, GEN ideal, long e)
2317 : {
2318 246786 : long court[] = {evaltyp(t_INT) | _evallg(3),0,0};
2319 246786 : affsi(e,court); return idealpow(nf,ideal,court);
2320 : }
2321 :
2322 : static GEN
2323 24041 : _idealmulred(GEN nf, GEN x, GEN y)
2324 24041 : { return idealred(nf,idealmul(nf,x,y)); }
2325 : static GEN
2326 27344 : _idealsqrred(GEN nf, GEN x)
2327 27344 : { return idealred(nf,idealsqr(nf,x)); }
2328 : static GEN
2329 7927 : _mul(void *data, GEN x, GEN y) { return _idealmulred((GEN)data,x,y); }
2330 : static GEN
2331 27344 : _sqr(void *data, GEN x) { return _idealsqrred((GEN)data, x); }
2332 :
2333 : /* compute x^n (x ideal, n integer), reducing along the way */
2334 : GEN
2335 75354 : idealpowred(GEN nf, GEN x, GEN n)
2336 : {
2337 75354 : pari_sp av = avma, av2;
2338 : long s;
2339 : GEN y;
2340 :
2341 75354 : if (typ(n) != t_INT) pari_err_TYPE("idealpowred",n);
2342 75354 : s = signe(n); if (s == 0) return idealpow(nf,x,n);
2343 75354 : y = gen_pow_i(x, n, (void*)nf, &_sqr, &_mul);
2344 75354 : av2 = avma;
2345 75354 : if (s < 0) y = idealinv(nf,y);
2346 75353 : if (s < 0 || is_pm1(n)) y = idealred(nf,y);
2347 75356 : return avma == av2? gerepilecopy(av,y): gerepileupto(av,y);
2348 : }
2349 :
2350 : GEN
2351 16114 : idealmulred(GEN nf, GEN x, GEN y)
2352 : {
2353 16114 : pari_sp av = avma;
2354 16114 : return gerepileupto(av, _idealmulred(nf,x,y));
2355 : }
2356 :
2357 : long
2358 91 : isideal(GEN nf,GEN x)
2359 : {
2360 91 : long N, i, j, lx, tx = typ(x);
2361 : pari_sp av;
2362 : GEN T, xZ;
2363 :
2364 91 : nf = checknf(nf); T = nf_get_pol(nf); lx = lg(x);
2365 91 : if (tx==t_VEC && lx==3) { x = gel(x,1); tx = typ(x); lx = lg(x); }
2366 91 : switch(tx)
2367 : {
2368 14 : case t_INT: case t_FRAC: return 1;
2369 7 : case t_POL: return varn(x) == varn(T);
2370 7 : case t_POLMOD: return RgX_equal_var(T, gel(x,1));
2371 14 : case t_VEC: return get_prid(x)? 1 : 0;
2372 42 : case t_MAT: break;
2373 7 : default: return 0;
2374 : }
2375 42 : N = degpol(T);
2376 42 : if (lx-1 != N) return (lx == 1);
2377 28 : if (nbrows(x) != N) return 0;
2378 :
2379 28 : av = avma; x = Q_primpart(x);
2380 28 : if (!ZM_ishnf(x)) return 0;
2381 14 : xZ = gcoeff(x,1,1);
2382 21 : for (j=2; j<=N; j++)
2383 14 : if (!dvdii(xZ, gcoeff(x,j,j))) return gc_long(av,0);
2384 14 : for (i=2; i<=N; i++)
2385 14 : for (j=2; j<=N; j++)
2386 7 : if (! hnf_invimage(x, zk_ei_mul(nf,gel(x,i),j))) return gc_long(av,0);
2387 7 : return gc_long(av,1);
2388 : }
2389 :
2390 : GEN
2391 36897 : idealdiv(GEN nf, GEN x, GEN y)
2392 : {
2393 36897 : pari_sp av = avma, tetpil;
2394 36897 : GEN z = idealinv(nf,y);
2395 36897 : tetpil = avma; return gerepile(av,tetpil, idealmul(nf,x,z));
2396 : }
2397 :
2398 : /* This routine computes the quotient x/y of two ideals in the number field nf.
2399 : * It assumes that the quotient is an integral ideal. The idea is to find an
2400 : * ideal z dividing y such that gcd(Nx/Nz, Nz) = 1. Then
2401 : *
2402 : * x + (Nx/Nz) x
2403 : * ----------- = ---
2404 : * y + (Ny/Nz) y
2405 : *
2406 : * Proof: we can assume x and y are integral. Let p be any prime ideal
2407 : *
2408 : * If p | Nz, then it divides neither Nx/Nz nor Ny/Nz (since Nx/Nz is the
2409 : * product of the integers N(x/y) and N(y/z)). Both the numerator and the
2410 : * denominator on the left will be coprime to p. So will x/y, since x/y is
2411 : * assumed integral and its norm N(x/y) is coprime to p.
2412 : *
2413 : * If instead p does not divide Nz, then v_p (Nx/Nz) = v_p (Nx) >= v_p(x).
2414 : * Hence v_p (x + Nx/Nz) = v_p(x). Likewise for the denominators. QED.
2415 : *
2416 : * Peter Montgomery. July, 1994. */
2417 : static void
2418 7 : err_divexact(GEN x, GEN y)
2419 7 : { pari_err_DOMAIN("idealdivexact","denominator(x/y)", "!=",
2420 0 : gen_1,mkvec2(x,y)); }
2421 : GEN
2422 3955 : idealdivexact(GEN nf, GEN x0, GEN y0)
2423 : {
2424 3955 : pari_sp av = avma;
2425 : GEN x, y, xZ, yZ, Nx, Ny, Nz, cy, q, r;
2426 :
2427 3955 : nf = checknf(nf);
2428 3955 : x = idealhnf_shallow(nf, x0);
2429 3955 : y = idealhnf_shallow(nf, y0);
2430 3955 : if (lg(y) == 1) pari_err_INV("idealdivexact", y0);
2431 3948 : if (lg(x) == 1) { set_avma(av); return cgetg(1, t_MAT); } /* numerator is zero */
2432 3948 : y = Q_primitive_part(y, &cy);
2433 3948 : if (cy) x = RgM_Rg_div(x,cy);
2434 3948 : xZ = gcoeff(x,1,1); if (typ(xZ) != t_INT) err_divexact(x,y);
2435 3941 : yZ = gcoeff(y,1,1); if (isint1(yZ)) return gerepilecopy(av, x);
2436 2877 : Nx = idealnorm(nf,x);
2437 2877 : Ny = idealnorm(nf,y);
2438 2877 : if (typ(Nx) != t_INT) err_divexact(x,y);
2439 2877 : q = dvmdii(Nx,Ny, &r);
2440 2877 : if (signe(r)) err_divexact(x,y);
2441 2877 : if (is_pm1(q)) { set_avma(av); return matid(nf_get_degree(nf)); }
2442 : /* Find a norm Nz | Ny such that gcd(Nx/Nz, Nz) = 1 */
2443 714 : for (Nz = Ny;;) /* q = Nx/Nz */
2444 714 : {
2445 1428 : GEN p1 = gcdii(Nz, q);
2446 1428 : if (is_pm1(p1)) break;
2447 714 : Nz = diviiexact(Nz,p1);
2448 714 : q = mulii(q,p1);
2449 : }
2450 714 : xZ = gcoeff(x,1,1); q = gcdii(q, xZ);
2451 714 : if (!equalii(xZ,q))
2452 : { /* Replace x/y by x+(Nx/Nz) / y+(Ny/Nz) */
2453 378 : x = ZM_hnfmodid(x, q);
2454 : /* y reduced to unit ideal ? */
2455 378 : if (Nz == Ny) return gerepileupto(av, x);
2456 :
2457 140 : yZ = gcoeff(y,1,1); q = gcdii(diviiexact(Ny,Nz), yZ);
2458 140 : y = ZM_hnfmodid(y, q);
2459 : }
2460 476 : yZ = gcoeff(y,1,1);
2461 476 : y = idealHNF_mul(nf,x, idealHNF_inv_Z(nf,y));
2462 476 : return gerepileupto(av, ZM_Z_divexact(y, yZ));
2463 : }
2464 :
2465 : GEN
2466 21 : idealintersect(GEN nf, GEN x, GEN y)
2467 : {
2468 21 : pari_sp av = avma;
2469 : long lz, lx, i;
2470 : GEN z, dx, dy, xZ, yZ;;
2471 :
2472 21 : nf = checknf(nf);
2473 21 : x = idealhnf_shallow(nf,x);
2474 21 : y = idealhnf_shallow(nf,y);
2475 21 : if (lg(x) == 1 || lg(y) == 1) { set_avma(av); return cgetg(1,t_MAT); }
2476 14 : x = Q_remove_denom(x, &dx);
2477 14 : y = Q_remove_denom(y, &dy);
2478 14 : if (dx) y = ZM_Z_mul(y, dx);
2479 14 : if (dy) x = ZM_Z_mul(x, dy);
2480 14 : xZ = gcoeff(x,1,1);
2481 14 : yZ = gcoeff(y,1,1);
2482 14 : dx = mul_denom(dx,dy);
2483 14 : z = ZM_lll(shallowconcat(x,y), 0.99, LLL_KER); lz = lg(z);
2484 14 : lx = lg(x);
2485 63 : for (i=1; i<lz; i++) setlg(z[i], lx);
2486 14 : z = ZM_hnfmodid(ZM_mul(x,z), lcmii(xZ, yZ));
2487 14 : if (dx) z = RgM_Rg_div(z,dx);
2488 14 : return gerepileupto(av,z);
2489 : }
2490 :
2491 : /*******************************************************************/
2492 : /* */
2493 : /* T2-IDEAL REDUCTION */
2494 : /* */
2495 : /*******************************************************************/
2496 :
2497 : static GEN
2498 21 : chk_vdir(GEN nf, GEN vdir)
2499 : {
2500 21 : long i, l = lg(vdir);
2501 : GEN v;
2502 21 : if (l != lg(nf_get_roots(nf))) pari_err_DIM("idealred");
2503 14 : switch(typ(vdir))
2504 : {
2505 0 : case t_VECSMALL: return vdir;
2506 14 : case t_VEC: break;
2507 0 : default: pari_err_TYPE("idealred",vdir);
2508 : }
2509 14 : v = cgetg(l, t_VECSMALL);
2510 56 : for (i = 1; i < l; i++) v[i] = itos(gceil(gel(vdir,i)));
2511 14 : return v;
2512 : }
2513 :
2514 : static void
2515 12471 : twistG(GEN G, long r1, long i, long v)
2516 : {
2517 12471 : long j, lG = lg(G);
2518 12471 : if (i <= r1) {
2519 37340 : for (j=1; j<lG; j++) gcoeff(G,i,j) = gmul2n(gcoeff(G,i,j), v);
2520 : } else {
2521 392 : long k = (i<<1) - r1;
2522 2142 : for (j=1; j<lG; j++)
2523 : {
2524 1750 : gcoeff(G,k-1,j) = gmul2n(gcoeff(G,k-1,j), v);
2525 1750 : gcoeff(G,k ,j) = gmul2n(gcoeff(G,k ,j), v);
2526 : }
2527 : }
2528 12471 : }
2529 :
2530 : GEN
2531 121850 : nf_get_Gtwist(GEN nf, GEN vdir)
2532 : {
2533 : long i, l, v, r1;
2534 : GEN G;
2535 :
2536 121850 : if (!vdir) return nf_get_roundG(nf);
2537 21 : if (typ(vdir) == t_MAT)
2538 : {
2539 0 : long N = nf_get_degree(nf);
2540 0 : if (lg(vdir) != N+1 || lgcols(vdir) != N+1) pari_err_DIM("idealred");
2541 0 : return vdir;
2542 : }
2543 21 : vdir = chk_vdir(nf, vdir);
2544 14 : G = RgM_shallowcopy(nf_get_G(nf));
2545 14 : r1 = nf_get_r1(nf);
2546 14 : l = lg(vdir);
2547 56 : for (i=1; i<l; i++)
2548 : {
2549 42 : v = vdir[i]; if (!v) continue;
2550 42 : twistG(G, r1, i, v);
2551 : }
2552 14 : return RM_round_maxrank(G);
2553 : }
2554 : GEN
2555 12429 : nf_get_Gtwist1(GEN nf, long i)
2556 : {
2557 12429 : GEN G = RgM_shallowcopy( nf_get_G(nf) );
2558 12429 : long r1 = nf_get_r1(nf);
2559 12429 : twistG(G, r1, i, 10);
2560 12429 : return RM_round_maxrank(G);
2561 : }
2562 :
2563 : GEN
2564 93958 : RM_round_maxrank(GEN G0)
2565 : {
2566 93958 : long e, r = lg(G0)-1;
2567 93958 : pari_sp av = avma;
2568 93958 : for (e = 4; ; e <<= 1, set_avma(av))
2569 0 : {
2570 93958 : GEN G = gmul2n(G0, e), H = ground(G);
2571 93960 : if (ZM_rank(H) == r) return H; /* maximal rank ? */
2572 : }
2573 : }
2574 :
2575 : GEN
2576 121843 : idealred0(GEN nf, GEN I, GEN vdir)
2577 : {
2578 121843 : pari_sp av = avma;
2579 121843 : GEN G, aI, IZ, J, y, yZ, my, c1 = NULL;
2580 : long N;
2581 :
2582 121843 : nf = checknf(nf);
2583 121843 : N = nf_get_degree(nf);
2584 : /* put first for sanity checks, unused when I obviously principal */
2585 121843 : G = nf_get_Gtwist(nf, vdir);
2586 121836 : switch (idealtyp(&I,&aI))
2587 : {
2588 34391 : case id_PRIME:
2589 34391 : if (pr_is_inert(I)) {
2590 585 : if (!aI) { set_avma(av); return matid(N); }
2591 585 : c1 = gel(I,1); I = matid(N);
2592 585 : goto END;
2593 : }
2594 33806 : IZ = pr_get_p(I);
2595 33806 : J = pr_inv_p(I);
2596 33808 : I = idealhnf_two(nf,I);
2597 33807 : break;
2598 87417 : case id_MAT:
2599 87417 : I = Q_primitive_part(I, &c1);
2600 87418 : IZ = gcoeff(I,1,1);
2601 87418 : if (is_pm1(IZ))
2602 : {
2603 8561 : if (!aI) { set_avma(av); return matid(N); }
2604 8519 : goto END;
2605 : }
2606 78857 : J = idealHNF_inv_Z(nf, I);
2607 78857 : break;
2608 21 : default: /* id_PRINCIPAL, silly case */
2609 21 : if (gequal0(I)) I = cgetg(1,t_MAT); else { c1 = I; I = matid(N); }
2610 21 : if (!aI) return I;
2611 14 : goto END;
2612 : }
2613 : /* now I integral, HNF; and J = (I\cap Z) I^(-1), integral */
2614 112664 : y = idealpseudomin(J, G); /* small elt in (I\cap Z)I^(-1), integral */
2615 112665 : if (equalii(ZV_content(y), IZ))
2616 : { /* already reduced */
2617 64335 : if (!aI) return gerepilecopy(av, I);
2618 60968 : goto END;
2619 : }
2620 :
2621 48330 : my = zk_multable(nf, y);
2622 48330 : I = ZM_Z_divexact(ZM_mul(my, I), IZ); /* y I / (I\cap Z), integral */
2623 48328 : c1 = mul_content(c1, IZ);
2624 48328 : my = ZM_gauss(my, col_ei(N,1)); /* y^-1 */
2625 48330 : yZ = Q_denom(my); /* (y) \cap Z */
2626 48330 : I = hnfmodid(I, yZ);
2627 48330 : if (!aI) return gerepileupto(av, I);
2628 45597 : c1 = RgC_Rg_mul(my, c1);
2629 115683 : END:
2630 115683 : if (c1) aI = ext_mul(nf, aI,c1);
2631 115681 : return gerepilecopy(av, mkvec2(I, aI));
2632 : }
2633 :
2634 : GEN
2635 7 : idealmin(GEN nf, GEN x, GEN vdir)
2636 : {
2637 7 : pari_sp av = avma;
2638 : GEN y, dx;
2639 7 : nf = checknf(nf);
2640 7 : switch( idealtyp(&x,&y) )
2641 : {
2642 0 : case id_PRINCIPAL: return gcopy(x);
2643 0 : case id_PRIME: x = pr_hnf(nf,x); break;
2644 7 : case id_MAT: if (lg(x) == 1) return gen_0;
2645 : }
2646 7 : x = Q_remove_denom(x, &dx);
2647 7 : y = idealpseudomin(x, nf_get_Gtwist(nf,vdir));
2648 7 : if (dx) y = RgC_Rg_div(y, dx);
2649 7 : return gerepileupto(av, y);
2650 : }
2651 :
2652 : /*******************************************************************/
2653 : /* */
2654 : /* APPROXIMATION THEOREM */
2655 : /* */
2656 : /*******************************************************************/
2657 : /* a = ppi(a,b) ppo(a,b), where ppi regroups primes common to a and b
2658 : * and ppo(a,b) = Z_ppo(a,b) */
2659 : /* return gcd(a,b),ppi(a,b),ppo(a,b) */
2660 : GEN
2661 455357 : Z_ppio(GEN a, GEN b)
2662 : {
2663 455357 : GEN x, y, d = gcdii(a,b);
2664 455357 : if (is_pm1(d)) return mkvec3(gen_1, gen_1, a);
2665 346353 : x = d; y = diviiexact(a,d);
2666 : for(;;)
2667 62965 : {
2668 409318 : GEN g = gcdii(x,y);
2669 409318 : if (is_pm1(g)) return mkvec3(d, x, y);
2670 62965 : x = mulii(x,g); y = diviiexact(y,g);
2671 : }
2672 : }
2673 : /* a = ppg(a,b)pple(a,b), where ppg regroups primes such that v(a) > v(b)
2674 : * and pple all others */
2675 : /* return gcd(a,b),ppg(a,b),pple(a,b) */
2676 : GEN
2677 0 : Z_ppgle(GEN a, GEN b)
2678 : {
2679 0 : GEN x, y, g, d = gcdii(a,b);
2680 0 : if (equalii(a, d)) return mkvec3(a, gen_1, a);
2681 0 : x = diviiexact(a,d); y = d;
2682 : for(;;)
2683 : {
2684 0 : g = gcdii(x,y);
2685 0 : if (is_pm1(g)) return mkvec3(d, x, y);
2686 0 : x = mulii(x,g); y = diviiexact(y,g);
2687 : }
2688 : }
2689 : static void
2690 0 : Z_dcba_rec(GEN L, GEN a, GEN b)
2691 : {
2692 : GEN x, r, v, g, h, c, c0;
2693 : long n;
2694 0 : if (is_pm1(b)) {
2695 0 : if (!is_pm1(a)) vectrunc_append(L, a);
2696 0 : return;
2697 : }
2698 0 : v = Z_ppio(a,b);
2699 0 : a = gel(v,2);
2700 0 : r = gel(v,3);
2701 0 : if (!is_pm1(r)) vectrunc_append(L, r);
2702 0 : v = Z_ppgle(a,b);
2703 0 : g = gel(v,1);
2704 0 : h = gel(v,2);
2705 0 : x = c0 = gel(v,3);
2706 0 : for (n = 1; !is_pm1(h); n++)
2707 : {
2708 : GEN d, y;
2709 : long i;
2710 0 : v = Z_ppgle(h,sqri(g));
2711 0 : g = gel(v,1);
2712 0 : h = gel(v,2);
2713 0 : c = gel(v,3); if (is_pm1(c)) continue;
2714 0 : d = gcdii(c,b);
2715 0 : x = mulii(x,d);
2716 0 : y = d; for (i=1; i < n; i++) y = sqri(y);
2717 0 : Z_dcba_rec(L, diviiexact(c,y), d);
2718 : }
2719 0 : Z_dcba_rec(L,diviiexact(b,x), c0);
2720 : }
2721 : static GEN
2722 3082737 : Z_cba_rec(GEN L, GEN a, GEN b)
2723 : {
2724 : GEN g;
2725 3082737 : if (lg(L) > 10)
2726 : { /* a few naive steps before switching to dcba */
2727 0 : Z_dcba_rec(L, a, b);
2728 0 : return gel(L, lg(L)-1);
2729 : }
2730 3082737 : if (is_pm1(a)) return b;
2731 1831711 : g = gcdii(a,b);
2732 1831711 : if (is_pm1(g)) { vectrunc_append(L, a); return b; }
2733 1368297 : a = diviiexact(a,g);
2734 1368297 : b = diviiexact(b,g);
2735 1368297 : return Z_cba_rec(L, Z_cba_rec(L, a, g), b);
2736 : }
2737 : GEN
2738 346143 : Z_cba(GEN a, GEN b)
2739 : {
2740 346143 : GEN L = vectrunc_init(expi(a) + expi(b) + 2);
2741 346143 : GEN t = Z_cba_rec(L, a, b);
2742 346143 : if (!is_pm1(t)) vectrunc_append(L, t);
2743 346143 : return L;
2744 : }
2745 : /* P = coprime base, extend it by b; TODO: quadratic for now */
2746 : GEN
2747 0 : ZV_cba_extend(GEN P, GEN b)
2748 : {
2749 0 : long i, l = lg(P);
2750 0 : GEN w = cgetg(l+1, t_VEC);
2751 0 : for (i = 1; i < l; i++)
2752 : {
2753 0 : GEN v = Z_cba(gel(P,i), b);
2754 0 : long nv = lg(v)-1;
2755 0 : gel(w,i) = vecslice(v, 1, nv-1); /* those divide P[i] but not b */
2756 0 : b = gel(v,nv);
2757 : }
2758 0 : gel(w,l) = b; return shallowconcat1(w);
2759 : }
2760 : GEN
2761 0 : ZV_cba(GEN v)
2762 : {
2763 0 : long i, l = lg(v);
2764 : GEN P;
2765 0 : if (l <= 2) return v;
2766 0 : P = Z_cba(gel(v,1), gel(v,2));
2767 0 : for (i = 3; i < l; i++) P = ZV_cba_extend(P, gel(v,i));
2768 0 : return P;
2769 : }
2770 :
2771 : /* write x = x1 x2, x2 maximal s.t. (x2,f) = 1, return x2 */
2772 : GEN
2773 3172817 : Z_ppo(GEN x, GEN f)
2774 : {
2775 : for (;;)
2776 : {
2777 3172817 : f = gcdii(x, f); if (is_pm1(f)) break;
2778 1890917 : x = diviiexact(x, f);
2779 : }
2780 1281898 : return x;
2781 : }
2782 : /* write x = x1 x2, x2 maximal s.t. (x2,f) = 1, return x2 */
2783 : ulong
2784 61697467 : u_ppo(ulong x, ulong f)
2785 : {
2786 : for (;;)
2787 : {
2788 61697467 : f = ugcd(x, f); if (f == 1) break;
2789 13327616 : x /= f;
2790 : }
2791 48369797 : return x;
2792 : }
2793 :
2794 : /* result known to be representable as an ulong */
2795 : static ulong
2796 3481481 : lcmuu(ulong a, ulong b) { ulong d = ugcd(a,b); return (a/d) * b; }
2797 :
2798 : /* assume 0 < x < N; return u in (Z/NZ)^* such that u x = gcd(x,N) (mod N);
2799 : * set *pd = gcd(x,N) */
2800 : ulong
2801 7539670 : Fl_invgen(ulong x, ulong N, ulong *pd)
2802 : {
2803 : ulong d, d0, e, v, v1;
2804 : long s;
2805 7539670 : *pd = d = xgcduu(N, x, 0, &v, &v1, &s);
2806 7539979 : if (s > 0) v = N - v;
2807 7539979 : if (d == 1) return v;
2808 : /* vx = gcd(x,N) (mod N), v coprime to N/d but need not be coprime to N */
2809 4569572 : e = N / d;
2810 4569572 : d0 = u_ppo(d, e); /* d = d0 d1, d0 coprime to N/d, rad(d1) | N/d */
2811 4569713 : if (d0 == 1) return v;
2812 3481419 : e = lcmuu(e, d / d0);
2813 3481475 : return u_chinese_coprime(v, 1, e, d0, e*d0);
2814 : }
2815 :
2816 : /* x t_INT, f ideal. Write x = x1 x2, sqf(x1) | f, (x2,f) = 1. Return x2 */
2817 : static GEN
2818 280 : nf_coprime_part(GEN nf, GEN x, GEN listpr)
2819 : {
2820 280 : long v, j, lp = lg(listpr), N = nf_get_degree(nf);
2821 : GEN x1, x2, ex;
2822 :
2823 : #if 0 /*1) via many gcds. Expensive ! */
2824 : GEN f = idealprodprime(nf, listpr);
2825 : f = ZM_hnfmodid(f, x); /* first gcd is less expensive since x in Z */
2826 : x = scalarmat(x, N);
2827 : for (;;)
2828 : {
2829 : if (gequal1(gcoeff(f,1,1))) break;
2830 : x = idealdivexact(nf, x, f);
2831 : f = ZM_hnfmodid(shallowconcat(f,x), gcoeff(x,1,1)); /* gcd(f,x) */
2832 : }
2833 : x2 = x;
2834 : #else /*2) from prime decomposition */
2835 280 : x1 = NULL;
2836 784 : for (j=1; j<lp; j++)
2837 : {
2838 504 : GEN pr = gel(listpr,j);
2839 504 : v = Z_pval(x, pr_get_p(pr)); if (!v) continue;
2840 :
2841 294 : ex = muluu(v, pr_get_e(pr)); /* = v_pr(x) > 0 */
2842 294 : x1 = x1? idealmulpowprime(nf, x1, pr, ex)
2843 294 : : idealpow(nf, pr, ex);
2844 : }
2845 280 : x = scalarmat(x, N);
2846 280 : x2 = x1? idealdivexact(nf, x, x1): x;
2847 : #endif
2848 280 : return x2;
2849 : }
2850 :
2851 : /* L0 in K^*, assume (L0,f) = 1. Return L integral, L0 = L mod f */
2852 : GEN
2853 6615 : make_integral(GEN nf, GEN L0, GEN f, GEN listpr)
2854 : {
2855 : GEN fZ, t, L, D2, d1, d2, d;
2856 :
2857 6615 : L = Q_remove_denom(L0, &d);
2858 6615 : if (!d) return L0;
2859 :
2860 : /* L0 = L / d, L integral */
2861 1106 : fZ = gcoeff(f,1,1);
2862 1106 : if (typ(L) == t_INT) return Fp_mul(L, Fp_inv(d, fZ), fZ);
2863 : /* Kill denom part coprime to fZ */
2864 630 : d2 = Z_ppo(d, fZ);
2865 630 : t = Fp_inv(d2, fZ); if (!is_pm1(t)) L = ZC_Z_mul(L,t);
2866 630 : if (equalii(d, d2)) return L;
2867 :
2868 280 : d1 = diviiexact(d, d2);
2869 : /* L0 = (L / d1) mod f. d1 not coprime to f
2870 : * write (d1) = D1 D2, D2 minimal, (D2,f) = 1. */
2871 280 : D2 = nf_coprime_part(nf, d1, listpr);
2872 280 : t = idealaddtoone_i(nf, D2, f); /* in D2, 1 mod f */
2873 280 : L = nfmuli(nf,t,L);
2874 :
2875 : /* if (L0, f) = 1, then L in D1 ==> in D1 D2 = (d1) */
2876 280 : return Q_div_to_int(L, d1); /* exact division */
2877 : }
2878 :
2879 : /* assume L is a list of prime ideals. Return the product */
2880 : GEN
2881 336 : idealprodprime(GEN nf, GEN L)
2882 : {
2883 336 : long l = lg(L), i;
2884 : GEN z;
2885 336 : if (l == 1) return matid(nf_get_degree(nf));
2886 336 : z = pr_hnf(nf, gel(L,1));
2887 371 : for (i=2; i<l; i++) z = idealHNF_mul_two(nf,z, gel(L,i));
2888 336 : return z;
2889 : }
2890 :
2891 : /* optimize for the frequent case I = nfhnf()[2]: lots of them are 1 */
2892 : GEN
2893 427 : idealprod(GEN nf, GEN I)
2894 : {
2895 427 : long i, l = lg(I);
2896 : GEN z;
2897 1043 : for (i = 1; i < l; i++)
2898 1036 : if (!equali1(gel(I,i))) break;
2899 427 : if (i == l) return gen_1;
2900 420 : z = gel(I,i);
2901 693 : for (i++; i<l; i++) z = idealmul(nf, z, gel(I,i));
2902 420 : return z;
2903 : }
2904 :
2905 : /* v_pr(idealprod(nf,I)) */
2906 : long
2907 2289 : idealprodval(GEN nf, GEN I, GEN pr)
2908 : {
2909 2289 : long i, l = lg(I), v = 0;
2910 13391 : for (i = 1; i < l; i++)
2911 11102 : if (!equali1(gel(I,i))) v += idealval(nf, gel(I,i), pr);
2912 2289 : return v;
2913 : }
2914 :
2915 : /* assume L is a list of prime ideals. Return prod L[i]^e[i] */
2916 : GEN
2917 55825 : factorbackprime(GEN nf, GEN L, GEN e)
2918 : {
2919 55825 : long l = lg(L), i;
2920 : GEN z;
2921 :
2922 55825 : if (l == 1) return matid(nf_get_degree(nf));
2923 42490 : z = idealpow(nf, gel(L,1), gel(e,1));
2924 64680 : for (i=2; i<l; i++)
2925 22190 : if (signe(gel(e,i))) z = idealmulpowprime(nf,z, gel(L,i),gel(e,i));
2926 42490 : return z;
2927 : }
2928 :
2929 : /* F in Z, divisible exactly by pr.p. Return F-uniformizer for pr, i.e.
2930 : * a t in Z_K such that v_pr(t) = 1 and (t, F/pr) = 1 */
2931 : GEN
2932 60597 : pr_uniformizer(GEN pr, GEN F)
2933 : {
2934 60597 : GEN p = pr_get_p(pr), t = pr_get_gen(pr);
2935 60597 : if (!equalii(F, p))
2936 : {
2937 32925 : long e = pr_get_e(pr);
2938 32925 : GEN u, v, q = (e == 1)? sqri(p): p;
2939 32925 : u = mulii(q, Fp_inv(q, diviiexact(F,p))); /* 1 mod F/p, 0 mod q */
2940 32924 : v = subui(1UL, u); /* 0 mod F/p, 1 mod q */
2941 32923 : if (pr_is_inert(pr))
2942 0 : t = addii(mulii(p, v), u);
2943 : else
2944 : {
2945 32923 : t = ZC_Z_mul(t, v);
2946 32924 : gel(t,1) = addii(gel(t,1), u); /* return u + vt */
2947 : }
2948 : }
2949 60596 : return t;
2950 : }
2951 : /* L = list of prime ideals, return lcm_i (L[i] \cap \ZM) */
2952 : GEN
2953 90204 : prV_lcm_capZ(GEN L)
2954 : {
2955 90204 : long i, r = lg(L);
2956 : GEN F;
2957 90204 : if (r == 1) return gen_1;
2958 70794 : F = pr_get_p(gel(L,1));
2959 125185 : for (i = 2; i < r; i++)
2960 : {
2961 54390 : GEN pr = gel(L,i), p = pr_get_p(pr);
2962 54390 : if (!dvdii(F, p)) F = mulii(F,p);
2963 : }
2964 70795 : return F;
2965 : }
2966 : /* v vector of prid. Return underlying list of rational primes */
2967 : GEN
2968 39746 : prV_primes(GEN v)
2969 : {
2970 39746 : long i, l = lg(v);
2971 39746 : GEN w = cgetg(l,t_VEC);
2972 109529 : for (i=1; i<l; i++) gel(w,i) = pr_get_p(gel(v,i));
2973 39746 : return ZV_sort_uniq(w);
2974 : }
2975 :
2976 : /* Given a prime ideal factorization with possibly zero or negative
2977 : * exponents, gives b such that v_p(b) = v_p(x) for all prime ideals pr | x
2978 : * and v_pr(b) >= 0 for all other pr.
2979 : * For optimal performance, all [anti-]uniformizers should be precomputed,
2980 : * but no support for this yet.
2981 : *
2982 : * If nored, do not reduce result.
2983 : * No garbage collecting */
2984 : static GEN
2985 63807 : idealapprfact_i(GEN nf, GEN x, int nored)
2986 : {
2987 : GEN z, d, L, e, e2, F;
2988 : long i, r;
2989 : int flagden;
2990 :
2991 63807 : nf = checknf(nf);
2992 63807 : L = gel(x,1);
2993 63807 : e = gel(x,2);
2994 63807 : F = prV_lcm_capZ(L);
2995 63808 : flagden = 0;
2996 63808 : z = NULL; r = lg(e);
2997 149654 : for (i = 1; i < r; i++)
2998 : {
2999 85846 : long s = signe(gel(e,i));
3000 : GEN pi, q;
3001 85846 : if (!s) continue;
3002 56880 : if (s < 0) flagden = 1;
3003 56880 : pi = pr_uniformizer(gel(L,i), F);
3004 56879 : q = nfpow(nf, pi, gel(e,i));
3005 56880 : z = z? nfmul(nf, z, q): q;
3006 : }
3007 63808 : if (!z) return gen_1;
3008 29145 : if (nored || typ(z) != t_COL) return z;
3009 10269 : e2 = cgetg(r, t_VEC);
3010 29077 : for (i=1; i<r; i++) gel(e2,i) = addiu(gel(e,i), 1);
3011 10268 : x = factorbackprime(nf, L,e2);
3012 10268 : if (flagden) /* denominator */
3013 : {
3014 10254 : z = Q_remove_denom(z, &d);
3015 10255 : d = diviiexact(d, Z_ppo(d, F));
3016 10253 : x = RgM_Rg_mul(x, d);
3017 : }
3018 : else
3019 14 : d = NULL;
3020 10267 : z = ZC_reducemodlll(z, x);
3021 10269 : return d? RgC_Rg_div(z,d): z;
3022 : }
3023 :
3024 : GEN
3025 0 : idealapprfact(GEN nf, GEN x) {
3026 0 : pari_sp av = avma;
3027 0 : return gerepileupto(av, idealapprfact_i(nf, x, 0));
3028 : }
3029 : GEN
3030 14 : idealappr(GEN nf, GEN x) {
3031 14 : pari_sp av = avma;
3032 14 : if (!is_nf_extfactor(x)) x = idealfactor(nf, x);
3033 14 : return gerepileupto(av, idealapprfact_i(nf, x, 0));
3034 : }
3035 :
3036 : /* OBSOLETE */
3037 : GEN
3038 14 : idealappr0(GEN nf, GEN x, long fl) { (void)fl; return idealappr(nf, x); }
3039 :
3040 : static GEN
3041 21 : mat_ideal_two_elt2(GEN nf, GEN x, GEN a)
3042 : {
3043 21 : GEN F = idealfactor(nf,a), P = gel(F,1), E = gel(F,2);
3044 21 : long i, r = lg(E);
3045 84 : for (i=1; i<r; i++) gel(E,i) = stoi( idealval(nf,x,gel(P,i)) );
3046 21 : return idealapprfact_i(nf,F,1);
3047 : }
3048 :
3049 : static void
3050 14 : not_in_ideal(GEN a) {
3051 14 : pari_err_DOMAIN("idealtwoelt2","element mod ideal", "!=", gen_0, a);
3052 0 : }
3053 : /* x integral in HNF, a an 'nf' */
3054 : static int
3055 28 : in_ideal(GEN x, GEN a)
3056 : {
3057 28 : switch(typ(a))
3058 : {
3059 14 : case t_INT: return dvdii(a, gcoeff(x,1,1));
3060 7 : case t_COL: return RgV_is_ZV(a) && !!hnf_invimage(x, a);
3061 7 : default: return 0;
3062 : }
3063 : }
3064 :
3065 : /* Given an integral ideal x and a in x, gives a b such that
3066 : * x = aZ_K + bZ_K using the approximation theorem */
3067 : GEN
3068 42 : idealtwoelt2(GEN nf, GEN x, GEN a)
3069 : {
3070 42 : pari_sp av = avma;
3071 : GEN cx, b;
3072 :
3073 42 : nf = checknf(nf);
3074 42 : a = nf_to_scalar_or_basis(nf, a);
3075 42 : x = idealhnf_shallow(nf,x);
3076 42 : if (lg(x) == 1)
3077 : {
3078 14 : if (!isintzero(a)) not_in_ideal(a);
3079 7 : set_avma(av); return gen_0;
3080 : }
3081 28 : x = Q_primitive_part(x, &cx);
3082 28 : if (cx) a = gdiv(a, cx);
3083 28 : if (!in_ideal(x, a)) not_in_ideal(a);
3084 21 : b = mat_ideal_two_elt2(nf, x, a);
3085 21 : if (typ(b) == t_COL)
3086 : {
3087 14 : GEN mod = idealhnf_principal(nf,a);
3088 14 : b = ZC_hnfrem(b,mod);
3089 14 : if (ZV_isscalar(b)) b = gel(b,1);
3090 : }
3091 : else
3092 : {
3093 7 : GEN aZ = typ(a) == t_COL? Q_denom(zk_inv(nf,a)): a; /* (a) \cap Z */
3094 7 : b = centermodii(b, aZ, shifti(aZ,-1));
3095 : }
3096 21 : b = cx? gmul(b,cx): gcopy(b);
3097 21 : return gerepileupto(av, b);
3098 : }
3099 :
3100 : /* Given 2 integral ideals x and y in nf, returns a beta in nf such that
3101 : * beta * x is an integral ideal coprime to y */
3102 : GEN
3103 44911 : idealcoprimefact(GEN nf, GEN x, GEN fy)
3104 : {
3105 44911 : GEN L = gel(fy,1), e;
3106 44911 : long i, r = lg(L);
3107 :
3108 44911 : e = cgetg(r, t_COL);
3109 84129 : for (i=1; i<r; i++) gel(e,i) = stoi( -idealval(nf,x,gel(L,i)) );
3110 44911 : return idealapprfact_i(nf, mkmat2(L,e), 0);
3111 : }
3112 : GEN
3113 70 : idealcoprime(GEN nf, GEN x, GEN y)
3114 : {
3115 70 : pari_sp av = avma;
3116 70 : return gerepileupto(av, idealcoprimefact(nf, x, idealfactor(nf,y)));
3117 : }
3118 :
3119 : GEN
3120 7 : nfmulmodpr(GEN nf, GEN x, GEN y, GEN modpr)
3121 : {
3122 7 : pari_sp av = avma;
3123 7 : GEN z, p, pr = modpr, T;
3124 :
3125 7 : nf = checknf(nf); modpr = nf_to_Fq_init(nf,&pr,&T,&p);
3126 0 : x = nf_to_Fq(nf,x,modpr);
3127 0 : y = nf_to_Fq(nf,y,modpr);
3128 0 : z = Fq_mul(x,y,T,p);
3129 0 : return gerepileupto(av, algtobasis(nf, Fq_to_nf(z,modpr)));
3130 : }
3131 :
3132 : GEN
3133 0 : nfdivmodpr(GEN nf, GEN x, GEN y, GEN modpr)
3134 : {
3135 0 : pari_sp av = avma;
3136 0 : nf = checknf(nf);
3137 0 : return gerepileupto(av, nfreducemodpr(nf, nfdiv(nf,x,y), modpr));
3138 : }
3139 :
3140 : GEN
3141 0 : nfpowmodpr(GEN nf, GEN x, GEN k, GEN modpr)
3142 : {
3143 0 : pari_sp av=avma;
3144 0 : GEN z, T, p, pr = modpr;
3145 :
3146 0 : nf = checknf(nf); modpr = nf_to_Fq_init(nf,&pr,&T,&p);
3147 0 : z = nf_to_Fq(nf,x,modpr);
3148 0 : z = Fq_pow(z,k,T,p);
3149 0 : return gerepileupto(av, algtobasis(nf, Fq_to_nf(z,modpr)));
3150 : }
3151 :
3152 : GEN
3153 0 : nfkermodpr(GEN nf, GEN x, GEN modpr)
3154 : {
3155 0 : pari_sp av = avma;
3156 0 : GEN T, p, pr = modpr;
3157 :
3158 0 : nf = checknf(nf); modpr = nf_to_Fq_init(nf, &pr,&T,&p);
3159 0 : if (typ(x)!=t_MAT) pari_err_TYPE("nfkermodpr",x);
3160 0 : x = nfM_to_FqM(x, nf, modpr);
3161 0 : return gerepilecopy(av, FqM_to_nfM(FqM_ker(x,T,p), modpr));
3162 : }
3163 :
3164 : GEN
3165 0 : nfsolvemodpr(GEN nf, GEN a, GEN b, GEN pr)
3166 : {
3167 0 : const char *f = "nfsolvemodpr";
3168 0 : pari_sp av = avma;
3169 : GEN T, p, modpr;
3170 :
3171 0 : nf = checknf(nf);
3172 0 : modpr = nf_to_Fq_init(nf, &pr,&T,&p);
3173 0 : if (typ(a)!=t_MAT) pari_err_TYPE(f,a);
3174 0 : a = nfM_to_FqM(a, nf, modpr);
3175 0 : switch(typ(b))
3176 : {
3177 0 : case t_MAT:
3178 0 : b = nfM_to_FqM(b, nf, modpr);
3179 0 : b = FqM_gauss(a,b,T,p);
3180 0 : if (!b) pari_err_INV(f,a);
3181 0 : a = FqM_to_nfM(b, modpr);
3182 0 : break;
3183 0 : case t_COL:
3184 0 : b = nfV_to_FqV(b, nf, modpr);
3185 0 : b = FqM_FqC_gauss(a,b,T,p);
3186 0 : if (!b) pari_err_INV(f,a);
3187 0 : a = FqV_to_nfV(b, modpr);
3188 0 : break;
3189 0 : default: pari_err_TYPE(f,b);
3190 : }
3191 0 : return gerepilecopy(av, a);
3192 : }
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