| 1 | r""" |
| 2 | Reduction of binary forms and hyperelliptic curve equations. |
| 3 | |
| 4 | This file contains functions that convert hyperelliptic curve equations into |
| 5 | equivalent ones with hopefully smaller coefficients. |
| 6 | |
| 7 | The first step is to reduce discriminants of hyperelliptic curve equations at |
| 8 | non-archimedean primes of a number field. Doing this globally is generally |
| 9 | impossible when the class number is > 1, so this is only approximate. |
| 10 | |
| 11 | This is followed by reduction as in Stoll-Cremona [SC2003]_, which |
| 12 | reduces the archimedean size of the coefficients. See |
| 13 | :module:`sage.schemes.hyperelliptic_curves.stoll_cremona`. |
| 14 | |
| 15 | For details see Bouyer-Streng [BS2013]_ |
| 16 | |
| 17 | REFERENCES: |
| 18 | .. [SC2003] M. Stoll, J. Cremona, "On the reduction theory of binary forms" |
| 19 | Journal fur die reine und angewandter Mathematik 565, pages 79 - 99 |
| 20 | .. [BS2013] F. Bouyer, M. Steng, "Examples of CM curves of genus two defined |
| 21 | over the reflex field", arXiv:1307.0486[math.NT] |
| 22 | |
| 23 | AUTHORS: |
| 24 | |
| 25 | - Florian Bouyer |
| 26 | - Marco Streng |
| 27 | |
| 28 | """ |
| 29 | #***************************************************************************** |
| 30 | # Copyright (C) 2011,2012,2013 |
| 31 | # Florian Bouyer <f.j.s.c.bouyer@gmail.com> |
| 32 | # Marco Streng <marco.streng@gmail.com> |
| 33 | # |
| 34 | # Distributed under the terms of the GNU General Public License (GPL) |
| 35 | # as published by the Free Software Foundation; either version 2 of |
| 36 | # the License, or (at your option) any later version. |
| 37 | # http://www.gnu.org/licenses/ |
| 38 | #***************************************************************************** |
| 39 | |
| 40 | from sage.rings.arith import gcd |
| 41 | from sage.rings.all import ZZ, QQ, RealField, FiniteField |
| 42 | from sage.misc.misc import prod |
| 43 | from sage.schemes.hyperelliptic_curves.stoll_cremona import stoll_cremona_reduction |
| 44 | |
| 45 | |
| 46 | def reduce_hyperelliptic_curve_polynomial(f, precision=3000, algorithm='default', primes=None): |
| 47 | r""" |
| 48 | |
| 49 | Reduces f by using the three functions reduce_gcd, reduce_discriminant, |
| 50 | stoll_cremona_reduction. |
| 51 | (See their respective documentations for details) |
| 52 | |
| 53 | Currently this is an internal function used by |
| 54 | :func: `HyperellipticCurve_from_invariants`. In the future it will |
| 55 | be a method of the HyperellipticCurve class. |
| 56 | |
| 57 | INPUT: |
| 58 | |
| 59 | - ``f`` - a polynomial |
| 60 | |
| 61 | - ``precision`` - an integer (default: 3000) the bit precision of the |
| 62 | numbers in `\CC` and `\RR` that is going to be used. |
| 63 | |
| 64 | - ``algorithm`` - default/magma. If set to magma in the stoll_cremona |
| 65 | reduction step, the function uses magma to calculate a 'better' covariant |
| 66 | of f |
| 67 | |
| 68 | - ``primes`` (default:None) -- a list of primes. If supplied, then reduce |
| 69 | the discriminant only at the given primes |
| 70 | |
| 71 | OUTPUT: |
| 72 | |
| 73 | A polynomial |
| 74 | |
| 75 | EXAMPLES: |
| 76 | |
| 77 | Over `\QQ`:: |
| 78 | |
| 79 | sage: P.<x> = QQ[] |
| 80 | sage: from sage.schemes.hyperelliptic_curves.reduction import reduce_hyperelliptic_curve_polynomial |
| 81 | sage: f = -16*x^6 - 1611*x^5 - 8640*x^4 + 69120*x^3 - 311040*x^2 + 746496*x - 746496 |
| 82 | sage: reduce_hyperelliptic_curve_polynomial(f) |
| 83 | -x^6 + 3*x - 2 |
| 84 | sage: f = -5*x^6 + 174*x^5 - 2700*x^4 + 21600*x^3 - 97200*x^2 + 233280*x - 233280 |
| 85 | sage: reduce_hyperelliptic_curve_polynomial(f) |
| 86 | -x^6 - x^5 - 5 |
| 87 | |
| 88 | Over number fields:: |
| 89 | |
| 90 | sage: X = var('X') |
| 91 | sage: k = NumberField(X^2 - 41,'a') |
| 92 | sage: a = k.an_element() |
| 93 | sage: P.<x> = k[] |
| 94 | sage: f = x^6 + 2*x^4 + (64*a + 410)*x^3 + 1311360*a + 8396801 |
| 95 | sage: reduce_hyperelliptic_curve_polynomial(f) |
| 96 | (-320*a + 2049)*x^6 + (-640*a + 4098)*x^4 + (-64*a + 410)*x^3 + 320*a + 2049 |
| 97 | sage: k = NumberField(X^2 - 73,'a') |
| 98 | sage: a = k.an_element() |
| 99 | sage: P.<x> = k[] |
| 100 | sage: f = (1/778034*a - 1/778034)*x^6 + (3/5329*a - 3/5329)*x^5 + (15/146*a - 9/146)*x^4 + (10*a + 2)*x^3 + (1095/2*a + 1533/2)*x^2 + (15987*a + 47961)*x + 389017/2*a + 2723119/2 |
| 101 | sage: reduce_hyperelliptic_curve_polynomial(f) |
| 102 | x^6 + 3*x^2 + 1/2*a - 1/2 |
| 103 | |
| 104 | """ |
| 105 | if precision is None: |
| 106 | precision = 3000 |
| 107 | k = f(0).parent() |
| 108 | f = reduce_gcd(f) |
| 109 | f = reduce_discriminant(f, primes=primes) |
| 110 | f = stoll_cremona_reduction(f, algorithm=algorithm, precision=precision) |
| 111 | try: |
| 112 | f = reduce_unit_in_disc(f) |
| 113 | except NotImplementedError: |
| 114 | pass |
| 115 | return f |
| 116 | |
| 117 | |
| 118 | def reduce_gcd(f): |
| 119 | r""" |
| 120 | Takes out the common factors of the coefficients of f. In the case of |
| 121 | number fields of class number larger than one, there may still be a small |
| 122 | prime ideal in the denominator after the reduction. |
| 123 | |
| 124 | This is an internal function called by :func: `reduce_hyperelliptic_curve_polynomial` |
| 125 | INPUT: |
| 126 | |
| 127 | - ``f`` - a polynomial |
| 128 | |
| 129 | OUTPUT: |
| 130 | |
| 131 | If the base ring is ZZ or QQ, then the output is an integer polynomial |
| 132 | whose coefficients have gcd 1 and which is a rational scaling of f. |
| 133 | |
| 134 | If the base ring is a number field, then it still removes many prime factors |
| 135 | from the gcd of the coefficients (all in the case of class number one). |
| 136 | |
| 137 | EXAMPLES:: |
| 138 | |
| 139 | sage: P.<x> = QQ[] |
| 140 | sage: from sage.schemes.hyperelliptic_curves.reduction import reduce_gcd |
| 141 | sage: reduce_gcd(3*x^2+6*x+321) |
| 142 | x^2 + 2*x + 107 |
| 143 | sage: reduce_gcd(321*x^6 + 1284*x^2 - 2247*x + 5136) |
| 144 | x^6 + 4*x^2 - 7*x + 16 |
| 145 | |
| 146 | sage: X = var('X') |
| 147 | sage: k = NumberField(X^2 + 41,'a') |
| 148 | sage: a = k.an_element() |
| 149 | sage: P.<x> = k[] |
| 150 | sage: reduce_gcd((a + 41)*x^6 + (41*a + 41)*x^4 + (42*a + 82)*x^2 + 16*a + 656) |
| 151 | (-a + 1)*x^6 + (-a + 41)*x^4 + (-2*a + 42)*x^2 - 16*a + 16 |
| 152 | |
| 153 | """ |
| 154 | k = f.base_ring() |
| 155 | f = f.denominator()*f |
| 156 | |
| 157 | if k is QQ or k is ZZ: |
| 158 | gcdofcoeffs = gcd([ZZ(coeff) for coeff in f.coeffs()]) |
| 159 | return f/gcdofcoeffs |
| 160 | |
| 161 | A = k.ideal(f.list()) |
| 162 | return f / gen_almost(A) |
| 163 | |
| 164 | |
| 165 | def gen_almost(A): |
| 166 | r""" |
| 167 | If A is a principal ideal, returns a generator. |
| 168 | |
| 169 | Otherwise, if A is a number field ideal, returns an element x of the number |
| 170 | field such that x divides A and (A/x).norm() is small. |
| 171 | |
| 172 | This is an internal function, used by :func:`reduce_at_infinity` |
| 173 | and :func:`reduce_gcd`. |
| 174 | |
| 175 | EXAMPLES:: |
| 176 | |
| 177 | sage: k = QuadraticField(-5,'a') |
| 178 | sage: p = k.ideal(next_prime(2000)).factor()[0][0]; p |
| 179 | Fractional ideal (2003, a + 1483) |
| 180 | sage: from sage.schemes.hyperelliptic_curves.reduction import gen_almost |
| 181 | sage: x = gen_almost(p); x |
| 182 | -27/2*a + 19/2 |
| 183 | sage: p/x |
| 184 | Fractional ideal (2, a + 1) |
| 185 | |
| 186 | """ |
| 187 | if A.is_principal(): |
| 188 | return A.gens_reduced()[0] |
| 189 | # Here is the obvious way: |
| 190 | #B = A.reduce_equiv() |
| 191 | #C = A/B |
| 192 | #if not C.is_principal(): |
| 193 | # raise RuntimeError |
| 194 | |
| 195 | # So we do something that looks stupid and slow, |
| 196 | # but seemed to be faster: |
| 197 | k = A.number_field() |
| 198 | for c in k.class_group(): |
| 199 | B = c.ideal() |
| 200 | C = A/B |
| 201 | if C.is_principal(): |
| 202 | break |
| 203 | |
| 204 | if not B.is_integral(): |
| 205 | raise RuntimeError |
| 206 | return C.gens_reduced()[0] |
| 207 | |
| 208 | |
| 209 | def reduce_at_infinity(f, primes=None): |
| 210 | r""" |
| 211 | If there is a 4-fold root at infinity modulo a prime, then try to remove |
| 212 | it. |
| 213 | |
| 214 | This works over any number field. If the class number is one, then the |
| 215 | discriminant of the output divides the discriminant of the input. |
| 216 | |
| 217 | This is an internal function, called by :func:`reduce_discriminant`. |
| 218 | |
| 219 | INPUT: |
| 220 | |
| 221 | - ``f`` -- a polynomial of degree 5 or 6 |
| 222 | - ``primes`` (default:None) -- a list of primes, |
| 223 | reduce only at the given primes |
| 224 | |
| 225 | OUTPUT: |
| 226 | |
| 227 | A polynomial ``g`` such that as many prime powers |
| 228 | as possible are removed from the discriminant by |
| 229 | transformations of the form `f(x/u^k)u^l` with |
| 230 | `k,l>=0` and `u` a uniformizer of a prime. |
| 231 | |
| 232 | If the class number of the base field is one, |
| 233 | then a small prime may remain or be introduced. |
| 234 | |
| 235 | EXAMPLES:: |
| 236 | |
| 237 | sage: P.<x> = QQ[] |
| 238 | sage: h = (prod([x+3^7*5^8*n for n in [1,2,3,4]])*(134*x^2+384*x+378)).reverse(); h |
| 239 | 4832126692668678984045982360839843750000*x^6 + 4908827128145711155933141708374023437500*x^5 + 1712976141118022078847492961120605468750*x^4 + 4177353776971384564386468750000*x^3 + 3422870579757550781628*x^2 + 1144757812884*x + 134 |
| 240 | sage: log(h.discriminant()).n() |
| 241 | 643.624696259543 + 3.14159265358979*I |
| 242 | sage: from sage.schemes.hyperelliptic_curves.reduction import reduce_at_infinity |
| 243 | sage: log(reduce_at_infinity(h).discriminant()).n() |
| 244 | 405.008434398004 + 3.14159265358979*I |
| 245 | |
| 246 | sage: X = var('X') |
| 247 | sage: k = NumberField(X^2 + 41,'a') |
| 248 | sage: a = k.an_element() |
| 249 | sage: P.<x> = k[] |
| 250 | sage: h = (prod([x+3^7*5^8*n for n in [1,2,3,4]])*(134*x^2+384*x+378)).reverse(); h |
| 251 | 4832126692668678984045982360839843750000*x^6 + 4908827128145711155933141708374023437500*x^5 + 1712976141118022078847492961120605468750*x^4 + 4177353776971384564386468750000*x^3 + 3422870579757550781628*x^2 + 1144757812884*x + 134 |
| 252 | sage: log(h.discriminant().norm()).n() |
| 253 | 1287.24939251909 |
| 254 | sage: log(reduce_at_infinity(h).discriminant().norm()).n() |
| 255 | 796.153925184809 |
| 256 | |
| 257 | """ |
| 258 | k = f.base_ring() |
| 259 | x = k['x'].gen() |
| 260 | |
| 261 | if k is ZZ or k is QQ: |
| 262 | A = gcd([ZZ(c) for c in f.list()[4:]]) |
| 263 | Bz = 1 |
| 264 | Cz = 1 |
| 265 | else: |
| 266 | A = k.ideal(f.list()[4:]) |
| 267 | Bz = k.ideal(1) |
| 268 | Cz = k.ideal(1) |
| 269 | By = 1 |
| 270 | Cy = 1 |
| 271 | if primes is None: |
| 272 | primes = [p for (p,e) in A.factor()] |
| 273 | for p in primes: |
| 274 | maxpow = ZZ(min([(f[6].valuation(p)-1)/3, \ |
| 275 | (f[5].valuation(p)-1)/2, \ |
| 276 | f[4].valuation(p)-1, \ |
| 277 | f[3].valuation(p)]).floor()) |
| 278 | if maxpow > 0: |
| 279 | powers = max([[3*pow + \ |
| 280 | min([f[indx].valuation(p)-indx*pow \ |
| 281 | for indx in range(7)]),pow] \ |
| 282 | for pow in range(maxpow+1)]) |
| 283 | if powers[0] > 0: |
| 284 | # The following two operations (taking powers of p) take way |
| 285 | # too long in Sage < 5.7 when p is an ideal, hence we |
| 286 | # use different powering functions, making use of |
| 287 | # the class group. |
| 288 | #B = B * p**powers[1] |
| 289 | #C = C * p**(3*powers[1]-powers[0]) |
| 290 | y, z = power_of_ideal(p, powers[1], factors=True) |
| 291 | Bz = Bz * z |
| 292 | By = By * y |
| 293 | y, z = power_of_ideal(p, 3*powers[1]-powers[0], factors=True) |
| 294 | Cz = Cz * z |
| 295 | Cy = Cy * y |
| 296 | # If we knew that B and C were principal, then |
| 297 | # the reduction step would be |
| 298 | # g = f(x/p**powers[1]) * p**(3*powers[1]-powers[0]) |
| 299 | |
| 300 | if k is ZZ or k is QQ: |
| 301 | return f(x/By/Bz)*Cy*Cz |
| 302 | |
| 303 | f = f(x/By/gen_almost(Bz)) * Cy*gen_almost(Cz) |
| 304 | return reduce_gcd(f) |
| 305 | |
| 306 | |
| 307 | def power_of_ideal(A, e, factors=False): |
| 308 | r""" |
| 309 | Return A**e. Uses the fact that the class group |
| 310 | is small, so goes via principal ideals. |
| 311 | |
| 312 | If factors is True, returns an element |
| 313 | and an ideal, such that A**e is the product. |
| 314 | |
| 315 | This is an internal function called by :func: `reduce_at_infinity` |
| 316 | """ |
| 317 | if A in QQ: |
| 318 | if factors: |
| 319 | return A**e, 1 |
| 320 | return A**e |
| 321 | |
| 322 | k = A.number_field() |
| 323 | m = k.class_group()(A).order() |
| 324 | q, r = ZZ(e).quo_rem(m) |
| 325 | assert (A**m).is_principal() |
| 326 | g = (A**m).gens_reduced()[0] |
| 327 | x = g**q |
| 328 | B = A**r |
| 329 | if factors: |
| 330 | return (x, B) |
| 331 | return x*B |
| 332 | |
| 333 | |
| 334 | def reduce_discriminant(f, primes=None): |
| 335 | r""" |
| 336 | Removes 10th powers from the discriminant by a `GL_2(QQ)` transformation. |
| 337 | |
| 338 | This is an internal function called by :func: `reduce_hyperelliptic_curve_polynomial` |
| 339 | |
| 340 | INPUT: |
| 341 | |
| 342 | - ``f`` - a univariate polynomial of degree at most 6, interpreted as the |
| 343 | binary sextic form `F(X,Y) = Y^6 f(X/Y)`. The polynomial ``f`` is not |
| 344 | allowed to have repeated roots. |
| 345 | |
| 346 | OUTPUT: |
| 347 | |
| 348 | a polynomial giving a `GL_2(QQ)` equivalent binary form with 10th powers |
| 349 | removed from the discriminant where possible |
| 350 | |
| 351 | EXAMPLES:: |
| 352 | |
| 353 | sage: from sage.schemes.hyperelliptic_curves.reduction import reduce_discriminant |
| 354 | sage: P.<x> = QQ[] |
| 355 | sage: f = 32*x^6 + 6*x^2 - 5 |
| 356 | sage: factor(f.discriminant()) |
| 357 | 2^33 * 3^6 * 5 * 13^2 |
| 358 | sage: g = reduce_discriminant(f); g |
| 359 | x^6 + 3*x^2 - 10 |
| 360 | sage: factor(g.discriminant()) |
| 361 | 2^13 * 3^6 * 5 * 13^2 |
| 362 | |
| 363 | An example where the degree of the input is 5:: |
| 364 | |
| 365 | sage: P.<y> = QQ[] |
| 366 | sage: f = y^5 + 3^20*y + 3^20 |
| 367 | sage: g = f(y+1) |
| 368 | sage: f |
| 369 | y^5 + 3486784401*y + 3486784401 |
| 370 | sage: reduce_discriminant(f) |
| 371 | x^5 + 81*x + 1 |
| 372 | sage: g |
| 373 | y^5 + 5*y^4 + 10*y^3 + 10*y^2 + 3486784406*y + 6973568803 |
| 374 | sage: reduce_discriminant(g) |
| 375 | x^5 + 5*x^4 + 10*x^3 + 10*x^2 + 86*x + 83 |
| 376 | |
| 377 | An example where a large power is removed:: |
| 378 | |
| 379 | sage: x = QQ['x'].gen() |
| 380 | sage: f = ((x-2)*(x-2+3^20)*(x-2+3^10)*(x-2+3^20*7)*(x^2+x+1)) |
| 381 | sage: reduce_discriminant(f) |
| 382 | 3486784401*x^6 + 1647132543836838*x^5 + 85105305481926930091*x^4 + 85110864542496497113*x^3 + 7206360479055108*x^2 + 170852435649*x |
| 383 | sage: reduce_discriminant(f).discriminant().factor() |
| 384 | -1 * 2^10 * 3^83 * 7^6 * 11^4 * 61^2 * 233^2 * 887^2 * 1549321^2 * 4607887^2 * 11920789^2 * 1964677987^2 * 3486489163^2 * 6188121169^2 |
| 385 | sage: f.discriminant().factor() |
| 386 | -1 * 2^10 * 3^183 * 7^6 * 11^4 * 61^2 * 233^2 * 887^2 * 1549321^2 * 4607887^2 * 11920789^2 * 1964677987^2 * 3486489163^2 * 6188121169^2 |
| 387 | sage: reduce_discriminant(f.reverse()).discriminant() == reduce_discriminant(f).discriminant() |
| 388 | True |
| 389 | |
| 390 | ALGORITHM: |
| 391 | |
| 392 | First we create a list of primes that are factors of the discriminant D of |
| 393 | f. For each prime `p` check that ``valuation(D, p)>=10`` |
| 394 | Then we try to maximize `e = min\{v(a_n)+(3-n)k\}` with 0 <= k <= |
| 395 | `min\{(v(a_{n})-1)/3,v(a_{n-1})-1)/2,v(a_{n-2})-1,v(a_{n-3})\} |
| 396 | (where `v(a_n)` is ``valuation(a_n, p)``). |
| 397 | Finally we do the substitution `f = f(x/p^e) and clear denominator |
| 398 | Next we do the same trick but looking at the coefficients in the reverse |
| 399 | order. Note that this comes down to seeing whether f has 0 as a fourfold |
| 400 | root mod `p`. So first we improve a bit by moving any fourfold root to 0 |
| 401 | before applying the same trick. (but replacing `a_{n-i}` with `a_i` in the |
| 402 | condition k <= min...). |
| 403 | |
| 404 | """ |
| 405 | k = f(0).parent() |
| 406 | OK = k.OK() |
| 407 | P = k['x'] |
| 408 | x = P.gen() |
| 409 | f = reduce_gcd(f) |
| 410 | f = reduce_at_infinity(f, primes=primes) |
| 411 | one = ZZ(1) |
| 412 | |
| 413 | from sage.schemes.hyperelliptic_curves.invariants import igusa_clebsch_invariants |
| 414 | |
| 415 | if k is QQ or k is ZZ: |
| 416 | D = gcd(igusa_clebsch_invariants(f)) |
| 417 | |
| 418 | if not (k is QQ or k is ZZ): |
| 419 | D = k.ideal(igusa_clebsch_invariants(f)) |
| 420 | c = coprime_representatives(k, D) |
| 421 | |
| 422 | if primes is None: |
| 423 | primes = [prime for (prime, power) in D.factor()] |
| 424 | for prime in primes: |
| 425 | if (D.valuation(prime)>=2): |
| 426 | if k is QQ or k is ZZ: |
| 427 | u_unif = prime |
| 428 | l_unif = prime |
| 429 | else: |
| 430 | u_unif = almost_generator(prime, c) |
| 431 | l_unif = one/almost_generator(one/prime, c) |
| 432 | f = reduce_locally(f, prime, u_unif, l_unif) |
| 433 | f = reduce_gcd(f) |
| 434 | f = reduce_at_infinity(f) |
| 435 | return f |
| 436 | |
| 437 | |
| 438 | def reduce_locally(f, prime, u_unif=None, l_unif=None): |
| 439 | r""" |
| 440 | Reduces `C : y^2 = f(x)` locally at ``prime``. Assumes that `C` has no |
| 441 | automorphisms over the algebraic closure other than the identity and the |
| 442 | hyperelliptic involution. |
| 443 | |
| 444 | This is an internal function called by :func: `reduce_discriminant` |
| 445 | INPUT: |
| 446 | |
| 447 | - ``f`` - a univariate polynomial of degree 2g+1 or 2g+2, interpreted as |
| 448 | the binary sextic form `F(X,Y) = Y^(2g+2) f(X/Y)`. The polynomial ``f`` |
| 449 | is not allowed to have repeated roots. |
| 450 | - ``prime`` - a prime number |
| 451 | - ``u_unif`` - a uniformizer such that u_unif / prime is integral |
| 452 | - ``l_unif`` - a uniformizer such that prime / l_unif is integral |
| 453 | |
| 454 | OUTPUT: |
| 455 | |
| 456 | a polynomial giving a `GL_2(QQ)` equivalent binary form with this prime |
| 457 | removed from the discriminant where possible |
| 458 | """ |
| 459 | g = ((ZZ(f.degree())-1)/2).floor() |
| 460 | |
| 461 | k = f.base_ring() |
| 462 | |
| 463 | if k is QQ: |
| 464 | F = FiniteField(prime) |
| 465 | if u_unif is None: |
| 466 | u_unif = QQ(prime) |
| 467 | if l_unif is None: |
| 468 | l_unif = QQ(prime) |
| 469 | else: |
| 470 | F = prime.residue_field() |
| 471 | if prime.is_principal(): |
| 472 | if u_unif is None: |
| 473 | u_unif = prime.gens_reduced()[0] |
| 474 | if l_unif is None: |
| 475 | l_unif = prime.gens_reduced()[0] |
| 476 | elif u_unif is None or l_unif is None: |
| 477 | raise ValueError, "Please supply u_unif and l_unif for non-principal ideal" |
| 478 | |
| 479 | x = f.parent().gen() |
| 480 | |
| 481 | try_this_prime = True |
| 482 | while try_this_prime: |
| 483 | try_this_prime = False |
| 484 | m = min([c.valuation(prime) for c in f.list()]) |
| 485 | f = f / l_unif**m |
| 486 | for k in range(g+2): |
| 487 | if f[2*g+2-k].valuation(prime) < g+2-k: |
| 488 | break |
| 489 | else: |
| 490 | # now val(f[2*g+2-k])>=g+2-k for all k |
| 491 | f = f(x/l_unif) |
| 492 | m = -min([c.valuation(prime) for c in f.list()]) |
| 493 | assert m <= g |
| 494 | f = f * u_unif ** m |
| 495 | |
| 496 | rts = f.roots(F) |
| 497 | for ind in range(len(rts)): |
| 498 | if rts[ind][1] >= 4: |
| 499 | root_mod_prime = F.lift(rts[ind][0]) |
| 500 | h = f(x + root_mod_prime) |
| 501 | for k in range(g+2): |
| 502 | if h[k].valuation(prime) < g+2-k: |
| 503 | break |
| 504 | else: |
| 505 | # now val(h[k])>=g+2-k for all k |
| 506 | f = h(x*u_unif) |
| 507 | try_this_prime = True |
| 508 | |
| 509 | return f |
| 510 | |
| 511 | |
| 512 | def coprime_generator_representatives(k, A): |
| 513 | r""" |
| 514 | Given a number field k and an ideal A, returns a list of integral ideals, |
| 515 | one for every generator of the class group, all of them coprime to A. |
| 516 | |
| 517 | This is an auxiliary function of :func:`coprime_representatives` and only |
| 518 | used for number fields of class number >1. |
| 519 | """ |
| 520 | c = k.class_group() |
| 521 | gens = c.gens() |
| 522 | gens_todo = [(gens[i], i) for i in range(len(gens))] |
| 523 | primes = [None for g in gens] |
| 524 | for p in k.primes_of_degree_one_iter(): |
| 525 | if p+A == 1: |
| 526 | for a in gens_todo: |
| 527 | if c(p) == a[0]: |
| 528 | primes[a[1]] = p |
| 529 | gens_todo.remove(a) |
| 530 | if len(gens_todo) == 0: |
| 531 | return primes |
| 532 | break |
| 533 | |
| 534 | |
| 535 | def coprime_representatives(k, A): |
| 536 | r""" |
| 537 | Given a number field k and an ideal A, returns a list of integral ideals, |
| 538 | one for every ideal class, all of them coprime to A. |
| 539 | |
| 540 | This provides data used by :func:`almost_generator` and only |
| 541 | used for number fields of class number >1. |
| 542 | """ |
| 543 | c = k.class_group() |
| 544 | l = coprime_generator_representatives(k, A) |
| 545 | ret = [] |
| 546 | for a in c: |
| 547 | m = a.list() |
| 548 | ret.append(prod([l[i]**m[i] for i in range(len(m))])) |
| 549 | return ret |
| 550 | |
| 551 | |
| 552 | def almost_generator(I, coprime_rep): |
| 553 | r""" |
| 554 | Given an ideal I and the output of :func:`coprime_representatives`, returns |
| 555 | a generator of I times an element of ``coprime_rep``. |
| 556 | |
| 557 | This is only used for number fields of class number >1. |
| 558 | """ |
| 559 | for J in coprime_rep: |
| 560 | K = I*J |
| 561 | if K.is_principal(): |
| 562 | return K.gens_reduced()[0] |
| 563 | |
| 564 | |
| 565 | def reduce_unit_in_disc(f, precision=100): |
| 566 | """ |
| 567 | Reduce the size of the coefficients of f by multiplying f by a unit. |
| 568 | |
| 569 | INPUT: |
| 570 | |
| 571 | -``f``- a polynomial in a number field `K` |
| 572 | |
| 573 | OUTPUT: |
| 574 | |
| 575 | a polynomial of the form u*f, where u is a unit of the maximal order of `K` |
| 576 | |
| 577 | EXAMPLES:: |
| 578 | |
| 579 | sage: from sage.schemes.hyperelliptic_curves.reduction import reduce_unit_in_disc |
| 580 | sage: X = var('X') |
| 581 | sage: k = NumberField(X^2-41,'a') |
| 582 | sage: a = k.an_element() |
| 583 | sage: P.<x> = k[] |
| 584 | sage: f = 2*x^4 + (64*a + 410)*x^3 + 1311360*a + 8396801 |
| 585 | sage: factor(f.discriminant()) |
| 586 | (-369836393348730718080*a - 2368108374136006451201) * (-118*a - 725) * (-118*a + 725) * (-1/2*a + 7/2)^4 * (1/2*a + 7/2)^4 |
| 587 | sage: g = reduce_unit_in_disc(f); g |
| 588 | (-640*a + 4098)*x^4 + (-64*a + 410)*x^3 + 320*a + 2049 |
| 589 | sage: factor(g.discriminant()) |
| 590 | (-1) * (-118*a - 725) * (-118*a + 725) * (-1/2*a + 7/2)^4 * (1/2*a + 7/2)^4 |
| 591 | sage: f = (-320*a + 2049)*x^6 + (-10374*a + 66426)*x^5 + (-74076*a + 474318)*x^4 + (228552*a - 1463448)*x^3 + (1510568*a - 9672351)*x^2 + (-4678674*a + 29958126)*x + 3024324*a - 19365120 |
| 592 | sage: factor(f.discriminant()) |
| 593 | (-6210885195014008862151805440*a + 39769069528107045198367948801) * (-a - 6)^4 * (-2*a + 1)^2 * (-a)^6 * (-a + 2)^2 * (4*a - 27)^2 * (3*a - 20)^2 * (-1/2*a + 7/2)^26 * (a - 6)^6 * (-a + 8)^2 * 3^2 * (1/2*a + 7/2)^22 |
| 594 | sage: g = reduce_unit_in_disc(f); g |
| 595 | x^6 + (-6*a - 6)*x^5 + (36*a + 462)*x^4 + (-312*a - 2712)*x^3 + (1512*a + 4961)*x^2 + (-2706*a - 2706)*x + 1476*a |
| 596 | sage: factor(g.discriminant()) |
| 597 | (1311360*a + 8396801) * (-a - 6)^4 * (-2*a + 1)^2 * (-a)^6 * (-a + 2)^2 * (4*a - 27)^2 * (3*a - 20)^2 * (-1/2*a + 7/2)^26 * (a - 6)^6 * (-a + 8)^2 * 3^2 * (1/2*a + 7/2)^22 |
| 598 | |
| 599 | Note that this does nothing over `\QQ`:: |
| 600 | |
| 601 | sage: P.<x> = QQ[] |
| 602 | sage: f = x^2 + 3 |
| 603 | sage: reduce_unit_in_disc(f) |
| 604 | x^2 + 3 |
| 605 | |
| 606 | This is not implemented for number fields of degree greater than 2:: |
| 607 | |
| 608 | sage: P.<y> = NumberField(x^3+2,'a')[] |
| 609 | sage: reduce_unit_in_disc(y^6+y+1) |
| 610 | Traceback (most recent call last): |
| 611 | ... |
| 612 | NotImplementedError |
| 613 | """ |
| 614 | k = f.base_ring() |
| 615 | if k is QQ or k.degree() == 1 or (k.degree() == 2 and k.is_totally_imaginary()): |
| 616 | return f |
| 617 | if k.degree() > 2: |
| 618 | raise NotImplementedError |
| 619 | epsilon = k(k.unit_group().gens()[1]) |
| 620 | Phi = k.embeddings(RealField(precision)) |
| 621 | d = f.discriminant() |
| 622 | # e = abs(Phi[0](epsilon))**30 |
| 623 | # while abs(Phi[0](d)/Phi[1](d)) < e: |
| 624 | # f = f*epsilon |
| 625 | # d = f.discriminant() |
| 626 | # e = e**-1 |
| 627 | # while abs(Phi[0](d)/Phi[1](d)) > e: |
| 628 | # f = f/epsilon |
| 629 | # d = f.discriminant() |
| 630 | # return f |
| 631 | |
| 632 | h = f*epsilon |
| 633 | while max([max([abs(Phi[0](coeff)),abs(Phi[1](coeff))]) for coeff in (h).coefficients()])\ |
| 634 | < max([max([abs(Phi[0](coeff)),abs(Phi[1](coeff))]) for coeff in (f).coefficients()]): |
| 635 | f = h |
| 636 | h = f*epsilon |
| 637 | h = f/epsilon |
| 638 | while max([max([abs(Phi[0](coeff)),abs(Phi[1](coeff))]) for coeff in (h).coefficients()])\ |
| 639 | < max([max([abs(Phi[0](coeff)),abs(Phi[1](coeff))]) for coeff in (f).coefficients()]): |
| 640 | f = h |
| 641 | h = f/epsilon |
| 642 | return f |
| 643 | No newline at end of file |