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# -*- coding: utf-8 -*- Global and semi-global minimal models for elliptic curves over number fields
When E is an elliptic curve defined over a number field K of class number 1, then it has a global minimal model, and we have a method to compute it, namely E.global_minimal_model(). Until Sage-6.7 this was done using Tate's algorithm to minimise one prime at a time without affecting the other primes. When the class number is not 1 a different approach is used.
In the general case global minimal models may or may not exist. This module includes functions to determine this, and to find a global minimal model when it does exist. The obstruction to the existence of a global minimal model is encoded in an ideal class, which is trivial if and only if a global minimal model exists: we provide a function which returns this class. When the obstruction is not trivial, there exist models which are minimal at all primes except at a single prime in the obstruction class, where the discriminant valuation is 12 more than the minimal valuation at that prime; we provide a function to return such a model.
The implementation of this functionality is based on work of Kraus [Kraus]_ which gives a local condition for when a pair of number field elements \(c_4\), \(c_6\) belong to a Weierstrass model which is integral at a prime \(P\), together with a global version. Only primes dividing 2 or 3 are hard to deal with. In order to compute the corresponding integral model one then needs to combine together the local transformations implicit in [Kraus]_ into a single global one.
Various utility functions relating to the testing and use of Kraus's conditions are included here.
AUTHORS:
- John Cremona (2015)
REFERENCES:
.. [Kraus] Kraus, Alain, Quelques remarques à propos des invariants \(c_4\), \(c_6\) et \(\Delta\) d'une courbe elliptique, Acta Arith. 54 (1989), 75-80. """
############################################################################## # Copyright (C) 2012-2014 John Cremona <john.cremona@gmail.com> # William Stein <wstein@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # # This code is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU # General Public License for more details. # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ ##############################################################################
r""" Check if c4, c6 are integral with valid associated discriminant.
INPUT:
- ``c4``, ``c6`` -- elements of a number field
OUTPUT:
Boolean, True if c4, c6 are both integral and c4^3-c6^2 is a nonzero multiple of 1728.
EXAMPLES:
Over `\QQ`::
sage: from sage.schemes.elliptic_curves.kraus import c4c6_nonsingular sage: c4c6_nonsingular(0,0) False sage: c4c6_nonsingular(0,1/2) False sage: c4c6_nonsingular(2,3) False sage: c4c6_nonsingular(4,8) False sage: all([c4c6_nonsingular(*E.c_invariants()) for E in cremona_curves([ 11..100])]) True
Over number fields::
sage: K.<a> = NumberField(x^2-10) sage: c4c6_nonsingular(-217728*a - 679104, 141460992*a + 409826304) True sage: K.<a> = NumberField(x^3-10) sage: c4c6_nonsingular(-217728*a - 679104, 141460992*a + 409826304) True """
r""" Return the elliptic curve [0,0,0,-c4/48,-c6/864] with given c-invariants.
INPUT:
- ``c4``, ``c6`` -- elements of a number field
- ``assume_nonsingular`` (boolean, default False) -- if True, check for integrality and nosingularity.
OUTPUT:
The elliptic curve with a-invariants [0,0,0,-c4/48,-c6/864], whose c-invariants are the given c4, c6. If the supplied invariants are singular, returns None when ``assume_nonsingular`` is False and raises an ArithmeticError otherwise.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.kraus import c4c6_model sage: K.<a> = NumberField(x^3-10) sage: c4c6_model(-217728*a - 679104, 141460992*a + 409826304) Elliptic Curve defined by y^2 = x^3 + (4536*a+14148)*x + (-163728*a-474336) over Number Field in a with defining polynomial x^3 - 10
sage: c4, c6 = EllipticCurve('389a1').c_invariants() sage: c4c6_model(c4,c6) Elliptic Curve defined by y^2 = x^3 - 7/3*x + 107/108 over Rational Field """ return None
# Arithmetic utility functions
r""" Returns b in O_K with P^e|(a-b), given a in O_{K,P}.
INPUT:
- ``a`` -- a number field element integral at ``P``
- ``P`` -- a prime ideal of the number field
- ``e`` -- a positive integer
OUTPUT:
A globally integral number field element `b` which is congruent to `a` modulo `P^e`.
ALGORITHM:
Totally naive, we simply test reisdues modulo `P^e` until one works. We will only use this when P is a prime dividing 2 and e is the ramification degree, so the number of residues to check is at worst `2^d` where `d` is the degree of the field.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.kraus import make_integral
sage: K.<a> = NumberField(x^2-10) sage: P = K.primes_above(2)[0] sage: e = P.ramification_index(); e 2 sage: x = 1/5 sage: b = make_integral(x,P,e) sage: b 1 sage: (b-x).valuation(P) >= e True sage: make_integral(1/a,P,e) Traceback (most recent call last): ... ArithmeticError: Cannot lift 1/10*a to O_K mod (Fractional ideal (2, a))^2 """
r""" Returns a local square root mod 4, if it exists.
INPUT:
- ``x`` -- an integral number field element
- ``P`` -- a prime ideal of the number field dividing 2
OUTPUT:
A pair (True, r) where that `r^2-x` has valuation at least `2e`, or (False, 0) if there is no such `r`. Note that `r^2\mod{P^{2e}}` only depends on `r\mod{P^e}`.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.kraus import sqrt_mod_4 sage: K.<a> = NumberField(x^2-10) sage: P = K.primes_above(2)[0] sage: sqrt_mod_4(1+2*a,P) (False, 0) sage: sqrt_mod_4(-1+2*a,P) (True, a + 1) sage: (1+a)^2 - (-1+2*a) 12 sage: e = P.ramification_index() sage: ((1+a)^2 - (-1+2*a)).mod(P**e) 0 """
# Kraus test and check for primes dividing 3:
r""" Test if b2 gives a valid model at a prime dividing 3.
INPUT:
- ``c4``, ``c6`` -- elements of a number field
- ``P`` - a prime ideal of the number field which divides 3
- ``b2`` -- an element of the number field
OUTPUT:
The elliptic curve which is the (b2/12,0,0)-transform of [0,0,0,-c4/48,-c6/864] if this is integral at P, else False.
EXAMPLES::
sage: K.<a> = NumberField(x^2-10) sage: c4 = -60544*a + 385796 sage: c6 = -55799680*a + 262126328 sage: P3a, P3b = K.primes_above(3) sage: from sage.schemes.elliptic_curves.kraus import test_b2_local
b2=0 works at the first prime but not the second::
sage: b2 = 0 sage: test_b2_local(c4,c6,P3a,b2) Elliptic Curve defined by y^2 = x^3 + (3784/3*a-96449/12)*x + (1743740/27*a-32765791/108) over Number Field in a with defining polynomial x^2 - 10 sage: test_b2_local(c4,c6,P3b,b2) False
b2=-a works at the second prime but not the first::
sage: b2 = -a sage: test_b2_local(c4,c6,P3a,b2,debug=True) test_b2_local: not integral at Fractional ideal (3, a + 1) False sage: test_b2_local(c4,c6,P3b,b2) Elliptic Curve defined by y^2 = x^3 + (-1/4*a)*x^2 + (3784/3*a-192893/24)*x + (56378369/864*a-32879311/108) over Number Field in a with defining polynomial x^2 - 10
Using CRT we can do both with the same b2::
sage: b2 = K.solve_CRT([0,-a],[P3a,P3b]); b2 a + 1 sage: test_b2_local(c4,c6,P3a,b2) Elliptic Curve defined by y^2 = x^3 + (1/4*a+1/4)*x^2 + (10091/8*a-128595/16)*x + (4097171/64*a-19392359/64) over Number Field in a with defining polynomial x^2 - 10 sage: test_b2_local(c4,c6,P3b,b2) Elliptic Curve defined by y^2 = x^3 + (1/4*a+1/4)*x^2 + (10091/8*a-128595/16)*x + (4097171/64*a-19392359/64) over Number Field in a with defining polynomial x^2 - 10 """ if debug: print("test_b2_local: wrong c-invariants at P=%s" % P) return False
r""" Test if b2 gives a valid model at all primes dividing 3.
INPUT:
- ``c4``, ``c6`` -- elements of a number field
- ``b2`` -- an element of the number field
OUTPUT:
The elliptic curve which is the (b2/12,0,0)-transform of [0,0,0,-c4/48,-c6/864] if this is integral at all primes P dividing 3, else False.
EXAMPLES::
sage: K.<a> = NumberField(x^2-10) sage: c4 = -60544*a + 385796 sage: c6 = -55799680*a + 262126328 sage: b2 = a+1 sage: from sage.schemes.elliptic_curves.kraus import test_b2_global sage: test_b2_global(c4,c6,b2) Elliptic Curve defined by y^2 = x^3 + (1/4*a+1/4)*x^2 + (10091/8*a-128595/16)*x + (4097171/64*a-19392359/64) over Number Field in a with defining polynomial x^2 - 10 sage: test_b2_global(c4,c6,0,debug=True) test_b2_global: not integral at all primes dividing 3 False sage: test_b2_global(c4,c6,-a,debug=True) test_b2_global: not integral at all primes dividing 3 False """ if debug: print("test_b2_global: wrong c-invariants") return False
r""" Test if c4,c6 satisfy Kraus's conditions at a prime P dividing 3.
INPUT:
- ``c4``, ``c6`` -- elements of a number field
- ``P`` - a prime ideal of the number field which divides 3
- ``assume_nonsingular`` (boolean, default False) -- if True, check for integrality and nosingularity.
OUTPUT:
Either (False, 0) if Kraus's conditions fail, or (True, b2) if they pass, in which case the elliptic curve which is the (b2/12,0,0)-transform of [0,0,0,-c4/48,-c6/864] is integral at P.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.kraus import check_Kraus_local_3 sage: K.<a> = NumberField(x^2-10) sage: c4 = -60544*a + 385796 sage: c6 = -55799680*a + 262126328 sage: P3a, P3b = K.primes_above(3) sage: check_Kraus_local_3(c4,c6,P3a) (True, 0) sage: check_Kraus_local_3(c4,c6,P3b) (True, -a)
An example in a field where 3 is ramified::
sage: K.<a> = NumberField(x^2-15) sage: c4 = -60504*a + 386001 sage: c6 = -55346820*a + 261045153 sage: P3 = K.primes_above(3)[0] sage: check_Kraus_local_3(c4,c6,P3) (True, a) """ return False, 0 assert test_b2_local(c4,c6,P,b2) assert test_b2_local(c4,c6,P,b2) # check for a solution x to x^3-3*x*c4-26=0 (27), such an x must # also satisfy x*c4+c6=0 (3) and x^2=c4 (3) and x^3=-c6 (9), and # if x is a solution then so is any x'=x (3) so it is enough to # check residues mod 3. assert test_b2_local(c4,c6,P,x) return False, 0
# Kraus test and check for primes dividing 2:
r""" Test if a1,a3 are valid at a prime P dividing 2.
INPUT:
- ``c4``, ``c6`` -- elements of a number field
- ``P`` - a prime ideal of the number field which divides 2
- ``a1``, ``a3`` -- elements of the number field
OUTPUT:
The elliptic curve which is the (a1^2/12,a1/2,a3/2)-transform of [0,0,0,-c4/48,-c6/864] if this is integral at P, else False.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.kraus import test_a1a3_local sage: K.<a> = NumberField(x^2-10) sage: c4 = -60544*a + 385796 sage: c6 = -55799680*a + 262126328 sage: P = K.primes_above(2)[0] sage: test_a1a3_local(c4,c6,P,a,0) Elliptic Curve defined by y^2 + a*x*y = x^3 + (3784/3*a-24106/3)*x + (1772120/27*a-2790758/9) over Number Field in a with defining polynomial x^2 - 10 sage: test_a1a3_local(c4,c6,P,a,a,debug=True) test_a1a3_local: not integral at Fractional ideal (2, a) False """ if debug: print("test_a1a3_local: wrong c-invariants at P=%s" % P) return False
r""" Test if a1,a3 are valid at all primes P dividing 2.
INPUT:
- ``c4``, ``c6`` -- elements of a number field
- ``a1``, ``a3`` -- elements of the number field
OUTPUT:
The elliptic curve which is the (a1^2/12,a1/2,a3/2)-transform of [0,0,0,-c4/48,-c6/864] if this is integral at all primes P dividing 2, else False.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.kraus import test_a1a3_global sage: K.<a> = NumberField(x^2-10) sage: c4 = -60544*a + 385796 sage: c6 = -55799680*a + 262126328 sage: test_a1a3_global(c4,c6,a,a,debug=False) False sage: test_a1a3_global(c4,c6,a,0) Elliptic Curve defined by y^2 + a*x*y = x^3 + (3784/3*a-24106/3)*x + (1772120/27*a-2790758/9) over Number Field in a with defining polynomial x^2 - 10 """ if debug: print("wrong c-invariants") return False print("not integral at all primes above 2")
r""" Test if the (r,s,t)-transform of the standard c4,c6-model is integral.
INPUT:
- ``c4``, ``c6`` -- elements of a number field
- ``r``, ``s``, ``t`` -- elements of the number field
OUTPUT:
The elliptic curve which is the (r,s,t)-transform of [0,0,0,-c4/48,-c6/864] if this is integral at all primes P, else False.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.kraus import test_rst_global sage: K.<a> = NumberField(x^2-10) sage: c4 = -60544*a + 385796 sage: c6 = -55799680*a + 262126328 sage: test_rst_global(c4,c6,1/3*a - 133/6, 3/2*a, -89/2*a + 5) Elliptic Curve defined by y^2 + 3*a*x*y + (-89*a+10)*y = x^3 + (a-89)*x^2 + (1202*a-5225)*x + (34881*a-151813) over Number Field in a with defining polynomial x^2 - 10 sage: test_rst_global(c4,c6,a, 3, -89*a, debug=False) False """ if debug: print("test_rst_global: wrong c-invariants") return False print("test_rst_global: not integral at some prime") print(E.ainvs()) K = E.base_field() for P in K.primes_above(2)+K.primes_above(3): if not E.is_local_integral_model(P): print(" -- not integral at P=%s" %P)
# When a1 is None this function finds a pair a1, a3 such that there is # a model with these invariants and a2=0 with the given c4, c6, # integral at P. The value of a1 is unique modulo 2 (i.e. mod P^e # where e is the ramification degree of 2); the value of a3 is unique # mod 2 once a1 is fixed, but not otherwise: when a1 is replaced by # a1+2s we must replace a3 by a3 + a1*s*(a1+s).
# Because of the latter point, and to fix the bug at #19665, we allow # the user to specify a1, in which case a3 is computed from it. This # is important in the global application where we need to put together # the transforms for all the primes above 2: we must first get the # local a1's, then CRT these to get a global a1, then go back to get # the local a3's and finally CRT these.
r""" Test if c4,c6 satisfy Kraus's conditions at a prime P dividing 2.
INPUT:
- ``c4``, ``c6`` -- integral elements of a number field
- ``P`` -- a prime ideal of the number field which divides 2
- ``a1`` -- an integral elements of a number field, or None (default)
- ``assume_nonsingular`` (boolean, default False) -- if True, check for integrality and nosingularity.
OUTPUT:
Either (False, 0, 0) if Kraus's condictions fail, or (True, a1, a3) if they pass, in which case the elliptic curve which is the (a1**2/12,a1/2,a3/2)-transform of [0,0,0,-c4/48,-c6/864] is integral at P. If a1 is provided and valid then the output will be (True, a1, a3) for suitable a3.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.kraus import check_Kraus_local_2 sage: K.<a> = NumberField(x^2-10) sage: c4 = -60544*a + 385796 # EllipticCurve([a,a,0,1263*a-8032,62956*a-305877]) sage: c6 = -55799680*a + 262126328 sage: P = K.primes_above(2)[0] sage: check_Kraus_local_2(c4,c6,P) (True, a, 0) """ return False,0,0
return False,0,0 # In the assignment to a1, a3 we divide by units at P, # (note that c6+a1**6 = 0 mod P**e so dividing by 4 is OK) # but the results, which are well-defined modulo P^e, may # not be globally integral else: raise RuntimeError("check_Kraus_local_2 fails")
else: raise RuntimeError("check_Kraus_local_2 fails") else: return False,0,0
# val(c4) strictly between 0 and 4e; a1 unique mod 2, with 3 conditions to be satisfied:
else: raise RuntimeError("check_Kraus_local_2 fails") # end of loop, but no a1 found return False,0,0
# Wrapper function for local Kraus check, outsources the real work to # other functions for primes dividing 2 or 3:
r""" Check Kraus's condictions locally at a prime P.
INPUT:
- ``c4``, ``c6`` -- elements of a number field
- ``P`` - a prime ideal of the number field
- ``assume_nonsingular`` (boolean, default False) -- if True, check for integrality and nosingularity.
OUTPUT:
Tuple: either (True,E) if there is a Weierstrass model E integral at P and with invariants c4, c6, or (False, None) if there is none.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.kraus import check_Kraus_local sage: K.<a> = NumberField(x^2-15) sage: P2 = K.primes_above(2)[0] sage: P3 = K.primes_above(3)[0] sage: P5 = K.primes_above(5)[0] sage: E = EllipticCurve([a,a,0,1263*a-8032,62956*a-305877]) sage: c4, c6 = E.c_invariants() sage: flag, E = check_Kraus_local(c4,c6,P2); flag True sage: E.is_local_integral_model(P2) and (c4,c6)==E.c_invariants() True sage: flag, E = check_Kraus_local(c4,c6,P3); flag True sage: E.is_local_integral_model(P3) and (c4,c6)==E.c_invariants() True sage: flag, E = check_Kraus_local(c4,c6,P5); flag True sage: E.is_local_integral_model(P5) and (c4,c6)==E.c_invariants() True
sage: c4 = 123+456*a sage: c6 = 789+101112*a sage: check_Kraus_local(c4,c6,P2) (False, None) sage: check_Kraus_local(c4,c6,P3) (False, None) sage: check_Kraus_local(c4,c6,P5) (False, None) """ return (False, None) return (False, None)
r""" Test if c4,c6 satisfy Kraus's conditions at all primes.
INPUT:
- ``c4``, ``c6`` -- elements of a number field
- ``assume_nonsingular`` (boolean, default False) -- if True, check for integrality and nosingularity.
OUTPUT:
Either False if Kraus's condictions fail, or, if they pass, an elliptic curve E which is integral and has c-invariants c4,c6.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.kraus import check_Kraus_global sage: K.<a> = NumberField(x^2-10) sage: E = EllipticCurve([a,a,0,1263*a-8032,62956*a-305877]) sage: c4, c6 = E.c_invariants() sage: check_Kraus_global(c4,c6,debug=True) Local Kraus conditions for (c4,c6)=(-60544*a + 385796,-55799680*a + 262126328) pass at all primes dividing 3 Using b2=a + 1 gives a model integral at 3: (0, 1/4*a + 1/4, 0, 10091/8*a - 128595/16, 4097171/64*a - 19392359/64) Local Kraus conditions for (c4,c6)=(-60544*a + 385796,-55799680*a + 262126328) pass at all primes dividing 2 Using (a1,a3)=(3*a,0) gives a model integral at 2: (3*a, 0, 0, 3784/3*a - 23606/3, 1999160/27*a - 9807634/27) (a1, b2, a3) = (3*a, a + 1, 0) Using (r, s, t)=(1/3*a - 133/6, 3/2*a, -89/2*a + 5) should give a global integral model... ...and it does! Elliptic Curve defined by y^2 + 3*a*x*y + (-89*a+10)*y = x^3 + (a-89)*x^2 + (1202*a-5225)*x + (34881*a-151813) over Number Field in a with defining polynomial x^2 - 10
sage: K.<a> = NumberField(x^2-15) sage: E = EllipticCurve([0, 0, 0, 4536*a + 14148, -163728*a - 474336]) sage: c4, c6 = E.c_invariants() sage: check_Kraus_global(c4,c6) Elliptic Curve defined by y^2 = x^3 + (4536*a+14148)*x + (-163728*a-474336) over Number Field in a with defining polynomial x^2 - 15
TESTS (see :trac:`17295`)::
sage: K.<a> =NumberField(x^3 - 7*x - 5) sage: E = EllipticCurve([a, 0, 1, 2*a^2 + 5*a + 3, -a^2 - 3*a - 2]) sage: assert E.conductor().norm() ==8 sage: G = K.galois_group(names='b') sage: def conj_curve(E,sigma): return EllipticCurve([sigma(a) for a in E.ainvs()]) sage: EL = conj_curve(E,G[0]) sage: L = EL.base_field() sage: assert L.class_number()== 2 sage: EL.isogeny_class() # long time (~10s) Isogeny class of Elliptic Curve defined by y^2 + (-1/90*b^4+7/18*b^2-1/2*b-98/45)*x*y + y = x^3 + (1/45*b^5-1/18*b^4-7/9*b^3+41/18*b^2+167/90*b-29/9)*x + (-1/90*b^5+1/30*b^4+7/18*b^3-4/3*b^2-61/90*b+11/5) over Number Field in b with defining polynomial x^6 - 42*x^4 + 441*x^2 - 697
""" return False
# Check all primes dividing 3; for each get the value of b2 if debug: print("Local Kraus condition for (c4,c6)=(%s,%s) fails at some prime dividing 3" % (c4,c6)) return False
# OK at all primes dividing 3; now use CRT to combine the b2 # values to get a single residue class for b2 mod 3:
# test that this b2 value works at all P|3: else: raise RuntimeError("Error in check_Kraus_global at some prime dividing 3")
# Check all primes dividing 2; for each get the value of a1, then # CRT these to get a single a1 (mod 2) and use these to obtain # local a3; finally CRT these if debug: print("Local Kraus condition for (c4,c6)=(%s,%s) fails at some prime dividing 2" % (c4,c6)) return False
# OK at all primes dividing 2; now use CRT to combine the a1 # values to get the residue classes of a1 mod 2: # See comment below: this is needed for when we combine with the primes above 3.
# Using this a1, recompute the local a3's: # Use CRT to combine these:
# test that these a1,a3 values work at all P|2: else: raise RuntimeError("Error in check_Kraus_global at some prime dividing 2")
# Now we put together the 2-adic and 3-adic transforms to get a # global (r,s,t)-transform from [0,0,0,-c4/48,-c6/864] to a global # integral model.
# We need the combined transform (r,s,t) to have both the forms # (r,s,t) = (a1^2/12,a1/2,a3/2)*(r2,0,0) with 2-integral r2, and # (r,s,t) = (b2/12,0,0,0)*(r3,s3,t3) with 3-integral r3,s3,t3.
# A solution (assuming that a1,a3,b2 are globally integral) is # r2=(b2-a1^2)/3, r3=(b2-a1^2)/4, s3=a1/2, t3=(a1*r2+a3)/2, # provided that a1 =0 (mod 3), to make t3 3-integral. Since a1 # was only determined mod 2 this can be fixed first, simply by # multiplying a1 by 3 if necessary. We did this above.
# Final computation of the curve E: if debug: print("Error in check_Kraus_global with combining mod-2 and mod-3 transforms") E = c4c6_model(c4,c6).rst_transform(r,s,t) print("Transformed model is %a" % (E.ainvs(),)) for P in Plist2+Plist3: if not E.is_local_integral_model(P): print("Not integral at P=%s" % P) raise RuntimeError("Error in check_Kraus_global combining transforms at 2 and 3")
# Success!
r""" Return a global minimal model for this elliptic curve if it exists, or a model minimal at all but one prime otherwise.
INPUT:
- ``E`` -- an elliptic curve over a number field
- ``debug`` (boolean, default False) -- if True, prints some messages about the progress of the computation.
OUTPUT:
A tuple (Emin,I) where Emin is an elliptic curve which is either a global minimal model of E if one exists (i.e., an integral model which is minimal at every prime), or a semin-global minimal model (i.e., an integral model which is minimal at every prime except one). I is the unit ideal of Emin is a global minimal model, else is the unique prime at which Emin is not minimal. Thus in all cases, Emin.minimal_discriminant_ideal() * I**12 == (E.discriminant()).
.. note::
This function is normally not called directly by users, who will use the elliptic curve method :meth:`global_minimal_model` instead; that method also applied various reductions after minimising the model.
EXAMPLES::
sage: K.<a> = NumberField(x^2-10) sage: K.class_number() 2 sage: E = EllipticCurve([0,0,0,-186408*a - 589491, 78055704*a + 246833838]) sage: from sage.schemes.elliptic_curves.kraus import semi_global_minimal_model sage: Emin, P = semi_global_minimal_model(E) sage: Emin Elliptic Curve defined by y^2 + 3*x*y + (2*a-11)*y = x^3 + (a-10)*x^2 + (-152*a-415)*x + (1911*a+5920) over Number Field in a with defining polynomial x^2 - 10 sage: E.minimal_discriminant_ideal()*P**12 == K.ideal(Emin.discriminant()) True
TESTS (see :trac:`20737`): a curve with no global minimal model whose non-minimality class has order 3 in the class group, which has order 3315. The smallest prime in that ideal class has norm 23567::
sage: K.<a> = NumberField(x^2-x+31821453) sage: ainvs = (0, 0, 0, -382586771000351226384*a - 2498023791133552294513515, 358777608829102441023422458989744*a + 1110881475104109582383304709231832166) sage: E = EllipticCurve(ainvs) sage: from sage.schemes.elliptic_curves.kraus import semi_global_minimal_model sage: Emin, p = semi_global_minimal_model(E) # long time (15s) sage: p # long time Fractional ideal (23567, a + 2270) sage: p.norm() # long time 23567 sage: Emin.discriminant().norm().factor() # long time 23567^12
""" else: print("No global minimal model, obstruction class = %s of order %s" % (c,c.order())) except RuntimeError: bound *=2 print("Using a prime in that class: %s" % P) raise RuntimeError("failed to compute global minimal model") |