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# -*- coding: utf-8 -*- Local data for elliptic curves over number fields
Let `E` be an elliptic curve over a number field `K` (including `\QQ`). There are several local invariants at a finite place `v` that can be computed via Tate's algorithm (see [Sil1994]_ IV.9.4 or [Tate1975]_).
These include the type of reduction (good, additive, multiplicative), a minimal equation of `E` over `K_v`, the Tamagawa number `c_v`, defined to be the index `[E(K_v):E^0(K_v)]` of the points with good reduction among the local points, and the exponent of the conductor `f_v`.
The functions in this file will typically be called by using ``local_data``.
EXAMPLES::
sage: K.<i> = NumberField(x^2+1) sage: E = EllipticCurve([(2+i)^2,(2+i)^7]) sage: pp = K.fractional_ideal(2+i) sage: da = E.local_data(pp) sage: da.has_bad_reduction() True sage: da.has_multiplicative_reduction() False sage: da.kodaira_symbol() I0* sage: da.tamagawa_number() 4 sage: da.minimal_model() Elliptic Curve defined by y^2 = x^3 + (4*i+3)*x + (-29*i-278) over Number Field in i with defining polynomial x^2 + 1
An example to show how the Neron model can change as one extends the field::
sage: E = EllipticCurve([0,-1]) sage: E.local_data(2) Local data at Principal ideal (2) of Integer Ring: Reduction type: bad additive Local minimal model: Elliptic Curve defined by y^2 = x^3 - 1 over Rational Field Minimal discriminant valuation: 4 Conductor exponent: 4 Kodaira Symbol: II Tamagawa Number: 1
sage: EK = E.base_extend(K) sage: EK.local_data(1+i) Local data at Fractional ideal (i + 1): Reduction type: bad additive Local minimal model: Elliptic Curve defined by y^2 = x^3 + (-1) over Number Field in i with defining polynomial x^2 + 1 Minimal discriminant valuation: 8 Conductor exponent: 2 Kodaira Symbol: IV* Tamagawa Number: 3
Or how the minimal equation changes::
sage: E = EllipticCurve([0,8]) sage: E.is_minimal() True sage: EK = E.base_extend(K) sage: da = EK.local_data(1+i) sage: da.minimal_model() Elliptic Curve defined by y^2 = x^3 + (-i) over Number Field in i with defining polynomial x^2 + 1
AUTHORS:
- John Cremona: First version 2008-09-21 (refactoring code from ``ell_number_field.py`` and ``ell_rational_field.py``)
- Chris Wuthrich: more documentation 2010-01
""" #***************************************************************************** # Copyright (C) 2005 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""" The class for the local reduction data of an elliptic curve.
Currently supported are elliptic curves defined over `\QQ`, and elliptic curves defined over a number field, at an arbitrary prime or prime ideal.
INPUT:
- ``E`` -- an elliptic curve defined over a number field, or `\QQ`.
- ``P`` -- a prime ideal of the field, or a prime integer if the field is `\QQ`.
- ``proof`` (bool)-- if True, only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.
- ``algorithm`` (string, default: "pari") -- Ignored unless the base field is `\QQ`. If "pari", use the PARI C-library ``ellglobalred`` implementation of Tate's algorithm over `\QQ`. If "generic", use the general number field implementation.
.. note::
This function is not normally called directly by users, who may access the data via methods of the EllipticCurve classes.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData sage: E = EllipticCurve('14a1') sage: EllipticCurveLocalData(E,2) Local data at Principal ideal (2) of Integer Ring: Reduction type: bad non-split multiplicative Local minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field Minimal discriminant valuation: 6 Conductor exponent: 1 Kodaira Symbol: I6 Tamagawa Number: 2
"""
r""" Initializes the reduction data for the elliptic curve `E` at the prime `P`.
INPUT:
- ``E`` -- an elliptic curve defined over a number field, or `\QQ`.
- ``P`` -- a prime ideal of the field, or a prime integer if the field is `\QQ`.
- ``proof`` (bool)-- if True, only use provably correct methods (default controlled by global proof module). Note that the proof module is number_field, not elliptic_curves, since the functions that actually need the flag are in number fields.
- ``algorithm`` (string, default: "pari") -- Ignored unless the base field is `\QQ`. If "pari", use the PARI C-library ``ellglobalred`` implementation of Tate's algorithm over `\QQ`. If "generic", use the general number field implementation.
- ``globally`` (bool, default: False) -- If True, the algorithm uses the generators of principal ideals rather than an arbitrary uniformizer.
.. note::
This function is not normally called directly by users, who may access the data via methods of the EllipticCurve classes.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData sage: E = EllipticCurve('14a1') sage: EllipticCurveLocalData(E,2) Local data at Principal ideal (2) of Integer Ring: Reduction type: bad non-split multiplicative Local minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field Minimal discriminant valuation: 6 Conductor exponent: 1 Kodaira Symbol: I6 Tamagawa Number: 2
::
sage: EllipticCurveLocalData(E,2,algorithm="generic") Local data at Principal ideal (2) of Integer Ring: Reduction type: bad non-split multiplicative Local minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field Minimal discriminant valuation: 6 Conductor exponent: 1 Kodaira Symbol: I6 Tamagawa Number: 2
::
sage: EllipticCurveLocalData(E,2,algorithm="pari") Local data at Principal ideal (2) of Integer Ring: Reduction type: bad non-split multiplicative Local minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field Minimal discriminant valuation: 6 Conductor exponent: 1 Kodaira Symbol: I6 Tamagawa Number: 2
::
sage: EllipticCurveLocalData(E,2,algorithm="unknown") Traceback (most recent call last): ... ValueError: algorithm must be one of 'pari', 'generic'
::
sage: EllipticCurveLocalData(E,3) Local data at Principal ideal (3) of Integer Ring: Reduction type: good Local minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field Minimal discriminant valuation: 0 Conductor exponent: 0 Kodaira Symbol: I0 Tamagawa Number: 1
::
sage: EllipticCurveLocalData(E,7) Local data at Principal ideal (7) of Integer Ring: Reduction type: bad split multiplicative Local minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field Minimal discriminant valuation: 3 Conductor exponent: 1 Kodaira Symbol: I3 Tamagawa Number: 3 """
else:
# We use a global minimal model since we can: else: else:
r""" Returns the string representation of this reduction data.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData sage: E = EllipticCurve('14a1') sage: EllipticCurveLocalData(E,2).__repr__() 'Local data at Principal ideal (2) of Integer Ring:\nReduction type: bad non-split multiplicative\nLocal minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field\nMinimal discriminant valuation: 6\nConductor exponent: 1\nKodaira Symbol: I6\nTamagawa Number: 2' sage: EllipticCurveLocalData(E,3).__repr__() 'Local data at Principal ideal (3) of Integer Ring:\nReduction type: good\nLocal minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field\nMinimal discriminant valuation: 0\nConductor exponent: 0\nKodaira Symbol: I0\nTamagawa Number: 1' sage: EllipticCurveLocalData(E,7).__repr__() 'Local data at Principal ideal (7) of Integer Ring:\nReduction type: bad split multiplicative\nLocal minimal model: Elliptic Curve defined by y^2 + x*y + y = x^3 + 4*x - 6 over Rational Field\nMinimal discriminant valuation: 3\nConductor exponent: 1\nKodaira Symbol: I3\nTamagawa Number: 3' """
""" Return the (local) minimal model from this local reduction data.
INPUT:
- ``reduce`` -- (default: True) if set to True and if the initial elliptic curve had globally integral coefficients, then the elliptic curve returned by Tate's algorithm will be "reduced" as specified in _reduce_model() for curves over number fields.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData sage: E = EllipticCurve([0,0,0,0,64]); E Elliptic Curve defined by y^2 = x^3 + 64 over Rational Field sage: data = EllipticCurveLocalData(E,2) sage: data.minimal_model() Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field sage: data.minimal_model() == E.local_minimal_model(2) True
To demonstrate the behaviour of the parameter ``reduce``::
sage: K.<a> = NumberField(x^3+x+1) sage: E = EllipticCurve(K, [0, 0, a, 0, 1]) sage: E.local_data(K.ideal(a-1)).minimal_model() Elliptic Curve defined by y^2 + a*y = x^3 + 1 over Number Field in a with defining polynomial x^3 + x + 1 sage: E.local_data(K.ideal(a-1)).minimal_model(reduce=False) Elliptic Curve defined by y^2 + (a+2)*y = x^3 + 3*x^2 + 3*x + (-a+1) over Number Field in a with defining polynomial x^3 + x + 1
sage: E = EllipticCurve([2, 1, 0, -2, -1]) sage: E.local_data(ZZ.ideal(2), algorithm="generic").minimal_model(reduce=False) Elliptic Curve defined by y^2 + 2*x*y + 2*y = x^3 + x^2 - 4*x - 2 over Rational Field sage: E.local_data(ZZ.ideal(2), algorithm="pari").minimal_model(reduce=False) Traceback (most recent call last): ... ValueError: the argument reduce must not be False if algorithm=pari is used sage: E.local_data(ZZ.ideal(2), algorithm="generic").minimal_model() Elliptic Curve defined by y^2 = x^3 - x^2 - 3*x + 2 over Rational Field sage: E.local_data(ZZ.ideal(2), algorithm="pari").minimal_model() Elliptic Curve defined by y^2 = x^3 - x^2 - 3*x + 2 over Rational Field
:trac:`14476`::
sage: t = QQ['t'].0 sage: K.<g> = NumberField(t^4 - t^3-3*t^2 - t +1) sage: E = EllipticCurve([-2*g^3 + 10/3*g^2 + 3*g - 2/3, -11/9*g^3 + 34/9*g^2 - 7/3*g + 4/9, -11/9*g^3 + 34/9*g^2 - 7/3*g + 4/9, 0, 0]) sage: vv = K.fractional_ideal(g^2 - g - 2) sage: E.local_data(vv).minimal_model() Elliptic Curve defined by y^2 + (-2*g^3+10/3*g^2+3*g-2/3)*x*y + (-11/9*g^3+34/9*g^2-7/3*g+4/9)*y = x^3 + (-11/9*g^3+34/9*g^2-7/3*g+4/9)*x^2 over Number Field in g with defining polynomial t^4 - t^3 - 3*t^2 - t + 1
""" # trac 14476 we only reduce if the coefficients are globally integral else: else:
""" Return the prime ideal associated with this local reduction data.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData sage: E = EllipticCurve([0,0,0,0,64]); E Elliptic Curve defined by y^2 = x^3 + 64 over Rational Field sage: data = EllipticCurveLocalData(E,2) sage: data.prime() Principal ideal (2) of Integer Ring """
""" Return the valuation of the conductor from this local reduction data.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData sage: E = EllipticCurve([0,0,0,0,64]); E Elliptic Curve defined by y^2 = x^3 + 64 over Rational Field sage: data = EllipticCurveLocalData(E,2) sage: data.conductor_valuation() 2 """
""" Return the valuation of the minimal discriminant from this local reduction data.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData sage: E = EllipticCurve([0,0,0,0,64]); E Elliptic Curve defined by y^2 = x^3 + 64 over Rational Field sage: data = EllipticCurveLocalData(E,2) sage: data.discriminant_valuation() 4 """
r""" Return the Kodaira symbol from this local reduction data.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData sage: E = EllipticCurve([0,0,0,0,64]); E Elliptic Curve defined by y^2 = x^3 + 64 over Rational Field sage: data = EllipticCurveLocalData(E,2) sage: data.kodaira_symbol() IV """
r""" Return the Tamagawa number from this local reduction data.
This is the index `[E(K_v):E^0(K_v)]`.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData sage: E = EllipticCurve([0,0,0,0,64]); E Elliptic Curve defined by y^2 = x^3 + 64 over Rational Field sage: data = EllipticCurveLocalData(E,2) sage: data.tamagawa_number() 3 """
r""" Return the Tamagawa index from this local reduction data.
This is the exponent of `E(K_v)/E^0(K_v)`; in most cases it is the same as the Tamagawa index.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData sage: E = EllipticCurve('816a1') sage: data = EllipticCurveLocalData(E,2) sage: data.kodaira_symbol() I2* sage: data.tamagawa_number() 4 sage: data.tamagawa_exponent() 2
sage: E = EllipticCurve('200c4') sage: data = EllipticCurveLocalData(E,5) sage: data.kodaira_symbol() I4* sage: data.tamagawa_number() 4 sage: data.tamagawa_exponent() 2 """
r""" Return the type of bad reduction of this reduction data.
OUTPUT:
(int or ``None``):
- +1 for split multiplicative reduction - -1 for non-split multiplicative reduction - 0 for additive reduction - ``None`` for good reduction
EXAMPLES::
sage: E=EllipticCurve('14a1') sage: [(p,E.local_data(p).bad_reduction_type()) for p in prime_range(15)] [(2, -1), (3, None), (5, None), (7, 1), (11, None), (13, None)]
sage: K.<a>=NumberField(x^3-2) sage: P17a, P17b = [P for P,e in K.factor(17)] sage: E = EllipticCurve([0,0,0,0,2*a+1]) sage: [(p,E.local_data(p).bad_reduction_type()) for p in [P17a,P17b]] [(Fractional ideal (4*a^2 - 2*a + 1), None), (Fractional ideal (2*a + 1), 0)] """
r""" Return True if there is good reduction.
EXAMPLES::
sage: E = EllipticCurve('14a1') sage: [(p,E.local_data(p).has_good_reduction()) for p in prime_range(15)] [(2, False), (3, True), (5, True), (7, False), (11, True), (13, True)]
sage: K.<a> = NumberField(x^3-2) sage: P17a, P17b = [P for P,e in K.factor(17)] sage: E = EllipticCurve([0,0,0,0,2*a+1]) sage: [(p,E.local_data(p).has_good_reduction()) for p in [P17a,P17b]] [(Fractional ideal (4*a^2 - 2*a + 1), True), (Fractional ideal (2*a + 1), False)] """
r""" Return True if there is bad reduction.
EXAMPLES::
sage: E = EllipticCurve('14a1') sage: [(p,E.local_data(p).has_bad_reduction()) for p in prime_range(15)] [(2, True), (3, False), (5, False), (7, True), (11, False), (13, False)]
::
sage: K.<a> = NumberField(x^3-2) sage: P17a, P17b = [P for P,e in K.factor(17)] sage: E = EllipticCurve([0,0,0,0,2*a+1]) sage: [(p,E.local_data(p).has_bad_reduction()) for p in [P17a,P17b]] [(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), True)] """
r""" Return True if there is multiplicative reduction.
.. note::
See also ``has_split_multiplicative_reduction()`` and ``has_nonsplit_multiplicative_reduction()``.
EXAMPLES::
sage: E = EllipticCurve('14a1') sage: [(p,E.local_data(p).has_multiplicative_reduction()) for p in prime_range(15)] [(2, True), (3, False), (5, False), (7, True), (11, False), (13, False)]
::
sage: K.<a> = NumberField(x^3-2) sage: P17a, P17b = [P for P,e in K.factor(17)] sage: E = EllipticCurve([0,0,0,0,2*a+1]) sage: [(p,E.local_data(p).has_multiplicative_reduction()) for p in [P17a,P17b]] [(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), False)] """
r""" Return True if there is split multiplicative reduction.
EXAMPLES::
sage: E = EllipticCurve('14a1') sage: [(p,E.local_data(p).has_split_multiplicative_reduction()) for p in prime_range(15)] [(2, False), (3, False), (5, False), (7, True), (11, False), (13, False)]
::
sage: K.<a> = NumberField(x^3-2) sage: P17a, P17b = [P for P,e in K.factor(17)] sage: E = EllipticCurve([0,0,0,0,2*a+1]) sage: [(p,E.local_data(p).has_split_multiplicative_reduction()) for p in [P17a,P17b]] [(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), False)] """
r""" Return True if there is non-split multiplicative reduction.
EXAMPLES::
sage: E = EllipticCurve('14a1') sage: [(p,E.local_data(p).has_nonsplit_multiplicative_reduction()) for p in prime_range(15)] [(2, True), (3, False), (5, False), (7, False), (11, False), (13, False)]
::
sage: K.<a> = NumberField(x^3-2) sage: P17a, P17b = [P for P,e in K.factor(17)] sage: E = EllipticCurve([0,0,0,0,2*a+1]) sage: [(p,E.local_data(p).has_nonsplit_multiplicative_reduction()) for p in [P17a,P17b]] [(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), False)] """
r""" Return True if there is additive reduction.
EXAMPLES::
sage: E = EllipticCurve('27a1') sage: [(p,E.local_data(p).has_additive_reduction()) for p in prime_range(15)] [(2, False), (3, True), (5, False), (7, False), (11, False), (13, False)]
::
sage: K.<a> = NumberField(x^3-2) sage: P17a, P17b = [P for P,e in K.factor(17)] sage: E = EllipticCurve([0,0,0,0,2*a+1]) sage: [(p,E.local_data(p).has_additive_reduction()) for p in [P17a,P17b]] [(Fractional ideal (4*a^2 - 2*a + 1), False), (Fractional ideal (2*a + 1), True)] """
r""" Tate's algorithm for an elliptic curve over a number field.
Computes both local reduction data at a prime ideal and a local minimal model.
The model is not required to be integral on input. If `P` is principal, uses a generator as uniformizer, so it will not affect integrality or minimality at other primes. If `P` is not principal, the minimal model returned will preserve integrality at other primes, but not minimality.
The optional argument globally, when set to True, tells the algorithm to use the generator of the prime ideal if it is principal. Otherwise just any uniformizer will be used.
.. note::
Called only by ``EllipticCurveLocalData.__init__()``.
OUTPUT:
(tuple) ``(Emin, p, val_disc, fp, KS, cp)`` where:
- ``Emin`` (EllipticCurve) is a model (integral and) minimal at P - ``p`` (int) is the residue characteristic - ``val_disc`` (int) is the valuation of the local minimal discriminant - ``fp`` (int) is the valuation of the conductor - ``KS`` (string) is the Kodaira symbol - ``cp`` (int) is the Tamagawa number
EXAMPLES (this raised a type error in sage prior to 4.4.4, see :trac:`7930`) ::
sage: E = EllipticCurve('99d1')
sage: R.<X> = QQ[] sage: K.<t> = NumberField(X^3 + X^2 - 2*X - 1) sage: L.<s> = NumberField(X^3 + X^2 - 36*X - 4)
sage: EK = E.base_extend(K) sage: toK = EK.torsion_order() sage: da = EK.local_data() # indirect doctest
sage: EL = E.base_extend(L) sage: da = EL.local_data() # indirect doctest
EXAMPLES:
The following example shows that the bug at :trac:`9324` is fixed::
sage: K.<a> = NumberField(x^2-x+6) sage: E = EllipticCurve([0,0,0,-53160*a-43995,-5067640*a+19402006]) sage: E.conductor() # indirect doctest Fractional ideal (18, 6*a)
The following example shows that the bug at :trac:`9417` is fixed::
sage: K.<a> = NumberField(x^2+18*x+1) sage: E = EllipticCurve(K, [0, -36, 0, 320, 0]) sage: E.tamagawa_number(K.ideal(2)) 4
This is to show that the bug :trac:`11630` is fixed. (The computation of the class group would produce a warning)::
sage: K.<t> = NumberField(x^7-2*x+177) sage: E = EllipticCurve([0,1,0,t,t]) sage: P = K.ideal(2,t^3 + t + 1) sage: E.local_data(P).kodaira_symbol() II
"""
# In case P is not principal we mostly use a uniformiser which # is globally integral (with positive valuation at some other # primes); for this to work, it is essential that we can # reduce (mod P) elements of K which are not integral (but are # P-integral). However, if the model is non-minimal and we # end up dividing a_i by pi^i then at that point we use a # uniformiser pi which has non-positive valuation at all other # primes, so that we can divide by it without losing # integrality at other primes.
else:
else:
# Since ResidueField is cached in a way that # does not care much about embeddings of number # fields, it can happen that F.p.ring() is different # from K. This is a problem: If F.p.ring() has no # embedding but K has, then there is no coercion # from F.p.ring().maximal_order() to K. But it is # no problem to do an explicit conversion in that # case (Simon King, trac ticket #8800).
except CoercionException: # the pushout does not exist, we need conversion pinv = lambda x: K(F.lift(~F(x))) proot = lambda x,e: K(F.lift(F(x).nth_root(e, extend = False, all = True)[0])) preduce = lambda x: K(F.lift(F(x)))
r""" Local function returning True iff `ax^2 + bx + c` has roots modulo `P` """ return (b != 0) or (c == 0) else: r""" Local function returning the number of roots of `x^3 + b*x^2 + c*x + d` modulo `P`, counting multiplicities """
else:
## Good reduction already
# Otherwise, we change coordinates so that p | a3, a4, a6 else: else: else: else: raise RuntimeError("Non-integral model after first transform!") raise RuntimeError("p does not divide a3 after first transform!") raise RuntimeError("p does not divide a4 after first transform!") raise RuntimeError("p does not divide a6 after first transform!")
# Now we test for Types In, II, III, IV # NB the c invariants never change.
# Multiplicative reduction: Type In (n = val_disc) else:
# Additive reduction
## Type II ## Type III ## Type IV
# If our curve is none of these types, we change coords so that # p | a1, a2; p^2 | a3, a4; p^3 | a6 else: raise RuntimeError("p does not divide a1 after second transform!") raise RuntimeError("p does not divide a2 after second transform!") raise RuntimeError("p^2 does not divide a3 after second transform!") raise RuntimeError("p^2 does not divide a4 after second transform!") raise RuntimeError("p^3 does not divide a6 after second transform!") raise RuntimeError("Non-integral model after second transform!")
# Analyze roots of the cubic T^3 + bT^2 + cT + d = 0 mod P, where # b = a2/p, c = a4/p^2, d = a6/p^3 else: else: ## Three distinct roots - Type I*0 ## One double root - Type I*m for some m ## Change coords so that the double root is T = 0 mod p else: r = (bc - 9*d)*pinv(2*x) # The rest of this branch is just to compute cp, fp, KS. # We use pi to keep transforms integral. else: else: else: else: else: else: else: # sw == 3 ## The cubic has a triple root ## First we change coordinates so that T = 0 mod p else: raise RuntimeError("Non-integral model after third transform!") raise RuntimeError("Cubic after transform does not have a triple root at 0") # We test for Type IV* # Now change coordinates so that p^3|a3, p^5|a6 else: # We test for types III* and II* ## Type III* ## Type II* else:
r""" Function to check that `P` determines a prime of `K`, and return that ideal.
INPUT:
- ``K`` -- a number field (including `\QQ`).
- ``P`` -- an element of ``K`` or a (fractional) ideal of ``K``.
OUTPUT:
- If ``K`` is `\QQ`: the prime integer equal to or which generates `P`.
- If ``K`` is not `\QQ`: the prime ideal equal to or generated by `P`.
.. note::
If `P` is not a prime and does not generate a prime, a TypeError is raised.
EXAMPLES::
sage: from sage.schemes.elliptic_curves.ell_local_data import check_prime sage: check_prime(QQ,3) 3 sage: check_prime(QQ,QQ(3)) 3 sage: check_prime(QQ,ZZ.ideal(31)) 31 sage: K.<a>=NumberField(x^2-5) sage: check_prime(K,a) Fractional ideal (a) sage: check_prime(K,a+1) Fractional ideal (a + 1) sage: [check_prime(K,P) for P in K.primes_above(31)] [Fractional ideal (5/2*a + 1/2), Fractional ideal (5/2*a - 1/2)] sage: L.<b> = NumberField(x^2+3) sage: check_prime(K, L.ideal(5)) Traceback (most recent call last): .. TypeError: The ideal Fractional ideal (5) is not a prime ideal of Number Field in a with defining polynomial x^2 - 5 sage: check_prime(K, L.ideal(b)) Traceback (most recent call last): TypeError: No compatible natural embeddings found for Number Field in a with defining polynomial x^2 - 5 and Number Field in b with defining polynomial x^2 + 3 """ else: raise TypeError("The element %s is not prime" % (P,) ) raise TypeError("The element %s is not prime" % (P,) ) else: raise TypeError("The ideal %s is not a prime ideal of %s" % (P, ZZ)) else: raise TypeError("%s is neither an element of QQ or an ideal of %s" % (P, ZZ))
raise TypeError("%s is not a number field" % (K,) )
# if P is an ideal, making sure it is an fractional ideal of K else:
raise TypeError("%s is not a valid prime of %s" % (P, K)) |