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ell_local_data.py
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ell_local_data.py
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# -*- coding: utf-8 -*-
r"""
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 [Sil2] IV.9.4 or [Ta]).
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
REFERENCES:
- [Sil2] Silverman, Joseph H., Advanced topics in the arithmetic of elliptic curves.
Graduate Texts in Mathematics, 151. Springer-Verlag, New York, 1994.
- [Ta] Tate, John, Algorithm for determining the type of a singular fiber in an elliptic pencil.
Modular functions of one variable, IV, pp. 33--52. Lecture Notes in Math., Vol. 476,
Springer, Berlin, 1975.
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
"""
from __future__ import absolute_import
#*****************************************************************************
# 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/
#*****************************************************************************
from sage.structure.sage_object import SageObject
from sage.misc.misc import verbose
from sage.rings.all import PolynomialRing, QQ, ZZ, Integer
from sage.rings.number_field.number_field_element import is_NumberFieldElement
from sage.rings.number_field.number_field_ideal import is_NumberFieldFractionalIdeal
from sage.rings.number_field.number_field import is_NumberField
from sage.rings.ideal import is_Ideal
from .constructor import EllipticCurve
from .kodaira_symbol import KodairaSymbol
class EllipticCurveLocalData(SageObject):
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
"""
def __init__(self, E, P, proof=None, algorithm="pari", globally=False):
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
"""
self._curve = E
K = E.base_field()
p = check_prime(K,P) # error handling done in that function
if algorithm != "pari" and algorithm != "generic":
raise ValueError("algorithm must be one of 'pari', 'generic'")
self._reduction_type = None
if K is QQ:
self._prime = ZZ.ideal(p)
else:
self._prime = p
if algorithm=="pari" and K is QQ:
Eint = E.integral_model()
data = Eint.pari_curve().elllocalred(p)
self._fp = data[0].sage()
self._KS = KodairaSymbol(data[1].sage())
self._cp = data[3].sage()
# We use a global minimal model since we can:
self._Emin_reduced = Eint.minimal_model()
self._val_disc = self._Emin_reduced.discriminant().valuation(p)
if self._fp>0:
self._reduction_type = Eint.ap(p) # = 0,-1 or +1
else:
self._Emin, ch, self._val_disc, self._fp, self._KS, self._cp, self._split = self._tate(proof, globally)
if self._fp>0:
if self._Emin.c4().valuation(p)>0:
self._reduction_type = 0
elif self._split:
self._reduction_type = +1
else:
self._reduction_type = -1
def __repr__(self):
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'
"""
red_type = "good"
if not self._reduction_type is None:
red_type = ["bad non-split multiplicative","bad additive","bad split multiplicative"][1+self._reduction_type]
return "Local data at %s:\nReduction type: %s\nLocal minimal model: %s\nMinimal discriminant valuation: %s\nConductor exponent: %s\nKodaira Symbol: %s\nTamagawa Number: %s"%(self._prime,red_type,self.minimal_model(),self._val_disc,self._fp,self._KS,self._cp)
def minimal_model(self, reduce=True):
"""
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
"""
if reduce:
try:
return self._Emin_reduced
except AttributeError:
pass
# trac 14476 we only reduce if the coefficients are globally integral
if all(a.is_integral() for a in self._Emin.a_invariants()):
self._Emin_reduced = self._Emin._reduce_model()
return self._Emin_reduced
else:
return self._Emin
else:
try:
return self._Emin
except AttributeError:
raise ValueError("the argument reduce must not be False if algorithm=pari is used")
def prime(self):
"""
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 self._prime
def conductor_valuation(self):
"""
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 self._fp
def discriminant_valuation(self):
"""
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
"""
return self._val_disc
def kodaira_symbol(self):
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
"""
return self._KS
def tamagawa_number(self):
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
"""
return self._cp
def tamagawa_exponent(self):
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
"""
cp = self._cp
if cp!=4:
return cp
ks = self._KS
if ks._roman==1 and ks._n%2==0 and ks._starred:
return ZZ(2)
return ZZ(4)
def bad_reduction_type(self):
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)]
"""
return self._reduction_type
def has_good_reduction(self):
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)]
"""
return self._reduction_type is None
def has_bad_reduction(self):
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)]
"""
return not self._reduction_type is None
def has_multiplicative_reduction(self):
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)]
"""
return self._reduction_type in (-1,+1)
def has_split_multiplicative_reduction(self):
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)]
"""
return self._reduction_type == +1
def has_nonsplit_multiplicative_reduction(self):
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)]
"""
return self._reduction_type == -1
def has_additive_reduction(self):
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)]
"""
return self._reduction_type == 0
def _tate(self, proof = None, globally = False):
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
"""
E = self._curve
P = self._prime
K = E.base_ring()
OK = K.maximal_order()
t = verbose("Running Tate's algorithm with P = %s"%P, level=1)
F = OK.residue_field(P)
p = F.characteristic()
# 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.
if globally:
principal_flag = P.is_principal()
else:
principal_flag = False
if (K is QQ) or principal_flag :
pi = P.gens_reduced()[0]
verbose("P is principal, generator pi = %s"%pi, t, 1)
else:
pi = K.uniformizer(P, 'positive')
verbose("uniformizer pi = %s"%pi, t, 1)
pi2 = pi*pi; pi3 = pi*pi2; pi4 = pi*pi3
pi_neg = None
prime = pi if K is QQ else P
pval = lambda x: x.valuation(prime)
pdiv = lambda x: x.is_zero() or pval(x) > 0
# 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).
from sage.categories.pushout import pushout, CoercionException
try:
if hasattr(F.p.ring(), 'maximal_order'): # it is not ZZ
_tmp_ = pushout(F.p.ring().maximal_order(),K)
pinv = lambda x: F.lift(~F(x))
proot = lambda x,e: F.lift(F(x).nth_root(e, extend = False, all = True)[0])
preduce = lambda x: F.lift(F(x))
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)))
def _pquadroots(a, b, c):
r"""
Local function returning True iff `ax^2 + bx + c` has roots modulo `P`
"""
(a, b, c) = (F(a), F(b), F(c))
if a == 0:
return (b != 0) or (c == 0)
elif p == 2:
return len(PolynomialRing(F, "x")([c,b,a]).roots()) > 0
else:
return (b**2 - 4*a*c).is_square()
def _pcubicroots(b, c, d):
r"""
Local function returning the number of roots of `x^3 +
b*x^2 + c*x + d` modulo `P`, counting multiplicities
"""
return sum([rr[1] for rr in PolynomialRing(F, 'x')([F(d), F(c), F(b), F(1)]).roots()],0)
if p == 2:
halfmodp = OK(Integer(0))
else:
halfmodp = pinv(Integer(2))
A = E.a_invariants()
A = [0, A[0], A[1], A[2], A[3], 0, A[4]]
indices = [1,2,3,4,6]
if min([pval(a) for a in A if a != 0]) < 0:
verbose("Non-integral model at P: valuations are %s; making integral"%([pval(a) for a in A if a != 0]), t, 1)
e = 0
for i in range(7):
if A[i] != 0:
e = max(e, (-pval(A[i])/i).ceil())
pie = pi**e
for i in range(7):
if A[i] != 0:
A[i] *= pie**i
verbose("P-integral model is %s, with valuations %s"%([A[i] for i in indices], [pval(A[i]) for i in indices]), t, 1)
split = None # only relevant for multiplicative reduction
(a1, a2, a3, a4, a6) = (A[1], A[2], A[3], A[4], A[6])
while True:
C = EllipticCurve([a1, a2, a3, a4, a6]);
(b2, b4, b6, b8) = C.b_invariants()
(c4, c6) = C.c_invariants()
delta = C.discriminant()
val_disc = pval(delta)
if val_disc == 0:
## Good reduction already
cp = 1
fp = 0
KS = KodairaSymbol("I0")
break #return
# Otherwise, we change coordinates so that p | a3, a4, a6
if p == 2:
if pdiv(b2):
r = proot(a4, 2)
t = proot(((r + a2)*r + a4)*r + a6, 2)
else:
temp = pinv(a1)
r = temp * a3
t = temp * (a4 + r*r)
elif p == 3:
if pdiv(b2):
r = proot(-b6, 3)
else:
r = -pinv(b2) * b4
t = a1 * r + a3
else:
if pdiv(c4):
r = -pinv(12) * b2
else:
r = -pinv(12*c4) * (c6 + b2 * c4)
t = -halfmodp * (a1 * r + a3)
r = preduce(r)
t = preduce(t)
verbose("Before first transform C = %s"%C)
verbose("[a1,a2,a3,a4,a6] = %s"%([a1, a2, a3, a4, a6]))
C = C.rst_transform(r, 0, t)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
if min([pval(a) for a in (a1, a2, a3, a4, a6) if a != 0]) < 0:
raise RuntimeError("Non-integral model after first transform!")
verbose("After first transform %s\n, [a1,a2,a3,a4,a6] = %s\n, valuations = %s"%([r, 0, t], [a1, a2, a3, a4, a6], [pval(a1), pval(a2), pval(a3), pval(a4), pval(a6)]), t, 2)
if pval(a3) == 0:
raise RuntimeError("p does not divide a3 after first transform!")
if pval(a4) == 0:
raise RuntimeError("p does not divide a4 after first transform!")
if pval(a6) == 0:
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.
if not pdiv(c4):
# Multiplicative reduction: Type In (n = val_disc)
split = False
if _pquadroots(1, a1, -a2):
cp = val_disc
split = True
elif Integer(2).divides(val_disc):
cp = 2
else:
cp = 1
KS = KodairaSymbol("I%s"%val_disc)
fp = 1
break #return
# Additive reduction
if pval(a6) < 2:
## Type II
KS = KodairaSymbol("II")
fp = val_disc
cp = 1
break #return
if pval(b8) < 3:
## Type III
KS = KodairaSymbol("III")
fp = val_disc - 1
cp = 2
break #return
if pval(b6) < 3:
## Type IV
cp = 1
a3t = preduce(a3/pi)
a6t = preduce(a6/pi2)
if _pquadroots(1, a3t, -a6t): cp = 3
KS = KodairaSymbol("IV")
fp = val_disc - 2
break #return
# If our curve is none of these types, we change coords so that
# p | a1, a2; p^2 | a3, a4; p^3 | a6
if p == 2:
s = proot(a2, 2) # so s^2=a2 (mod pi)
t = pi*proot(a6/pi2, 2) # so t^2=a6 (mod pi^3)
elif p == 3:
s = a1 # so a1'=2s+a1=3a1=0 (mod pi)
t = a3 # so a3'=2t+a3=3a3=0 (mod pi^2)
else:
s = -a1*halfmodp # so a1'=2s+a1=0 (mod pi)
t = -a3*halfmodp # so a3'=2t+a3=0 (mod pi^2)
C = C.rst_transform(0, s, t)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
verbose("After second transform %s\n[a1, a2, a3, a4, a6] = %s\nValuations: %s"%([0, s, t], [a1,a2,a3,a4,a6],[pval(a1),pval(a2),pval(a3),pval(a4),pval(a6)]), t, 2)
if pval(a1) == 0:
raise RuntimeError("p does not divide a1 after second transform!")
if pval(a2) == 0:
raise RuntimeError("p does not divide a2 after second transform!")
if pval(a3) < 2:
raise RuntimeError("p^2 does not divide a3 after second transform!")
if pval(a4) < 2:
raise RuntimeError("p^2 does not divide a4 after second transform!")
if pval(a6) < 3:
raise RuntimeError("p^3 does not divide a6 after second transform!")
if min(pval(a1), pval(a2), pval(a3), pval(a4), pval(a6)) < 0:
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
b = preduce(a2/pi)
c = preduce(a4/pi2)
d = preduce(a6/pi3)
bb = b*b
cc = c*c
bc = b*c
w = 27*d*d - bb*cc + 4*b*bb*d - 18*bc*d + 4*c*cc
x = 3*c - bb
if pdiv(w):
if pdiv(x):
sw = 3
else:
sw = 2
else:
sw = 1
verbose("Analyzing roots of cubic T^3 + %s*T^2 + %s*T + %s, case %s"%(b, c, d, sw), t, 1)
if sw == 1:
## Three distinct roots - Type I*0
verbose("Distinct roots", t, 1)
KS = KodairaSymbol("I0*")
cp = 1 + _pcubicroots(b, c, d)
fp = val_disc - 4
break #return
elif sw == 2:
## One double root - Type I*m for some m
verbose("One double root", t, 1)
## Change coords so that the double root is T = 0 mod p
if p == 2:
r = proot(c, 2)
elif p == 3:
r = c * pinv(b)
else:
r = (bc - 9*d)*pinv(2*x)
r = pi * preduce(r)
C = C.rst_transform(r, 0, 0)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
# The rest of this branch is just to compute cp, fp, KS.
# We use pi to keep transforms integral.
ix = 3; iy = 3; mx = pi2; my = mx
while True:
a2t = preduce(a2 / pi)
a3t = preduce(a3 / my)
a4t = preduce(a4 / (pi*mx))
a6t = preduce(a6 / (mx*my))
if pdiv(a3t*a3t + 4*a6t):
if p == 2:
t = my*proot(a6t, 2)
else:
t = my*preduce(-a3t*halfmodp)
C = C.rst_transform(0, 0, t)
(a1, a2, a3, a4, a6) = C.a_invariants()
(b2, b4, b6, b8) = C.b_invariants()
my *= pi
iy += 1
a2t = preduce(a2 / pi)
a3t = preduce(a3/my)
a4t = preduce(a4/(pi*mx))
a6t = preduce(a6/(mx*my))
if pdiv(a4t*a4t - 4*a6t*a2t):
if p == 2:
r = mx*proot(a6t*pinv(a2t), 2)