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: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) + Integer(1), names=('i',)); (i,) = K._first_ngens(1)
>>> E = EllipticCurve([(Integer(2)+i)**Integer(2), (Integer(2)+i)**Integer(7)])
>>> pp = K.fractional_ideal(Integer(2)+i)
>>> da = E.local_data(pp)
>>> da.has_bad_reduction()
True
>>> da.has_multiplicative_reduction()
False
>>> da.kodaira_symbol()
I0*
>>> da.tamagawa_number()
4
>>> 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: # needs sage.rings.number_field
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

>>> from sage.all import *
>>> E = EllipticCurve([Integer(0),-Integer(1)])
>>> E.local_data(Integer(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

>>> # needs sage.rings.number_field
>>> EK = E.base_extend(K)
>>> EK.local_data(Integer(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: # needs sage.rings.number_field
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

>>> from sage.all import *
>>> E = EllipticCurve([Integer(0),Integer(8)])
>>> E.is_minimal()
True

>>> # needs sage.rings.number_field
>>> EK = E.base_extend(K)
>>> da = EK.local_data(Integer(1)+i)
>>> 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

class sage.schemes.elliptic_curves.ell_local_data.EllipticCurveLocalData(E, P, proof=None, algorithm='pari', globally=False)[source]

Bases: SageObject

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 – boolean; 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

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData
>>> E = EllipticCurve('14a1')
>>> EllipticCurveLocalData(E,Integer(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
bad_reduction_type()[source]

Return the type of bad reduction of this reduction data.

OUTPUT:

integer 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: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
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)]

>>> from sage.all import *
>>> E = EllipticCurve('14a1')
>>> [(p,E.local_data(p).bad_reduction_type()) for p in prime_range(Integer(15))]
[(2, -1), (3, None), (5, None), (7, 1), (11, None), (13, None)]

>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> P17a, P17b = [P for P,e in K.factor(Integer(17))]
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(0), Integer(2)*a+Integer(1)])
>>> [(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)]
conductor_valuation()[source]

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

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),Integer(64)]); E
Elliptic Curve defined by y^2 = x^3 + 64 over Rational Field
>>> data = EllipticCurveLocalData(E,Integer(2))
>>> data.conductor_valuation()
2
discriminant_valuation()[source]

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

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),Integer(64)]); E
Elliptic Curve defined by y^2 = x^3 + 64 over Rational Field
>>> data = EllipticCurveLocalData(E,Integer(2))
>>> data.discriminant_valuation()
4
has_additive_reduction()[source]

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)]

>>> from sage.all import *
>>> E = EllipticCurve('27a1')
>>> [(p, E.local_data(p).has_additive_reduction()) for p in prime_range(Integer(15))]
[(2, False), (3, True), (5, False), (7, False), (11, False), (13, False)]


sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
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)]

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> P17a, P17b = [P for P,e in K.factor(Integer(17))]
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(0), Integer(2)*a+Integer(1)])
>>> [(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)]
has_bad_reduction()[source]

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)]

>>> from sage.all import *
>>> E = EllipticCurve('14a1')
>>> [(p,E.local_data(p).has_bad_reduction()) for p in prime_range(Integer(15))]
[(2, True), (3, False), (5, False), (7, True), (11, False), (13, False)]


sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
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)]

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> P17a, P17b = [P for P,e in K.factor(Integer(17))]
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(0), Integer(2)*a+Integer(1)])
>>> [(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)]
has_good_reduction()[source]

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: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
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)]

>>> from sage.all import *
>>> E = EllipticCurve('14a1')
>>> [(p,E.local_data(p).has_good_reduction()) for p in prime_range(Integer(15))]
[(2, False), (3, True), (5, True), (7, False), (11, True), (13, True)]

>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> P17a, P17b = [P for P,e in K.factor(Integer(17))]
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(0), Integer(2)*a+Integer(1)])
>>> [(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)]
has_multiplicative_reduction()[source]

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)]

>>> from sage.all import *
>>> E = EllipticCurve('14a1')
>>> [(p, E.local_data(p).has_multiplicative_reduction()) for p in prime_range(Integer(15))]
[(2, True), (3, False), (5, False), (7, True), (11, False), (13, False)]


sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
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)]

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> P17a, P17b = [P for P,e in K.factor(Integer(17))]
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(0), Integer(2)*a+Integer(1)])
>>> [(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)]
has_nonsplit_multiplicative_reduction()[source]

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)]

>>> from sage.all import *
>>> E = EllipticCurve('14a1')
>>> [(p, E.local_data(p).has_nonsplit_multiplicative_reduction())
...  for p in prime_range(Integer(15))]
[(2, True), (3, False), (5, False), (7, False), (11, False), (13, False)]


sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
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)]

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> P17a, P17b = [P for P,e in K.factor(Integer(17))]
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(0), Integer(2)*a+Integer(1)])
>>> [(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)]
has_split_multiplicative_reduction()[source]

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)]

>>> from sage.all import *
>>> E = EllipticCurve('14a1')
>>> [(p, E.local_data(p).has_split_multiplicative_reduction())
...  for p in prime_range(Integer(15))]
[(2, False), (3, False), (5, False), (7, True), (11, False), (13, False)]


sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
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)]

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) - Integer(2), names=('a',)); (a,) = K._first_ngens(1)
>>> P17a, P17b = [P for P,e in K.factor(Integer(17))]
>>> E = EllipticCurve([Integer(0), Integer(0), Integer(0), Integer(0), Integer(2)*a+Integer(1)])
>>> [(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)]
kodaira_symbol()[source]

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

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),Integer(64)]); E
Elliptic Curve defined by y^2 = x^3 + 64 over Rational Field
>>> data = EllipticCurveLocalData(E,Integer(2))
>>> data.kodaira_symbol()
IV
minimal_model(reduce=True)[source]

Return the (local) minimal model from this local reduction data.

INPUT:

  • reduce – boolean (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

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),Integer(64)]); E
Elliptic Curve defined by y^2 = x^3 + 64 over Rational Field
>>> data = EllipticCurveLocalData(E, Integer(2))
>>> data.minimal_model()
Elliptic Curve defined by y^2 = x^3 + 1 over Rational Field
>>> data.minimal_model() == E.local_minimal_model(Integer(2))
True

To demonstrate the behaviour of the parameter reduce:

sage: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(3) + x + Integer(1), names=('a',)); (a,) = K._first_ngens(1)
>>> E = EllipticCurve(K, [Integer(0), Integer(0), a, Integer(0), Integer(1)])
>>> E.local_data(K.ideal(a-Integer(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
>>> E.local_data(K.ideal(a-Integer(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

>>> E = EllipticCurve([Integer(2), Integer(1), Integer(0), -Integer(2), -Integer(1)])
>>> E.local_data(ZZ.ideal(Integer(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
>>> E.local_data(ZZ.ideal(Integer(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
>>> E.local_data(ZZ.ideal(Integer(2)), algorithm='generic').minimal_model()
Elliptic Curve defined by y^2 = x^3 - x^2 - 3*x + 2 over Rational Field
>>> E.local_data(ZZ.ideal(Integer(2)), algorithm='pari').minimal_model()
Elliptic Curve defined by y^2 = x^3 - x^2 - 3*x + 2 over Rational Field

Issue #14476:

sage: # needs sage.rings.number_field
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

>>> from sage.all import *
>>> # needs sage.rings.number_field
>>> t = QQ['t'].gen(0)
>>> K = NumberField(t**Integer(4) - t**Integer(3)-Integer(3)*t**Integer(2) - t +Integer(1), names=('g',)); (g,) = K._first_ngens(1)
>>> E = EllipticCurve([-Integer(2)*g**Integer(3) + Integer(10)/Integer(3)*g**Integer(2) + Integer(3)*g - Integer(2)/Integer(3),
...                    -Integer(11)/Integer(9)*g**Integer(3) + Integer(34)/Integer(9)*g**Integer(2) - Integer(7)/Integer(3)*g + Integer(4)/Integer(9),
...                    -Integer(11)/Integer(9)*g**Integer(3) + Integer(34)/Integer(9)*g**Integer(2) - Integer(7)/Integer(3)*g + Integer(4)/Integer(9), Integer(0), Integer(0)])
>>> vv = K.fractional_ideal(g**Integer(2) - g - Integer(2))
>>> 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
prime()[source]

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

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),Integer(64)]); E
Elliptic Curve defined by y^2 = x^3 + 64 over Rational Field
>>> data = EllipticCurveLocalData(E,Integer(2))
>>> data.prime()
Principal ideal (2) of Integer Ring
tamagawa_exponent()[source]

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

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData
>>> E = EllipticCurve('816a1')
>>> data = EllipticCurveLocalData(E, Integer(2))
>>> data.kodaira_symbol()
I2*
>>> data.tamagawa_number()
4
>>> data.tamagawa_exponent()
2

>>> E = EllipticCurve('200c4')
>>> data = EllipticCurveLocalData(E, Integer(5))
>>> data.kodaira_symbol()
I4*
>>> data.tamagawa_number()
4
>>> data.tamagawa_exponent()
2
tamagawa_number()[source]

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

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_local_data import EllipticCurveLocalData
>>> E = EllipticCurve([Integer(0),Integer(0),Integer(0),Integer(0),Integer(64)]); E
Elliptic Curve defined by y^2 = x^3 + 64 over Rational Field
>>> data = EllipticCurveLocalData(E,Integer(2))
>>> data.tamagawa_number()
3
sage.schemes.elliptic_curves.ell_local_data.check_prime(K, P)[source]

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: # needs sage.rings.number_field
sage: x = polygen(ZZ, 'x')
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

>>> from sage.all import *
>>> from sage.schemes.elliptic_curves.ell_local_data import check_prime
>>> check_prime(QQ, Integer(3))
3
>>> check_prime(QQ, QQ(Integer(3)))
3
>>> check_prime(QQ, ZZ.ideal(Integer(31)))
31

>>> # needs sage.rings.number_field
>>> x = polygen(ZZ, 'x')
>>> K = NumberField(x**Integer(2) - Integer(5), names=('a',)); (a,) = K._first_ngens(1)
>>> check_prime(K, a)
Fractional ideal (a)
>>> check_prime(K, a + Integer(1))
Fractional ideal (a + 1)
>>> [check_prime(K, P) for P in K.primes_above(Integer(31))]
[Fractional ideal (5/2*a + 1/2), Fractional ideal (5/2*a - 1/2)]
>>> L = NumberField(x**Integer(2) + Integer(3), names=('b',)); (b,) = L._first_ngens(1)
>>> check_prime(K, L.ideal(Integer(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
>>> 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