Local data for elliptic curves over number fields

Let $E$ be an elliptic curve over a number field $K$ (including $\mathbf{Q}$). 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

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

Python

>>> 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

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

Python

>>> 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

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

Python

>>> 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)

Bases: SageObject

The class for the local reduction data of an elliptic curve.

Currently supported are elliptic curves defined over $\mathbf{Q}$, 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 $\mathbf{Q}$

• P – a prime ideal of the field, or a prime integer if the field is $\mathbf{Q}$

• 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 $\mathbf{Q}$. If “pari”, use the PARI C-library ellglobalred implementation of Tate’s algorithm over $\mathbf{Q}$. 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

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

Python

>>> 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()

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

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)]

Python

>>> 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()

Return the valuation of the conductor from this local reduction data.

EXAMPLES:

Sage

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

Python

>>> 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()

Return the valuation of the minimal discriminant from this local reduction data.

EXAMPLES:

Sage

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

Python

>>> 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()

Return True if there is additive reduction.

EXAMPLES:

Sage

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)]

Python

>>> 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

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)]

Python

>>> 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()

Return True if there is bad reduction.

EXAMPLES:

Sage

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)]

Python

>>> 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

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)]

Python

>>> 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()

Return True if there is good reduction.

EXAMPLES:

Sage

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)]

Python

>>> 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()

Return True if there is multiplicative reduction.

Note

See also has_split_multiplicative_reduction() and has_nonsplit_multiplicative_reduction().

EXAMPLES:

Sage

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)]

Python

>>> 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

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)]

Python

>>> 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()

Return True if there is non-split multiplicative reduction.

EXAMPLES:

Sage

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)]

Python

>>> 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

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)]

Python

>>> 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()

Return True if there is split multiplicative reduction.

EXAMPLES:

Sage

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)]

Python

>>> 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

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)]

Python

>>> 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()

Return the Kodaira symbol from this local reduction data.

EXAMPLES:

Sage

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

Python

>>> 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)

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

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

Python

>>> 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

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

Python

>>> 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

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

Python

>>> 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()

Return the prime ideal associated with this local reduction data.

EXAMPLES:

Sage

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

Python

>>> 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()

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

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

Python

>>> 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()

Return the Tamagawa number from this local reduction data.

This is the index $[E(K_v):E^0(K_v)]$.

EXAMPLES:

Sage

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

Python

>>> 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)

Function to check that $P$ determines a prime of $K$, and return that ideal.

INPUT:

• K – a number field (including $\mathbf{Q}$)

• P – an element of K or a (fractional) ideal of K

OUTPUT: if K is $\mathbf{Q}$: the prime integer equal to or which generates $P$

• If K is not $\mathbf{Q}$: 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

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

Python

>>> 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