Local data for elliptic curves over number fields - Elliptic curves

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# 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

    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 \(\QQ\), 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

      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 \(\QQ\))

  - `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

      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