Local data for elliptic curves over number fields#
Let \(E\) be an elliptic curve over a number field \(K\) (including \(\QQ\)). There are several local invariants at a finite place \(v\) that can be computed via Tate’s algorithm (see [Sil1994] IV.9.4 or [Tate1975]).
These include the type of reduction (good, additive, multiplicative), a minimal equation of \(E\) over \(K_v\), the Tamagawa number \(c_v\), defined to be the index \([E(K_v):E^0(K_v)]\) of the points with good reduction among the local points, and the exponent of the conductor \(f_v\).
The functions in this file will typically be called by using local_data
.
EXAMPLES:
sage: K.<i> = NumberField(x^2+1)
sage: E = EllipticCurve([(2+i)^2,(2+i)^7])
sage: pp = K.fractional_ideal(2+i)
sage: da = E.local_data(pp)
sage: da.has_bad_reduction()
True
sage: da.has_multiplicative_reduction()
False
sage: da.kodaira_symbol()
I0*
sage: da.tamagawa_number()
4
sage: da.minimal_model()
Elliptic Curve defined by y^2 = x^3 + (4*i+3)*x + (-29*i-278) over Number Field in i with defining polynomial x^2 + 1
An example to show how the Neron model can change as one extends the field:
sage: E = EllipticCurve([0,-1])
sage: E.local_data(2)
Local data at Principal ideal (2) of Integer Ring:
Reduction type: bad additive
Local minimal model: Elliptic Curve defined by y^2 = x^3 - 1 over Rational Field
Minimal discriminant valuation: 4
Conductor exponent: 4
Kodaira Symbol: II
Tamagawa Number: 1
sage: EK = E.base_extend(K)
sage: EK.local_data(1+i)
Local data at Fractional ideal (i + 1):
Reduction type: bad additive
Local minimal model: Elliptic Curve defined by y^2 = x^3 + (-1) over Number Field in i with defining polynomial x^2 + 1
Minimal discriminant valuation: 8
Conductor exponent: 2
Kodaira Symbol: IV*
Tamagawa Number: 3
Or how the minimal equation changes:
sage: E = EllipticCurve([0,8])
sage: E.is_minimal()
True
sage: EK = E.base_extend(K)
sage: da = EK.local_data(1+i)
sage: da.minimal_model()
Elliptic Curve defined by y^2 = x^3 + (-i) over Number Field in i with defining polynomial x^2 + 1
AUTHORS:
John Cremona: First version 2008-09-21 (refactoring code from
ell_number_field.py
andell_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:
sage.structure.sage_object.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
(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-libraryellglobalred
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
- bad_reduction_type()#
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)]
- conductor_valuation()#
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
- discriminant_valuation()#
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
- has_additive_reduction()#
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)]
- has_bad_reduction()#
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)]
- has_good_reduction()#
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)]
- has_multiplicative_reduction()#
Return
True
if there is multiplicative reduction.Note
See also
has_split_multiplicative_reduction()
andhas_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)]
- has_nonsplit_multiplicative_reduction()#
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)]
- has_split_multiplicative_reduction()#
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)]
- kodaira_symbol()#
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
- minimal_model(reduce=True)#
Return the (local) minimal model from this local reduction data.
INPUT:
reduce
– (default:True
) if set toTrue
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
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
- prime()#
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
- 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: 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
- tamagawa_number()#
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
- 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 ofK
or a (fractional) ideal ofK
.
OUTPUT:
If
K
is \(\QQ\): the prime integer equal to or which generates \(P\).If
K
is not \(\QQ\): the prime ideal equal to or generated by \(P\).
Note
If \(P\) is not a prime and does not generate a prime, a
TypeError
is raised.EXAMPLES:
sage: from sage.schemes.elliptic_curves.ell_local_data import check_prime sage: check_prime(QQ,3) 3 sage: check_prime(QQ,QQ(3)) 3 sage: check_prime(QQ,ZZ.ideal(31)) 31 sage: K.<a> = NumberField(x^2-5) sage: check_prime(K,a) Fractional ideal (a) sage: check_prime(K,a+1) Fractional ideal (a + 1) sage: [check_prime(K,P) for P in K.primes_above(31)] [Fractional ideal (5/2*a + 1/2), Fractional ideal (5/2*a - 1/2)] sage: L.<b> = NumberField(x^2+3) sage: check_prime(K, L.ideal(5)) Traceback (most recent call last): ... TypeError: The ideal Fractional ideal (5) is not a prime ideal of Number Field in a with defining polynomial x^2 - 5 sage: check_prime(K, L.ideal(b)) Traceback (most recent call last): ... TypeError: No compatible natural embeddings found for Number Field in a with defining polynomial x^2 - 5 and Number Field in b with defining polynomial x^2 + 3