Base field
Generator , with minimal polynomial ; class number .
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))
gp: K = nfinit(Polrev([1, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([3,-36]),K([-47,84])])
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([3,-36]),Polrev([-47,84])], K);
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![3,-36],K![-47,84]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Mordell-Weil group structure
Mordell-Weil generators
Invariants
Conductor: | = | = | |||
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||||
Conductor norm: | = | = | |||
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||||
Discriminant: | = | ||||
Discriminant ideal: | = | = | |||
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||||
Discriminant norm: | = | = | |||
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||||
j-invariant: | = | ||||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||||
Endomorphism ring: | = | ||||
Geometric endomorphism ring: | = | (no potential complex multiplication) | |||
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||||
Sato-Tate group: | = |
BSD invariants
Analytic rank: | = | ||
sage: E.rank()
magma: Rank(E);
|
|||
Mordell-Weil rank: | = | ||
Regulator: | ≈ | ||
Néron-Tate Regulator: | ≈ | ||
Global period: | ≈ | ||
Tamagawa product: | = | = | |
Torsion order: | = | ||
Special value: | ≈ | ||
Analytic order of Ш: | Ш | = | (rounded) |
BSD formula
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
This elliptic curve is not semistable. There are 4 primes of bad reduction.
Tamagawa number | Kodaira symbol | Reduction type | Root number | ) | ) | |||
---|---|---|---|---|---|---|---|---|
Additive | ||||||||
Additive | ||||||||
Split multiplicative | ||||||||
Split multiplicative |
Galois Representations
The mod Galois Representation has maximal image for all primes except those listed.
prime | Image of Galois Representation |
---|---|
3B.1.1[2] |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree for
3.
Its isogeny class
138384.2-b
consists of curves linked by isogenies of
degree 3.
Base change
This elliptic curve is a -curve.
It is not the base change of an elliptic curve defined over any subfield.