Base field 4.4.6125.1
Generator \(a\), with minimal polynomial \( x^{4} - x^{3} - 9 x^{2} + 9 x + 11 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([11, 9, -9, -1, 1]))
gp: K = nfinit(Polrev([11, 9, -9, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![11, 9, -9, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([0,0,0,0]),K([-13,-12,3,2]),K([-8,-6,2,1]),K([-18,-27,4,4]),K([103,134,-25,-21])])
gp: E = ellinit([Polrev([0,0,0,0]),Polrev([-13,-12,3,2]),Polrev([-8,-6,2,1]),Polrev([-18,-27,4,4]),Polrev([103,134,-25,-21])], K);
magma: E := EllipticCurve([K![0,0,0,0],K![-13,-12,3,2],K![-8,-6,2,1],K![-18,-27,4,4],K![103,134,-25,-21]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((7a^3+9a^2-41a-31)\) | = | \((7a^3+9a^2-41a-31)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 71 \) | = | \(71\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((4a^3+5a^2-24a-28)\) | = | \((7a^3+9a^2-41a-31)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 5041 \) | = | \(71^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{92210540544}{5041} a^{3} + \frac{107272101888}{5041} a^{2} - \frac{598652993536}{5041} a - \frac{468105109504}{5041} \) | ||
sage: E.j_invariant()
gp: E.j
magma: jInvariant(E);
| |||
| Endomorphism ring: | \(\Z\) | ||
| Geometric endomorphism ring: | \(\Z\) | (no potential complex multiplication) | |
sage: E.has_cm(), E.cm_discriminant()
magma: HasComplexMultiplication(E);
| |||
| Sato-Tate group: | $\mathrm{SU}(2)$ | ||
Mordell-Weil group
| Rank: | \(1\) |
| Generator | $\left(-a^{3} - a^{2} + 6 a + 6 : -a^{3} - a^{2} + 5 a + 5 : 1\right)$ |
| Height | \(0.021909019988006405124582135582321888585\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.021909019988006405124582135582321888585 \) | ||
| Period: | \( 1115.1697044655710840083886686594120101 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.49747330362330 \) | ||
| Analytic order of Ш: | \( 1 \) (rounded) | ||
Local data at primes of bad reduction
sage: E.local_data()
magma: LocalInformation(E);
| prime | Norm | Tamagawa number | Kodaira symbol | Reduction type | Root number | ord(\(\mathfrak{N}\)) | ord(\(\mathfrak{D}\)) | ord\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((7a^3+9a^2-41a-31)\) | \(71\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 71.2-h consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.