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([-12,-11,3,2]),K([-4,1,1,0]),K([1,0,0,0]),K([-58,-69,17,12]),K([-62,-75,18,13])])
gp: E = ellinit([Polrev([-12,-11,3,2]),Polrev([-4,1,1,0]),Polrev([1,0,0,0]),Polrev([-58,-69,17,12]),Polrev([-62,-75,18,13])], K);
magma: E := EllipticCurve([K![-12,-11,3,2],K![-4,1,1,0],K![1,0,0,0],K![-58,-69,17,12],K![-62,-75,18,13]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+2a^2-7a-9)\) | = | \((a^3+2a^2-7a-9)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 59 \) | = | \(59\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((12a^3+11a^2-59a-61)\) | = | \((a^3+2a^2-7a-9)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -205379 \) | = | \(-59^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{30886657696}{205379} a^{3} + \frac{263733970778}{205379} a^{2} - \frac{91149969253}{205379} a - \frac{1666178595071}{205379} \) | ||
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 | $\left(-a^{3} - 2 a^{2} + 5 a + 8 : 4 a^{3} + 5 a^{2} - 23 a - 16 : 1\right)$ |
| Height | \(0.062194364252432044227068110986727272931\) |
| 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: | \( 0.062194364252432044227068110986727272931 \) | ||
| Period: | \( 271.92996515815070178103777634240379646 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.59320171061156 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+2a^2-7a-9)\) | \(59\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
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 59.4-b 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.