Base field 3.3.1957.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 9 x + 10 \); class number \(2\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([10, -9, -1, 1]))
gp: K = nfinit(Polrev([10, -9, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10, -9, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0]),K([-6,0,1]),K([0,1,0]),K([-80088908,44572268,23317913]),K([1380453494659,-768270441835,-401919492463])])
gp: E = ellinit([Polrev([1,1,0]),Polrev([-6,0,1]),Polrev([0,1,0]),Polrev([-80088908,44572268,23317913]),Polrev([1380453494659,-768270441835,-401919492463])], K);
magma: E := EllipticCurve([K![1,1,0],K![-6,0,1],K![0,1,0],K![-80088908,44572268,23317913],K![1380453494659,-768270441835,-401919492463]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((11,a-4)\) | = | \((11,a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 11 \) | = | \(11\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-78432a^2+77121a+694319)\) | = | \((11,a-4)^{16}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -45949729863572161 \) | = | \(-11^{16}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{52028892561854339324}{45949729863572161} a^{2} + \frac{184047527372651237376}{45949729863572161} a - \frac{86097660316480487429}{45949729863572161} \) | ||
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 | $\left(\frac{44563}{16} a^{2} + \frac{85191}{16} a - \frac{153049}{16} : \frac{32054567}{64} a^{2} + \frac{61272379}{64} a - \frac{110096357}{64} : 1\right)$ |
| Height | \(3.0838184486394974690009033885897681373\) |
| Torsion structure: | \(\Z/2\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(\frac{4275}{4} a^{2} + \frac{4087}{2} a - \frac{14681}{4} : -\frac{4181}{2} a^{2} - \frac{7993}{2} a + \frac{57431}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 3.0838184486394974690009033885897681373 \) | ||
| Period: | \( 2.7653467736644736139366071600251171096 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 1.1566294317998090015866604447952603614 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((11,a-4)\) | \(11\) | \(2\) | \(I_{16}\) | Non-split multiplicative | \(1\) | \(1\) | \(16\) | \(16\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
11.1-b
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.