Base field 5.5.179024.1
Generator \(a\), with minimal polynomial \( x^{5} - 8 x^{3} + 6 x - 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 6, 0, -8, 0, 1]))
gp: K = nfinit(Polrev([-2, 6, 0, -8, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 6, 0, -8, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([10,-8,-16,1,2]),K([22,-25,-46,3,6]),K([-5,8,15,-1,-2]),K([-20,33,39,-5,-5]),K([38,-44,-82,6,11])])
gp: E = ellinit([Polrev([10,-8,-16,1,2]),Polrev([22,-25,-46,3,6]),Polrev([-5,8,15,-1,-2]),Polrev([-20,33,39,-5,-5]),Polrev([38,-44,-82,6,11])], K);
magma: E := EllipticCurve([K![10,-8,-16,1,2],K![22,-25,-46,3,6],K![-5,8,15,-1,-2],K![-20,33,39,-5,-5],K![38,-44,-82,6,11]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-4a^4-2a^3+31a^2+16a-17)\) | = | \((2a^4+a^3-15a^2-7a+7)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 9 \) | = | \(3^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^3+2a^2+7a-11)\) | = | \((2a^4+a^3-15a^2-7a+7)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 6561 \) | = | \(3^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{31616}{9} a^{4} + \frac{9088}{9} a^{3} + \frac{325760}{9} a^{2} + \frac{158336}{9} a - \frac{135424}{9} \) | ||
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: | \(0\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-2 a^{4} - a^{3} + 16 a^{2} + 9 a - 8 : 2 a^{4} + 2 a^{3} - 14 a^{2} - 10 a + 7 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 7633.8769516288736668459820867746350569 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 1.00234452 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2a^4+a^3-15a^2-7a+7)\) | \(3\) | \(2\) | \(I_{2}^{*}\) | Additive | \(-1\) | \(2\) | \(8\) | \(2\) |
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 |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
9.1-c
consists of curves linked by isogenies of
degrees dividing 12.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.