Base field 3.3.316.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 4 x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -4, -1, 1]))
gp: K = nfinit(Polrev([2, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0]),K([1,1,0]),K([-3,0,1]),K([-17919757326708,-2232268991762,4217266330899]),K([-23956533748648438226,-2984271854928490894,5637971616536361013])])
gp: E = ellinit([Polrev([1,1,0]),Polrev([1,1,0]),Polrev([-3,0,1]),Polrev([-17919757326708,-2232268991762,4217266330899]),Polrev([-23956533748648438226,-2984271854928490894,5637971616536361013])], K);
magma: E := EllipticCurve([K![1,1,0],K![1,1,0],K![-3,0,1],K![-17919757326708,-2232268991762,4217266330899],K![-23956533748648438226,-2984271854928490894,5637971616536361013]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^2-a-7)\) | = | \((-a+1)^{4}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 16 \) | = | \(2^{4}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((46a^2-13a-63)\) | = | \((-a+1)^{20}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1048576 \) | = | \(2^{20}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{130050481}{256} a^{2} + \frac{87514925}{128} a - \frac{54929731}{128} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(\frac{3051731}{4} a^{2} - \frac{807667}{2} a - \frac{12967237}{4} : -\frac{1122033}{2} a^{2} + \frac{2375647}{8} a + \frac{19070711}{8} : 1\right)$ | $\left(359846 a^{2} - 190473 a - 1529037 : -264610 a^{2} + 140063 a + 1124366 : 1\right)$ |
| 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: | \( 50.147603731874905070594376538076334645 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 0.70525577773956474808140129581548975701 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+1)\) | \(2\) | \(4\) | \(I_{12}^{*}\) | Additive | \(-1\) | \(4\) | \(20\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 4 and 8.
Its isogeny class
16.5-b
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.