Base field \(\Q(\sqrt{217}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 54 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-54, -1, 1]))
gp: K = nfinit(Polrev([-54, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-54, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([0,1]),K([-263319434670,33477944960]),K([70827044374677700,-9004819170532378])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,0]),Polrev([0,1]),Polrev([-263319434670,33477944960]),Polrev([70827044374677700,-9004819170532378])], K);
magma: E := EllipticCurve([K![1,0],K![0,0],K![0,1],K![-263319434670,33477944960],K![70827044374677700,-9004819170532378]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-996a+7834)\) | = | \((-a+8)\cdot(-a-7)\cdot(-498a+3917)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 28 \) | = | \(2\cdot2\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((44736a+307136)\) | = | \((-a+8)^{6}\cdot(-a-7)^{12}\cdot(-498a+3917)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 1835008 \) | = | \(2^{6}\cdot2^{12}\cdot7\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{341762709}{28672} a + \frac{1369351027}{14336} \) | ||
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 | $\left(-23576 a + 185436 : -3867384 a + 30418750 : 1\right)$ |
| Height | \(0.29812351543591234475014258460215508223\) |
| 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: | $\left(-\frac{103751}{4} a + \frac{816049}{4} : \frac{103747}{8} a - \frac{816049}{8} : 1\right)$ |
| 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: | \( 0.29812351543591234475014258460215508223 \) | ||
| Period: | \( 19.304899526767692533432644351310637718 \) | ||
| Tamagawa product: | \( 12 \) = \(( 2 \cdot 3 )\cdot2\cdot1\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.3441487289522040673643404290047398609 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+8)\) | \(2\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
| \((-a-7)\) | \(2\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
| \((-498a+3917)\) | \(7\) | \(1\) | \(I_{1}\) | Non-split multiplicative | \(1\) | \(1\) | \(1\) | \(1\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
28.1-g
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.