Base field \(\Q(\sqrt{457}) \)
Generator \(a\), with minimal polynomial \( x^{2} - x - 114 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-114, -1, 1]))
gp: K = nfinit(Polrev([-114, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-114, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,0]),K([0,1]),K([1,0]),K([-102655956047261842,-10075393175452795]),K([-18261363907664053540688320,-1792301473526103661546294])])
gp: E = ellinit([Polrev([1,0]),Polrev([0,1]),Polrev([1,0]),Polrev([-102655956047261842,-10075393175452795]),Polrev([-18261363907664053540688320,-1792301473526103661546294])], K);
magma: E := EllipticCurve([K![1,0],K![0,1],K![1,0],K![-102655956047261842,-10075393175452795],K![-18261363907664053540688320,-1792301473526103661546294]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-53a-540)\) | = | \((6987a-78176)\cdot(-196a+2193)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 6 \) | = | \(2\cdot3\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-29267a+327462)\) | = | \((6987a-78176)^{4}\cdot(-196a+2193)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -144 \) | = | \(-2^{4}\cdot3^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1399157}{144} a + \frac{14336917}{144} \) | ||
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 | $\left(-\frac{969434906}{81} a - \frac{3292452634}{27} : -\frac{7160300007059}{729} a - \frac{24318238505290}{243} : 1\right)$ |
| Height | \(1.8323957108589342020944433706507945019\) |
| 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: | $\left(-\frac{47473175}{4} a - \frac{483693683}{4} : \frac{47473175}{8} a + \frac{483693679}{8} : 1\right)$ |
| 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: | \( 1.8323957108589342020944433706507945019 \) | ||
| Period: | \( 18.039948946002181955426086893478814742 \) | ||
| Tamagawa product: | \( 4 \) = \(2\cdot2\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 3.0926193317313108731318833608166397661 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((6987a-78176)\) | \(2\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
| \((-196a+2193)\) | \(3\) | \(2\) | \(I_{2}\) | Split multiplicative | \(-1\) | \(1\) | \(2\) | \(2\) |
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 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2.
Its isogeny class
6.1-a
consists of curves linked by isogenies of
degree 2.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.