Base field \(\Q(\sqrt{46}) \)
Generator \(a\), with minimal polynomial \( x^{2} - 46 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-46, 0, 1]))
gp: K = nfinit(Polrev([-46, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-46, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([1,1]),K([-16285948,2401231]),K([35696899378,-5263220672])])
gp: E = ellinit([Polrev([1,1]),Polrev([0,-1]),Polrev([1,1]),Polrev([-16285948,2401231]),Polrev([35696899378,-5263220672])], K);
magma: E := EllipticCurve([K![1,1],K![0,-1],K![1,1],K![-16285948,2401231],K![35696899378,-5263220672]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-5a+34)\) | = | \((-23a+156)\cdot(a+7)\) |
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: | \((-160a-1168)\) | = | \((-23a+156)^{8}\cdot(a+7)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 186624 \) | = | \(2^{8}\cdot3^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{3535318439}{5832} a - \frac{47959064233}{11664} \) | ||
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 | $\left(-295 a + 1998 : 4530 a - 30717 : 1\right)$ |
| Height | \(0.23114559295140330995317094288326034283\) |
| 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: | $\left(486 a - 3299 : 1406 a - 9529 : 1\right)$ |
| 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: | \( 0.23114559295140330995317094288326034283 \) | ||
| Period: | \( 23.696821987697670537614431236460045277 \) | ||
| Tamagawa product: | \( 12 \) = \(2\cdot( 2 \cdot 3 )\) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 2.4228027756117493743436077585167595224 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-23a+156)\) | \(2\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
| \((a+7)\) | \(3\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
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
6.2-b
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.