Base field 3.3.1825.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 8 x + 7 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([7, -8, -1, 1]))
gp: K = nfinit(Polrev([7, -8, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![7, -8, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-5,1,1]),K([-5,1,1]),K([1,1,0]),K([98,-76,-40]),K([-622,497,259])])
gp: E = ellinit([Polrev([-5,1,1]),Polrev([-5,1,1]),Polrev([1,1,0]),Polrev([98,-76,-40]),Polrev([-622,497,259])], K);
magma: E := EllipticCurve([K![-5,1,1],K![-5,1,1],K![1,1,0],K![98,-76,-40],K![-622,497,259]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2)\) | = | \((2)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 8 \) | = | \(8\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-32)\) | = | \((2)^{5}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -32768 \) | = | \(-8^{5}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{121945}{32} \) | ||
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 | $\left(a^{2} + 3 a + 1 : 2 a^{2} + 3 a - 6 : 1\right)$ |
| Height | \(1.3247515694991825831566633179094913275\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 1.3247515694991825831566633179094913275 \) | ||
| Period: | \( 34.128163477957967967611687718240764297 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.1749521287132293518572514463755508588 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2)\) | \(8\) | \(1\) | \(I_{5}\) | Non-split multiplicative | \(1\) | \(1\) | \(5\) | \(5\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3B |
| \(5\) | 5B.4.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
3, 5 and 15.
Its isogeny class
8.1-b
consists of curves linked by isogenies of
degrees dividing 15.
Base change
This elliptic curve is a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.