Base field 3.3.733.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 7 x + 8 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([8, -7, -1, 1]))
gp: K = nfinit(Polrev([8, -7, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![8, -7, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-5,0,1]),K([0,0,0]),K([1,1,0]),K([-128731016,3380608,18957864]),K([-491718787736,12904890246,72417315372])])
gp: E = ellinit([Polrev([-5,0,1]),Polrev([0,0,0]),Polrev([1,1,0]),Polrev([-128731016,3380608,18957864]),Polrev([-491718787736,12904890246,72417315372])], K);
magma: E := EllipticCurve([K![-5,0,1],K![0,0,0],K![1,1,0],K![-128731016,3380608,18957864],K![-491718787736,12904890246,72417315372]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2-6)\) | = | \((a^2-6)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 2 \) | = | \(2\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^2+16a+64)\) | = | \((a^2-6)^{18}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -262144 \) | = | \(-2^{18}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1308241604941929}{262144} a^{2} - \frac{233087636858447}{262144} a + \frac{8883119257726807}{262144} \) | ||
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 | $\left(-\frac{172251}{25} a^{2} - \frac{32982}{25} a + \frac{1175351}{25} : \frac{155165794}{125} a^{2} + \frac{27422908}{125} a - \frac{1053013219}{125} : 1\right)$ |
| Height | \(1.6512868987226162541344758122566922855\) |
| 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: | $\left(\frac{4447}{4} a^{2} + \frac{661}{4} a - \frac{29865}{4} : \frac{15863}{8} a^{2} + \frac{3121}{8} a - \frac{108465}{8} : 1\right)$ |
| 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: | \( 1.6512868987226162541344758122566922855 \) | ||
| Period: | \( 9.3293269566393914904633731243606414525 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(2\) | ||
| Leading coefficient: | \( 0.85351689191865735834263246428729022860 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-6)\) | \(2\) | \(2\) | \(I_{18}\) | Non-split multiplicative | \(1\) | \(1\) | \(18\) | \(18\) |
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.1.2 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
2.1-b
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.