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([1,1,0]),K([-5,0,1]),K([-5,1,1]),K([-18521362200115984,486063540319786,2727719912262131]),K([848274944932820465801609,-22261619768770883846309,-124929064793722765572661])])
gp: E = ellinit([Polrev([1,1,0]),Polrev([-5,0,1]),Polrev([-5,1,1]),Polrev([-18521362200115984,486063540319786,2727719912262131]),Polrev([848274944932820465801609,-22261619768770883846309,-124929064793722765572661])], K);
magma: E := EllipticCurve([K![1,1,0],K![-5,0,1],K![-5,1,1],K![-18521362200115984,486063540319786,2727719912262131],K![848274944932820465801609,-22261619768770883846309,-124929064793722765572661]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^2+a-6)\) | = | \((a^2-6)\cdot(a^2+2a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 14 \) | = | \(2\cdot7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^2-8a+16)\) | = | \((a^2-6)^{4}\cdot(a^2+2a-3)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -784 \) | = | \(-2^{4}\cdot7^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1129805449}{784} a^{2} - \frac{201001775}{784} a + \frac{7672803655}{784} \) | ||
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: | \(0\) |
| Torsion structure: | \(\Z/8\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(3713910 a^{2} + 661797 a - 25217647 : -184195643785 a^{2} - 32822573297 a + 1250698144979 : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 0 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(0\) | ||
| Regulator: | \( 1 \) | ||
| Period: | \( 233.91800181197995351023480391304222135 \) | ||
| Tamagawa product: | \( 8 \) = \(2^{2}\cdot2\) | ||
| Torsion order: | \(8\) | ||
| Leading coefficient: | \( 1.0799948166390446304772952349780471732 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^2-6)\) | \(2\) | \(4\) | \(I_{4}\) | Split multiplicative | \(-1\) | \(1\) | \(4\) | \(4\) |
| \((a^2+2a-3)\) | \(7\) | \(2\) | \(I_{2}\) | Non-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, 4 and 8.
Its isogeny class
14.1-a
consists of curves linked by isogenies of
degrees dividing 8.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.