Base field 5.5.153424.1
Generator \(a\), with minimal polynomial \( x^{5} - 2 x^{4} - 4 x^{3} + 8 x^{2} - 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-2, 0, 8, -4, -2, 1]))
gp: K = nfinit(Polrev([-2, 0, 8, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2, 0, 8, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([6,3,-10,-1,2]),K([0,7,-4,-2,1]),K([3,-1,-5,0,1]),K([-68,-38,234,-16,-69]),K([-929,-637,3067,-155,-893])])
gp: E = ellinit([Polrev([6,3,-10,-1,2]),Polrev([0,7,-4,-2,1]),Polrev([3,-1,-5,0,1]),Polrev([-68,-38,234,-16,-69]),Polrev([-929,-637,3067,-155,-893])], K);
magma: E := EllipticCurve([K![6,3,-10,-1,2],K![0,7,-4,-2,1],K![3,-1,-5,0,1],K![-68,-38,234,-16,-69],K![-929,-637,3067,-155,-893]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a+1)\) | = | \((a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 7 \) | = | \(7\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((5a^4-a^3-20a^2-a-1)\) | = | \((a+1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 117649 \) | = | \(7^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{8004305085704}{117649} a^{4} + \frac{4141088345280}{117649} a^{3} + \frac{37991329701812}{117649} a^{2} - \frac{7545938455168}{117649} a - \frac{10474018356248}{117649} \) | ||
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: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(9 a^{4} - 14 a^{3} - 46 a^{2} + 48 a + 31 : 53 a^{4} - 66 a^{3} - 252 a^{2} + 242 a + 155 : 1\right)$ |
| 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: | \( 3502.6115511987247155257020611078769636 \) | ||
| Tamagawa product: | \( 6 \) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 1.49036991 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a+1)\) | \(7\) | \(6\) | \(I_{6}\) | 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 |
| \(3\) | 3B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
7.1-a
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.