Base field 6.6.592661.1
Generator \(a\), with minimal polynomial \( x^{6} - x^{5} - 5 x^{4} + 4 x^{3} + 5 x^{2} - 2 x - 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -2, 5, 4, -5, -1, 1]))
gp: K = nfinit(Polrev([-1, -2, 5, 4, -5, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -2, 5, 4, -5, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,2,5,-4,-1,1]),K([0,2,0,-1,0,0]),K([-1,2,4,-4,-1,1]),K([0,2,0,-1,0,0]),K([0,0,0,0,0,0])])
gp: E = ellinit([Polrev([-3,2,5,-4,-1,1]),Polrev([0,2,0,-1,0,0]),Polrev([-1,2,4,-4,-1,1]),Polrev([0,2,0,-1,0,0]),Polrev([0,0,0,0,0,0])], K);
magma: E := EllipticCurve([K![-3,2,5,-4,-1,1],K![0,2,0,-1,0,0],K![-1,2,4,-4,-1,1],K![0,2,0,-1,0,0],K![0,0,0,0,0,0]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-a^4-5a^3+5a^2+5a-3)\) | = | \((a^5-a^4-5a^3+5a^2+5a-3)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 67 \) | = | \(67\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((-a^5+a^4+4a^3-3a^2-1)\) | = | \((a^5-a^4-5a^3+5a^2+5a-3)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -67 \) | = | \(-67\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{293657}{67} a^{5} + \frac{844369}{67} a^{4} - \frac{137481}{67} a^{3} - \frac{966413}{67} a^{2} + \frac{347718}{67} a + \frac{215682}{67} \) | ||
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 | $\left(-a^{2} + 2 : -3 a^{5} + 4 a^{4} + 14 a^{3} - 15 a^{2} - 10 a + 6 : 1\right)$ |
| Height | \(0.0088248633042439913228218977246167853033\) |
| 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: | \( 0.0088248633042439913228218977246167853033 \) | ||
| Period: | \( 38394.882441067072297130643770062678508 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 2.64076 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-a^4-5a^3+5a^2+5a-3)\) | \(67\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 67.1-a consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.