Base field 6.6.485125.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([2,3,-3,-1,1,0]),K([7,-6,-11,8,3,-2]),K([-2,5,4,-5,-1,1]),K([22,-19,-39,24,11,-6]),K([15,-12,-27,15,8,-4])])
gp: E = ellinit([Polrev([2,3,-3,-1,1,0]),Polrev([7,-6,-11,8,3,-2]),Polrev([-2,5,4,-5,-1,1]),Polrev([22,-19,-39,24,11,-6]),Polrev([15,-12,-27,15,8,-4])], K);
magma: E := EllipticCurve([K![2,3,-3,-1,1,0],K![7,-6,-11,8,3,-2],K![-2,5,4,-5,-1,1],K![22,-19,-39,24,11,-6],K![15,-12,-27,15,8,-4]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
Conductor: | \((-a^4+a^3+3a^2-3a-1)\) | = | \((-a^4+a^3+3a^2-3a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
Conductor norm: | \( 59 \) | = | \(59\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
Discriminant: | \((-2a^5+3a^4+8a^3-11a^2-4a+4)\) | = | \((-a^4+a^3+3a^2-3a-1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( -59 \) | = | \(-59\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( \frac{11048}{59} a^{5} - \frac{11375}{59} a^{4} - \frac{30730}{59} a^{3} + \frac{29362}{59} a^{2} - \frac{55427}{59} a + \frac{52669}{59} \) | ||
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 | $\left(-1 : a : 1\right)$ |
Height | \(0.011019161973449524225845028798795874919\) |
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: | \( 0.011019161973449524225845028798795874919 \) | ||
Period: | \( 25511.378948931881591934538847291066172 \) | ||
Tamagawa product: | \( 1 \) | ||
Torsion order: | \(1\) | ||
Leading coefficient: | \( 2.42163 \) | ||
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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((-a^4+a^3+3a^2-3a-1)\) | \(59\) | \(1\) | \(I_{1}\) | 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 59.2-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.