Base field \(\Q(\zeta_{20})^+\)
Generator \(a\), with minimal polynomial \( x^{4} - 5 x^{2} + 5 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([5, 0, -5, 0, 1]))
gp: K = nfinit(Polrev([5, 0, -5, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![5, 0, -5, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([1,1,0,0]),K([2,1,-1,0]),K([-3,-2,1,1]),K([3,6,-1,-2]),K([-5,-5,4,3])])
gp: E = ellinit([Polrev([1,1,0,0]),Polrev([2,1,-1,0]),Polrev([-3,-2,1,1]),Polrev([3,6,-1,-2]),Polrev([-5,-5,4,3])], K);
magma: E := EllipticCurve([K![1,1,0,0],K![2,1,-1,0],K![-3,-2,1,1],K![3,6,-1,-2],K![-5,-5,4,3]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^3+a^2-4a-1)\) | = | \((a^3+a^2-4a-1)\) |
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: | \((-13a^3-8a^2+33a+36)\) | = | \((a^3+a^2-4a-1)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -205379 \) | = | \(-59^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{3559234608}{205379} a^{3} - \frac{7434481239}{205379} a^{2} - \frac{3401019009}{205379} a + \frac{8873009136}{205379} \) | ||
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 | $\left(-a^{3} + a^{2} + a - 2 : a^{3} - a^{2} + 2 : 1\right)$ |
| Height | \(0.0031523408913066039099401569475851514177\) |
| 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: | \( 0.0031523408913066039099401569475851514177 \) | ||
| Period: | \( 1465.9293006815241145897773934434963308 \) | ||
| Tamagawa product: | \( 3 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 1.23997362999756 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^3+a^2-4a-1)\) | \(59\) | \(3\) | \(I_{3}\) | Split multiplicative | \(-1\) | \(1\) | \(3\) | \(3\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(3\) | 3Nn |
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 59.3-b 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.