Base field 3.3.316.1
Generator \(a\), with minimal polynomial \( x^{3} - x^{2} - 4 x + 2 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, -4, -1, 1]))
gp: K = nfinit(Polrev([2, -4, -1, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, -4, -1, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-3,0,1]),K([3,0,-1]),K([0,0,0]),K([-11326059353,-1410890261,2665494175]),K([433135270894324,53955777236750,-101934795285681])])
gp: E = ellinit([Polrev([-3,0,1]),Polrev([3,0,-1]),Polrev([0,0,0]),Polrev([-11326059353,-1410890261,2665494175]),Polrev([433135270894324,53955777236750,-101934795285681])], K);
magma: E := EllipticCurve([K![-3,0,1],K![3,0,-1],K![0,0,0],K![-11326059353,-1410890261,2665494175],K![433135270894324,53955777236750,-101934795285681]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^2-3a-1)\) | = | \((-a+1)^{6}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 64 \) | = | \(2^{6}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((76a^2-105a-249)\) | = | \((-a+1)^{22}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 4194304 \) | = | \(2^{22}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{200481}{16} a^{2} - \frac{325027}{8} a + \frac{233973}{8} \) | ||
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 | $\left(-\frac{35347}{2} a^{2} + 9355 a + \frac{150195}{2} : \frac{2223279}{4} a^{2} - \frac{1176819}{4} a - \frac{4723513}{2} : 1\right)$ | |
| Height | \(0.81026412734311478958373883853510748600\) | |
| Torsion structure: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(28603 a^{2} - 15140 a - 121538 : 39736 a^{2} - 21033 a - 168844 : 1\right)$ | $\left(-\frac{44763}{4} a^{2} + \frac{11847}{2} a + \frac{190205}{4} : -\frac{124373}{8} a^{2} + 8229 a + \frac{528477}{8} : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| ||
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.81026412734311478958373883853510748600 \) | ||
| Period: | \( 75.058615256171572543621122153228743827 \) | ||
| Tamagawa product: | \( 4 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 2.5659304579191107995173056971277748399 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-a+1)\) | \(2\) | \(4\) | \(I_{12}^{*}\) | Additive | \(1\) | \(6\) | \(22\) | \(4\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(2\) | 2Cs |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2 and 4.
Its isogeny class
64.7-d
consists of curves linked by isogenies of
degrees dividing 16.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.