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([3,-8,-10,9,3,-2]),K([-1,-5,1,5,0,-1]),K([0,0,0,0,0,0]),K([-4,0,0,0,0,0]),K([12,-9,-33,9,8,-2])])
gp: E = ellinit([Polrev([3,-8,-10,9,3,-2]),Polrev([-1,-5,1,5,0,-1]),Polrev([0,0,0,0,0,0]),Polrev([-4,0,0,0,0,0]),Polrev([12,-9,-33,9,8,-2])], K);
magma: E := EllipticCurve([K![3,-8,-10,9,3,-2],K![-1,-5,1,5,0,-1],K![0,0,0,0,0,0],K![-4,0,0,0,0,0],K![12,-9,-33,9,8,-2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((2a^5-3a^4-8a^3+10a^2+5a-4)\) | = | \((2a^5-3a^4-8a^3+10a^2+5a-4)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 31 \) | = | \(31\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((4a^5-5a^4-19a^3+19a^2+19a-11)\) | = | \((2a^5-3a^4-8a^3+10a^2+5a-4)^{2}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 961 \) | = | \(31^{2}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{350193349920473908}{961} a^{5} + \frac{614989688312525919}{961} a^{4} + \frac{1550742937477348718}{961} a^{3} - \frac{2423388078335799313}{961} a^{2} - \frac{1291346530388574699}{961} a + \frac{1436064511168210027}{961} \) | ||
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: | \(\Z/2\Z\oplus\Z/4\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(-2 : -2 a^{5} + 3 a^{4} + 9 a^{3} - 10 a^{2} - 8 a + 3 : 1\right)$ | $\left(2 a^{5} - a^{4} - 11 a^{3} + 4 a^{2} + 12 a - 1 : 9 a^{5} - 12 a^{4} - 46 a^{3} + 45 a^{2} + 51 a - 15 : 1\right)$ |
| 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: | \( 8206.3430774766224292553541054757555920 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(8\) | ||
| Leading coefficient: | \( 1.47276 \) | ||
| Analytic order of Ш: | \( 4 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2a^5-3a^4-8a^3+10a^2+5a-4)\) | \(31\) | \(2\) | \(I_{2}\) | Non-split multiplicative | \(1\) | \(1\) | \(2\) | \(2\) |
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 |
| \(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6 and 12.
Its isogeny class
31.1-a
consists of curves linked by isogenies of
degrees dividing 24.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.