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([16,-45,-165,45,40,-10]),K([57,-36,-678,-31,153,6])])
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([16,-45,-165,45,40,-10]),Polrev([57,-36,-678,-31,153,6])], 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![16,-45,-165,45,40,-10],K![57,-36,-678,-31,153,6]]);
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: | \((-5a^5+2a^4+23a^3-9a^2-14a+14)\) | = | \((2a^5-3a^4-8a^3+10a^2+5a-4)^{4}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 923521 \) | = | \(31^{4}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{161977561667364262506242}{923521} a^{5} - \frac{474485436612723225705736}{923521} a^{4} - \frac{206957557465744779382665}{923521} a^{3} + \frac{1488151984849774698530977}{923521} a^{2} - \frac{1059026459496257469864311}{923521} a + \frac{174295328969264546943196}{923521} \) | ||
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: | \(\Z/2\Z\oplus\Z/2\Z\) | |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| ||
| Torsion generators: | $\left(4 a^{5} - 2 a^{4} - 22 a^{3} + 8 a^{2} + 24 a - 4 : 6 a^{5} - 9 a^{4} - 30 a^{3} + 31 a^{2} + 37 a - 11 : 1\right)$ | $\left(-\frac{1}{2} a^{5} + 2 a^{4} + \frac{9}{4} a^{3} - \frac{33}{4} a^{2} - \frac{9}{4} a - 1 : -\frac{11}{2} a^{5} + \frac{61}{8} a^{4} + \frac{195}{8} a^{3} - \frac{99}{4} a^{2} - \frac{83}{4} a + \frac{27}{4} : 1\right)$ |
| 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: | \( 128.22411058557222545711490789805868112 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(4\) | ||
| Leading coefficient: | \( 1.47276 \) | ||
| Analytic order of Ш: | \( 64 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2a^5-3a^4-8a^3+10a^2+5a-4)\) | \(31\) | \(2\) | \(I_{4}\) | Non-split multiplicative | \(1\) | \(1\) | \(4\) | \(4\) |
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 |
| \(3\) | 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.