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([0,-2,0,1,0,0]),K([-3,-1,4,1,-1,0]),K([-2,7,7,-9,-2,2]),K([-79,330,-563,-258,139,36]),K([-1392,7810,-4432,-9619,784,1873])])
gp: E = ellinit([Polrev([0,-2,0,1,0,0]),Polrev([-3,-1,4,1,-1,0]),Polrev([-2,7,7,-9,-2,2]),Polrev([-79,330,-563,-258,139,36]),Polrev([-1392,7810,-4432,-9619,784,1873])], K);
magma: E := EllipticCurve([K![0,-2,0,1,0,0],K![-3,-1,4,1,-1,0],K![-2,7,7,-9,-2,2],K![-79,330,-563,-258,139,36],K![-1392,7810,-4432,-9619,784,1873]]);
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: | \((-21a^5+74a^4+14a^3-250a^2+157a+112)\) | = | \((2a^5-3a^4-8a^3+10a^2+5a-4)^{8}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
Discriminant norm: | \( 852891037441 \) | = | \(31^{8}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
j-invariant: | \( -\frac{1033073743235841200494976959934}{852891037441} a^{5} + \frac{3933929786659871173277923782815}{852891037441} a^{4} - \frac{2975097977593569902108674723259}{852891037441} a^{3} - \frac{2901568778559675791252921941164}{852891037441} a^{2} + \frac{3183623527721495073846445577852}{852891037441} a - \frac{573680380524522897838041437515}{852891037441} \) | ||
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: | \(\Z/2\Z\) |
sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
Torsion generator: | $\left(-\frac{9}{2} a^{5} - 5 a^{4} + 29 a^{3} + \frac{39}{2} a^{2} - \frac{145}{4} a + \frac{5}{4} : \frac{9}{4} a^{5} + \frac{29}{8} a^{4} - \frac{91}{8} a^{3} - 15 a^{2} + \frac{43}{4} a - 3 : 1\right)$ |
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: | \( 2.0035017278995660227674204359071668926 \) | ||
Tamagawa product: | \( 2 \) | ||
Torsion order: | \(2\) | ||
Leading coefficient: | \( 1.47276 \) | ||
Analytic order of Ш: | \( 1024 \) (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\((j)_{-}\) |
---|---|---|---|---|---|---|---|---|
\((2a^5-3a^4-8a^3+10a^2+5a-4)\) | \(31\) | \(2\) | \(I_{8}\) | Non-split multiplicative | \(1\) | \(1\) | \(8\) | \(8\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
prime | Image of Galois Representation |
---|---|
\(2\) | 2B |
\(3\) | 3B |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3, 4, 6, 8, 12 and 24.
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.