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([-1,5,4,-5,-1,1]),K([2,7,0,-6,0,1]),K([0,2,3,-4,-1,1]),K([3176,-12508,-10918,11588,3083,-2340]),K([118082,-428431,-382074,403418,108051,-82115])])
gp: E = ellinit([Polrev([-1,5,4,-5,-1,1]),Polrev([2,7,0,-6,0,1]),Polrev([0,2,3,-4,-1,1]),Polrev([3176,-12508,-10918,11588,3083,-2340]),Polrev([118082,-428431,-382074,403418,108051,-82115])], K);
magma: E := EllipticCurve([K![-1,5,4,-5,-1,1],K![2,7,0,-6,0,1],K![0,2,3,-4,-1,1],K![3176,-12508,-10918,11588,3083,-2340],K![118082,-428431,-382074,403418,108051,-82115]]);
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: | \((983081a^5-1160204a^4-4935963a^3+4682995a^2+4917165a-2633621)\) | = | \((2a^5-3a^4-8a^3+10a^2+5a-4)^{24}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 620412660965527688188300451573157121 \) | = | \(31^{24}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{308629524306573180705764124400765214989728416}{620412660965527688188300451573157121} a^{5} - \frac{96678377635333373686040740401662681795239150}{620412660965527688188300451573157121} a^{4} - \frac{1398478163616726416481971891411691120456947555}{620412660965527688188300451573157121} a^{3} + \frac{111294770463329935355252104822968002094080011}{620412660965527688188300451573157121} a^{2} + \frac{807036468907095340993839638491373208865016181}{620412660965527688188300451573157121} a - \frac{182510843354625433266392408147536567243503960}{620412660965527688188300451573157121} \) | ||
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: | $\left(-\frac{79}{4} a^{5} + \frac{75}{4} a^{4} + \frac{399}{4} a^{3} - 63 a^{2} - \frac{411}{4} a + 15 : 26 a^{5} - 32 a^{4} - 130 a^{3} + \frac{231}{2} a^{2} + \frac{275}{2} a - \frac{291}{8} : 1\right)$ |
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2a^5-3a^4-8a^3+10a^2+5a-4)\) | \(31\) | \(2\) | \(I_{24}\) | Non-split multiplicative | \(1\) | \(1\) | \(24\) | \(24\) |
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.