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([-2,7,7,-9,-2,2]),K([3,-4,-9,8,3,-2]),K([1,-2,-6,4,2,-1]),K([-13,20,-3,28,-75,29]),K([59,-417,522,482,-1030,343])])
gp: E = ellinit([Polrev([-2,7,7,-9,-2,2]),Polrev([3,-4,-9,8,3,-2]),Polrev([1,-2,-6,4,2,-1]),Polrev([-13,20,-3,28,-75,29]),Polrev([59,-417,522,482,-1030,343])], K);
magma: E := EllipticCurve([K![-2,7,7,-9,-2,2],K![3,-4,-9,8,3,-2],K![1,-2,-6,4,2,-1],K![-13,20,-3,28,-75,29],K![59,-417,522,482,-1030,343]]);
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: | \((984a^5-1559a^4-4321a^3+5273a^2+3734a-2029)\) | = | \((2a^5-3a^4-8a^3+10a^2+5a-4)^{12}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 787662783788549761 \) | = | \(31^{12}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -\frac{1127683403680349712780740748443904004}{787662783788549761} a^{5} + \frac{354688222610355323886840898021118087}{787662783788549761} a^{4} + \frac{5108550614403361265287641142493913090}{787662783788549761} a^{3} - \frac{411149191401225017208777070394047609}{787662783788549761} a^{2} - \frac{2948347178168149967480663778467655563}{787662783788549761} a + \frac{669060970718259450595665810647045219}{787662783788549761} \) | ||
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: | $\left(4 a^{4} - 7 a^{3} - 8 a^{2} + 9 a + 1 : 2 a^{5} - 3 a^{4} - 3 a^{3} + 3 a^{2} - 5 a + 1 : 1\right)$ | $\left(-4 a^{4} + 5 a^{3} + 12 a^{2} - 11 a + 1 : -2 a^{5} + 3 a^{4} + 5 a^{3} - 7 a^{2} + 5 a - 1 : 1\right)$ |
| 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: | \( 128.22411058557222545711490789805868113 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((2a^5-3a^4-8a^3+10a^2+5a-4)\) | \(31\) | \(2\) | \(I_{12}\) | Non-split multiplicative | \(1\) | \(1\) | \(12\) | \(12\) |
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.