Base field 6.6.703493.1
Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 5 x^{4} + 11 x^{3} + 2 x^{2} - 9 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -9, 2, 11, -5, -2, 1]))
gp: K = nfinit(Polrev([1, -9, 2, 11, -5, -2, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -9, 2, 11, -5, -2, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-4,9,10,-11,-2,2]),K([2,4,-1,-5,0,1]),K([-3,14,6,-12,-1,2]),K([8,-21,-14,22,3,-4]),K([5,-8,-12,11,2,-2])])
gp: E = ellinit([Polrev([-4,9,10,-11,-2,2]),Polrev([2,4,-1,-5,0,1]),Polrev([-3,14,6,-12,-1,2]),Polrev([8,-21,-14,22,3,-4]),Polrev([5,-8,-12,11,2,-2])], K);
magma: E := EllipticCurve([K![-4,9,10,-11,-2,2],K![2,4,-1,-5,0,1],K![-3,14,6,-12,-1,2],K![8,-21,-14,22,3,-4],K![5,-8,-12,11,2,-2]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-2a^5+a^4+12a^3-4a^2-12a-1)\) | = | \((-2a^5+a^4+12a^3-4a^2-12a-1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 71 \) | = | \(71\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^3-3a-1)\) | = | \((-2a^5+a^4+12a^3-4a^2-12a-1)\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 71 \) | = | \(71\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{1061831}{71} a^{5} - \frac{19223}{71} a^{4} - \frac{5239532}{71} a^{3} + \frac{1358533}{71} a^{2} + \frac{4577495}{71} a - \frac{444281}{71} \) | ||
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: | \(1\) |
| Generator | $\left(3 a^{5} - a^{4} - 19 a^{3} + 4 a^{2} + 24 a + 3 : 8 a^{5} - 8 a^{4} - 45 a^{3} + 39 a^{2} + 40 a - 16 : 1\right)$ |
| Height | \(0.22095699775900382026933421020447041490\) |
| Torsion structure: | \(\Z/5\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-a^{4} + a^{3} + 5 a^{2} - 5 a - 1 : -a^{5} - a^{4} + 8 a^{3} + 4 a^{2} - 16 a : 1\right)$ |
| sage: T.gens()
gp: T[3]
magma: [piT(P) : P in Generators(T)];
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.22095699775900382026933421020447041490 \) | ||
| Period: | \( 41839.532971909374145793439845048731629 \) | ||
| Tamagawa product: | \( 1 \) | ||
| Torsion order: | \(5\) | ||
| Leading coefficient: | \( 2.64531 \) | ||
| Analytic order of Ш: | \( 1 \) (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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^5+a^4+12a^3-4a^2-12a-1)\) | \(71\) | \(1\) | \(I_{1}\) | Split multiplicative | \(-1\) | \(1\) | \(1\) | \(1\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) except those listed.
| prime | Image of Galois Representation |
|---|---|
| \(5\) | 5B.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
5.
Its isogeny class
71.1-d
consists of curves linked by isogenies of
degree 5.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.