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([-1,14,5,-12,-1,2]),K([4,-7,-6,6,1,-1]),K([-5,9,11,-11,-2,2]),K([-2,-31,-1,29,-2,-5]),K([5,-49,-13,55,-2,-10])])
gp: E = ellinit([Polrev([-1,14,5,-12,-1,2]),Polrev([4,-7,-6,6,1,-1]),Polrev([-5,9,11,-11,-2,2]),Polrev([-2,-31,-1,29,-2,-5]),Polrev([5,-49,-13,55,-2,-10])], K);
magma: E := EllipticCurve([K![-1,14,5,-12,-1,2],K![4,-7,-6,6,1,-1],K![-5,9,11,-11,-2,2],K![-2,-31,-1,29,-2,-5],K![5,-49,-13,55,-2,-10]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((a^5-a^4-6a^3+4a^2+7a+1)\) | = | \((a^5-a^4-6a^3+4a^2+7a+1)\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 41 \) | = | \(41\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((8a^5+10a^4-37a^3-56a^2+28a+26)\) | = | \((a^5-a^4-6a^3+4a^2+7a+1)^{6}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( -4750104241 \) | = | \(-41^{6}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( \frac{2361415517723691336}{4750104241} a^{5} + \frac{312405531641926157}{4750104241} a^{4} - \frac{11145383118536595673}{4750104241} a^{3} + \frac{2208505528549032067}{4750104241} a^{2} + \frac{9451172168707485868}{4750104241} a - \frac{1100146408697507811}{4750104241} \) | ||
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 | $\left(-3 a^{5} + 4 a^{4} + 15 a^{3} - 20 a^{2} - 11 a + 15 : -7 a^{5} + 14 a^{4} + 34 a^{3} - 70 a^{2} - 21 a + 55 : 1\right)$ |
| Height | \(0.49259532543800074765744061664164434652\) |
| Torsion structure: | \(\Z/6\Z\) |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
| Torsion generator: | $\left(-a^{4} + 4 a^{2} - 1 : -6 a^{5} + 6 a^{4} + 35 a^{3} - 29 a^{2} - 35 a + 10 : 1\right)$ |
| 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: | \( 0.49259532543800074765744061664164434652 \) | ||
| Period: | \( 4356.8269132172565763499349366187791454 \) | ||
| Tamagawa product: | \( 6 \) | ||
| Torsion order: | \(6\) | ||
| Leading coefficient: | \( 2.55877 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((a^5-a^4-6a^3+4a^2+7a+1)\) | \(41\) | \(6\) | \(I_{6}\) | Split multiplicative | \(-1\) | \(1\) | \(6\) | \(6\) |
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.1.1 |
Isogenies and isogeny class
This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\)
2, 3 and 6.
Its isogeny class
41.4-e
consists of curves linked by isogenies of
degrees dividing 6.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.