Base field 6.6.1241125.1
Generator \(a\), with minimal polynomial \( x^{6} - 7 x^{4} - 2 x^{3} + 11 x^{2} + 7 x + 1 \); class number \(1\).
sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 7, 11, -2, -7, 0, 1]))
gp: K = nfinit(Polrev([1, 7, 11, -2, -7, 0, 1]));
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);
Weierstrass equation
sage: E = EllipticCurve([K([-6,-10,2,7,0,-1]),K([5,20,2,-13,-1,2]),K([-19,-53,-2,34,2,-5]),K([-21,-60,0,40,2,-6]),K([-24,-55,3,35,1,-5])])
gp: E = ellinit([Polrev([-6,-10,2,7,0,-1]),Polrev([5,20,2,-13,-1,2]),Polrev([-19,-53,-2,34,2,-5]),Polrev([-21,-60,0,40,2,-6]),Polrev([-24,-55,3,35,1,-5])], K);
magma: E := EllipticCurve([K![-6,-10,2,7,0,-1],K![5,20,2,-13,-1,2],K![-19,-53,-2,34,2,-5],K![-21,-60,0,40,2,-6],K![-24,-55,3,35,1,-5]]);
This is a global minimal model.
sage: E.is_global_minimal_model()
Invariants
| Conductor: | \((-a^5+a^4+6a^3-4a^2-8a)\) | = | \((-2a^5+a^4+13a^3-2a^2-19a-5)^{2}\) |
sage: E.conductor()
gp: ellglobalred(E)[1]
magma: Conductor(E);
| |||
| Conductor norm: | \( 25 \) | = | \(5^{2}\) |
sage: E.conductor().norm()
gp: idealnorm(ellglobalred(E)[1])
magma: Norm(Conductor(E));
| |||
| Discriminant: | \((a^3-a^2-3a+2)\) | = | \((-2a^5+a^4+13a^3-2a^2-19a-5)^{3}\) |
sage: E.discriminant()
gp: E.disc
magma: Discriminant(E);
| |||
| Discriminant norm: | \( 125 \) | = | \(5^{3}\) |
sage: E.discriminant().norm()
gp: norm(E.disc)
magma: Norm(Discriminant(E));
| |||
| j-invariant: | \( -2708 a^{5} + 733 a^{4} + 12894 a^{3} - 3948 a^{2} - 9886 a - 1842 \) | ||
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 | $\left(-2 a^{5} + a^{4} + 13 a^{3} - 2 a^{2} - 20 a - 5 : -a^{5} + a^{4} + 6 a^{3} - 3 a^{2} - 9 a + 1 : 1\right)$ |
| Height | \(0.0089644533804967780913918391767011938020\) |
| Torsion structure: | trivial |
| sage: T = E.torsion_subgroup(); T.invariants()
gp: T = elltors(E); T[2]
magma: T,piT := TorsionSubgroup(E); Invariants(T);
| |
BSD invariants
| Analytic rank: | \( 1 \) | ||
|
sage: E.rank()
magma: Rank(E);
|
|||
| Mordell-Weil rank: | \(1\) | ||
| Regulator: | \( 0.0089644533804967780913918391767011938020 \) | ||
| Period: | \( 31253.004797552320406718629967556060877 \) | ||
| Tamagawa product: | \( 2 \) | ||
| Torsion order: | \(1\) | ||
| Leading coefficient: | \( 3.01779 \) | ||
| 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\((j)_{-}\) |
|---|---|---|---|---|---|---|---|---|
| \((-2a^5+a^4+13a^3-2a^2-19a-5)\) | \(5\) | \(2\) | \(III\) | Additive | \(-1\) | \(2\) | \(3\) | \(0\) |
Galois Representations
The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .
Isogenies and isogeny class
This curve has no rational isogenies. Its isogeny class 25.2-a consists of this curve only.
Base change
This elliptic curve is not a \(\Q\)-curve.
It is not the base change of an elliptic curve defined over any subfield.