Properties

Label 6.6.1241125.1-25.2-d1
Base field 6.6.1241125.1
Conductor \((-a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 8 a)\)
Conductor norm \( 25 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 1 \)

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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: x = polygen(QQ); K.<a> = NumberField(x^6 - 7*x^4 - 2*x^3 + 11*x^2 + 7*x + 1)
 
gp: K = nfinit(a^6 - 7*a^4 - 2*a^3 + 11*a^2 + 7*a + 1);
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 7, 11, -2, -7, 0, 1]);
 

Weierstrass equation

\(y^2+\left(a^{2}+a-3\right)xy+\left(-5a^{5}+2a^{4}+34a^{3}-2a^{2}-52a-19\right)y=x^{3}+\left(a^{4}-a^{3}-5a^{2}+3a+3\right)x^{2}+\left(-8a^{5}+3a^{4}+53a^{3}-5a^{2}-79a-22\right)x-8a^{5}+3a^{4}+54a^{3}-4a^{2}-83a-27\)
sage: E = EllipticCurve(K, [a^2 + a - 3, a^4 - a^3 - 5*a^2 + 3*a + 3, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 52*a - 19, -8*a^5 + 3*a^4 + 53*a^3 - 5*a^2 - 79*a - 22, -8*a^5 + 3*a^4 + 54*a^3 - 4*a^2 - 83*a - 27])
 
gp: E = ellinit([a^2 + a - 3, a^4 - a^3 - 5*a^2 + 3*a + 3, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 52*a - 19, -8*a^5 + 3*a^4 + 53*a^3 - 5*a^2 - 79*a - 22, -8*a^5 + 3*a^4 + 54*a^3 - 4*a^2 - 83*a - 27],K)
 
magma: E := ChangeRing(EllipticCurve([a^2 + a - 3, a^4 - a^3 - 5*a^2 + 3*a + 3, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 52*a - 19, -8*a^5 + 3*a^4 + 53*a^3 - 5*a^2 - 79*a - 22, -8*a^5 + 3*a^4 + 54*a^3 - 4*a^2 - 83*a - 27]),K);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((-a^{5} + a^{4} + 6 a^{3} - 4 a^{2} - 8 a)\) = \( \left(5, a - 2\right)^{2} \)
sage: E.conductor()
 
magma: Conductor(E);
 
Conductor norm: \( 25 \) = \( 5^{2} \)
sage: E.conductor().norm()
 
magma: Norm(Conductor(E));
 
Discriminant: \((5 a^{5} - 7 a^{4} - 26 a^{3} + 24 a^{2} + 25 a + 14)\) = \( \left(5, a - 2\right)^{9} \)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1953125 \) = \( 5^{9} \)
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 $\left(-a^{4} + 5 a^{2} + a - 2 : 2 a^{5} - 11 a^{3} - 2 a^{2} + 8 a + 1 : 1\right)$
Height \(0.0202071928319971\)
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: not available
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.0202071928319971 \)
Period: not available
Tamagawa product: \( 2 \)
Torsion order: \(1\)
Leading coefficient: not available
Analytic order of Ш: not available

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\((j)_{-}\)
\( \left(5, a - 2\right) \) \(5\) \(2\) \(III^*\) Additive \(-1\) \(2\) \(9\) \(0\)

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-d consists of this curve only.

Base change

This curve is not the base change of an elliptic curve defined over \(\Q\). It is not a \(\Q\)-curve.