Properties

Label 6.6.1241125.1-25.2-a1
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

\(y^2+\left(-a^{5}+7a^{3}+2a^{2}-10a-6\right)xy+\left(-5a^{5}+2a^{4}+34a^{3}-2a^{2}-53a-19\right)y=x^{3}+\left(2a^{5}-a^{4}-13a^{3}+2a^{2}+20a+5\right)x^{2}+\left(-6a^{5}+2a^{4}+40a^{3}-60a-21\right)x-5a^{5}+a^{4}+35a^{3}+3a^{2}-55a-24\)
sage: E = EllipticCurve(K, [-a^5 + 7*a^3 + 2*a^2 - 10*a - 6, 2*a^5 - a^4 - 13*a^3 + 2*a^2 + 20*a + 5, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 53*a - 19, -6*a^5 + 2*a^4 + 40*a^3 - 60*a - 21, -5*a^5 + a^4 + 35*a^3 + 3*a^2 - 55*a - 24])
 
gp: E = ellinit([-a^5 + 7*a^3 + 2*a^2 - 10*a - 6, 2*a^5 - a^4 - 13*a^3 + 2*a^2 + 20*a + 5, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 53*a - 19, -6*a^5 + 2*a^4 + 40*a^3 - 60*a - 21, -5*a^5 + a^4 + 35*a^3 + 3*a^2 - 55*a - 24],K)
 
magma: E := ChangeRing(EllipticCurve([-a^5 + 7*a^3 + 2*a^2 - 10*a - 6, 2*a^5 - a^4 - 13*a^3 + 2*a^2 + 20*a + 5, -5*a^5 + 2*a^4 + 34*a^3 - 2*a^2 - 53*a - 19, -6*a^5 + 2*a^4 + 40*a^3 - 60*a - 21, -5*a^5 + a^4 + 35*a^3 + 3*a^2 - 55*a - 24]),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: \((-3 a^{5} + a^{4} + 21 a^{3} - a^{2} - 35 a - 9)\) = \( \left(5, a - 2\right)^{3} \)
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 \(0.00896445338049678\)
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: \( 0.00896445338049678 \)
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 \(-1\) \(2\) \(3\) \(0\)

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