Properties

Label 6.6.485125.1-31.1-a5
Base field 6.6.485125.1
Conductor norm \( 31 \)
CM no
Base change no
Q-curve no
Torsion order \( 8 \)
Rank \( 0 \)

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Base field 6.6.485125.1

Generator \(a\), with minimal polynomial \( x^{6} - 2 x^{5} - 4 x^{4} + 8 x^{3} + 2 x^{2} - 5 x + 1 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -5, 2, 8, -4, -2, 1]))
 
gp: K = nfinit(Polrev([1, -5, 2, 8, -4, -2, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -5, 2, 8, -4, -2, 1]);
 

Weierstrass equation

\({y}^2+\left(-2a^{5}+3a^{4}+9a^{3}-10a^{2}-8a+3\right){x}{y}={x}^{3}+\left(-a^{5}+5a^{3}+a^{2}-5a-1\right){x}^{2}+{x}\)
sage: E = EllipticCurve([K([3,-8,-10,9,3,-2]),K([-1,-5,1,5,0,-1]),K([0,0,0,0,0,0]),K([1,0,0,0,0,0]),K([0,0,0,0,0,0])])
 
gp: E = ellinit([Polrev([3,-8,-10,9,3,-2]),Polrev([-1,-5,1,5,0,-1]),Polrev([0,0,0,0,0,0]),Polrev([1,0,0,0,0,0]),Polrev([0,0,0,0,0,0])], K);
 
magma: E := EllipticCurve([K![3,-8,-10,9,3,-2],K![-1,-5,1,5,0,-1],K![0,0,0,0,0,0],K![1,0,0,0,0,0],K![0,0,0,0,0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^5-3a^4-8a^3+10a^2+5a-4)\) = \((2a^5-3a^4-8a^3+10a^2+5a-4)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 31 \) = \(31\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((a^2-2a-1)\) = \((2a^5-3a^4-8a^3+10a^2+5a-4)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( -31 \) = \(-31\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{349745746}{31} a^{5} - \frac{614724517}{31} a^{4} - \frac{1548446203}{31} a^{3} + \frac{2422385572}{31} a^{2} + \frac{1289061786}{31} a - \frac{1434751947}{31} \)
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: \(0\)
Torsion structure: \(\Z/8\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(a^{4} - a^{3} - 3 a^{2} + a + 1 : -a^{5} + a^{4} + 3 a^{3} - a^{2} - a : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 65650.744619812979434042832843806044736 \)
Tamagawa product: \( 1 \)
Torsion order: \(8\)
Leading coefficient: \( 1.47276 \)
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\((j)_{-}\)
\((2a^5-3a^4-8a^3+10a^2+5a-4)\) \(31\) \(1\) \(I_{1}\) Non-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
\(2\) 2B
\(3\) 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6, 8, 12 and 24.
Its isogeny class 31.1-a consists of curves linked by isogenies of degrees dividing 24.

Base change

This elliptic curve is not a \(\Q\)-curve.

It is not the base change of an elliptic curve defined over any subfield.