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Properties

Label 6.6.966125.1-31.2-a1
Base field 6.6.966125.1
Conductor norm \( 31 \)
CM no
Base change no
Q-curve no
Torsion order \( 1 \)
Rank \( 0 \)

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Base field 6.6.966125.1

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, 0, 8, 4, -6, -1, 1]))
 
gp: K = nfinit(Polrev([-1, 0, 8, 4, -6, -1, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 0, 8, 4, -6, -1, 1]);
 

Weierstrass equation

\({y}^2+\left(2a^{5}-2a^{4}-9a^{3}+9a^{2}+4a-2\right){x}{y}+\left(a^{5}-a^{4}-4a^{3}+5a^{2}-3\right){y}={x}^{3}+\left(2a^{5}-3a^{4}-10a^{3}+13a^{2}+8a-2\right){x}^{2}+\left(11a^{5}-7a^{4}-46a^{3}+33a^{2}+12a-5\right){x}+16a^{5}-6a^{4}-64a^{3}+34a^{2}+12a-7\)
sage: E = EllipticCurve([K([-2,4,9,-9,-2,2]),K([-2,8,13,-10,-3,2]),K([-3,0,5,-4,-1,1]),K([-5,12,33,-46,-7,11]),K([-7,12,34,-64,-6,16])])
 
gp: E = ellinit([Polrev([-2,4,9,-9,-2,2]),Polrev([-2,8,13,-10,-3,2]),Polrev([-3,0,5,-4,-1,1]),Polrev([-5,12,33,-46,-7,11]),Polrev([-7,12,34,-64,-6,16])], K);
 
magma: E := EllipticCurve([K![-2,4,9,-9,-2,2],K![-2,8,13,-10,-3,2],K![-3,0,5,-4,-1,1],K![-5,12,33,-46,-7,11],K![-7,12,34,-64,-6,16]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((a^2-2)\) = \((a^2-2)\)
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: \((a^2-2)\) = \((a^2-2)\)
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: \( \frac{188987486}{31} a^{5} - \frac{272082083}{31} a^{4} - \frac{1014613018}{31} a^{3} + \frac{1202373611}{31} a^{2} + \frac{985030371}{31} a - \frac{434306805}{31} \)
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: 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: \( 0 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(0\)
Regulator: \( 1 \)
Period: \( 2487.2294882046993522585334279619234265 \)
Tamagawa product: \( 1 \)
Torsion order: \(1\)
Leading coefficient: \( 2.53046 \)
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\((j)_{-}\)
\((a^2-2)\) \(31\) \(1\) \(I_{1}\) Non-split multiplicative \(1\) \(1\) \(1\) \(1\)

Galois Representations

The mod \( p \) Galois Representation has maximal image for all primes \( p < 1000 \) .

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 31.2-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.