Properties

Label 2.0.8.1-11552.2-a1
Base field \(\Q(\sqrt{-2}) \)
Conductor \((76a)\)
Conductor norm \( 11552 \)
CM no
Base change yes: 1216.a1,608.a1
Q-curve yes
Torsion order \( 1 \)
Rank \( 2 \)

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Show commands: Magma / Pari/GP / SageMath

Base field \(\Q(\sqrt{-2}) \)

Generator \(a\), with minimal polynomial \( x^{2} + 2 \); class number \(1\).

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 1]))
 
gp: K = nfinit(Pol(Vecrev([2, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
 

Weierstrass equation

\({y}^2+a{x}{y}+a{y}={x}^{3}-{x}^{2}+3{x}\)
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,1]),K([3,0]),K([0,0])])
 
gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([3,0])),Pol(Vecrev([0,0]))], K);
 
magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,1],K![3,0],K![0,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((76a)\) = \((a)^{5}\cdot(-3a+1)\cdot(3a+1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 11552 \) = \(2^{5}\cdot19\cdot19\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-152)\) = \((a)^{6}\cdot(-3a+1)\cdot(3a+1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 23104 \) = \(2^{6}\cdot19\cdot19\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{27000}{19} \)
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: \(2\)
Generators $\left(1 : -a - 1 : 1\right)$ $\left(0 : 0 : 1\right)$
Heights \(0.225942131685295\) \(0.314996586909048\)
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: \( 2 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(2\)
Regulator: \( 0.0711710003198227 \)
Period: \( 5.96377584296626 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 2.40103986396434 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(III\) Additive \(-1\) \(5\) \(6\) \(0\)
\((-3a+1)\) \(19\) \(1\) \(I_{1}\) Split multiplicative \(-1\) \(1\) \(1\) \(1\)
\((3a+1)\) \(19\) \(1\) \(I_{1}\) 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 11552.2-a consists of this curve only.

Base change

This curve is the base change of 1216.a1, 608.a1, defined over \(\Q\), so it is also a \(\Q\)-curve.