Properties

Label 2.0.8.1-23328.4-s1
Base field \(\Q(\sqrt{-2}) \)
Conductor norm \( 23328 \)
CM no
Base change yes
Q-curve yes
Torsion order \( 1 \)
Rank \( 1 \)

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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(Polrev([2, 0, 1]));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);
 

Weierstrass equation

\({y}^2={x}^{3}-6{x}+6\)
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([-6,0]),K([6,0])])
 
gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-6,0]),Polrev([6,0])], K);
 
magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![-6,0],K![6,0]]);
 

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((108a)\) = \((a)^{5}\cdot(-a-1)^{3}\cdot(a-1)^{3}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 23328 \) = \(2^{5}\cdot3^{3}\cdot3^{3}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((-1728)\) = \((a)^{12}\cdot(-a-1)^{3}\cdot(a-1)^{3}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 2985984 \) = \(2^{12}\cdot3^{3}\cdot3^{3}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( -13824 \)
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 $\left(1 : -1 : 1\right)$
Height \(0.55503222229670979713879781924606427697\)
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: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.55503222229670979713879781924606427697 \)
Period: \( 3.5196330480059061612594250365281087728 \)
Tamagawa product: \( 2 \)  =  \(2\cdot1\cdot1\)
Torsion order: \(1\)
Leading coefficient: \( 5.5253599718718841175871584510991361532 \)
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\((j)_{-}\)
\((a)\) \(2\) \(2\) \(I_{3}^{*}\) Additive \(1\) \(5\) \(12\) \(0\)
\((-a-1)\) \(3\) \(1\) \(II\) Additive \(-1\) \(3\) \(3\) \(0\)
\((a-1)\) \(3\) \(1\) \(II\) Additive \(-1\) \(3\) \(3\) \(0\)

Galois Representations

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

prime Image of Galois Representation
\(3\) 3Nn

Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 23328.4-s consists of this curve only.

Base change

This elliptic curve is a \(\Q\)-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
\(\Q\) 864.l1
\(\Q\) 1728.e1