Properties

Label 2.2.24.1-23.2-a2
Base field \(\Q(\sqrt{6}) \)
Conductor \(\left(2a + 1\right)\)
Conductor norm \( 23 \)
CM no
Base change no
Q-curve no
Torsion order \( 2 \)
Rank \( 0 \)

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Base field \(\Q(\sqrt{6}) \)

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

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

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \(\left(2a + 1\right)\) = \(\left(2a + 1\right)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 23 \) = \(23\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \(( 2 a + 1 )\) = \(\left(2a + 1\right)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 23 \) = \(23\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( \frac{66417408}{23} a + \frac{162689472}{23} \)
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: \(\Z/2\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generator: $\left(-\frac{1}{2} : -\frac{1}{4} a - \frac{1}{2} : 1\right)$
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: \( 10.3857798207435 \)
Tamagawa product: \( 1 \)
Torsion order: \(2\)
Leading coefficient: \( 2.11998842847632 \)
Analytic order of Ш: \( 4 \) (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\((j)_{-}\)
\(\left(2a + 1\right)\) \(23\) \(1\) \(I_{1}\) 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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2.
Its isogeny class 23.2-a consists of curves linked by isogenies of degree 2.

Base change

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