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Properties

Label 4.4.2304.1-1.1-a12
Base field \(\Q(\sqrt{2}, \sqrt{3})\)
Conductor norm \( 1 \)
CM yes (\(-8\))
Base change yes
Q-curve yes
Torsion order \( 12 \)
Rank \( 0 \)

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Show commands: Magma / PariGP / SageMath

Base field \(\Q(\sqrt{2}, \sqrt{3})\)

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

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

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((1)\) = \((1)\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 1 \) = 1
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((1)\) = \((1)\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 1 \) = 1
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 8000 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[\sqrt{-2}]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(0\)
Torsion structure: \(\Z/2\Z\oplus\Z/6\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-\frac{1}{2} a^{2} - a + \frac{1}{2} : \frac{1}{2} a + \frac{1}{2} : 1\right)$ $\left(-a^{3} + 5 a + 3 : -7 a^{3} - 6 a^{2} + 21 a + 12 : 1\right)$
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: \( 2576.5722942826273015649122433889662772 \)
Tamagawa product: \( 1 \)
Torsion order: \(12\)
Leading coefficient: \( 0.372767982390426 \)
Analytic order of Ш: \( 1 \) (rounded)

Local data at primes of bad reduction

sage: E.local_data()
 
magma: LocalInformation(E);
 
No primes of bad reduction.

Galois Representations

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

prime Image of Galois Representation
\(2\) 2Cs
\(3\) 3Cs.1.1

For all other primes \(p\), the image is the normalizer of a split Cartan subgroup if \(\left(\frac{ -2 }{p}\right)=+1\) or the normalizer of a nonsplit Cartan subgroup if \(\left(\frac{ -2 }{p}\right)=-1\).

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2, 3, 4, 6 and 12.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 72.

Base change

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

Base field Curve
\(\Q(\sqrt{6}) \) 2.2.24.1-1.1-a1
\(\Q(\sqrt{6}) \) 2.2.24.1-16.1-a2