Properties

Label 4.4.2304.1-64.1-b2
Base field \(\Q(\sqrt{2}, \sqrt{3})\)
Conductor \((2a^3-6a)\)
Conductor norm \( 64 \)
CM yes (\(-4\))
Base change yes: 32.a3,288.d3,64.a3,576.c3
Q-curve yes
Torsion order \( 8 \)
Rank \( 1 \)

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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(Pol(Vecrev([1, 0, -4, 0, 1])));
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, -4, 0, 1]);
 

Weierstrass equation

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

This is a global minimal model.

sage: E.is_global_minimal_model()
 

Invariants

Conductor: \((2a^3-6a)\) = \((a^3-4a+1)^{6}\)
sage: E.conductor()
 
gp: ellglobalred(E)[1]
 
magma: Conductor(E);
 
Conductor norm: \( 64 \) = \(2^{6}\)
sage: E.conductor().norm()
 
gp: idealnorm(ellglobalred(E)[1])
 
magma: Norm(Conductor(E));
 
Discriminant: \((8)\) = \((a^3-4a+1)^{12}\)
sage: E.discriminant()
 
gp: E.disc
 
magma: Discriminant(E);
 
Discriminant norm: \( 4096 \) = \(2^{12}\)
sage: E.discriminant().norm()
 
gp: norm(E.disc)
 
magma: Norm(Discriminant(E));
 
j-invariant: \( 1728 \)
sage: E.j_invariant()
 
gp: E.j
 
magma: jInvariant(E);
 
Endomorphism ring: \(\Z\)
Geometric endomorphism ring: \(\Z[\sqrt{-1}]\) (potential complex multiplication)
sage: E.has_cm(), E.cm_discriminant()
 
magma: HasComplexMultiplication(E);
 
Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

Rank: \(1\)
Generator $\left(4 a^{3} + 7 a^{2} - 2 a - 2 : -34 a^{3} - 66 a^{2} + 8 a + 17 : 1\right)$
Height \(0.222156468709905\)
Torsion structure: \(\Z/2\Z\times\Z/4\Z\)
sage: T = E.torsion_subgroup(); T.invariants()
 
gp: T = elltors(E); T[2]
 
magma: T,piT := TorsionSubgroup(E); Invariants(T);
 
Torsion generators: $\left(-\frac{1}{2} a^{2} + \frac{1}{2} : \frac{1}{2} a^{2} + \frac{1}{2} a : 1\right)$ $\left(a^{3} + a^{2} - a : a^{3} + 2 a^{2} - a - 1 : 1\right)$
sage: T.gens()
 
gp: T[3]
 
magma: [piT(P) : P in Generators(T)];
 

BSD invariants

Analytic rank: \( 1 \)
sage: E.rank()
 
magma: Rank(E);
 
Mordell-Weil rank: \(1\)
Regulator: \( 0.222156468709905 \)
Period: \( 1512.58176103387 \)
Tamagawa product: \( 4 \)
Torsion order: \(8\)
Leading coefficient: \( 1.75015532638695 \)
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\((j)_{-}\)
\((a^3-4a+1)\) \(2\) \(4\) \(I_2^{*}\) Additive \(-1\) \(6\) \(12\) \(0\)

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

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

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree \(d\) for \(d=\) 2 and 4.
Its isogeny class 64.1-b consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base change of elliptic curves 32.a3, 288.d3, 64.a3, 576.c3, defined over \(\Q\), so it is also a \(\Q\)-curve.