# 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 $$1$$

# Related objects

Show commands for: Magma / Pari/GP / 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(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.