# Properties

 Label 2.0.8.1-1024.1-b2 Base field $$\Q(\sqrt{-2})$$ Conductor $$(32)$$ Conductor norm $$1024$$ CM yes ($$-4$$) Base change yes: 256.c2,256.b2 Q-curve yes Torsion order $$4$$ Rank $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## 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(Pol(Vecrev([2, 0, 1])));

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

## Weierstrass equation

$${y}^2={x}^{3}+2{x}$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([2,0]),K([0,0])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([2,0])),Pol(Vecrev([0,0]))], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![2,0],K![0,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(32)$$ = $$(a)^{10}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$1024$$ = $$2^{10}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-512)$$ = $$(a)^{18}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$262144$$ = $$2^{18}$$ 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 $\left(-4 : 6 a : 1\right)$ Height $$1.21741806395396$$ Torsion structure: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-a : 0 : 1\right)$ $\left(0 : 0 : 1\right)$ 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: $$1.21741806395396$$ Period: $$4.86149051383978$$ Tamagawa product: $$4$$ Torsion order: $$4$$ Leading coefficient: $$2.09249385197453$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$4$$ $$I_4^{*}$$ Additive $$1$$ $$10$$ $$18$$ $$0$$

## 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.
Its isogeny class 1024.1-b consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base change of elliptic curves 256.c2, 256.b2, defined over $$\Q$$, so it is also a $$\Q$$-curve.