# Properties

 Label 2.0.8.1-2592.3-e1 Base field $$\Q(\sqrt{-2})$$ Conductor norm $$2592$$ CM yes ($$-4$$) Base change yes Q-curve yes Torsion order $$2$$ Rank $$0$$

# Related objects

Show commands: Magma / PariGP / 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(Polrev([2, 0, 1]));

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

## Weierstrass equation

$${y}^2={x}^{3}-27{x}$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([-27,0]),K([0,0])])

gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([-27,0]),Polrev([0,0])], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![-27,0],K![0,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(36a)$$ = $$(a)^{5}\cdot(-a-1)^{2}\cdot(a-1)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$2592$$ = $$2^{5}\cdot3^{2}\cdot3^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(1259712)$$ = $$(a)^{12}\cdot(-a-1)^{9}\cdot(a-1)^{9}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1586874322944$$ = $$2^{12}\cdot3^{9}\cdot3^{9}$$ 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: $$0$$ 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: $\left(0 : 0 : 1\right)$ 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: $$1.3231301275875865554387560110854884008$$ Tamagawa product: $$8$$  =  $$2\cdot2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$1.8711885712188085490736288453408046822$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$2$$ $$III^{*}$$ Additive $$-1$$ $$5$$ $$12$$ $$0$$
$$(-a-1)$$ $$3$$ $$2$$ $$III^{*}$$ Additive $$1$$ $$2$$ $$9$$ $$0$$
$$(a-1)$$ $$3$$ $$2$$ $$III^{*}$$ Additive $$1$$ $$2$$ $$9$$ $$0$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ .

The image is a Borel subgroup if $$p=2$$, 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 2592.3-e consists of curves linked by isogenies of degree 2.

## Base change

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

Base field Curve
$$\Q$$ 288.e1
$$\Q$$ 576.a1