# Properties

 Label 2.0.8.1-23328.4-j1 Base field $$\Q(\sqrt{-2})$$ Conductor $$(108a)$$ Conductor norm $$23328$$ CM no Base change yes: 1728.g1,864.j1 Q-curve yes Torsion order $$1$$ Rank $$0$$

# Related objects

Show commands: 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

$${y}^2={x}^{3}+54{x}+54$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([54,0]),K([54,0])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([54,0])),Pol(Vecrev([54,0]))], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![54,0],K![54,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(108a)$$ = $$(a)^{5}\cdot(-a-1)^{3}\cdot(a-1)^{3}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$23328$$ = $$2^{5}\cdot3^{3}\cdot3^{3}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-11337408)$$ = $$(a)^{12}\cdot(-a-1)^{11}\cdot(a-1)^{11}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$128536820158464$$ = $$2^{12}\cdot3^{11}\cdot3^{11}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$1536$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

## Mordell-Weil group

 Rank: $$0$$ Torsion structure: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$0.918503032239555$$ Tamagawa product: $$4$$  =  $$2^{2}\cdot1\cdot1$$ Torsion order: $$1$$ Leading coefficient: $$2.59791889054798$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$4$$ $$I_3^{*}$$ Additive $$-1$$ $$5$$ $$12$$ $$0$$
$$(-a-1)$$ $$3$$ $$1$$ $$II^{*}$$ Additive $$-1$$ $$3$$ $$11$$ $$0$$
$$(a-1)$$ $$3$$ $$1$$ $$II^{*}$$ Additive $$-1$$ $$3$$ $$11$$ $$0$$

## Galois Representations

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

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 23328.4-j consists of this curve only.

## Base change

This curve is the base change of 1728.g1, 864.j1, defined over $$\Q$$, so it is also a $$\Q$$-curve.