# Properties

 Label 2.0.8.1-1458.4-b3 Base field $$\Q(\sqrt{-2})$$ Conductor $$(27a)$$ Conductor norm $$1458$$ CM no Base change yes: 54.a3,1728.d3 Q-curve yes Torsion order $$3$$ Rank $$1$$

# 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

$${y}^2+{x}{y}={x}^{3}-{x}^{2}+12{x}+8$$
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([0,0]),K([12,0]),K([8,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([12,0])),Pol(Vecrev([8,0]))], K);

magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,0],K![12,0],K![8,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(27a)$$ = $$(a)\cdot(-a-1)^{3}\cdot(a-1)^{3}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$1458$$ = $$2\cdot3^{3}\cdot3^{3}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-157464)$$ = $$(a)^{6}\cdot(-a-1)^{9}\cdot(a-1)^{9}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$24794911296$$ = $$2^{6}\cdot3^{9}\cdot3^{9}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{9261}{8}$$ 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: $$1$$ Generator $\left(-8 : -18 a + 4 : 1\right)$ Height $$0.338560565002408$$ Torsion structure: $$\Z/3\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(1 : -5 : 1\right)$ 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: $$0.338560565002408$$ Period: $$1.87837840894579$$ Tamagawa product: $$18$$  =  $$2\cdot3\cdot3$$ Torsion order: $$3$$ Leading coefficient: $$1.79872367891558$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$
$$(-a-1)$$ $$3$$ $$3$$ $$IV^{*}$$ Additive $$-1$$ $$3$$ $$9$$ $$0$$
$$(a-1)$$ $$3$$ $$3$$ $$IV^{*}$$ Additive $$-1$$ $$3$$ $$9$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3Cs.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 1458.4-b consists of curves linked by isogenies of degrees dividing 9.

## Base change

This curve is the base change of 54.a3, 1728.d3, defined over $$\Q$$, so it is also a $$\Q$$-curve.