# Properties

 Base field $$\Q(\sqrt{-2})$$ Label 2.0.8.1-9.3-CMa1 Conductor $$(-2 a - 1)$$ Conductor norm $$9$$ CM yes ($$-8$$) base-change no Q-curve yes Torsion order $$6$$ Rank $$0$$

# 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: x = polygen(QQ); K.<a> = NumberField(x^2 + 2)

gp: K = nfinit(a^2 + 2);

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

## Weierstrass equation

$$y^2 + a x y + y = x^{3} + \left(a + 1\right) x^{2} + \left(-a - 1\right) x$$
sage: E = EllipticCurve(K, [a, a + 1, 1, -a - 1, 0])

gp: E = ellinit([a, a + 1, 1, -a - 1, 0],K)

magma: E := ChangeRing(EllipticCurve([a, a + 1, 1, -a - 1, 0]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-2 a - 1)$$ = $$\left(a - 1\right)^{2}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$9$$ = $$3^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(10 a + 23)$$ = $$\left(a - 1\right)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$729$$ = $$3^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$8000$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z[\sqrt{-2}]$$ ( Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{U}(1)$

## Mordell-Weil group

Rank: $$0$$

sage: E.rank()

magma: Rank(E);

sage: gens = E.gens(); gens

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: $$\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(-a : a - 1 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## 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$$(j)_{-}$$
$$\left(a - 1\right)$$ $$3$$ $$2$$ $$I_{0}^*$$ Additive $$-1$$ $$2$$ $$6$$ $$0$$

## 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

For all other primes $$p$$, the image is a Borel subgroup if $$p=2$$, a split Cartan subgroup if $$\left(\frac{ -2 }{p}\right)=+1$$ or a nonsplit Cartan subgroup if $$\left(\frac{ -2 }{p}\right)=-1$$.

## Isogenies and isogeny class

This curve has no rational isogenies other than endomorphisms. Its isogeny class 9.3-CMa consists of this curve only.

## Base change

This curve is not the base-change of an elliptic curve defined over $$\Q$$. It is a $$\Q$$-curve.