# Properties

 Base field $$\Q(\sqrt{-2})$$ Label 2.0.8.1-32.1-a1 Conductor $$(4 a)$$ Conductor norm $$32$$ CM yes ($$-4$$) base-change yes: 32.a3,64.a3 Q-curve yes Torsion order $$4$$ Rank $$0$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{-2})$$

Generator $$a$$, with minimal polynomial $$x^{2} + 2$$; class number $$1$$.

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

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

gp: K = nfinit(a^2 + 2);

## Weierstrass equation

$$y^2 = x^{3} - x$$
magma: E := ChangeRing(EllipticCurve([0, 0, 0, -1, 0]),K);

sage: E = EllipticCurve(K, [0, 0, 0, -1, 0])

gp: E = ellinit([0, 0, 0, -1, 0],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(4 a)$$ = $$\left(a\right)^{5}$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$32$$ = $$2^{5}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(64)$$ = $$\left(a\right)^{12}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$4096$$ = $$2^{12}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$1728$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: E.j $$\text{End} (E)$$ = $$\Z[\sqrt{-1}]$$ ( Complex Multiplication ) magma: HasComplexMultiplication(E);  sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $N(\mathrm{U}(1))$

## Mordell-Weil group

Rank: $$0$$

magma: Rank(E);

sage: E.rank()

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

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/2\Z\times\Z/2\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(-1 : 0 : 1\right)$,$\left(0 : 0 : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[3]

## Local data at primes of bad reduction

magma: LocalInformation(E);

sage: E.local_data()

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$2$$ $$I_{3}^*$$ Additive $$-1$$ $$5$$ $$12$$ $$0$$

## 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 32.1-a consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base-change of elliptic curves 32.a3, 64.a3, defined over $$\Q$$, so it is also a $$\Q$$-curve.