Properties

 Base field $$\Q(\sqrt{2})$$ Label 2.2.8.1-32.1-a6 Conductor $$(4 a)$$ Conductor norm $$32$$ CM yes ($$-16$$) base-change yes: 32.a2,64.a2 Q-curve yes Torsion order $$8$$ 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 (2.8): K = nfinit(a^2 - 2);

Weierstrass equation

$$y^2 + a x y + a y = x^{3} + x^{2} - 3 x$$
magma: E := ChangeRing(EllipticCurve([a, 1, a, -3, 0]),K);
sage: E = EllipticCurve(K, [a, 1, a, -3, 0])
gp (2.8): E = ellinit([a, 1, a, -3, 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() $$\mathfrak{D}$$ = $$(8)$$ = $$\left(a\right)^{6}$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$64$$ = $$2^{6}$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$287496$$ magma: jInvariant(E); sage: E.j_invariant() gp (2.8): E.j $$\text{End} (E)$$ = $$\Z[\sqrt{-4}]$$ ( 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: Generators(E); // includes torsion
sage: E.gens()
magma: Regulator(Generators(E));
sage: E.regulator_of_points(E.gens())

Torsion subgroup

Structure: $$\Z/2\Z\times\Z/4\Z$$ magma: TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E)); sage: E.torsion_order() gp (2.8): elltors(E)[1] $\left(0 : -a : 1\right)$,$\left(a - 1 : -1 : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E); sage: E.torsion_subgroup().gens() gp (2.8): elltors(E)[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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$2$$ $$III$$ Additive $$1$$ $$5$$ $$6$$ $$0$$

Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ 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, 4 and 8.
Its isogeny class 32.1-a consists of curves linked by isogenies of degrees dividing 16.

Base change

This curve is the base-change of elliptic curves 32.a2, 64.a2, defined over $$\Q$$, so it is also a $$\Q$$-curve.