Properties

 Label 2.2.8.1-32.1-a4 Base field $$\Q(\sqrt{2})$$ Conductor $$(4a)$$ Conductor norm $$32$$ CM yes ($$-4$$) Base change yes: 32.a3,64.a3 Q-curve yes Torsion order $$8$$ 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: 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

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

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

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

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(4a)$$ = $$(a)^{5}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$32$$ = $$2^{5}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(64)$$ = $$(a)^{12}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$4096$$ = $$2^{12}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$1728$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[\sqrt{-1}]$$ (potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/2\Z\times\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(0 : 0 : 1\right)$ $\left(-a + 1 : a - 2 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$27.5007432720815$$ Tamagawa product: $$4$$ Torsion order: $$8$$ Leading coefficient: $$0.607686314229973$$ 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 $$1$$ $$5$$ $$12$$ $$0$$

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 and 4.
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.a3, 64.a3, defined over $$\Q$$, so it is also a $$\Q$$-curve.