# Properties

 Label 2.0.4.1-64.1-CMa2 Base field $$\Q(\sqrt{-1})$$ Conductor norm $$64$$ CM yes ($$-16$$) Base change yes Q-curve yes Torsion order $$4$$ Rank $$0$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{-1})$$

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))

gp: K = nfinit(Pol(Vecrev([1, 0, 1])));

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

## Weierstrass equation

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

gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([2,0])),Pol(Vecrev([0,3]))], K);

magma: E := EllipticCurve([K![1,1],K![0,1],K![0,0],K![2,0],K![0,3]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(8)$$ = $$(i+1)^{6}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$64$$ = $$2^{6}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-8)$$ = $$(i+1)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$64$$ = $$2^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$287496$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z[\sqrt{-4}]$$ (complex multiplication) Geometric endomorphism ring: $$\Z[\sqrt{-4}]$$ sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{U}(1)$

## Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-i : -1 : 1\right)$ 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: $$6.87518581802037$$ Tamagawa product: $$1$$ Torsion order: $$4$$ Leading coefficient: $$0.429699113626273$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$1$$ $$II$$ Additive $$-1$$ $$6$$ $$6$$ $$0$$

## Galois Representations

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

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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies (excluding endomorphisms) of degree $$d$$ for $$d=$$ 2.
Its isogeny class 64.1-CMa consists of curves linked by isogenies of degree 2.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 32.a1
$$\Q$$ 32.a2