# Properties

 Label 2.0.3.1-4096.1-c2 Base field $$\Q(\sqrt{-3})$$ Conductor norm $$4096$$ CM yes ($$-4$$) Base change yes Q-curve yes Torsion order $$4$$ Rank $$1$$

# Related objects

Show commands: Magma / PariGP / SageMath

## Base field$$\Q(\sqrt{-3})$$

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

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

gp: K = nfinit(Polrev([1, -1, 1]));

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

## Weierstrass equation

$${y}^2={x}^{3}+\left(-4a+4\right){x}$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([4,-4]),K([0,0])])

gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,0]),Polrev([4,-4]),Polrev([0,0])], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![4,-4],K![0,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(64)$$ = $$(2)^{6}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$4096$$ = $$4^{6}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(4096)$$ = $$(2)^{12}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$16777216$$ = $$4^{12}$$ 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: $$1$$ Generator $\left(-a : -2 a + 1 : 1\right)$ Height $$0.88862587483961923979418733498068928657$$ Torsion structure: $$\Z/2\Z\oplus\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-2 a : 0 : 1\right)$ $\left(0 : 0 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.88862587483961923979418733498068928657$$ Period: $$3.4375929090101864137450478899052785989$$ Tamagawa product: $$4$$ Torsion order: $$4$$ Leading coefficient: $$1.7636515007312641893769342998137403163$$ 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$$(j)_{-}$$
$$(2)$$ $$4$$ $$4$$ $$I_{2}^{*}$$ Additive $$1$$ $$6$$ $$12$$ $$0$$

## 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 4096.1-c consists of curves linked by isogenies of degrees dividing 4.

## Base change

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

Base field Curve
$$\Q$$ 64.a3
$$\Q$$ 576.c3