# Properties

 Label 2.2.12.1-1024.1-j5 Base field $$\Q(\sqrt{3})$$ Conductor $$(32)$$ Conductor norm $$1024$$ CM yes ($$-12$$) Base change no Q-curve yes Torsion order $$4$$ Rank $$1$$

# Learn more about

Show commands for: Magma / Pari/GP / SageMath

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

Generator $$a$$, with minimal polynomial $$x^{2} - 3$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - 3)

gp: K = nfinit(a^2 - 3);

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);

## Weierstrass equation

$$y^2=x^{3}-ax^{2}+\left(-10a-19\right)x+31a+54$$
sage: E = EllipticCurve(K, [0, -a, 0, -10*a - 19, 31*a + 54])

gp: E = ellinit([0, -a, 0, -10*a - 19, 31*a + 54],K)

magma: E := ChangeRing(EllipticCurve([0, -a, 0, -10*a - 19, 31*a + 54]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(32)$$ = $$\left(a + 1\right)^{10}$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$1024$$ = $$2^{10}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(2048)$$ = $$\left(a + 1\right)^{22}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$4194304$$ = $$2^{22}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$54000$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[\sqrt{-3}]$$ (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 $\left(a + 3 : -a : 1\right)$ Height $$0.859914975775406$$ Torsion structure: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-2 a - 3 : 0 : 1\right)$ $\left(a + 2 : 0 : 1\right)$ 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: $$0.859914975775406$$ Period: $$21.6718995789899$$ Tamagawa product: $$4$$ Torsion order: $$4$$ Leading coefficient: $$2.68987360532914$$ 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$$(j)_{-}$$
$$\left(a + 1\right)$$ $$2$$ $$4$$ $$I_{8}^*$$ Additive $$-1$$ $$10$$ $$22$$ $$0$$

## 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 a Borel subgroup if $$p=3$$, the normalizer of a split Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=-1$$.

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 1024.1-j consists of curves linked by isogenies of degrees dividing 12.

## Base change

This curve is not the base change of an elliptic curve defined over $$\Q$$. It is a $$\Q$$-curve.