# Properties

 Base field $$\Q(\sqrt{3})$$ Label 2.2.12.1-1024.1-k6 Conductor $$(32)$$ Conductor norm $$1024$$ CM yes ($$-12$$) base-change no Q-curve yes Torsion order $$4$$ Rank not available

# Related objects

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

$$y^2 = x^{3} + a x^{2} + \left(10 a - 19\right) x - 31 a + 54$$
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

 $$\mathfrak{N}$$ = $$(32)$$ = $$\left(a + 1\right)^{10}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$1024$$ = $$2^{10}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(2048)$$ = $$\left(a + 1\right)^{22}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$4194304$$ = $$2^{22}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$54000$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z[\sqrt{-3}]$$ ( Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $N(\mathrm{U}(1))$

## Mordell-Weil group

Rank not available.

sage: E.rank()

magma: Rank(E);

Regulator: not available

sage: gens = E.gens(); gens

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

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); $\left(-2 a + 1 : 0 : 1\right)$,$\left(-a + 2 : 0 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## 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-k 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.