# Properties

 Label 2.0.3.1-112896.2-a3 Base field $$\Q(\sqrt{-3})$$ Conductor $$\left(336\right)$$ Conductor norm $$112896$$ CM no Base change yes: 1008.i2,1008.c2 Q-curve yes Torsion order $$4$$ Rank $$2$$

# Learn more about

Show commands for: Magma / Pari/GP / 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(Pol(Vecrev([1, -1, 1])));

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

## Weierstrass equation

$$y^2=x^{3}-9ax-26$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([0,-9]),K([-26,0])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,-9])),Pol(Vecrev([-26,0]))], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![0,-9],K![-26,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$\left(336\right)$$ = $$\left(-2a + 1\right)^{2}\cdot\left(2\right)^{4}\cdot\left(-3a + 1\right)\cdot\left(3a - 2\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$112896$$ = $$3^{2}\cdot4^{4}\cdot7\cdot7$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$( -338688 )$$ = $$\left(-2a + 1\right)^{6}\cdot\left(2\right)^{8}\cdot\left(-3a + 1\right)^{2}\cdot\left(3a - 2\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$114709561344$$ = $$3^{6}\cdot4^{8}\cdot7^{2}\cdot7^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{11664}{49}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

## Mordell-Weil group

 Rank: $$2$$ Generators $\left(a - 3 : 6 a : 1\right)$ $\left(a + 1 : -6 a + 4 : 1\right)$ Heights $$0.437819058744267$$ $$0.437819058744267$$ 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: $\left(-3 a - 1 : 0 : 1\right)$ $\left(2 a - 2 : 0 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$2$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$2$$ Regulator: $$0.186964756300001$$ Period: $$1.66978647048161$$ Tamagawa product: $$64$$  =  $$2^{2}\cdot2^{2}\cdot2\cdot2$$ Torsion order: $$4$$ Leading coefficient: $$5.76779792671005$$ 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$$(j)_{-}$$
$$\left(-2a + 1\right)$$ $$3$$ $$4$$ $$I_{0}^*$$ Additive $$-1$$ $$2$$ $$6$$ $$0$$
$$\left(2\right)$$ $$4$$ $$4$$ $$I_{0}^*$$ Additive $$1$$ $$4$$ $$8$$ $$0$$
$$\left(-3a + 1\right)$$ $$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$\left(3a - 2\right)$$ $$7$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$

## 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 112896.2-a consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base change of elliptic curves 1008.i2, 1008.c2, defined over $$\Q$$, so it is also a $$\Q$$-curve.