# Properties

 Label 2.0.3.1-126976.2-i1 Base field $$\Q(\sqrt{-3})$$ Conductor $$(384a-320)$$ Conductor norm $$126976$$ CM no Base change no Q-curve no Torsion order $$1$$ Rank $$0$$

# Related objects

Show commands: 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

$${y}^2={x}^{3}+\left(a+1\right){x}^{2}+13a{x}-6a-13$$
sage: E = EllipticCurve([K([0,0]),K([1,1]),K([0,0]),K([0,13]),K([-13,-6])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,1])),Pol(Vecrev([0,0])),Pol(Vecrev([0,13])),Pol(Vecrev([-13,-6]))], K);

magma: E := EllipticCurve([K![0,0],K![1,1],K![0,0],K![0,13],K![-13,-6]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(384a-320)$$ = $$(2)^{6}\cdot(6a-5)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$126976$$ = $$4^{6}\cdot31$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-196608a+163840)$$ = $$(2)^{15}\cdot(6a-5)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$33285996544$$ = $$4^{15}\cdot31$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{34992}{31} a - \frac{5832}{31}$$ 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: $$0$$ Torsion structure: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$1.81729055383045$$ Tamagawa product: $$2$$  =  $$2\cdot1$$ Torsion order: $$1$$ Leading coefficient: $$4.19685276179909$$ 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$$(j)_{-}$$
$$(2)$$ $$4$$ $$2$$ $$I_{5}^{*}$$ Additive $$-1$$ $$6$$ $$15$$ $$0$$
$$(6a-5)$$ $$31$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

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

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 126976.2-i consists of this curve only.

## Base change

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