# Properties

 Label 2.0.3.1-77841.1-CMb1 Base field $$\Q(\sqrt{-3})$$ Conductor $$(-315a+99)$$ Conductor norm $$77841$$ CM yes ($$-3$$) Base change no Q-curve yes Torsion order $$1$$ Rank $$0$$

# Related objects

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

$${y}^2+{y}={x}^{3}-3a+8$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([1,0]),K([0,0]),K([8,-3])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([8,-3]))], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![1,0],K![0,0],K![8,-3]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-315a+99)$$ = $$(-2a+1)^{4}\cdot(-6a+1)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$77841$$ = $$3^{4}\cdot31^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(17496a-25515)$$ = $$(-2a+1)^{12}\cdot(-6a+1)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$510714801$$ = $$3^{12}\cdot31^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$0$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z[(1+\sqrt{-3})/2]$$ (complex multiplication) Geometric endomorphism ring: $$\Z[(1+\sqrt{-3})/2]$$ sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{U}(1)$

## 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: $$2.64135321544326$$ Tamagawa product: $$1$$  =  $$1\cdot1$$ Torsion order: $$1$$ Leading coefficient: $$3.04997197992210$$ 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$$(j)_{-}$$
$$(-2a+1)$$ $$3$$ $$1$$ $$II^{*}$$ Additive $$1$$ $$4$$ $$12$$ $$0$$
$$(-6a+1)$$ $$31$$ $$1$$ $$II$$ Additive $$-1$$ $$2$$ $$2$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$31$$ 31Cs.7.1

For all other primes $$p$$, the image is a Borel subgroup if $$p=3$$, a split Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=+1$$ or a nonsplit Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=-1$$.

## Isogenies and isogeny class

This curve has no rational isogenies other than endomorphisms. Its isogeny class 77841.1-CMb consists of this curve only.

## Base change

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