# Properties

 Label 2.0.3.1-81.1-CMa1 Base field $$\Q(\sqrt{-3})$$ Conductor $$(9)$$ Conductor norm $$81$$ CM yes ($$-3$$) Base change yes: 27.a4,27.a3 Q-curve yes Torsion order $$9$$ 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: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 1)

gp: K = nfinit(a^2 - a + 1);

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

## Weierstrass equation

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

gp: E = ellinit([0, 0, 1, 0, 0],K)

magma: E := ChangeRing(EllipticCurve([0, 0, 1, 0, 0]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(9)$$ = $$\left(-2 a + 1\right)^{4}$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$81$$ = $$3^{4}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(27)$$ = $$\left(-2 a + 1\right)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$729$$ = $$3^{6}$$ 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: $$\Z/3\Z\times\Z/3\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-1 : -a : 1\right)$ $\left(0 : -1 : 1\right)$ sage: T.gens()  gp: T  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$8.10862826406269$$ Tamagawa product: $$3$$ Torsion order: $$9$$ Leading coefficient: $$0.346779163778904$$ 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$$(j)_{-}$$
$$\left(-2 a + 1\right)$$ $$3$$ $$3$$ $$IV$$ Additive $$-1$$ $$4$$ $$6$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3Cs.1.1

For all other primes $$p$$, the image is 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 non-trivial cyclic isogenies (excluding endomorphisms) of degree $$d$$ for $$d=$$ 3.
Its isogeny class 81.1-CMa consists of curves linked by isogenies of degree 3.

## Base change

This curve is the base change of elliptic curves 27.a4, 27.a3, defined over $$\Q$$, so it is also a $$\Q$$-curve.

This is first curve (with respect to norm of the conductor) over $\Q(\sqrt{-3})$ whose torsion subgroup is $C_3 \times C_3$.