# Properties

 Label 3.3.81.1-27.1-a4 Base field $$\Q(\zeta_{9})^+$$ Conductor $$(3)$$ Conductor norm $$27$$ CM yes ($$-3$$) Base change yes: 27.a4 Q-curve yes Torsion order $$9$$ Rank $$0$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\zeta_{9})^+$$

Generator $$a$$, with minimal polynomial $$x^{3} - 3 x - 1$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-1, -3, 0, 1]))

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -3, 0, 1]);

## Weierstrass equation

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

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

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

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(3)$$ = $$(-a^2+1)^{3}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$27$$ = $$3^{3}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-27)$$ = $$(-a^2+1)^{9}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-19683$$ = $$3^{9}$$ 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: $$\Z$$ Geometric endomorphism ring: $$\Z[(1+\sqrt{-3})/2]$$ (potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$

## Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/9\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(a + 1 : a^{2} + a - 1 : 1\right)$ sage: T.gens()  gp: T[3]  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: $$148.869942571185$$ Tamagawa product: $$3$$ Torsion order: $$9$$ Leading coefficient: $$0.612633508523396$$ 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$$(j)_{-}$$
$$(-a^2+1)$$ $$3$$ $$3$$ $$IV^{*}$$ Additive $$-1$$ $$3$$ $$9$$ $$0$$

## 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 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 $$d=$$ 3 and 9.
Its isogeny class 27.1-a consists of curves linked by isogenies of degrees dividing 27.

## Base change

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