# Properties

 Base field $$\Q(\zeta_{9})^+$$ Label 3.3.81.1-27.1-a2 Conductor $$(3,3)$$ Conductor norm $$27$$ CM yes ($$-3$$) base-change yes: 27.a3 Q-curve yes Torsion order $$3$$ Rank not available

# Related objects

Show commands for: Magma / SageMath / Pari/GP

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

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

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

sage: x = polygen(QQ); K.<a> = NumberField(x^3 - 3*x - 1)

gp (2.8): K = nfinit(a^3 - 3*a - 1);

## Weierstrass equation

$$y^2 + \left(a^{2} - 1\right) y = x^{3} + \left(a^{2} + a - 2\right) x^{2} + \left(a + 2\right) x + 12 a^{2} - 4 a - 35$$
magma: E := ChangeRing(EllipticCurve([0, a^2 + a - 2, a^2 - 1, a + 2, 12*a^2 - 4*a - 35]),K);

sage: E = EllipticCurve(K, [0, a^2 + a - 2, a^2 - 1, a + 2, 12*a^2 - 4*a - 35])

gp (2.8): E = ellinit([0, a^2 + a - 2, a^2 - 1, a + 2, 12*a^2 - 4*a - 35],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(3,3)$$ = $$\left(-a^{2} + 1\right)^{3}$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$27$$ = $$3^{3}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(3,3 a,3 a^{2} - 6)$$ = $$\left(-a^{2} + 1\right)^{3}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$27$$ = $$3^{3}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$0$$ magma: jInvariant(E);  sage: E.j_invariant()  gp (2.8): E.j $$\text{End} (E)$$ = $$\Z[(1+\sqrt{-3})/2]$$ ( Complex Multiplication ) magma: HasComplexMultiplication(E);  sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $N(\mathrm{U}(1))$

## Mordell-Weil group

Rank not available.
magma: Rank(E);

sage: E.rank()

magma: Generators(E); // includes torsion

sage: E.gens()

Regulator: not available

magma: Regulator(Generators(E));

sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $$\Z/3\Z$$ magma: TorsionSubgroup(E);  sage: E.torsion_subgroup().gens()  gp (2.8): elltors(E)[2] magma: Order(TorsionSubgroup(E));  sage: E.torsion_order()  gp (2.8): elltors(E)[1] $\left(-2 a^{2} + 6 : 3 a^{2} - a - 10 : 1\right)$ magma: [f(P): P in Generators(T)] where T,f:=TorsionSubgroup(E);  sage: E.torsion_subgroup().gens()  gp (2.8): elltors(E)[3]

## Local data at primes of bad reduction

magma: LocalInformation(E);

sage: E.local_data()

prime Norm Tamagawa number Kodaira symbol Reduction type Root number ord($$\mathfrak{N}$$) ord($$\mathfrak{D}$$) ord$$(j)_{-}$$
$$\left(-a^{2} + 1\right)$$ $$3$$ $$1$$ $$II$$ Additive $$-1$$ $$3$$ $$3$$ $$0$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ 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.a3, defined over $$\Q$$, so it is also a $$\Q$$-curve.