# Properties

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

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

gp: K = nfinit(a^3 - 3*a - 1);

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

## Weierstrass equation

$$y^2 + \left(a + 1\right) y = x^{3} + \left(a^{2} - 1\right) x^{2} + \left(17 a^{2} - 3 a - 53\right) x - 56 a^{2} + 17 a + 164$$
sage: E = EllipticCurve(K, [0, a^2 - 1, a + 1, 17*a^2 - 3*a - 53, -56*a^2 + 17*a + 164])

gp: E = ellinit([0, a^2 - 1, a + 1, 17*a^2 - 3*a - 53, -56*a^2 + 17*a + 164],K)

magma: E := ChangeRing(EllipticCurve([0, a^2 - 1, a + 1, 17*a^2 - 3*a - 53, -56*a^2 + 17*a + 164]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(3,3)$$ = $$\left(-a^{2} + 1\right)^{3}$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$27$$ = $$3^{3}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(3,3 a,3 a^{2} - 6)$$ = $$\left(-a^{2} + 1\right)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$27$$ = $$3^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-12288000$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z[(1+\sqrt{-27})/2]$$ ( Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $N(\mathrm{U}(1))$

## Mordell-Weil group

Rank not available.

sage: E.rank()

magma: Rank(E);

Regulator: not available

sage: gens = E.gens(); gens

magma: gens := [P:P in Generators(E)|Order(P) eq 0]; gens;

sage: E.regulator_of_points(gens)

magma: Regulator(gens);

## Torsion subgroup

Structure: $$\Z/9\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(a^{2} - a - 2 : 6 a^{2} - 3 a - 17 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## 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$$(j)_{-}$$
$$\left(-a^{2} + 1\right)$$ $$3$$ $$1$$ $$II$$ Additive $$-1$$ $$3$$ $$3$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3B.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, 9 and 27.
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.a2, defined over $$\Q$$, so it is also a $$\Q$$-curve.