# 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 / 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: K = nfinit(a^3 - 3*a - 1);

## Weierstrass equation

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

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)

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: E.disc $$N(\mathfrak{D})$$ = $$27$$ = $$3^{3}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$-12288000$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: E.j $$\text{End} (E)$$ = $$\Z[(1+\sqrt{-27})/2]$$ ( 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()

Regulator: not available

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

sage: gens = E.gens(); gens

magma: Regulator(gens);

sage: E.regulator_of_points(gens)

## Torsion subgroup

Structure: $$\Z/9\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(a^{2} - a - 2 : 6 a^{2} - 3 a - 17 : 1\right)$ magma: [piT(P) : P in Generators(T)];  sage: T.gens()  gp: T[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 $$p < 1000$$ 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.