# Properties

 Base field $$\Q(\sqrt{-11})$$ Label 2.0.11.1-32400.8-e1 Conductor $$(180)$$ Conductor norm $$32400$$ CM no base-change yes: 21780.q2,180.a2 Q-curve yes Torsion order $$6$$ Rank not available

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field $$\Q(\sqrt{-11})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x + 3$$; class number $$1$$.

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 3)

gp: K = nfinit(a^2 - a + 3);

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

## Weierstrass equation

$$y^2 = x^{3} - 327 x + 3454$$
sage: E = EllipticCurve(K, [0, 0, 0, -327, 3454])

gp: E = ellinit([0, 0, 0, -327, 3454],K)

magma: E := ChangeRing(EllipticCurve([0, 0, 0, -327, 3454]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(180)$$ = $$\left(2\right)^{2} \cdot \left(-a\right)^{2} \cdot \left(a - 1\right)^{2} \cdot \left(-a - 1\right) \cdot \left(a - 2\right)$$ sage: E.conductor()  magma: Conductor(E); $$N(\mathfrak{N})$$ = $$32400$$ = $$3^{4} \cdot 4^{2} \cdot 5^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); $$\mathfrak{D}$$ = $$(2916000000)$$ = $$\left(2\right)^{8} \cdot \left(-a\right)^{6} \cdot \left(a - 1\right)^{6} \cdot \left(-a - 1\right)^{6} \cdot \left(a - 2\right)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); $$N(\mathfrak{D})$$ = $$8503056000000000000$$ = $$3^{12} \cdot 4^{8} \cdot 5^{12}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); $$j$$ = $$-\frac{20720464}{15625}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## 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: $$\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); $\left(23 : 90 : 1\right)$ 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$$(j)_{-}$$
$$\left(-a\right)$$ $$3$$ $$2$$ $$I_{0}^*$$ Additive $$-1$$ $$2$$ $$6$$ $$0$$
$$\left(a - 1\right)$$ $$3$$ $$2$$ $$I_{0}^*$$ Additive $$-1$$ $$2$$ $$6$$ $$0$$
$$\left(-a - 1\right)$$ $$5$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(a - 2\right)$$ $$5$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$\left(2\right)$$ $$4$$ $$3$$ $$IV^*$$ Additive $$1$$ $$2$$ $$8$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B
$$3$$ 3B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 32400.8-e consists of curves linked by isogenies of degrees dividing 6.

## Base change

This curve is the base-change of elliptic curves 21780.q2, 180.a2, defined over $$\Q$$, so it is also a $$\Q$$-curve.