# Properties

 Base field $$\Q(\sqrt{-2})$$ Label 2.0.8.1-108.2-a5 Conductor $$(-6 a - 6)$$ Conductor norm $$108$$ CM no base-change no Q-curve yes Torsion order $$12$$ Rank $$0$$

# Related objects

Show commands for: Magma / SageMath / Pari/GP

## Base field $$\Q(\sqrt{-2})$$

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

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2, 0, 1]);

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

gp (2.8): K = nfinit(a^2 + 2);

## Weierstrass equation

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

sage: E = EllipticCurve(K, [a, -a + 1, a, -3*a + 4, -2*a + 4])

gp (2.8): E = ellinit([a, -a + 1, a, -3*a + 4, -2*a + 4],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(-6 a - 6)$$ = $$\left(a\right)^{2} \cdot \left(-a - 1\right)^{2} \cdot \left(a - 1\right)$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$108$$ = $$2^{2} \cdot 3^{3}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(5832 a - 2916)$$ = $$\left(a\right)^{4} \cdot \left(-a - 1\right)^{8} \cdot \left(a - 1\right)^{6}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$76527504$$ = $$2^{4} \cdot 3^{14}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$-\frac{855712}{729} a + \frac{467888}{729}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp (2.8): E.j $$\text{End} (E)$$ = $$\Z$$ (no Complex Multiplication ) magma: HasComplexMultiplication(E);  sage: E.has_cm(), E.cm_discriminant() $$\text{ST} (E)$$ = $\mathrm{SU}(2)$

## Mordell-Weil group

Rank: $$0$$
magma: Rank(E);

sage: E.rank()

magma: Generators(E); // includes torsion

sage: E.gens()

magma: Regulator(Generators(E));

sage: E.regulator_of_points(E.gens())

## Torsion subgroup

Structure: $$\Z/2\Z\times\Z/6\Z$$ 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] $\left(-a + 4 : -7 a + 8 : 1\right)$,$\left(-a - \frac{1}{2} : -\frac{1}{4} a - 1 : 1\right)$ 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$3$$ $$IV$$ Additive $$-1$$ $$2$$ $$4$$ $$0$$
$$\left(-a - 1\right)$$ $$3$$ $$4$$ $$I_{2}^*$$ Additive $$-1$$ $$2$$ $$8$$ $$2$$
$$\left(a - 1\right)$$ $$3$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs
$$3$$ 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 108.2-a consists of curves linked by isogenies of degrees dividing 12.

## Base change

This curve is not the base-change of an elliptic curve defined over $$\Q$$. It is a $$\Q$$-curve.