# Properties

 Base field $$\Q(\sqrt{-2})$$ Label 2.0.8.1-1458.4-e2 Conductor $$(27 a)$$ Conductor norm $$1458$$ CM no base-change yes: 1728.z2,54.b2 Q-curve yes Torsion order $$9$$ Rank $$0$$

# Learn more about

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

$$y^2 + x y + y = x^{3} - x^{2} - 14 x + 29$$
magma: E := ChangeRing(EllipticCurve([1, -1, 1, -14, 29]),K);

sage: E = EllipticCurve(K, [1, -1, 1, -14, 29])

gp (2.8): E = ellinit([1, -1, 1, -14, 29],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(27 a)$$ = $$\left(a\right) \cdot \left(-a - 1\right)^{3} \cdot \left(a - 1\right)^{3}$$ magma: Conductor(E);  sage: E.conductor() $$N(\mathfrak{N})$$ = $$1458$$ = $$2 \cdot 3^{6}$$ magma: Norm(Conductor(E));  sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(124416)$$ = $$\left(a\right)^{18} \cdot \left(-a - 1\right)^{5} \cdot \left(a - 1\right)^{5}$$ magma: Discriminant(E);  sage: E.discriminant()  gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$15479341056$$ = $$2^{18} \cdot 3^{10}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp (2.8): norm(E.disc) $$j$$ = $$-\frac{1167051}{512}$$ 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: $$\Z/9\Z$$ 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] $\left(-3 : -5 : 1\right)$ 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$18$$ $$I_{18}$$ Split multiplicative $$-1$$ $$1$$ $$18$$ $$18$$
$$\left(-a - 1\right)$$ $$3$$ $$3$$ $$IV$$ Additive $$1$$ $$3$$ $$5$$ $$0$$
$$\left(a - 1\right)$$ $$3$$ $$3$$ $$IV$$ Additive $$1$$ $$3$$ $$5$$ $$0$$

## Galois Representations

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

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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3 and 9.
Its isogeny class 1458.4-e consists of curves linked by isogenies of degrees dividing 9.

## Base change

This curve is the base-change of elliptic curves 1728.z2, 54.b2, defined over $$\Q$$, so it is also a $$\Q$$-curve.