# Properties

 Base field $$\Q(\sqrt{2})$$ Label 2.2.8.1-450.1-a2 Conductor $$(15 a)$$ Conductor norm $$450$$ CM no base-change yes: 30.a8,960.e8 Q-curve yes Torsion order $$6$$ 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

$$y^2 + x y + y = x^{3} + x + 2$$
magma: E := ChangeRing(EllipticCurve([1, 0, 1, 1, 2]),K);
sage: E = EllipticCurve(K, [1, 0, 1, 1, 2])
gp (2.8): E = ellinit([1, 0, 1, 1, 2],K)

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 $$\mathfrak{N}$$ = $$(15 a)$$ = $$\left(a\right) \cdot \left(3\right) \cdot \left(5\right)$$ magma: Conductor(E); sage: E.conductor() $$N(\mathfrak{N})$$ = $$450$$ = $$2 \cdot 9 \cdot 25$$ magma: Norm(Conductor(E)); sage: E.conductor().norm() $$\mathfrak{D}$$ = $$(2160)$$ = $$\left(a\right)^{8} \cdot \left(3\right)^{3} \cdot \left(5\right)$$ magma: Discriminant(E); sage: E.discriminant() gp (2.8): E.disc $$N(\mathfrak{D})$$ = $$4665600$$ = $$2^{8} \cdot 9^{3} \cdot 25$$ magma: Norm(Discriminant(E)); sage: E.discriminant().norm() gp (2.8): norm(E.disc) $$j$$ = $$\frac{357911}{2160}$$ 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: $$\Z/6\Z$$ 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] $\left(3 : 4 : 1\right)$ 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$2$$ $$I_{8}$$ Non-split multiplicative $$1$$ $$1$$ $$8$$ $$8$$
$$\left(3\right)$$ $$9$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$\left(5\right)$$ $$25$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p$$ 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, 4, 6 and 12.
Its isogeny class 450.1-a consists of curves linked by isogenies of degrees dividing 12.

## Base change

This curve is the base-change of elliptic curves 30.a8, 960.e8, defined over $$\Q$$, so it is also a $$\Q$$-curve.