# Properties

 Base field $$\Q(\sqrt{2})$$ Label 2.2.8.1-450.1-a3 Conductor $$(15 a)$$ Conductor norm $$450$$ CM no base-change yes: 30.a2,960.e2 Q-curve yes Torsion order $$4$$ 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: K = nfinit(a^2 - 2);

## Weierstrass equation

$$y^2 + x y + y = x^{3} - 454 x - 544$$
magma: E := ChangeRing(EllipticCurve([1, 0, 1, -454, -544]),K);

sage: E = EllipticCurve(K, [1, 0, 1, -454, -544])

gp: E = ellinit([1, 0, 1, -454, -544],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() $$\mathfrak{D}$$ = $$(5859375000)$$ = $$\left(a\right)^{6} \cdot \left(3\right) \cdot \left(5\right)^{12}$$ magma: Discriminant(E);  sage: E.discriminant()  gp: E.disc $$N(\mathfrak{D})$$ = $$34332275390625000000$$ = $$2^{6} \cdot 9 \cdot 25^{12}$$ magma: Norm(Discriminant(E));  sage: E.discriminant().norm()  gp: norm(E.disc) $$j$$ = $$\frac{10316097499609}{5859375000}$$ magma: jInvariant(E);  sage: E.j_invariant()  gp: 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: 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: $$\Z/4\Z$$ magma: T,piT := TorsionSubgroup(E); Invariants(T);  sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2] $\left(53 : -250 a - 27 : 1\right)$ 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$
$$\left(3\right)$$ $$9$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$
$$\left(5\right)$$ $$25$$ $$12$$ $$I_{12}$$ Split multiplicative $$-1$$ $$1$$ $$12$$ $$12$$

## 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.1.2

## 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.a2, 960.e2, defined over $$\Q$$, so it is also a $$\Q$$-curve.