# Properties

 Label 2.0.8.1-288.2-a3 Base field $$\Q(\sqrt{-2})$$ Conductor $$(12 a)$$ Conductor norm $$288$$ CM no Base change yes: 192.a4,96.a4 Q-curve yes Torsion order $$4$$ Rank $$1$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

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

gp: K = nfinit(a^2 + 2);

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

## Weierstrass equation

$$y^2+axy=x^{3}+x^{2}+2x+1$$
sage: E = EllipticCurve(K, [a, 1, 0, 2, 1])

gp: E = ellinit([a, 1, 0, 2, 1],K)

magma: E := ChangeRing(EllipticCurve([a, 1, 0, 2, 1]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(12 a)$$ = $$\left(a\right)^{5} \cdot \left(-a - 1\right) \cdot \left(a - 1\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$288$$ = $$2^{5} \cdot 3^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(648)$$ = $$\left(a\right)^{6} \cdot \left(-a - 1\right)^{4} \cdot \left(a - 1\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$419904$$ = $$2^{6} \cdot 3^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{97336}{81}$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z$$ (no potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{SU}(2)$

## Mordell-Weil group

 Rank: $$1$$ Generator $\left(0 : -1 : 1\right)$ Height $$0.337900933603021$$ Torsion structure: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-\frac{1}{2} : \frac{1}{4} a : 1\right)$ $\left(-a : -1 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.337900933603021$$ Period: $$4.69072859700528$$ Tamagawa product: $$8$$  =  $$2\cdot2\cdot2$$ Torsion order: $$4$$ Leading coefficient: $$1.12076535989854$$ Analytic order of Ш: $$1$$ (rounded)

## 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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$2$$ $$III$$ Additive $$1$$ $$5$$ $$6$$ $$0$$
$$\left(-a - 1\right)$$ $$3$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$
$$\left(a - 1\right)$$ $$3$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 288.2-a consists of curves linked by isogenies of degrees dividing 8.

## Base change

This curve is the base change of elliptic curves 192.a4, 96.a4, defined over $$\Q$$, so it is also a $$\Q$$-curve.