# Properties

 Label 2.2.8.1-288.1-c1 Base field $$\Q(\sqrt{2})$$ Conductor $$(12 a)$$ Conductor norm $$288$$ CM no Base change yes: 96.a4,192.c4 Q-curve yes Torsion order $$4$$ Rank $$0$$

# 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: $$(12 a)$$ = $$\left(a\right)^{5} \cdot \left(3\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$288$$ = $$2^{5} \cdot 9$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(648)$$ = $$\left(a\right)^{6} \cdot \left(3\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$419904$$ = $$2^{6} \cdot 9^{4}$$ 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: $$0$$ Torsion structure: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(2 : -a - 3 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$8.01724737328688$$ Tamagawa product: $$8$$  =  $$2\cdot2^{2}$$ Torsion order: $$4$$ Leading coefficient: $$1.41726249602530$$ 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 $$-1$$ $$5$$ $$6$$ $$0$$
$$\left(3\right)$$ $$9$$ $$4$$ $$I_{4}$$ 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
$$2$$ 2B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 288.1-c consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base change of elliptic curves 96.a4, 192.c4, defined over $$\Q$$, so it is also a $$\Q$$-curve.