# Properties

 Label 2.0.8.1-7056.2-f4 Base field $$\Q(\sqrt{-2})$$ Conductor $$\left(84\right)$$ Conductor norm $$7056$$ CM no Base change yes: 1344.q2,336.d2 Q-curve yes Torsion order $$8$$ 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: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([2, 0, 1]))

gp: K = nfinit(Pol(Vecrev([2, 0, 1])));

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

## Weierstrass equation

$$y^2+axy+ay=x^{3}-3655x-83662$$
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([0,1]),K([-3655,0]),K([-83662,0])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([0,1])),Pol(Vecrev([-3655,0])),Pol(Vecrev([-83662,0]))], K);

magma: E := EllipticCurve([K![0,1],K![0,0],K![0,1],K![-3655,0],K![-83662,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$\left(84\right)$$ = $$\left(a\right)^{4}\cdot\left(-a - 1\right)\cdot\left(a - 1\right)\cdot\left(7\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$7056$$ = $$2^{4}\cdot3\cdot3\cdot49$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$( 59769456768 )$$ = $$\left(a\right)^{14}\cdot\left(-a - 1\right)^{4}\cdot\left(a - 1\right)^{4}\cdot\left(7\right)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$3572387962341821005824$$ = $$2^{14}\cdot3^{4}\cdot3^{4}\cdot49^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{84448510979617}{933897762}$$ 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: $$\Z/8\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-209 : -1954 a : 1\right)$ 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: $$0.171272958266504$$ Tamagawa product: $$512$$  =  $$2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{3}$$ Torsion order: $$8$$ Leading coefficient: $$3.87546464717201$$ Analytic order of Ш: $$4$$ (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$$(j)_{-}$$
$$\left(a\right)$$ $$2$$ $$4$$ $$I_{6}^*$$ Additive $$1$$ $$4$$ $$14$$ $$2$$
$$\left(-a - 1\right)$$ $$3$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(a - 1\right)$$ $$3$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(7\right)$$ $$49$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$

## 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, 4 and 8.
Its isogeny class 7056.2-f consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is the base change of elliptic curves 1344.q2, 336.d2, defined over $$\Q$$, so it is also a $$\Q$$-curve.