# Properties

 Label 2.0.8.1-5776.2-d1 Base field $$\Q(\sqrt{-2})$$ Conductor $$(76)$$ Conductor norm $$5776$$ CM no Base change yes: 1216.n1,304.d1 Q-curve yes Torsion order $$1$$ Rank $$1$$

# Related objects

Show commands: 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

$${y}^2+a{x}{y}+a{y}={x}^{3}-279{x}-1950$$
sage: E = EllipticCurve([K([0,1]),K([0,0]),K([0,1]),K([-279,0]),K([-1950,0])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([0,1])),Pol(Vecrev([-279,0])),Pol(Vecrev([-1950,0]))], K);

magma: E := EllipticCurve([K![0,1],K![0,0],K![0,1],K![-279,0],K![-1950,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(76)$$ = $$(a)^{4}\cdot(-3a+1)\cdot(3a+1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$5776$$ = $$2^{4}\cdot19\cdot19$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-316940672)$$ = $$(a)^{14}\cdot(-3a+1)^{5}\cdot(3a+1)^{5}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$100451389567811584$$ = $$2^{14}\cdot19^{5}\cdot19^{5}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{37966934881}{4952198}$$ 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 $\left(15 : -46 a : 1\right)$ Height $$0.0655851879501470$$ Torsion structure: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$0.0655851879501470$$ Period: $$0.482527981338880$$ Tamagawa product: $$100$$  =  $$2^{2}\cdot5\cdot5$$ Torsion order: $$1$$ Leading coefficient: $$4.47551758649682$$ 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$$(j)_{-}$$
$$(a)$$ $$2$$ $$4$$ $$I_{6}^{*}$$ Additive $$1$$ $$4$$ $$14$$ $$2$$
$$(-3a+1)$$ $$19$$ $$5$$ $$I_{5}$$ Split multiplicative $$-1$$ $$1$$ $$5$$ $$5$$
$$(3a+1)$$ $$19$$ $$5$$ $$I_{5}$$ Split multiplicative $$-1$$ $$1$$ $$5$$ $$5$$

## Galois Representations

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

prime Image of Galois Representation
$$5$$ 5B.4.2

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 5776.2-d consists of curves linked by isogenies of degree 5.

## Base change

This curve is the base change of 1216.n1, 304.d1, defined over $$\Q$$, so it is also a $$\Q$$-curve.