# Properties

 Label 2.0.4.1-92416.1-d3 Base field $$\Q(\sqrt{-1})$$ Conductor $$(304)$$ Conductor norm $$92416$$ CM no Base change yes: 1216.e3,1216.m3 Q-curve yes Torsion order $$1$$ Rank $$2$$

# Related objects

Show commands: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{-1})$$

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

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, 0, 1]))

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

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

## Weierstrass equation

$${y}^2={x}^{3}-i{x}^{2}-152{x}+5776i$$
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([-152,0]),K([0,5776])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([-152,0])),Pol(Vecrev([0,5776]))], K);

magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![-152,0],K![0,5776]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(304)$$ = $$(i+1)^{8}\cdot(19)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$92416$$ = $$2^{8}\cdot361$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(14384365568)$$ = $$(i+1)^{42}\cdot(19)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$206909972793863962624$$ = $$2^{42}\cdot361^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{94196375}{3511808}$$ 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: $$2$$ Generators $\left(228 i : 2432 i - 2432 : 1\right)$ $\left(-28 i : -128 i - 128 : 1\right)$ Heights $$0.566135023124893$$ $$0.834776094913618$$ 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: $$2$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$2$$ Regulator: $$0.472595983798029$$ Period: $$0.284215849978369$$ Tamagawa product: $$12$$  =  $$2^{2}\cdot3$$ Torsion order: $$1$$ Leading coefficient: $$6.44732492311299$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$4$$ $$I_{30}^{*}$$ Additive $$1$$ $$8$$ $$42$$ $$18$$
$$(19)$$ $$361$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$

## Galois Representations

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

prime Image of Galois Representation
$$3$$ 3Cs

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 92416.1-d consists of curves linked by isogenies of degrees dividing 9.

## Base change

This curve is the base change of 1216.e3, 1216.m3, defined over $$\Q$$, so it is also a $$\Q$$-curve.