# Properties

 Label 2.0.4.1-92416.1-h1 Base field $$\Q(\sqrt{-1})$$ Conductor $$(304)$$ Conductor norm $$92416$$ CM no Base change yes: 1216.c1,1216.q1 Q-curve yes Torsion order $$1$$ Rank $$0$$

# 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

$${y}^2={x}^{3}-i{x}^{2}+21{x}+31i$$
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([21,0]),K([0,31])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([21,0])),Pol(Vecrev([0,31]))], K);

magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![21,0],K![0,31]]);

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: $$(4864)$$ = $$(i+1)^{16}\cdot(19)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$23658496$$ = $$2^{16}\cdot361$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{4194304}{19}$$ 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: 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: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$2.41539188674506$$ Tamagawa product: $$1$$  =  $$1\cdot1$$ Torsion order: $$1$$ Leading coefficient: $$2.41539188674506$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$1$$ $$II^{*}$$ Additive $$1$$ $$8$$ $$16$$ $$0$$
$$(19)$$ $$361$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

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

## Isogenies and isogeny class

This curve has no rational isogenies. Its isogeny class 92416.1-h consists of this curve only.

## Base change

This curve is the base change of 1216.c1, 1216.q1, defined over $$\Q$$, so it is also a $$\Q$$-curve.