# Properties

 Label 2.0.4.1-92416.1-b2 Base field $$\Q(\sqrt{-1})$$ Conductor $$(304)$$ Conductor norm $$92416$$ CM no Base change yes: 1216.o2,1216.b2 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

$${y}^2={x}^{3}-{x}^{2}-37{x}-81$$
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([0,0]),K([-37,0]),K([-81,0])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-37,0])),Pol(Vecrev([-81,0]))], K);

magma: E := EllipticCurve([K![0,0],K![-1,0],K![0,0],K![-37,0],K![-81,0]]);

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: $$(-438976)$$ = $$(i+1)^{12}\cdot(19)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$192699928576$$ = $$2^{12}\cdot361^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{89915392}{6859}$$ 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 $\left(-8 : 19 i : 1\right)$ $\left(-19 i - 8 : -19 i - 95 : 1\right)$ Heights $$0.198634777456924$$ $$0.813602971000253$$ 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: $$0.151745901379087$$ Period: $$1.40296351266077$$ Tamagawa product: $$6$$  =  $$2\cdot3$$ Torsion order: $$1$$ Leading coefficient: $$5.10945510793628$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$2$$ $$I_0^{*}$$ Additive $$1$$ $$8$$ $$12$$ $$0$$
$$(19)$$ $$361$$ $$3$$ $$I_{3}$$ 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-b consists of curves linked by isogenies of degrees dividing 9.

## Base change

This curve is the base change of 1216.o2, 1216.b2, defined over $$\Q$$, so it is also a $$\Q$$-curve.