# Properties

 Label 2.0.4.1-58482.1-f3 Base field $$\Q(\sqrt{-1})$$ Conductor $$(171i+171)$$ Conductor norm $$58482$$ CM no Base change yes: 2736.n3,342.e3 Q-curve yes Torsion order $$3$$ Rank $$1$$

# 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

$${y}^2+i{x}{y}+i{y}={x}^{3}+{x}^{2}+86{x}+2437$$
sage: E = EllipticCurve([K([0,1]),K([1,0]),K([0,1]),K([86,0]),K([2437,0])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([86,0])),Pol(Vecrev([2437,0]))], K);

magma: E := EllipticCurve([K![0,1],K![1,0],K![0,1],K![86,0],K![2437,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(171i+171)$$ = $$(i+1)\cdot(3)^{2}\cdot(19)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$58482$$ = $$2\cdot9^{2}\cdot361$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-2560108032)$$ = $$(i+1)^{18}\cdot(3)^{6}\cdot(19)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$6554153135510913024$$ = $$2^{18}\cdot9^{6}\cdot361^{3}$$ 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: $$1$$ Generator $\left(-38 i - 57 : 389 i - 456 : 1\right)$ Height $$1.51316276216989$$ Torsion structure: $$\Z/3\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-19 : -67 i : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$1.51316276216989$$ Period: $$0.378954466637826$$ Tamagawa product: $$54$$  =  $$( 2 \cdot 3^{2} )\cdot1\cdot3$$ Torsion order: $$3$$ Leading coefficient: $$6.88103744969174$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$18$$ $$I_{18}$$ Split multiplicative $$-1$$ $$1$$ $$18$$ $$18$$
$$(3)$$ $$9$$ $$1$$ $$I_0^{*}$$ Additive $$1$$ $$2$$ $$6$$ $$0$$
$$(19)$$ $$361$$ $$3$$ $$I_{3}$$ 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.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 58482.1-f consists of curves linked by isogenies of degrees dividing 9.

## Base change

This curve is the base change of 2736.n3, 342.e3, defined over $$\Q$$, so it is also a $$\Q$$-curve.