# Properties

 Label 2.0.4.1-58482.1-a1 Base field $$\Q(\sqrt{-1})$$ Conductor $$(171i+171)$$ Conductor norm $$58482$$ CM no Base change yes: 2736.f2,342.a2 Q-curve yes Torsion order $$2$$ 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

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

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-3,0])),Pol(Vecrev([1,0]))], K);

magma: E := EllipticCurve([K![1,0],K![-1,0],K![0,0],K![-3,0],K![1,0]]);

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: $$(2052)$$ = $$(i+1)^{4}\cdot(3)^{3}\cdot(19)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$4210704$$ = $$2^{4}\cdot9^{3}\cdot361$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{132651}{76}$$ 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 $\left(-1 : -1 : 1\right)$ $\left(-2 : -2 i + 1 : 1\right)$ Heights $$0.311372269638457$$ $$0.763972800958920$$ Torsion structure: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(2 : -1 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$2$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$2$$ Regulator: $$0.237879944976628$$ Period: $$3.85556479164268$$ Tamagawa product: $$4$$  =  $$2\cdot2\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$3.66864616195914$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$2$$ $$I_{4}$$ Non-split multiplicative $$1$$ $$1$$ $$4$$ $$4$$
$$(3)$$ $$9$$ $$2$$ $$III$$ Additive $$1$$ $$2$$ $$3$$ $$0$$
$$(19)$$ $$361$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 58482.1-a consists of curves linked by isogenies of degree 2.

## Base change

This curve is the base change of 2736.f2, 342.a2, defined over $$\Q$$, so it is also a $$\Q$$-curve.