# Properties

 Label 2.0.4.1-29241.1-b2 Base field $$\Q(\sqrt{-1})$$ Conductor norm $$29241$$ CM no Base change yes Q-curve yes Torsion order $$3$$ Rank $$2$$

# Related objects

Show commands: Magma / PariGP / 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(Polrev([1, 0, 1]));

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

## Weierstrass equation

$${y}^2+i{y}={x}^{3}-84{x}-315$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,1]),K([-84,0]),K([-315,0])])

gp: E = ellinit([Polrev([0,0]),Polrev([0,0]),Polrev([0,1]),Polrev([-84,0]),Polrev([-315,0])], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![0,1],K![-84,0],K![-315,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(171)$$ = $$(3)^{2}\cdot(19)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$29241$$ = $$9^{2}\cdot361$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-5000211)$$ = $$(3)^{6}\cdot(19)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$25002110044521$$ = $$9^{6}\cdot361^{3}$$ 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 $\left(-\frac{21}{2} i - 11 : \frac{61}{4} i - \frac{225}{4} : 1\right)$ $\left(-5 : -5 i : 1\right)$ Heights $$2.4442015483114639942548861672933640890$$ $$0.67790060647708463625367879465644085186$$ 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: $\left(-3 : -10 i : 1\right)$ 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: $$0.62293993936307260541383631961123108874$$ Period: $$0.93530900844051168958806055461294363412$$ Tamagawa product: $$12$$  =  $$2^{2}\cdot3$$ Torsion order: $$3$$ Leading coefficient: $$3.1074204640195622165456095010284428275$$ 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$$(j)_{-}$$
$$(3)$$ $$9$$ $$4$$ $$I_0^{*}$$ 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 29241.1-b consists of curves linked by isogenies of degrees dividing 9.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 171.b2
$$\Q$$ 2736.c2