# Properties

 Label 2.0.4.1-3249.1-b2 Base field $$\Q(\sqrt{-1})$$ Conductor norm $$3249$$ CM no Base change yes Q-curve yes Torsion order $$5$$ Rank $$1$$

# 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

$${y}^2+i{y}={x}^{3}-{x}^{2}+20{x}+32$$
sage: E = EllipticCurve([K([0,0]),K([-1,0]),K([0,1]),K([20,0]),K([32,0])])

gp: E = ellinit([Polrev([0,0]),Polrev([-1,0]),Polrev([0,1]),Polrev([20,0]),Polrev([32,0])], K);

magma: E := EllipticCurve([K![0,0],K![-1,0],K![0,1],K![20,0],K![32,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(57)$$ = $$(3)\cdot(19)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$3249$$ = $$9\cdot361$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-1121931)$$ = $$(3)^{10}\cdot(19)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1258729168761$$ = $$9^{10}\cdot361$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{841232384}{1121931}$$ 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 $\left(-6 i + 1 : 7 i + 6 : 1\right)$ Height $$1.1314245811679054280346669140891203517$$ Torsion structure: $$\Z/5\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-11 : -41 i : 1\right)$ 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: $$1.1314245811679054280346669140891203517$$ Period: $$1.3588291502046804185980242187002843408$$ Tamagawa product: $$10$$  =  $$( 2 \cdot 5 )\cdot1$$ Torsion order: $$5$$ Leading coefficient: $$1.2299301617192571175538189806165440492$$ 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$$(j)_{-}$$
$$(3)$$ $$9$$ $$10$$ $$I_{10}$$ Split multiplicative $$-1$$ $$1$$ $$10$$ $$10$$
$$(19)$$ $$361$$ $$1$$ $$I_{1}$$ 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
$$5$$ 5B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 5.
Its isogeny class 3249.1-b consists of curves linked by isogenies of degree 5.

## Base change

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

Base field Curve
$$\Q$$ 57.b2
$$\Q$$ 912.d2