# Properties

 Label 2.0.4.1-12544.1-a3 Base field $$\Q(\sqrt{-1})$$ Conductor $$(112)$$ Conductor norm $$12544$$ CM no Base change yes: 448.e3,448.d3 Q-curve yes Torsion order $$4$$ 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

$${y}^2={x}^{3}+19{x}+30i$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([19,0]),K([0,30])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([19,0])),Pol(Vecrev([0,30]))], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![19,0],K![0,30]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(112)$$ = $$(i+1)^{8}\cdot(7)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$12544$$ = $$2^{8}\cdot49$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-50176)$$ = $$(i+1)^{20}\cdot(7)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$2517630976$$ = $$2^{20}\cdot49^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{740772}{49}$$ 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 $\left(-4 i : 3 i + 3 : 1\right)$ $\left(-2 i + 1 : -i + 3 : 1\right)$ Heights $$1.39347716695542$$ $$0.882923580137468$$ Torsion structure: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-3 i : 0 : 1\right)$ $\left(-2 i : 0 : 1\right)$ 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: $$0.744889195381569$$ Period: $$2.00814785915744$$ Tamagawa product: $$8$$  =  $$2^{2}\cdot2$$ Torsion order: $$4$$ Leading coefficient: $$2.99169528603001$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$4$$ $$I_{8}^{*}$$ Additive $$1$$ $$8$$ $$20$$ $$0$$
$$(7)$$ $$49$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$

## Galois Representations

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

prime Image of Galois Representation
$$2$$ 2Cs

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 12544.1-a consists of curves linked by isogenies of degrees dividing 4.

## Base change

This curve is the base change of 448.e3, 448.d3, defined over $$\Q$$, so it is also a $$\Q$$-curve.