# Properties

 Label 2.0.4.1-57600.2-z1 Base field $$\Q(\sqrt{-1})$$ Conductor norm $$57600$$ CM no Base change yes Q-curve yes Torsion order $$4$$ 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

$${y}^2={x}^{3}+i{x}^{2}-24{x}-24i$$
sage: E = EllipticCurve([K([0,0]),K([0,1]),K([0,0]),K([-24,0]),K([0,-24])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([-24,0])),Pol(Vecrev([0,-24]))], K);

magma: E := EllipticCurve([K![0,0],K![0,1],K![0,0],K![-24,0],K![0,-24]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(240)$$ = $$(i+1)^{8}\cdot(-i-2)\cdot(2i+1)\cdot(3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$57600$$ = $$2^{8}\cdot5\cdot5\cdot9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(960000)$$ = $$(i+1)^{18}\cdot(-i-2)^{4}\cdot(2i+1)^{4}\cdot(3)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$921600000000$$ = $$2^{18}\cdot5^{4}\cdot5^{4}\cdot9$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{2863288}{1875}$$ 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 $\left(-i - 5 : 5 i - 5 : 1\right)$ Height $$0.443150775611460$$ Torsion structure: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(4 i : 10 i - 10 : 1\right)$ 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: $$0.443150775611460$$ Period: $$1.38615777652593$$ Tamagawa product: $$64$$  =  $$2^{2}\cdot2^{2}\cdot2^{2}\cdot1$$ Torsion order: $$4$$ Leading coefficient: $$4.91421515029857$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$4$$ $$I_{6}^{*}$$ Additive $$-1$$ $$8$$ $$18$$ $$0$$
$$(-i-2)$$ $$5$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$(2i+1)$$ $$5$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$(3)$$ $$9$$ $$1$$ $$I_{1}$$ 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 and 4.
Its isogeny class 57600.2-z consists of curves linked by isogenies of degrees dividing 4.

## Base change

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

Base field Curve
$$\Q$$ 960.f4
$$\Q$$ 960.o4