# Properties

 Label 2.0.4.1-76050.5-m2 Base field $$\Q(\sqrt{-1})$$ Conductor $$(195i+195)$$ Conductor norm $$76050$$ CM no Base change yes: 390.e1,3120.o1 Q-curve yes Torsion order $$2$$ 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

$${y}^2+i{x}{y}+i{y}={x}^{3}-{x}^{2}-45{x}+127$$
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,1]),K([-45,0]),K([127,0])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([-45,0])),Pol(Vecrev([127,0]))], K);

magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,1],K![-45,0],K![127,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(195i+195)$$ = $$(i+1)\cdot(-i-2)\cdot(2i+1)\cdot(3)\cdot(-3i-2)\cdot(2i+3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$76050$$ = $$2\cdot5\cdot5\cdot9\cdot13\cdot13$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(1232010)$$ = $$(i+1)^{2}\cdot(-i-2)\cdot(2i+1)\cdot(3)^{6}\cdot(-3i-2)^{2}\cdot(2i+3)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1517848640100$$ = $$2^{2}\cdot5\cdot5\cdot9^{6}\cdot13^{2}\cdot13^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{10779215329}{1232010}$$ 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 $\left(1 : -i - 9 : 1\right)$ Height $$0.225607083762179$$ 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: $\left(\frac{13}{4} : -\frac{17}{8} i : 1\right)$ 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: $$0.225607083762179$$ Period: $$1.21906926899628$$ Tamagawa product: $$48$$  =  $$2\cdot1\cdot1\cdot( 2 \cdot 3 )\cdot2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$6.60073590437623$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$(-i-2)$$ $$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$(2i+1)$$ $$5$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$
$$(3)$$ $$9$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$
$$(-3i-2)$$ $$13$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$(2i+3)$$ $$13$$ $$2$$ $$I_{2}$$ Non-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
$$2$$ 2B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2.
Its isogeny class 76050.5-m consists of curves linked by isogenies of degree 2.

## Base change

This curve is the base change of 390.e1, 3120.o1, defined over $$\Q$$, so it is also a $$\Q$$-curve.