# Properties

 Label 2.0.4.1-40000.3-CMc2 Base field $$\Q(\sqrt{-1})$$ Conductor $$(200)$$ Conductor norm $$40000$$ CM yes ($$-16$$) Base change yes: 800.d1,800.d2 Q-curve yes Torsion order $$2$$ Rank $$2$$

# Related objects

Show commands for: 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

$${y}^2+\left(i+1\right){x}{y}={x}^{3}+i{x}^{2}+68{x}+253i$$
sage: E = EllipticCurve([K([1,1]),K([0,1]),K([0,0]),K([68,0]),K([0,253])])

gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,1])),Pol(Vecrev([0,0])),Pol(Vecrev([68,0])),Pol(Vecrev([0,253]))], K);

magma: E := EllipticCurve([K![1,1],K![0,1],K![0,0],K![68,0],K![0,253]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(200)$$ = $$(i+1)^{6}\cdot(-i-2)^{2}\cdot(2i+1)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$40000$$ = $$2^{6}\cdot5^{2}\cdot5^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-125000)$$ = $$(i+1)^{6}\cdot(-i-2)^{6}\cdot(2i+1)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$15625000000$$ = $$2^{6}\cdot5^{6}\cdot5^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$287496$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z[\sqrt{-4}]$$ (complex multiplication) Geometric endomorphism ring: $$\Z[\sqrt{-4}]$$ sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{U}(1)$

## Mordell-Weil group

 Rank: $$2$$ Generators $\left(-7 i - 2 : 6 i - 13 : 1\right)$ $\left(-\frac{9}{2} i - \frac{3}{4} : \frac{53}{8} i - \frac{7}{4} : 1\right)$ Heights $$0.949741086265898$$ $$1.89948217253180$$ 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: $\left(-\frac{11}{2} i : \frac{11}{4} i - \frac{11}{4} : 1\right)$ 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: $$0.902008130941528$$ Period: $$1.37503716360407$$ Tamagawa product: $$4$$  =  $$1\cdot2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$4.96117880767060$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$1$$ $$II$$ Additive $$-1$$ $$6$$ $$6$$ $$0$$
$$(-i-2)$$ $$5$$ $$2$$ $$I_0^{*}$$ Additive $$1$$ $$2$$ $$6$$ $$0$$
$$(2i+1)$$ $$5$$ $$2$$ $$I_0^{*}$$ Additive $$1$$ $$2$$ $$6$$ $$0$$

## Galois Representations

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

The image is a Borel subgroup if $$p=2$$, a split Cartan subgroup if $$\left(\frac{ -1 }{p}\right)=+1$$ or a nonsplit Cartan subgroup if $$\left(\frac{ -1 }{p}\right)=-1$$.

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies (excluding endomorphisms) of degree $$d$$ for $$d=$$ 2.
Its isogeny class 40000.3-CMc consists of curves linked by isogenies of degree 2.

## Base change

This curve is the base change of elliptic curves 800.d1, 800.d2, defined over $$\Q$$, so it is also a $$\Q$$-curve.