# Properties

 Label 2.0.4.1-25.1-CMa1 Base field $$\Q(\sqrt{-1})$$ Conductor norm $$25$$ CM yes ($$-4$$) Base change no Q-curve yes Torsion order $$10$$ Rank $$0$$

# 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

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

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

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

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(4i+3)$$ = $$(-i-2)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$25$$ = $$5^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-2i+11)$$ = $$(-i-2)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$125$$ = $$5^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$1728$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z[\sqrt{-1}]$$ (complex multiplication) Geometric endomorphism ring: $$\Z[\sqrt{-1}]$$ sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $\mathrm{U}(1)$

## Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/10\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(1 : -i - 2 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

## BSD invariants

 Analytic rank: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$9.1954277214250313093576845952971768982$$ Tamagawa product: $$2$$ Torsion order: $$10$$ Leading coefficient: $$0.18390855442850062618715369190594353797$$ 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$$(j)_{-}$$
$$(-i-2)$$ $$5$$ $$2$$ $$III$$ Additive $$-1$$ $$2$$ $$3$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$5$$ 5Cs.1.1

For all other primes $$p$$, 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 no rational isogenies other than endomorphisms. Its isogeny class 25.1-CMa consists of this curve only.

## Base change

This elliptic curve is a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.