# Properties

 Label 2.0.4.1-225.2-a5 Base field $$\Q(\sqrt{-1})$$ Conductor norm $$225$$ CM no Base change yes Q-curve yes Torsion order $$16$$ 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

$${y}^2+i{x}{y}+i{y}={x}^{3}-{x}^{2}+36{x}+28$$
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,1]),K([36,0]),K([28,0])])

gp: E = ellinit([Polrev([0,1]),Polrev([-1,0]),Polrev([0,1]),Polrev([36,0]),Polrev([28,0])], K);

magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,1],K![36,0],K![28,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(15)$$ = $$(-i-2)\cdot(2i+1)\cdot(3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$225$$ = $$5\cdot5\cdot9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-3515625)$$ = $$(-i-2)^{8}\cdot(2i+1)^{8}\cdot(3)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$12359619140625$$ = $$5^{8}\cdot5^{8}\cdot9^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{4733169839}{3515625}$$ 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: $$0$$ Torsion structure: $$\Z/2\Z\oplus\Z/8\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-6 i + 1 : -i - 3 : 1\right)$ $\left(-32 : -172 i : 1\right)$ 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: $$1.1178508563087645508833713910561208612$$ Tamagawa product: $$128$$  =  $$2^{3}\cdot2^{3}\cdot2$$ Torsion order: $$16$$ Leading coefficient: $$0.55892542815438227544168569552806043058$$ 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$$(j)_{-}$$
$$(-i-2)$$ $$5$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$
$$(2i+1)$$ $$5$$ $$8$$ $$I_{8}$$ Split multiplicative $$-1$$ $$1$$ $$8$$ $$8$$
$$(3)$$ $$9$$ $$2$$ $$I_{2}$$ 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, 4 and 8.
Its isogeny class 225.2-a consists of curves linked by isogenies of degrees dividing 16.

## Base change

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

Base field Curve
$$\Q$$ 15.a8
$$\Q$$ 240.d8