# Properties

 Label 2.0.4.1-38025.5-a7 Base field $$\Q(\sqrt{-1})$$ Conductor $$(195)$$ Conductor norm $$38025$$ CM no Base change yes: 195.a4,3120.k4 Q-curve yes Torsion order $$16$$ Rank $$0$$

# 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: x = polygen(QQ); K.<i> = NumberField(x^2 + 1)

gp: K = nfinit(i^2 + 1);

magma: R<x> := PolynomialRing(Rationals()); K<i> := NumberField(R![1, 0, 1]);

## Weierstrass equation

$$y^2+ixy=x^{3}-520x+4225$$
sage: E = EllipticCurve(K, [i, 0, 0, -520, 4225])

gp: E = ellinit([i, 0, 0, -520, 4225],K)

magma: E := ChangeRing(EllipticCurve([i, 0, 0, -520, 4225]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(195)$$ = $$\left(3\right) \cdot \left(-i - 2\right) \cdot \left(2 i + 1\right) \cdot \left(-3 i - 2\right) \cdot \left(2 i + 3\right)$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$38025$$ = $$5^{2} \cdot 9 \cdot 13^{2}$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(1445900625)$$ = $$\left(3\right)^{4} \cdot \left(-i - 2\right)^{4} \cdot \left(2 i + 1\right)^{4} \cdot \left(-3 i - 2\right)^{4} \cdot \left(2 i + 3\right)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$2090628617375390625$$ = $$5^{8} \cdot 9^{4} \cdot 13^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{15551989015681}{1445900625}$$ 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: $$\Z/4\Z\times\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(-65 : -455 i : 1\right)$ $\left(-15 i + 10 : -95 i - 45 : 1\right)$ 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: $$0.370634167434087$$ Tamagawa product: $$1024$$  =  $$2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{2}\cdot2^{2}$$ Torsion order: $$16$$ Leading coefficient: $$1.48253666973635$$ 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$$(j)_{-}$$
$$\left(-i - 2\right)$$ $$5$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(2 i + 1\right)$$ $$5$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(-3 i - 2\right)$$ $$13$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(2 i + 3\right)$$ $$13$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$
$$\left(3\right)$$ $$9$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$

## 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 and 4.
Its isogeny class 38025.5-a consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is the base change of elliptic curves 195.a4, 3120.k4, defined over $$\Q$$, so it is also a $$\Q$$-curve.