# Properties

 Label 2.0.4.1-5625.3-b10 Base field $$\Q(\sqrt{-1})$$ Conductor $$(75)$$ Conductor norm $$5625$$ CM no Base change yes: 75.b1,1200.e1 Q-curve yes Torsion order $$4$$ Rank $$0$$

# 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

$${y}^2+{x}{y}+{y}={x}^{3}-54001{x}-4834477$$
sage: E = EllipticCurve([K([1,0]),K([0,0]),K([1,0]),K([-54001,0]),K([-4834477,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([-54001,0])),Pol(Vecrev([-4834477,0]))], K);

magma: E := EllipticCurve([K![1,0],K![0,0],K![1,0],K![-54001,0],K![-4834477,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(75)$$ = $$(-i-2)^{2}\cdot(2i+1)^{2}\cdot(3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$5625$$ = $$5^{2}\cdot5^{2}\cdot9$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(6328125)$$ = $$(-i-2)^{7}\cdot(2i+1)^{7}\cdot(3)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$40045166015625$$ = $$5^{7}\cdot5^{7}\cdot9^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{1114544804970241}{405}$$ 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: $$\Z/4\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-133 : -25 i + 66 : 1\right)$ 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: $$0.111785085630876$$ Tamagawa product: $$64$$  =  $$2^{2}\cdot2^{2}\cdot2^{2}$$ Torsion order: $$4$$ Leading coefficient: $$1.78856137009402$$ Analytic order of Ш: $$4$$ (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$$(j)_{-}$$
$$(-i-2)$$ $$5$$ $$4$$ $$I_1^{*}$$ Additive $$1$$ $$2$$ $$7$$ $$1$$
$$(2i+1)$$ $$5$$ $$4$$ $$I_1^{*}$$ Additive $$1$$ $$2$$ $$7$$ $$1$$
$$(3)$$ $$9$$ $$4$$ $$I_{4}$$ 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
$$2$$ 2B

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 4, 8 and 16.
Its isogeny class 5625.3-b consists of curves linked by isogenies of degrees dividing 16.

## Base change

This curve is the base change of 75.b1, 1200.e1, defined over $$\Q$$, so it is also a $$\Q$$-curve.