Properties

 Label 2.0.4.1-61250.3-b2 Base field $$\Q(\sqrt{-1})$$ Conductor $$(175i+175)$$ Conductor norm $$61250$$ CM no Base change yes: 350.a2,2800.x2 Q-curve yes Torsion order $$1$$ Rank $$2$$

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

$${y}^2+i{x}{y}={x}^{3}-{x}^{2}+5{x}-5$$
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,0]),K([5,0]),K([-5,0])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([5,0])),Pol(Vecrev([-5,0]))], K);

magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,0],K![5,0],K![-5,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(175i+175)$$ = $$(i+1)\cdot(-i-2)^{2}\cdot(2i+1)^{2}\cdot(7)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$61250$$ = $$2\cdot5^{2}\cdot5^{2}\cdot49$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-9800)$$ = $$(i+1)^{6}\cdot(-i-2)^{2}\cdot(2i+1)^{2}\cdot(7)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$96040000$$ = $$2^{6}\cdot5^{2}\cdot5^{2}\cdot49^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{397535}{392}$$ 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: $$2$$ Generators $\left(-1 : -3 i : 1\right)$ $\left(-2 i : i - 2 : 1\right)$ Heights $$0.217698566798014$$ $$0.640123204783338$$ Torsion structure: trivial sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T);

BSD invariants

 Analytic rank: $$2$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$2$$ Regulator: $$0.0919612382695749$$ Period: $$2.98722015367149$$ Tamagawa product: $$4$$  =  $$2\cdot1\cdot1\cdot2$$ Torsion order: $$1$$ Leading coefficient: $$4.39533542904735$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$2$$ $$I_{6}$$ Non-split multiplicative $$1$$ $$1$$ $$6$$ $$6$$
$$(-i-2)$$ $$5$$ $$1$$ $$II$$ Additive $$1$$ $$2$$ $$2$$ $$0$$
$$(2i+1)$$ $$5$$ $$1$$ $$II$$ Additive $$1$$ $$2$$ $$2$$ $$0$$
$$(7)$$ $$49$$ $$2$$ $$I_{2}$$ 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
$$3$$ 3B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 61250.3-b consists of curves linked by isogenies of degree 3.

Base change

This curve is the base change of 350.a2, 2800.x2, defined over $$\Q$$, so it is also a $$\Q$$-curve.