Properties

 Label 2.0.4.1-97344.2-a3 Base field $$\Q(\sqrt{-1})$$ Conductor $$(312)$$ Conductor norm $$97344$$ CM no Base change yes: 1248.g2,1248.a2 Q-curve yes Torsion order $$4$$ 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

$${y}^2={x}^{3}-i{x}^{2}-8{x}-10i$$
sage: E = EllipticCurve([K([0,0]),K([0,-1]),K([0,0]),K([-8,0]),K([0,-10])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([-8,0])),Pol(Vecrev([0,-10]))], K);

magma: E := EllipticCurve([K![0,0],K![0,-1],K![0,0],K![-8,0],K![0,-10]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(312)$$ = $$(i+1)^{6}\cdot(3)\cdot(-3i-2)\cdot(2i+3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$97344$$ = $$2^{6}\cdot9\cdot13\cdot13$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(97344)$$ = $$(i+1)^{12}\cdot(3)^{2}\cdot(-3i-2)^{2}\cdot(2i+3)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$9475854336$$ = $$2^{12}\cdot9^{2}\cdot13^{2}\cdot13^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{778688}{1521}$$ 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 $\left(-3 i + 3 : 10 i - 2 : 1\right)$ $\left(2 i + 2 : 5 i - 1 : 1\right)$ Heights $$0.852284451683303$$ $$0.416607960420835$$ Torsion structure: $$\Z/2\Z\times\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(i - 3 : 0 : 1\right)$ $\left(-i : 0 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

BSD invariants

 Analytic rank: $$2$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$2$$ Regulator: $$0.355068487114171$$ Period: $$2.04745307656231$$ Tamagawa product: $$32$$  =  $$2^{2}\cdot2\cdot2\cdot2$$ Torsion order: $$4$$ Leading coefficient: $$5.81588853065787$$ 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$$(j)_{-}$$
$$(i+1)$$ $$2$$ $$4$$ $$I_{2}^{*}$$ Additive $$1$$ $$6$$ $$12$$ $$0$$
$$(3)$$ $$9$$ $$2$$ $$I_{2}$$ Split multiplicative $$-1$$ $$1$$ $$2$$ $$2$$
$$(-3i-2)$$ $$13$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$
$$(2i+3)$$ $$13$$ $$2$$ $$I_{2}$$ Non-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.
Its isogeny class 97344.2-a consists of curves linked by isogenies of degrees dividing 4.

Base change

This curve is the base change of 1248.g2, 1248.a2, defined over $$\Q$$, so it is also a $$\Q$$-curve.