Properties

 Label 2.2.12.1-1296.1-e3 Base field $$\Q(\sqrt{3})$$ Conductor $$(36)$$ Conductor norm $$1296$$ CM yes ($$-3$$) Base change no Q-curve yes Torsion order $$1$$ Rank $$0$$

Related objects

Show commands: Magma / Pari/GP / SageMath

Base field$$\Q(\sqrt{3})$$

Generator $$a$$, with minimal polynomial $$x^{2} - 3$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-3, 0, 1]))

gp: K = nfinit(Pol(Vecrev([-3, 0, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 0, 1]);

Weierstrass equation

$${y}^2={x}^{3}-6a$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([0,0]),K([0,-6])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,-6]))], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![0,0],K![0,-6]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(36)$$ = $$(a+1)^{4}\cdot(a)^{4}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$1296$$ = $$2^{4}\cdot3^{4}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-46656)$$ = $$(a+1)^{12}\cdot(a)^{12}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$2176782336$$ = $$2^{12}\cdot3^{12}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$0$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[(1+\sqrt{-3})/2]$$ (potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$

Mordell-Weil group

 Rank: $$0$$ 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: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$4.68151871101520$$ Tamagawa product: $$1$$  =  $$1\cdot1$$ Torsion order: $$1$$ Leading coefficient: $$1.35143804401045$$ 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$$(j)_{-}$$
$$(a+1)$$ $$2$$ $$1$$ $$II^{*}$$ Additive $$1$$ $$4$$ $$12$$ $$0$$
$$(a)$$ $$3$$ $$1$$ $$II^{*}$$ Additive $$1$$ $$4$$ $$12$$ $$0$$

Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$3$$ 3Cs

For all other primes $$p$$, the image is the normalizer of a split Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=-1$$.

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3 and 9.
Its isogeny class 1296.1-e consists of curves linked by isogenies of degrees dividing 27.

Base change

This curve is not the base change of an elliptic curve defined over $$\Q$$. It is a $$\Q$$-curve.