Properties

 Label 2.0.3.1-324.1-a5 Base field $$\Q(\sqrt{-3})$$ Conductor $$(18)$$ Conductor norm $$324$$ CM no Base change yes: 54.a3,54.b3 Q-curve yes Torsion order $$9$$ Rank $$0$$

Related objects

Show commands for: Magma / Pari/GP / SageMath

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

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

sage: x = polygen(QQ); K.<a> = NumberField(x^2 - x + 1)

gp: K = nfinit(a^2 - a + 1);

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

Weierstrass equation

$$y^2+\left(a+1\right)xy+y=x^{3}+\left(a-2\right)x-1$$
sage: E = EllipticCurve(K, [a + 1, 0, 1, a - 2, -1])

gp: E = ellinit([a + 1, 0, 1, a - 2, -1],K)

magma: E := ChangeRing(EllipticCurve([a + 1, 0, 1, a - 2, -1]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(18)$$ = $$\left(2\right) \cdot \left(-2 a + 1\right)^{4}$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$324$$ = $$3^{4} \cdot 4$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(216)$$ = $$\left(2\right)^{3} \cdot \left(-2 a + 1\right)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$46656$$ = $$3^{6} \cdot 4^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{9261}{8}$$ 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: $$\Z/3\Z\times\Z/3\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generators: $\left(a - 2 : -a : 1\right)$ $\left(-a : a - 2 : 1\right)$ 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: $$5.63513522683738$$ Tamagawa product: $$9$$  =  $$3\cdot3$$ Torsion order: $$9$$ Leading coefficient: $$0.722988186696557$$ 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$$(j)_{-}$$
$$\left(-2 a + 1\right)$$ $$3$$ $$3$$ $$IV$$ Additive $$1$$ $$4$$ $$6$$ $$0$$
$$\left(2\right)$$ $$4$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$

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.1.1[2]

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 324.1-a consists of curves linked by isogenies of degrees dividing 9.

Base change

This curve is the base change of elliptic curves 54.a3, 54.b3, defined over $$\Q$$, so it is also a $$\Q$$-curve.