Properties

 Label 2.0.3.1-14400.1-e1 Base field $$\Q(\sqrt{-3})$$ Conductor $$(120)$$ Conductor norm $$14400$$ CM no Base change yes: 360.c2,360.d2 Q-curve yes Torsion order $$2$$ 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: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([1, -1, 1]))

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

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

Weierstrass equation

$${y}^2={x}^{3}-3{x}-18$$
sage: E = EllipticCurve([K([0,0]),K([0,0]),K([0,0]),K([-3,0]),K([-18,0])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-3,0])),Pol(Vecrev([-18,0]))], K);

magma: E := EllipticCurve([K![0,0],K![0,0],K![0,0],K![-3,0],K![-18,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(120)$$ = $$(-2a+1)^{2}\cdot(2)^{3}\cdot(5)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$14400$$ = $$3^{2}\cdot4^{3}\cdot25$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-138240)$$ = $$(-2a+1)^{6}\cdot(2)^{10}\cdot(5)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$19110297600$$ = $$3^{6}\cdot4^{10}\cdot25$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{108}{5}$$ 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: $$\Z/2\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(3 : 0 : 1\right)$ 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: $$1.94870920223201$$ Tamagawa product: $$4$$  =  $$2\cdot2\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$2.25017556496191$$ 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$$(j)_{-}$$
$$(-2a+1)$$ $$3$$ $$2$$ $$I_0^{*}$$ Additive $$-1$$ $$2$$ $$6$$ $$0$$
$$(2)$$ $$4$$ $$2$$ $$III^{*}$$ Additive $$-1$$ $$3$$ $$10$$ $$0$$
$$(5)$$ $$25$$ $$1$$ $$I_{1}$$ Split multiplicative $$-1$$ $$1$$ $$1$$ $$1$$

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.
Its isogeny class 14400.1-e consists of curves linked by isogenies of degree 2.

Base change

This curve is the base change of elliptic curves 360.c2, 360.d2, defined over $$\Q$$, so it is also a $$\Q$$-curve.