Properties

 Label 2.2.12.1-3072.1-cd2 Base field $$\Q(\sqrt{3})$$ Conductor norm $$3072$$ CM no Base change no Q-curve no Torsion order $$2$$ Rank $$1$$

Related objects

Show commands: Magma / PariGP / 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(Polrev([-3, 0, 1]));

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

Weierstrass equation

$${y}^2={x}^{3}+\left(-a-1\right){x}^{2}+\left(1500096a-2598240\right){x}+1316558104a-2280345528$$
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([0,0]),K([-2598240,1500096]),K([-2280345528,1316558104])])

gp: E = ellinit([Polrev([0,0]),Polrev([-1,-1]),Polrev([0,0]),Polrev([-2598240,1500096]),Polrev([-2280345528,1316558104])], K);

magma: E := EllipticCurve([K![0,0],K![-1,-1],K![0,0],K![-2598240,1500096],K![-2280345528,1316558104]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$(32a)$$ = $$(a+1)^{10}\cdot(a)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$3072$$ = $$2^{10}\cdot3$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(4096a)$$ = $$(a+1)^{24}\cdot(a)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$-50331648$$ = $$-2^{24}\cdot3$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{132636728}{3} a + 76579552$$ 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: $$1$$ Generator $\left(-873 a + 1513 : 24848 a - 43038 : 1\right)$ Height $$1.8699872378927468140711698136545388189$$ 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: $\left(381 a - 659 : 0 : 1\right)$ sage: T.gens()  gp: T[3]  magma: [piT(P) : P in Generators(T)];

BSD invariants

 Analytic rank: $$1$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$1$$ Regulator: $$1.8699872378927468140711698136545388189$$ Period: $$3.0751643582262279617685076266357027056$$ Tamagawa product: $$4$$  =  $$2^{2}\cdot1$$ Torsion order: $$2$$ Leading coefficient: $$3.3200631755007039335032232920015800537$$ 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$$(j)_{-}$$
$$(a+1)$$ $$2$$ $$4$$ $$I_{10}^{*}$$ Additive $$-1$$ $$10$$ $$24$$ $$0$$
$$(a)$$ $$3$$ $$1$$ $$I_{1}$$ Non-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, 4 and 8.
Its isogeny class 3072.1-cd consists of curves linked by isogenies of degrees dividing 8.

Base change

This elliptic curve is not a $$\Q$$-curve.

It is not the base change of an elliptic curve defined over any subfield.