Properties

 Label 2.2.12.1-3072.1-bw2 Base field $$\Q(\sqrt{3})$$ Conductor $$(32a)$$ Conductor norm $$3072$$ CM no Base change no Q-curve yes Torsion order $$2$$ Rank $$1$$

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

$${y}^2={x}^{3}+{x}^{2}+\left(150a-305\right){x}-1434a+2499$$
sage: E = EllipticCurve([K([0,0]),K([1,0]),K([0,0]),K([-305,150]),K([2499,-1434])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,0])),Pol(Vecrev([0,0])),Pol(Vecrev([-305,150])),Pol(Vecrev([2499,-1434]))], K);

magma: E := EllipticCurve([K![0,0],K![1,0],K![0,0],K![-305,150],K![2499,-1434]]);

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: $$(6144a-18432)$$ = $$(a+1)^{23}\cdot(a)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$226492416$$ = $$2^{23}\cdot3^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{1291466278840}{9} a + 248542803096$$ 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 $\left(-4 a + 7 : -2 a : 1\right)$ Height $$0.543379137034940$$ 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: $\left(-6 a + 3 : 0 : 1\right)$ 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: $$0.543379137034940$$ Period: $$8.80783365999859$$ Tamagawa product: $$6$$  =  $$2\cdot3$$ Torsion order: $$2$$ Leading coefficient: $$4.14479156650866$$ 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$$(j)_{-}$$
$$(a+1)$$ $$2$$ $$2$$ $$I_9^{*}$$ Additive $$1$$ $$10$$ $$23$$ $$0$$
$$(a)$$ $$3$$ $$3$$ $$I_{3}$$ 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
$$2$$ 2B

Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2 and 4.
Its isogeny class 3072.1-bw consists of curves linked by isogenies of degrees dividing 4.

Base change

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