# Properties

 Label 2.2.12.1-3072.1-d1 Base field $$\Q(\sqrt{3})$$ Conductor norm $$3072$$ CM no Base change yes 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

$${y}^2={x}^{3}+\left(a+1\right){x}^{2}+\left(-6a+13\right){x}+23a-39$$
sage: E = EllipticCurve([K([0,0]),K([1,1]),K([0,0]),K([13,-6]),K([-39,23])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([1,1])),Pol(Vecrev([0,0])),Pol(Vecrev([13,-6])),Pol(Vecrev([-39,23]))], K);

magma: E := EllipticCurve([K![0,0],K![1,1],K![0,0],K![13,-6],K![-39,23]]);

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: $$(-1152)$$ = $$(a+1)^{14}\cdot(a)^{4}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1327104$$ = $$2^{14}\cdot3^{4}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{4000}{9}$$ 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 $\left(-2 a + 3 : 5 a - 9 : 1\right)$ Height $$0.603762616876009$$ 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: $\left(-a + 1 : 0 : 1\right)$ 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: $$0.603762616876009$$ Period: $$6.40509292372428$$ Tamagawa product: $$8$$  =  $$2\cdot2^{2}$$ Torsion order: $$2$$ Leading coefficient: $$4.46540672832774$$ 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$$(j)_{-}$$
$$(a+1)$$ $$2$$ $$2$$ $$I_0^{*}$$ Additive $$1$$ $$10$$ $$14$$ $$0$$
$$(a)$$ $$3$$ $$4$$ $$I_{4}$$ Split multiplicative $$-1$$ $$1$$ $$4$$ $$4$$

## 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-d consists of curves linked by isogenies of degrees dividing 4.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 384.g2
$$\Q$$ 1152.j2