# Properties

 Label 2.2.24.1-722.1-g3 Base field $$\Q(\sqrt{6})$$ Conductor norm $$722$$ CM no Base change yes Q-curve yes Torsion order $$3$$ Rank $$1$$

# Related objects

Show commands: Magma / PariGP / SageMath

## Base field$$\Q(\sqrt{6})$$

Generator $$a$$, with minimal polynomial $$x^{2} - 6$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-6, 0, 1]))

gp: K = nfinit(Polrev([-6, 0, 1]));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6, 0, 1]);

## Weierstrass equation

$${y}^2+\left(a+1\right){x}{y}={x}^{3}+\left(-189a+466\right){x}+17680a-43306$$
sage: E = EllipticCurve([K([1,1]),K([0,0]),K([0,0]),K([466,-189]),K([-43306,17680])])

gp: E = ellinit([Polrev([1,1]),Polrev([0,0]),Polrev([0,0]),Polrev([466,-189]),Polrev([-43306,17680])], K);

magma: E := EllipticCurve([K![1,1],K![0,0],K![0,0],K![466,-189],K![-43306,17680]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(-19a+38)$$ = $$(-a+2)\cdot(a+5)\cdot(a-5)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$722$$ = $$2\cdot19\cdot19$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-3511808)$$ = $$(-a+2)^{18}\cdot(a+5)^{3}\cdot(a-5)^{3}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$12332795428864$$ = $$2^{18}\cdot19^{3}\cdot19^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{94196375}{3511808}$$ 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 $\left(-28 a + 68 : 309 a - 756 : 1\right)$ Height $$0.56613502312489363059399202396476133603$$ Torsion structure: $$\Z/3\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(-\frac{38}{3} a + \frac{92}{3} : \frac{755}{9} a - \frac{616}{3} : 1\right)$ 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: $$0.56613502312489363059399202396476133603$$ Period: $$1.4463135242216651559100740810240158370$$ Tamagawa product: $$162$$  =  $$( 2 \cdot 3^{2} )\cdot3\cdot3$$ Torsion order: $$3$$ Leading coefficient: $$6.0169908333288472661435336551301982303$$ 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$$(j)_{-}$$
$$(-a+2)$$ $$2$$ $$18$$ $$I_{18}$$ Split multiplicative $$-1$$ $$1$$ $$18$$ $$18$$
$$(a+5)$$ $$19$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$(a-5)$$ $$19$$ $$3$$ $$I_{3}$$ 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

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3.
Its isogeny class 722.1-g consists of curves linked by isogenies of degrees dividing 9.

## Base change

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

Base field Curve
$$\Q$$ 342.e3
$$\Q$$ 1216.m3