# Properties

 Label 2.2.24.1-1.1-a4 Base field $$\Q(\sqrt{6})$$ Conductor $$(1)$$ Conductor norm $$1$$ CM yes ($$-72$$) Base change no Q-curve yes Torsion order $$6$$ Rank $$0$$

# Learn more about

Show commands for: Magma / Pari/GP / 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(Pol(Vecrev([-6, 0, 1])));

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

## Weierstrass equation

$$y^2+axy+\left(a+1\right)y=x^{3}+\left(-a+1\right)x^{2}+\left(-18a-41\right)x+56a+138$$
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([1,1]),K([-41,-18]),K([138,56])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,-1])),Pol(Vecrev([1,1])),Pol(Vecrev([-41,-18])),Pol(Vecrev([138,56]))], K);

magma: E := EllipticCurve([K![0,1],K![1,-1],K![1,1],K![-41,-18],K![138,56]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(1)$$ = $$(1)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$1$$ = 1 sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$( 1 )$$ = $$( 1 )$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$1$$ = 1 sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-77092288000 a + 188837384000$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[\sqrt{-18}]$$ (potential complex multiplication) sage: E.has_cm(), E.cm_discriminant()  magma: HasComplexMultiplication(E); Sato-Tate group: $N(\mathrm{U}(1))$

## Mordell-Weil group

 Rank: $$0$$ Torsion structure: $$\Z/6\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(a + 2 : -a - 2 : 1\right)$ 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: $$50.7599477371936$$ Tamagawa product: $$1$$ Torsion order: $$6$$ Leading coefficient: $$0.287814748440893$$ Analytic order of Ш: $$1$$ (rounded)

## Local data at primes of bad reduction

sage: E.local_data()

magma: LocalInformation(E);

No primes of bad reduction.

## Galois Representations

The mod $$p$$ Galois Representation has maximal image for all primes $$p < 1000$$ except those listed.

prime Image of Galois Representation
$$3$$ 3B.1.1

For all other primes $$p$$, the image is a Borel subgroup if $$p=2$$, the normalizer of a split Cartan subgroup if $$\left(\frac{ -2 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -2 }{p}\right)=-1$$.

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3, 6, 9 and 18.
Its isogeny class 1.1-a consists of curves linked by isogenies of degrees dividing 18.

## Base change

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