Properties

 Label 2.2.24.1-24.1-a5 Base field $$\Q(\sqrt{6})$$ Conductor $$\left(-2a\right)$$ Conductor norm $$24$$ CM no Base change yes: 72.a2,192.b2 Q-curve yes Torsion order $$2$$ Rank $$0$$

Related objects

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

$$y^2+axy+ay=x^{3}-x^{2}-19x-29$$
sage: E = EllipticCurve([K([0,1]),K([-1,0]),K([0,1]),K([-19,0]),K([-29,0])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([-1,0])),Pol(Vecrev([0,1])),Pol(Vecrev([-19,0])),Pol(Vecrev([-29,0]))], K);

magma: E := EllipticCurve([K![0,1],K![-1,0],K![0,1],K![-19,0],K![-29,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

Invariants

 Conductor: $$\left(-2a\right)$$ = $$\left(-a + 2\right)^{3}\cdot\left(a + 3\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$24$$ = $$2^{3}\cdot3$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$( 48 )$$ = $$\left(-a + 2\right)^{8}\cdot\left(a + 3\right)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$2304$$ = $$2^{8}\cdot3^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{28756228}{3}$$ 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: $$0$$ 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: $\left(-\frac{5}{2} : \frac{3}{4} a : 1\right)$ 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: $$5.68350851750159$$ Tamagawa product: $$4$$  =  $$2\cdot2$$ Torsion order: $$2$$ Leading coefficient: $$1.16014131805341$$ 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$$(j)_{-}$$
$$\left(-a + 2\right)$$ $$2$$ $$2$$ $$I_{1}^*$$ Additive $$1$$ $$3$$ $$8$$ $$0$$
$$\left(a + 3\right)$$ $$3$$ $$2$$ $$I_{2}$$ Non-split multiplicative $$1$$ $$1$$ $$2$$ $$2$$

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 24.1-a consists of curves linked by isogenies of degrees dividing 8.

Base change

This curve is the base change of elliptic curves 72.a2, 192.b2, defined over $$\Q$$, so it is also a $$\Q$$-curve.