# Properties

 Label 2.2.21.1-16.1-a3 Base field $$\Q(\sqrt{21})$$ Conductor $$\left(4\right)$$ Conductor norm $$16$$ CM yes ($$-12$$) Base change no Q-curve yes Torsion order $$2$$ Rank $$0$$

# Related objects

Show commands for: Magma / Pari/GP / SageMath

## Base field$$\Q(\sqrt{21})$$

Generator $$a$$, with minimal polynomial $$x^{2} - x - 5$$; class number $$1$$.

sage: R.<x> = PolynomialRing(QQ); K.<a> = NumberField(R([-5, -1, 1]))

gp: K = nfinit(Pol(Vecrev([-5, -1, 1])));

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5, -1, 1]);

## Weierstrass equation

$$y^2=x^{3}+\left(-a-1\right)x^{2}+\left(6a-13\right)x+11a-31$$
sage: E = EllipticCurve([K([0,0]),K([-1,-1]),K([0,0]),K([-13,6]),K([-31,11])])

gp: E = ellinit([Pol(Vecrev([0,0])),Pol(Vecrev([-1,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([-13,6])),Pol(Vecrev([-31,11]))], K);

magma: E := EllipticCurve([K![0,0],K![-1,-1],K![0,0],K![-13,6],K![-31,11]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$\left(4\right)$$ = $$\left(2\right)^{2}$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$16$$ = $$4^{2}$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$( 256 )$$ = $$\left(2\right)^{8}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$65536$$ = $$4^{8}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$54000$$ sage: E.j_invariant()  gp: E.j  magma: jInvariant(E); Endomorphism ring: $$\Z$$ Geometric endomorphism ring: $$\Z[\sqrt{-3}]$$ (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: $$\Z/2\Z$$ 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: $$0$$ sage: E.rank()  magma: Rank(E); Mordell-Weil rank: $$0$$ Regulator: $$1$$ Period: $$17.6950319084543$$ Tamagawa product: $$1$$ Torsion order: $$2$$ Leading coefficient: $$0.965343132680366$$ 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$$(j)_{-}$$
$$\left(2\right)$$ $$4$$ $$1$$ $$IV^*$$ Additive $$1$$ $$2$$ $$8$$ $$0$$

## Galois Representations

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

prime Image of Galois Representation
$$7$$ 7Ns.2.1

For all other primes $$p$$, the image is a Borel subgroup if $$p\in \{ 2, 3\}$$, the normalizer of a split Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=+1$$ or the normalizer of a nonsplit Cartan subgroup if $$\left(\frac{ -3 }{p}\right)=-1$$.

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 2, 3 and 6.
Its isogeny class 16.1-a consists of curves linked by isogenies of degrees dividing 6.

## Base change

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