# Properties

 Label 2.2.21.1-2268.1-v3 Base field $$\Q(\sqrt{21})$$ Conductor $$(18a+54)$$ Conductor norm $$2268$$ CM no Base change yes: 2646.f3,378.e3 Q-curve yes Torsion order $$3$$ Rank $$1$$

# Related objects

Show commands: 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

$${y}^2+{x}{y}+{y}={x}^{3}-{x}^{2}+10{x}+5$$
sage: E = EllipticCurve([K([1,0]),K([-1,0]),K([1,0]),K([10,0]),K([5,0])])

gp: E = ellinit([Pol(Vecrev([1,0])),Pol(Vecrev([-1,0])),Pol(Vecrev([1,0])),Pol(Vecrev([10,0])),Pol(Vecrev([5,0]))], K);

magma: E := EllipticCurve([K![1,0],K![-1,0],K![1,0],K![10,0],K![5,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(18a+54)$$ = $$(-a+2)^{4}\cdot(2)\cdot(a+3)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$2268$$ = $$3^{4}\cdot4\cdot7$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-96768)$$ = $$(-a+2)^{6}\cdot(2)^{9}\cdot(a+3)^{2}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$9364045824$$ = $$3^{6}\cdot4^{9}\cdot7^{2}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{4492125}{3584}$$ 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 $\left(0 : a - 1 : 1\right)$ Height $$0.424276273106024$$ 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: $\left(1 : -5 : 1\right)$ 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: $$0.424276273106024$$ Period: $$4.72065527940385$$ Tamagawa product: $$54$$  =  $$3\cdot3^{2}\cdot2$$ Torsion order: $$3$$ Leading coefficient: $$5.24472391568351$$ 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$$(j)_{-}$$
$$(-a+2)$$ $$3$$ $$3$$ $$IV$$ Additive $$-1$$ $$4$$ $$6$$ $$0$$
$$(2)$$ $$4$$ $$9$$ $$I_{9}$$ Split multiplicative $$-1$$ $$1$$ $$9$$ $$9$$
$$(a+3)$$ $$7$$ $$2$$ $$I_{2}$$ 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
$$3$$ 3B.1.1

## Isogenies and isogeny class

This curve has non-trivial cyclic isogenies of degree $$d$$ for $$d=$$ 3 and 9.
Its isogeny class 2268.1-v consists of curves linked by isogenies of degrees dividing 9.

## Base change

This curve is the base change of 2646.f3, 378.e3, defined over $$\Q$$, so it is also a $$\Q$$-curve.