# Properties

 Label 2.2.21.1-2268.1-v2 Base field $$\Q(\sqrt{21})$$ Conductor $$(18a+54)$$ Conductor norm $$2268$$ CM no Base change yes: 378.e2,2646.f2 Q-curve yes Torsion order $$3$$ Rank $$1$$

# Learn more

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

$${y}^2+a{x}{y}={x}^{3}+\left(-a+1\right){x}^{2}+\left(60a-168\right){x}+576a-1608$$
sage: E = EllipticCurve([K([0,1]),K([1,-1]),K([0,0]),K([-168,60]),K([-1608,576])])

gp: E = ellinit([Pol(Vecrev([0,1])),Pol(Vecrev([1,-1])),Pol(Vecrev([0,0])),Pol(Vecrev([-168,60])),Pol(Vecrev([-1608,576]))], K);

magma: E := EllipticCurve([K![0,1],K![1,-1],K![0,0],K![-168,60],K![-1608,576]]);

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: $$(-74088)$$ = $$(-a+2)^{6}\cdot(2)^{3}\cdot(a+3)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$5489031744$$ = $$3^{6}\cdot4^{3}\cdot7^{6}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$-\frac{7414875}{2744}$$ 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 $\left(-13 a + 39 : -118 a + 330 : 1\right)$ Height $$1.27282881931807$$ 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: $\left(-6 a + 18 : -34 a + 92 : 1\right)$ 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: $$1.27282881931807$$ Period: $$1.57355175980128$$ Tamagawa product: $$54$$  =  $$3\cdot3\cdot( 2 \cdot 3 )$$ 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 $$-1$$ $$4$$ $$6$$ $$0$$
$$(2)$$ $$4$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$
$$(a+3)$$ $$7$$ $$6$$ $$I_{6}$$ Split multiplicative $$-1$$ $$1$$ $$6$$ $$6$$

## 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 2268.1-v consists of curves linked by isogenies of degrees dividing 9.

## Base change

This curve is the base change of 378.e2, 2646.f2, defined over $$\Q$$, so it is also a $$\Q$$-curve.