# Properties

 Label 2.2.21.1-324.1-b3 Base field $$\Q(\sqrt{21})$$ Conductor $$(18)$$ Conductor norm $$324$$ CM no Base change yes: 54.a3,2646.bd3 Q-curve yes Torsion order $$3$$ 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: x = polygen(QQ); K.<a> = NumberField(x^2 - x - 5)

gp: K = nfinit(a^2 - a - 5);

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

## Weierstrass equation

$$y^2+axy+y=x^{3}+\left(-a+1\right)x^{2}+\left(-8a+21\right)x-18a+50$$
sage: E = EllipticCurve(K, [a, -a + 1, 1, -8*a + 21, -18*a + 50])

gp: E = ellinit([a, -a + 1, 1, -8*a + 21, -18*a + 50],K)

magma: E := ChangeRing(EllipticCurve([a, -a + 1, 1, -8*a + 21, -18*a + 50]),K);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(18)$$ = $$\left(2\right) \cdot \left(-a + 2\right)^{4}$$ sage: E.conductor()  magma: Conductor(E); Conductor norm: $$324$$ = $$3^{4} \cdot 4$$ sage: E.conductor().norm()  magma: Norm(Conductor(E)); Discriminant: $$(216)$$ = $$\left(2\right)^{3} \cdot \left(-a + 2\right)^{6}$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$46656$$ = $$3^{6} \cdot 4^{3}$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{9261}{8}$$ 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: $$\Z/3\Z$$ sage: T = E.torsion_subgroup(); T.invariants()  gp: T = elltors(E); T[2]  magma: T,piT := TorsionSubgroup(E); Invariants(T); Torsion generator: $\left(0 : -2 a + 5 : 1\right)$ 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: $$13.2895953592672$$ Tamagawa product: $$9$$  =  $$3\cdot3$$ Torsion order: $$3$$ Leading coefficient: $$2.90002746138933$$ 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$$(j)_{-}$$
$$\left(-a + 2\right)$$ $$3$$ $$3$$ $$IV$$ Additive $$-1$$ $$4$$ $$6$$ $$0$$
$$\left(2\right)$$ $$4$$ $$3$$ $$I_{3}$$ Split multiplicative $$-1$$ $$1$$ $$3$$ $$3$$

## 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 324.1-b consists of curves linked by isogenies of degrees dividing 9.

## Base change

This curve is the base change of elliptic curves 54.a3, 2646.bd3, defined over $$\Q$$, so it is also a $$\Q$$-curve.