# Properties

 Label 2.2.21.1-17.1-a1 Base field $$\Q(\sqrt{21})$$ Conductor $$\left(-2a + 3\right)$$ Conductor norm $$17$$ CM no Base change no Q-curve no Torsion order $$3$$ Rank $$0$$

# Learn more about

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

$$y^2+\left(a+1\right)xy+\left(a+1\right)y=x^{3}-ax^{2}-7x-2a+2$$
sage: E = EllipticCurve([K([1,1]),K([0,-1]),K([1,1]),K([-7,0]),K([2,-2])])

gp: E = ellinit([Pol(Vecrev([1,1])),Pol(Vecrev([0,-1])),Pol(Vecrev([1,1])),Pol(Vecrev([-7,0])),Pol(Vecrev([2,-2]))], K);

magma: E := EllipticCurve([K![1,1],K![0,-1],K![1,1],K![-7,0],K![2,-2]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$\left(-2a + 3\right)$$ = $$\left(-2a + 3\right)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$17$$ = $$17$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$( 2 a - 3 )$$ = $$\left(-2a + 3\right)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$17$$ = $$17$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{20811}{17} a - \frac{19591}{17}$$ 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: $\left(0 : -2 : 1\right)$ 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: $$45.1644838550263$$ Tamagawa product: $$1$$ Torsion order: $$3$$ Leading coefficient: $$1.09507759782682$$ 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$$(j)_{-}$$
$$\left(-2a + 3\right)$$ $$17$$ $$1$$ $$I_{1}$$ Non-split multiplicative $$1$$ $$1$$ $$1$$ $$1$$

## 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.
Its isogeny class 17.1-a consists of curves linked by isogenies of degree 3.

## Base change

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