# Properties

 Label 2.2.21.1-1936.1-a2 Base field $$\Q(\sqrt{21})$$ Conductor norm $$1936$$ CM no Base change yes Q-curve yes Torsion order $$3$$ Rank $$1$$

# Learn more

Show commands: Magma / PariGP / 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(Polrev([-5, -1, 1]));

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

## Weierstrass equation

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

gp: E = ellinit([Polrev([0,0]),Polrev([1,0]),Polrev([0,0]),Polrev([3,0]),Polrev([-1,0])], K);

magma: E := EllipticCurve([K![0,0],K![1,0],K![0,0],K![3,0],K![-1,0]]);

This is a global minimal model.

sage: E.is_global_minimal_model()

## Invariants

 Conductor: $$(44)$$ = $$(2)^{2}\cdot(11)$$ sage: E.conductor()  gp: ellglobalred(E)[1]  magma: Conductor(E); Conductor norm: $$1936$$ = $$4^{2}\cdot121$$ sage: E.conductor().norm()  gp: idealnorm(ellglobalred(E)[1])  magma: Norm(Conductor(E)); Discriminant: $$(-2816)$$ = $$(2)^{8}\cdot(11)$$ sage: E.discriminant()  gp: E.disc  magma: Discriminant(E); Discriminant norm: $$7929856$$ = $$4^{8}\cdot121$$ sage: E.discriminant().norm()  gp: norm(E.disc)  magma: Norm(Discriminant(E)); j-invariant: $$\frac{8192}{11}$$ 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 $\left(\frac{10}{27} : -\frac{58}{243} a + \frac{29}{243} : 1\right)$ Height $$3.9980292544271781596578159653549397233$$ 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: $\left(1 : -2 : 1\right)$ 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: $$3.9980292544271781596578159653549397233$$ Period: $$5.8271008329690240617443803756006953776$$ Tamagawa product: $$3$$  =  $$3\cdot1$$ Torsion order: $$3$$ Leading coefficient: $$3.3892030958849423310662473537048783567$$ 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$$(j)_{-}$$
$$(2)$$ $$4$$ $$3$$ $$IV^{*}$$ Additive $$1$$ $$2$$ $$8$$ $$0$$
$$(11)$$ $$121$$ $$1$$ $$I_{1}$$ 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 1936.1-a consists of curves linked by isogenies of degree 3.

## Base change

This elliptic curve is a $$\Q$$-curve. It is the base change of the following 2 elliptic curves:

Base field Curve
$$\Q$$ 44.a2
$$\Q$$ 19404.g2