Isogenous of the Elliptic Curves over the Rationals

Abstract

An elliptic curve is a pair $(E,O),$ where $E$ is a smooth projective curve of genus 1 and $O$ is a point of $E$, called the point at infinity. Every elliptic curve can be given by a Weierstrass equation $$E:y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ Let $\mathbb{Q}$ be the set of rationals. $E$ is said to be defined over $\mathbb{Q}$ if the coefficients $a_i, i=1,2,3,4,6$ are rationals and $O$ is defined over $\mathbb{Q}$.

Let $E/ \mathbb{Q}$ be an elliptic curve and let $E(\mathbb{Q})_{tors}$ be the torsion group of points of $E$ defined over $\mathbb{Q}$. The theorem of Mazur asserts that $E (\mathbb{Q})_{tors}$ is one of the following 15 groups $$E(\mathbb{Q})_{tors}=\begin{cases} \mathbb{Z}/m\mathbb{Z}, & m=1,2,\ldots,10,12 \\ \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/ 2m \mathbb{Z}, & m=1,2,3,4.\end{cases}.$$ We say that an elliptic curve $E'/\mathbb{Q}$ is isogenous to the elliptic curve $E$ if there is an isogeny, i.e. a morphism $\phi:E\rightarrow E'$ such that $\phi(O)=O$ , where $O$ is the point at infinity.

We give an explicit model of all elliptic curves for which $E(\mathbb{Q})_{tors}$ is in the form $\mathbb{Z}/m\mathbb{Z}$ where $m$ = 9,10,12 or $\mathbb{Z}/ 2 \mathbb{Z}\times \mathbb{Z}/ 2m \mathbb{Z} \ {\rm where} \ m=4$, according to Mazur's theorem. Morever, for every family of such elliptic curves, we give an explicit model of all their isogenous curves with cyclic kernels consisting of rationals points.