ISOGENOUS OF THE ELLIPTIC CURVES OVER THE RATIONALS

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 $$E(mathbb{Q})_{tors}=begin{cases} mathbb{Z}/mmathbb{Z}, & m=1,2,ldots,10,12 mathbb{Z}/2mathbb{Z}times mathbb{Z}/ 2m mathbb{Z}, & m=1,2,3,4.end{cases}.$$ 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 $phi:Erightarrow E'$ 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 $mathbb{Z}/mmathbb{Z}$ 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.