Abderrahmane Nitaj
Journal of Computational Mathematics. 2002, 20(4): 337-348.
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
$$E:y^2+a_1xy+a_3y=x^2+a_2x^2+a_4x+a_6.$$
Let Q be the set of rationals. E issaid to be difined over Q if the coefficients a_i, i=1,2,3,4,6 are rationals and O is defined over Q.\par Let E/Q be an elliptic curve and let $E(Q)_{tors}$ is one of the following 15 groups $$E(Q)_{tors}=\left\{
\begin{array}{ll}
Z/mZ, & m=1,2,\ldots,10,12 \ Z/2Z\times Z/2Z, & m=1,2,3,4.
\end{array}
\right.$$We say that an elliptic curve E'/Q is isogenous to the elliptic curve E if there os an isogeny, i.e. a morphism $\phi:E\rightarrow E'$ such that $\phi(O)=O$ , where O is the point at infinity.\par We give an expicit model of all elliptic curves for which $E(Q)_{tors}$ is in the form $Z/mZ$ where m=9,10,12 or $Z/2Z\times Z/2Z$ 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 rationsl points.
