(untitled) (Journal of Computational Mathematics, 2002)
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 Q be the set of rationals. E is said to be defined over Q if the coefficients a_i, i=1,2,3,4,6 are rationals and O is defined over Q. Let E/ Q be an elliptic curve and let E(Q)_tors be the torsion group of points of E defined over Q. The theorem of Mazur asserts that E (Q)_tors is one of the following 15 groups E(Q)_tors=cases Z/mZ, & m=1,2,…,10,12 Z/2Z× Z/ 2m Z, & m=1,2,3,4.cases. We say that an elliptic curve E'/Q is isogenous to the elliptic curve E if there is an isogeny, i.e. a morphism φ:E→ E' such that φ(O)=O , where O is the point at infinity. We give an explicit model of all elliptic curves for which E(Q)_tors is in the form Z/mZ where m = 9,10,12 or Z/ 2 Z× Z/ 2m Z 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.