document: Isogenous of the elliptic curves over the rationals An elliptic curve is a pair -LRB- E , O -RRB- , 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/sup 2 / + a/sub 1/xy + a/sub 3/y = x/sup 3 / + a/sub 2/x/sup 2 / + a/sub 4/x + a/sub 6 / . Let Q be the set of rationals . E is said to be defined over Q if the coefficients a/sub 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 -LRB- Q -RRB- / sub tors / be the torsion group of points of E defined over Q . The theorem of Mazur asserts that E -LRB- Q -RRB- / sub tors / is one of the following 15 groups E -LRB- Q -RRB- / sub tors / -LCB- Z/mZ , Z/mZ * Z/2mZ , m , = 1 , 2 , ... , 10 , 12 , m = 1 , 2 , 3 , 4 . We say that an elliptic curve E ' / Q is isogenous to the elliptic curve E if there is an isogeny , i.e. a morphism phi : E to E ' such that phi -LRB- O -RRB- = O , where O is the point at infinity . We give an explicit model of all elliptic curves for which E -LRB- Q -RRB- / sub tors / is in the form Z/mZ where m = 9,10,12 or Z/2Z * Z/2mZ where m = 4 , according to Mazur 's theorem . Moreover , for every family of such elliptic curves , we give an explicit model of all their isogenous curves with cyclic kernels consisting of rational points
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