ISOGENOUS OF THE ELLIPTIC CURVES OVER THE RATIONALS

作     者：Abderrahmane Nitaj 

作者机构：MathematikUniversitat des SaarlandesPostfach 15 1150D-66041SaarbrockenGermany 

出 版 物：《Journal of Computational Mathematics》 (计算数学（英文）)

年 卷 期：2002年第20卷第4期

页      面：337-348页

核心收录：

学科分类：07[理学] 070102[理学-计算数学] 0701[理学-数学] 

基　　金：This research was supported by the TMR programme of the European Community under contract ERBFMBICT960848 

主　　题：Courbe elliptique Isogenie 

摘      要：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 equationE:y2+a1xy+a3y=x2+a2x2+a4x+*** Q be the set of rationals. E is said to be dinned over Q if the coefficients ai, i = 1, 2, 3, 4, 6 are rationals and O is defined over *** 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 groupsWe 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 *** 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/2Z × Z/2mZ 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 rational points.