Volume 20, Issue 4
Isogenous of the Elliptic Curves Over the Rationals
DOI:

J. Comp. Math., 20 (2002), pp. 337-348

Published online: 2002-08

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• Abstract

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.

• Keywords

Courbe elliptique Isogenie

• AMS Subject Headings

@Article{JCM-20-337, author = {}, title = {Isogenous of the Elliptic Curves Over the Rationals}, journal = {Journal of Computational Mathematics}, year = {2002}, volume = {20}, number = {4}, pages = {337--348}, abstract = { 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. }, issn = {1991-7139}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/jcm/8922.html} }
TY - JOUR T1 - Isogenous of the Elliptic Curves Over the Rationals JO - Journal of Computational Mathematics VL - 4 SP - 337 EP - 348 PY - 2002 DA - 2002/08 SN - 20 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/jcm/8922.html KW - Courbe elliptique KW - Isogenie AB - 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.