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

Abderrahmane Nitaj

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

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@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.
Abderrahmane Nitaj. (1970). Isogenous of the Elliptic Curves Over the Rationals. Journal of Computational Mathematics. 20 (4). 337-348. doi:
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