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

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 $$E:y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ Let $\mathbb{Q}$ be the set of rationals. $E$ is said to be defined over $\mathbb{Q}$ if the coefficients $a_i, i=1,2,3,4,6$ are rationals and $O$ is defined over $\mathbb{Q}$.

Let $E/ \mathbb{Q}$ be an elliptic curve and let $E(\mathbb{Q})_{tors}$ be the torsion group of points of $E$ defined over $\mathbb{Q}$. The theorem of Mazur asserts that $E (\mathbb{Q})_{tors}$ is one of the following 15 groups $$E(\mathbb{Q})_{tors}=\begin{cases} \mathbb{Z}/m\mathbb{Z}, & m=1,2,\ldots,10,12 \\ \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/ 2m \mathbb{Z}, & m=1,2,3,4.\end{cases}.$$ We say that an elliptic curve $E'/\mathbb{Q}$ is isogenous to the elliptic curve $E$ if there is an isogeny, i.e. a morphism $\phi:E\rightarrow E'$ such that $\phi(O)=O$ , where $O$ is the point at infinity.

We give an explicit model of all elliptic curves for which $E(\mathbb{Q})_{tors}$ is in the form $\mathbb{Z}/m\mathbb{Z}$ where $m$ = 9,10,12 or $\mathbb{Z}/ 2 \mathbb{Z}\times \mathbb{Z}/ 2m \mathbb{Z} \ {\rm where} \ m=4$, 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.

• 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 $$E:y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ Let $\mathbb{Q}$ be the set of rationals. $E$ is said to be defined over $\mathbb{Q}$ if the coefficients $a_i, i=1,2,3,4,6$ are rationals and $O$ is defined over $\mathbb{Q}$.

Let $E/ \mathbb{Q}$ be an elliptic curve and let $E(\mathbb{Q})_{tors}$ be the torsion group of points of $E$ defined over $\mathbb{Q}$. The theorem of Mazur asserts that $E (\mathbb{Q})_{tors}$ is one of the following 15 groups $$E(\mathbb{Q})_{tors}=\begin{cases} \mathbb{Z}/m\mathbb{Z}, & m=1,2,\ldots,10,12 \\ \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/ 2m \mathbb{Z}, & m=1,2,3,4.\end{cases}.$$ We say that an elliptic curve $E'/\mathbb{Q}$ is isogenous to the elliptic curve $E$ if there is an isogeny, i.e. a morphism $\phi:E\rightarrow E'$ such that $\phi(O)=O$ , where $O$ is the point at infinity.

We give an explicit model of all elliptic curves for which $E(\mathbb{Q})_{tors}$ is in the form $\mathbb{Z}/m\mathbb{Z}$ where $m$ = 9,10,12 or $\mathbb{Z}/ 2 \mathbb{Z}\times \mathbb{Z}/ 2m \mathbb{Z} \ {\rm where} \ m=4$, 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.

}, 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, 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 $$E:y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6.$$ Let $\mathbb{Q}$ be the set of rationals. $E$ is said to be defined over $\mathbb{Q}$ if the coefficients $a_i, i=1,2,3,4,6$ are rationals and $O$ is defined over $\mathbb{Q}$.

Let $E/ \mathbb{Q}$ be an elliptic curve and let $E(\mathbb{Q})_{tors}$ be the torsion group of points of $E$ defined over $\mathbb{Q}$. The theorem of Mazur asserts that $E (\mathbb{Q})_{tors}$ is one of the following 15 groups $$E(\mathbb{Q})_{tors}=\begin{cases} \mathbb{Z}/m\mathbb{Z}, & m=1,2,\ldots,10,12 \\ \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/ 2m \mathbb{Z}, & m=1,2,3,4.\end{cases}.$$ We say that an elliptic curve $E'/\mathbb{Q}$ is isogenous to the elliptic curve $E$ if there is an isogeny, i.e. a morphism $\phi:E\rightarrow E'$ such that $\phi(O)=O$ , where $O$ is the point at infinity.

We give an explicit model of all elliptic curves for which $E(\mathbb{Q})_{tors}$ is in the form $\mathbb{Z}/m\mathbb{Z}$ where $m$ = 9,10,12 or $\mathbb{Z}/ 2 \mathbb{Z}\times \mathbb{Z}/ 2m \mathbb{Z} \ {\rm where} \ m=4$, 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.

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