ASKSAGE: Sage Q&A Forum - RSS feedhttps://ask.sagemath.org/questions/Q&A Forum for SageenCopyright Sage, 2010. Some rights reserved under creative commons license.Sun, 23 May 2021 19:37:31 +0200Computing the endomorphism ring of an elliptic curve over a finite fieldhttps://ask.sagemath.org/question/57238/computing-the-endomorphism-ring-of-an-elliptic-curve-over-a-finite-field/$
\newcommand{\End}{\mathrm{End}}
\newcommand{\Gal}{\mathrm{Gal}}
\newcommand{\kb}{\overline{k}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\Q}{\mathbb{Q}}
$
I would like to have an algorithm (possibly very inefficient) that computes
the endomorphism ring of a given elliptic curve $E$ over a finite field $k$.
For simplicity, we shall assume that $E$ is ordinary (to avoid maximal orders
in quaternion algebras...), so it is enough to compute the conductor of $\End(E)$
in the imaginary quadratic field $K := \Q(\pi)$, where $q = |k|$
and $\pi = \sqrt{a_q^2 - 4q}$. We know that this conductor divides $[O_K : \Z[\pi]]$,
the latter being quite easy to compute in Sage I suppose.
But now, is there a way to check whether, for a given $f \mid [O_K : \Z[\pi]]$,
we have $\Z + f O_K = \End(E)$ ? This is where I don't know how to proceed.
I am aware of [Kohel's thesis](http://iml.univ-mrs.fr/~kohel/pub/thesis.pdf),
which involves isogeny graphs, but I'm not sure if one can implement this in Sage easily.
Ideally, I want to reproduce the table on page 303
in [Edixhoven, van der Geer and Moonen's *Abelian Varieties*](http://van-der-geer.nl/~gerard/AV.pdf),
which lists $\End_{\Bbb F_7}(E)$ for all (isomorphism classes of)
elliptic curves over $\Bbb F_7$.Sun, 23 May 2021 11:02:33 +0200https://ask.sagemath.org/question/57238/computing-the-endomorphism-ring-of-an-elliptic-curve-over-a-finite-field/Comment by David Loeffler for <p>$
\newcommand{\End}{\mathrm{End}}
\newcommand{\Gal}{\mathrm{Gal}}
\newcommand{\kb}{\overline{k}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\Q}{\mathbb{Q}}
$</p>
<p>I would like to have an algorithm (possibly very inefficient) that computes
the endomorphism ring of a given elliptic curve $E$ over a finite field $k$. </p>
<p>For simplicity, we shall assume that $E$ is ordinary (to avoid maximal orders
in quaternion algebras...), so it is enough to compute the conductor of $\End(E)$
in the imaginary quadratic field $K := \Q(\pi)$, where $q = |k|$
and $\pi = \sqrt{a_q^2 - 4q}$. We know that this conductor divides $[O_K : \Z[\pi]]$,
the latter being quite easy to compute in Sage I suppose.</p>
<p>But now, is there a way to check whether, for a given $f \mid [O_K : \Z[\pi]]$,
we have $\Z + f O_K = \End(E)$ ? This is where I don't know how to proceed. </p>
<p>I am aware of <a href="http://iml.univ-mrs.fr/~kohel/pub/thesis.pdf">Kohel's thesis</a>,
which involves isogeny graphs, but I'm not sure if one can implement this in Sage easily.</p>
<p>Ideally, I want to reproduce the table on page 303
in <a href="http://van-der-geer.nl/~gerard/AV.pdf">Edixhoven, van der Geer and Moonen's <em>Abelian Varieties</em></a>,
which lists $\End_{\Bbb F_7}(E)$ for all (isomorphism classes of)
elliptic curves over $\Bbb F_7$.</p>
https://ask.sagemath.org/question/57238/computing-the-endomorphism-ring-of-an-elliptic-curve-over-a-finite-field/?comment=57243#post-id-57243Now answered on math.SE.Sun, 23 May 2021 19:37:31 +0200https://ask.sagemath.org/question/57238/computing-the-endomorphism-ring-of-an-elliptic-curve-over-a-finite-field/?comment=57243#post-id-57243Comment by Watson for <p>$
\newcommand{\End}{\mathrm{End}}
\newcommand{\Gal}{\mathrm{Gal}}
\newcommand{\kb}{\overline{k}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\Q}{\mathbb{Q}}
$</p>
<p>I would like to have an algorithm (possibly very inefficient) that computes
the endomorphism ring of a given elliptic curve $E$ over a finite field $k$. </p>
<p>For simplicity, we shall assume that $E$ is ordinary (to avoid maximal orders
in quaternion algebras...), so it is enough to compute the conductor of $\End(E)$
in the imaginary quadratic field $K := \Q(\pi)$, where $q = |k|$
and $\pi = \sqrt{a_q^2 - 4q}$. We know that this conductor divides $[O_K : \Z[\pi]]$,
the latter being quite easy to compute in Sage I suppose.</p>
<p>But now, is there a way to check whether, for a given $f \mid [O_K : \Z[\pi]]$,
we have $\Z + f O_K = \End(E)$ ? This is where I don't know how to proceed. </p>
<p>I am aware of <a href="http://iml.univ-mrs.fr/~kohel/pub/thesis.pdf">Kohel's thesis</a>,
which involves isogeny graphs, but I'm not sure if one can implement this in Sage easily.</p>
<p>Ideally, I want to reproduce the table on page 303
in <a href="http://van-der-geer.nl/~gerard/AV.pdf">Edixhoven, van der Geer and Moonen's <em>Abelian Varieties</em></a>,
which lists $\End_{\Bbb F_7}(E)$ for all (isomorphism classes of)
elliptic curves over $\Bbb F_7$.</p>
https://ask.sagemath.org/question/57238/computing-the-endomorphism-ring-of-an-elliptic-curve-over-a-finite-field/?comment=57240#post-id-57240@slelievre : thanks for the update, and for your comment! I actually asked the same question there (https://math.stackexchange.com/questions/4147940), but I might have more chance to get an answer here.Sun, 23 May 2021 11:21:56 +0200https://ask.sagemath.org/question/57238/computing-the-endomorphism-ring-of-an-elliptic-curve-over-a-finite-field/?comment=57240#post-id-57240Comment by slelievre for <p>$
\newcommand{\End}{\mathrm{End}}
\newcommand{\Gal}{\mathrm{Gal}}
\newcommand{\kb}{\overline{k}}
\newcommand{\Z}{\mathbb{Z}}
\newcommand{\F}{\mathbb{F}}
\newcommand{\Q}{\mathbb{Q}}
$</p>
<p>I would like to have an algorithm (possibly very inefficient) that computes
the endomorphism ring of a given elliptic curve $E$ over a finite field $k$. </p>
<p>For simplicity, we shall assume that $E$ is ordinary (to avoid maximal orders
in quaternion algebras...), so it is enough to compute the conductor of $\End(E)$
in the imaginary quadratic field $K := \Q(\pi)$, where $q = |k|$
and $\pi = \sqrt{a_q^2 - 4q}$. We know that this conductor divides $[O_K : \Z[\pi]]$,
the latter being quite easy to compute in Sage I suppose.</p>
<p>But now, is there a way to check whether, for a given $f \mid [O_K : \Z[\pi]]$,
we have $\Z + f O_K = \End(E)$ ? This is where I don't know how to proceed. </p>
<p>I am aware of <a href="http://iml.univ-mrs.fr/~kohel/pub/thesis.pdf">Kohel's thesis</a>,
which involves isogeny graphs, but I'm not sure if one can implement this in Sage easily.</p>
<p>Ideally, I want to reproduce the table on page 303
in <a href="http://van-der-geer.nl/~gerard/AV.pdf">Edixhoven, van der Geer and Moonen's <em>Abelian Varieties</em></a>,
which lists $\End_{\Bbb F_7}(E)$ for all (isomorphism classes of)
elliptic curves over $\Bbb F_7$.</p>
https://ask.sagemath.org/question/57238/computing-the-endomorphism-ring-of-an-elliptic-curve-over-a-finite-field/?comment=57239#post-id-57239Welcome to Ask Sage!
Thank you for your question.Sun, 23 May 2021 11:16:41 +0200https://ask.sagemath.org/question/57238/computing-the-endomorphism-ring-of-an-elliptic-curve-over-a-finite-field/?comment=57239#post-id-57239