examples of elliptic curves with complex multiplicationExamplesOfEllipticCurvesWithComplexMultiplication2004-05-26 10:12:372004-05-26 10:12:37Examplecomplex multiplication% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsthm}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
%\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newtheorem{thm}{Theorem}
\newtheorem{defn}{Definition}
\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{cor}{Corollary}
% Some sets
\newcommand{\Nats}{\mathbb{N}}
\newcommand{\Ints}{\mathbb{Z}}
\newcommand{\Reals}{\mathbb{R}}
\newcommand{\Complex}{\mathbb{C}}
\newcommand{\Rats}{\mathbb{Q}}Here we show some elliptic curves defined over $\Rats$ which have complex multiplication by a quadratic imaginary field $K$ of class number $1$ (with $\operatorname{End}(E)$ exactly isomorphic to the full ring of integers $\mathcal{O}_K$).
\begin{center}
\begin{tabular}{|c|c|}
\hline
% after \\: \hline or \cline{col1-col2} \cline{col3-col4} ...
$K$ & {\bf Curve} \\
\hline
$\Rats(\sqrt{-1})$ & $y^2=x^3+x$ \\
$\Rats(\sqrt{-2})$ & $y^2=x^3+4x^2+2x$ \\
$\Rats(\sqrt{-3})$ & $y^2+y=x^3$ \\
$\Rats(\sqrt{-7})$ & $y^2+xy=x^3-x^2-2x-1$ \\
$\Rats(\sqrt{-11})$ & $y^2+y=x^3-x^2-7x+10$ \\
$\Rats(\sqrt{-19})$ & $y^2+y=x^3-38x+90$ \\
$\Rats(\sqrt{-43})$ & $y^2+y=x^3-860x+9707$ \\
$\Rats(\sqrt{-67})$ & $y^2+y=x^3-7370x+243528$ \\
$\Rats(\sqrt{-163})$ & $y^2+y=x^3-2174420x+1234136692$ \\
\hline
\end{tabular}
\end{center}