%
% elliptic.tex -- elliptic curve
%
% (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule
%
\documentclass[tikz]{standalone}
\usepackage{amsmath}
\usepackage{times}
\usepackage{txfonts}
\usepackage{pgfplots}
\usepackage{csvsimple}
\usetikzlibrary{arrows,intersections,math}
\begin{document}
\def\skala{4}
\begin{tikzpicture}[>=latex,thick,scale=\skala]

\def\uone{-1.1}
\def\utwo{0.3}
\pgfmathparse{-(\uone+\utwo)}
\xdef\uthree{\pgfmathresult}
\xdef\r{0.017}

\def\gone{-1.05}
\def\gtwo{0.2}
\def\gthree{1.105}

\pgfmathparse{-sqrt((\gone-\uone)*(\gone-\utwo)*(\gone-\uthree))}
\xdef\yone{\pgfmathresult}
\pgfmathparse{sqrt((\gtwo-\uone)*(\gtwo-\utwo)*(\gtwo-\uthree))}
\xdef\ytwo{\pgfmathresult}
\pgfmathparse{sqrt((\gthree-\uone)*(\gthree-\utwo)*(\gthree-\uthree))}
\xdef\ythree{\pgfmathresult}

\begin{scope}
\clip (-1.5,-1.5) rectangle (1.5,1.5);
\draw[color=blue]
	({\gone-(\gtwo-\gone)},{\yone-(\ytwo-\yone)})
	--
	({\gone+2*(\gtwo-\gone)},{\yone+2*(\ytwo-\yone)});
\draw[color=blue] (\gthree,-2) -- (\gthree,2);
\end{scope}

\draw[line width=1.4pt,color=red]
	(\uone,0) --
	plot[domain={\uone+0.001}:{\utwo-0.001},samples=100]
		(\x,{sqrt((\x-\uone)*(\x-\utwo)*(\x-\uthree))})
	-- (\utwo,0);
\draw[line width=1.4pt,color=red]
	(\uone,0) --
	plot[domain={\uone+0.001}:{\utwo-0.001},samples=100]
		(\x,{-sqrt((\x-\uone)*(\x-\utwo)*(\x-\uthree))})
	-- (\utwo,0);

\draw[line width=1.4pt,color=red]
	(\uthree,0) --
	plot[domain={\uthree}:1.5,samples=100]
		(\x,{sqrt((\x-\uone)*(\x-\utwo)*(\x-\uthree))}) ;
\draw[line width=1.4pt,color=red]
	(\uthree,0) --
	plot[domain={\uthree}:1.5,samples=100]
		(\x,{-sqrt((\x-\uone)*(\x-\utwo)*(\x-\uthree))}) ;

\draw[->] (-1.5,0) -- (1.5,0) coordinate[label={$u$}];
\draw[->] (0,-1.5) -- (0,1.5) coordinate[label={right:$v$}];
\node at (0,0) [below left] {$O$};

\fill[color=white] (\uone,0) circle[radius=\r];
\draw (\uone,0) circle[radius=\r];
\node at ({\uone+0.01},-0.01) [above left] {$u_1$};

\fill[color=white] (\utwo,0) circle[radius=\r];
\draw (\utwo,0) circle[radius=\r];
\node at ({\utwo-0.01},-0.01) [above right] {$u_2$};

\fill[color=white] (\uthree,0) circle[radius=\r];
\draw (\uthree,0) circle[radius=\r];
\node at ({\uthree+0.01},-0.01) [above left] {$u_3$};

\fill[color=white] (\gone,\yone) circle[radius=\r];
\draw[color=blue] (\gone,\yone) circle[radius=\r];

\fill[color=white] (\gtwo,\ytwo) circle[radius=\r];
\draw[color=blue] (\gtwo,\ytwo) circle[radius=\r];

\fill[color=white] (\gthree,\ythree) circle[radius=\r];
\draw[color=blue] (\gthree,\ythree) circle[radius=\r];
\fill[color=white] (\gthree,-\ythree) circle[radius=\r];
\draw[color=blue] (\gthree,-\ythree) circle[radius=\r];

\node[color=blue] at (\gone,{\yone-0.03}) [above left] {$g_1$};
\node[color=blue] at ({\gtwo+0.03},{\ytwo+0.01}) [above] {$g_2$};
\node[color=blue] at (\gthree,{\ythree+0.02}) [below right] {$g_3$};
\node[color=blue] at (\gthree,{-\ythree+0.03}) [below left] {$g_1g_2=g_3^{-1}$};

\end{tikzpicture}
\end{document}