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\begin{document}

\title{Algebra 6.62}
\author{Michael Vaughan-Lee}
\date{June 2013}
\maketitle

Algebra 6.62 has two parameters $x,y$, where $x,y$ are integers with $y\neq 0%
\func{mod}p$. Parameter pairs $(x,y)$ and $(z,t)$ give isomorphic algebras
if and only if%
\[
\left( 
\begin{array}{ll}
1 & 0 \\ 
z & t%
\end{array}%
\right) =\left( 
\begin{array}{ll}
\mu  & \nu  \\ 
\omega \nu  & \mu 
\end{array}%
\right) \left( 
\begin{array}{ll}
1 & 0 \\ 
x & y%
\end{array}%
\right) \left( 
\begin{array}{ll}
\mu +\nu x & \nu y \\ 
\omega \nu y & \mu +\nu x%
\end{array}%
\right) ^{-1}\func{mod}p
\]%
for some matrix $\left( 
\begin{array}{ll}
\mu  & \nu  \\ 
\omega \nu  & \mu 
\end{array}%
\right) $ with determinant coprime to $p$. (Here, as elsewhere, $\omega $ is
a primitive element modulo $p$.) So we need to compute representatives for
the orbits of non-singular matrices $\left( 
\begin{array}{ll}
1 & 0 \\ 
x & y%
\end{array}%
\right) \in \,$GL$(2,p)$ under the action of the group of non-singular
matrices $\left( 
\begin{array}{ll}
\mu  & \nu  \\ 
\omega \nu  & \mu 
\end{array}%
\right) \in \,$GL$(2,p)$ given above. There are $p$ orbits.

It is easy enough to generate the $p$ orbit representatives with a simple
loop over all non-singular matrices $\left( 
\begin{array}{ll}
\mu & \nu \\ 
\omega \nu & \mu%
\end{array}%
\right) $ and $\left( 
\begin{array}{ll}
1 & 0 \\ 
x & y%
\end{array}%
\right) $. However this method has complexity $p^{4}$ for output of size $p$%
, which is not very satisfactory! Can we do better? Multiplying $\left( 
\begin{array}{ll}
\mu & \nu \\ 
\omega \nu & \mu%
\end{array}%
\right) $ through by a non-zero constant has no effect on the action, so we
can assume that $\mu =0,1$, and that if $\mu =0$ then $\nu =1$. This reduces
the complexity to $p^{3}$.

\end{document}