  An algorithm that uses the cycle structure of the rows, or the columns, of a
Latin square to compute its autotopy group is introduced. As a result, a bound
for the size of the autotopy group is obtained. This bound is used to show that
the computation time for the autotopy group of Latin squares that have two rows
or two columns that map from one to the other by a permutation which decomposes
into a bounded number of disjoint cycles, is polynomial in the order $n$.
