Isomorphy classes of finite order automorphisms of SL(2,k)

In this talk, we consider the order m k-automorphisms of Sl(2,k). We first characterize the form that order $m$ k-automorphisms of Sl(2,k) will take and then we find simple conditions on matrices $A$ and $B$ involving eigenvalues and the field that the entries of $A$ and $B$ lie in that are equivalent to isomorphy of the order $m$ k-automorphism $Inn_A$ and $Inn_B$. We examine the number of isomorphy classes, and consider a few examples.