 By Jordan normal form, matrices are classified up to conjugacy (in GL("n",C)) by eigenvalues and nilpotence (concretely, nilpotence means where 1s occur in the Jordan blocks).Thus elements of SL(2) are classified up to conjugacy in GL(2) (or indeed SL±(2)) by trace (since determinant is fixed, and trace and determinant determine eigenvalues), except if the eigenvalues are equal, so ±I and the parabolic elements of trace +2 and trace -2 are not conjugate (the former have no off-diagonal entries in Jordan form, while the latter do).Up to conjugacy in SL(2) (instead of GL(2)), there is an additional datum, corresponding to orientation