  In this paper we determine the group of rational automorphisms of binary
cubic and quartic forms with integer coefficients and non-zero discriminant in
terms of certain quadratic covariants of cubic and quartic forms. This allows
one to give precise asymptotic formulae for the number of integers in an
interval representable by a binary cubic or quartic form and extends work of
Hooley. Further, we give the field of definition of lines contained in certain
cubic and quartic surfaces related to binary cubic and quartic forms.
