in joint work with daniel hast and joseph we count degree 3 and 4 covers of the projective line over finite fields. this is a geometric analogue of the number field side of counting cubic and quartic fields. we take a geometric approach, by using a vector bundle parametrization of these curves which is different from the recent work of manjul bhargava, arul shankar, xiaoheng wang "geometry of numbers methods over global fields: prehomogeneous vector spaces" in which the authors extend the geometry of numbers methods to global fields. our count is just for $s_3$ and $s_4$ covers, and we put the rest of the curves in our error term.