  In this paper, we present efficient algorithms for computing the number of
points and the order of the Jacobian group of a superelliptic curve over finite
fields of prime order p. Our method employs the Hasse-Weil bounds in
conjunction with the Hasse-Witt matrix for superelliptic curves, whose entries
we express in terms of multinomial coefficients. We present a fast algorithm
for counting points on specific trinomial superelliptic curves and a slower,
more general method for all superelliptic curves. For the first case, we reduce
the problem of simplifying the entries of the Hasse-Witt matrix modulo p to a
problem of solving quadratic Diophantine equations. For the second case, we
extend Bostan et al.'s method for hyperelliptic curves to general superelliptic
curves. We believe the methods we describe are asymptotically the most
efficient known point-counting algorithms for certain families of trinomial
superelliptic curves.
