On the curve Y^n = X^l (X^m + 1) over finite fields

Let X be the nonsingular model of a plane curve of type y^n = f (x) over the finite field F of order q^2 , where f(x) is a separable polynomial of degree coprime to n. If the number of F-rational points of X attains the Hasse-Weil bound, then the condition n divides q + 1 is equivalente to the solubility of f(x) in F. In this paper, we investigate this condition for f(x) = x^l (x^m + 1)