On the Dickson–Guralnick–Zieve curve

The Dickson–Guralnick–Zieve curve, briefly DGZ curve, defined over the finite field Fq arises naturally from the classical Dickson invariant of the projective linear group PGL(3,Fq). The DGZ curve is an (absolutely irreducible, singular) plane curve of degree q3−q2 and genus [Formula presented]q(q−1)(q3−2q−2)+1. In this paper we show that the DGZ curve has several remarkable features, those appearing most interesting are: the DGZ curve has a large automorphism group compared to its genus albeit its Hasse–Witt invariant is positive; the Fermat curve of degree q−1 is a quotient curve of the DGZ curve; among the plane curves with the same degree and genus of the DGZ curve and defined over Fq3, the DGZ curve is optimal with respect the number of its Fq3-rational points.