A note on some Picard curves over finite fields

In this note we provide a complete classification for the Newton polygon of the Picard curvesy3=x4−xandy3=x4−1 defined over a finite field of characteristic p>3p>3. In fact, we complete the results obtained in [19]. Using these results, we determine all finite fields over which these curves attain the Hasse–Weil upper bound.