Optimal curves over finite fields with discriminant −19

In this work we study the properties of maximal and minimal curves of genus 3 over finite fields with discriminant −19. We prove that any such curve can be given by an explicit equation of certain form (see Theorem 5.1). Using these equations we obtain a table of maximal and minimal curves over prime finite fields with discriminant −19 of cardinality up to 997. We also show that existence of a maximal curve implies that there is no minimal curve and vice versa.