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.
- Curves over finite fields
- Explicit equations of curves over finite fields
- Optimal curves
- The Hasse-Weil-Serre bound