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In this work we study the properties of maximal and minimal curves of genus 3over finite fields with discriminant -19. We prove that any such curve can begiven by an explicit equation of certain form. Using these equations we obtaina table of maximal and minimal curves over finite fields with discriminant -19of cardinality up to 997. We also show that existence of a maximal curveimplies that there is no minimal curve and vice versa.

Optimal Curves of Genus 3 over Finite Fields with Discriminant -19