A Quotient Curve of the Fermat Curve of Degree Twelve Attaining the Serre Bound

Abstract

We find a new curve of genus four attaining the Serre bound over prime fields. It is defined by the equation y12=x4(1-x), which attains the bound over the finite field [Formula: see text] if and only if the prime number p satisfies p ≡ 1 mod 12, [Formula: see text] and [Formula: see text] with an integer n. Furthermore, we show that if a standard conjecture of prime numbers is true then infinitely many prime numbers satisfy these conditions.