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# The Maximum Number of Points on a Curve of Genus $4$ over $\mathbb{F}_8$ is $25$

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Published:2003-04-01
Printed: Apr 2003
• David Savitt
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## Abstract

We prove that the maximum number of rational points on a smooth, geometrically irreducible genus 4 curve over the field of 8 elements is 25. The body of the paper shows that 27 points is not possible by combining techniques from algebraic geometry with a computer verification. The appendix shows that 26 points is not possible by examining the zeta functions.
 MSC Classifications: 11G20 - Curves over finite and local fields [See also 14H25] 14H25 - Arithmetic ground fields [See also 11Dxx, 11G05, 14Gxx]

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