Abstract
We define two families of sextics. By computer search on one family, we find new curves of genus 5 attaining the Hasse–Weil–Serre bound over \(\mathbb {F}_{71}\), \(\mathbb {F}_{191}\) and \(\mathbb {F}_{11^5}\), and we update 3 entries of genus 5 in manYPoints.org. Among another family, we find new curves of genus 7 attaining the Hasse–Weil–Serre bound over \(\mathbb {F}_{p^3}\) for some primes p. We determine the precise condition on the finite field over which the sextics attain the Hasse–Weil–Serre bound.
Partially supported by JSPS Grant-in-Aid for Scientific Research (C) 17K05344.
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Acknowledgements
The author wishes to express her thanks to Massimo Giulietti, Gary McGuire, Maria Montanucci and Carlos Moreno for their valuable comments on this research.
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Kawakita, M.Q. (2018). Some Sextics of Genera Five and Seven Attaining the Serre Bound. In: Budaghyan, L., Rodríguez-Henríquez, F. (eds) Arithmetic of Finite Fields. WAIFI 2018. Lecture Notes in Computer Science(), vol 11321. Springer, Cham. https://doi.org/10.1007/978-3-030-05153-2_15
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