manYPoints

Submitted by	Everett Howe
Date	08-30-2011
Reference	Everett W. Howe New bounds on the maximum number of points on genus-4 curves over small finite fields Arithmetic, Geometry, Cryptography and Coding Theory (Y. Aubry, C. Ritzenthaler, and A. Zykin, eds.), Contemporary Mathematics 574, American Mathematical Society, Providence, RI, 2012, pp. 69–86
Comments	Here is a genus-4 curve with 102 points: y^2 = x^3 + x, z^2 = x^3 + x^2 + x + 4
Tags	Explicit curves

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Submitted by	Everett Howe
Date	07-16-2016
Reference	Momonari Kudo, Shushi Harashita Superspecial curves of genus $4$ in small characteristic Finite Fields Appl. 45 (2017), 131–169
Comments	Theorem B of the cited reference shows that defect 0 is impossible. Then other methods show that defects 1, 2, and 3 are also impossible — for example, the Magma programs in IsogenyClasses.magma (http://alumnus.caltech.edu/~however/papers/paper35.html) can prove this.
Tags	Analysis and enumeration

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