Entry details for q = 72 = 49, g = 4
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Lower bound Nmin = 102

Submitted by Everett Howe
Date 02/16/2014
Reference Everett W. Howe
Quickly constructing curves of genus 4 with many points
Frobenius Distributions: Sato–Tate and Lang–Trotter conjectures (D. Kohel and I. Shparlinski, eds.), Contemporary Mathematics 663, American Mathematical Society, Providence, RI, 2016, pp. 149–173
Comments
Explicit curve: y^2 = x^3 + 3*x, z^2 = x^3 + x^2 + 6*x + 1.
Tags Explicit curves

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Explicit Curve     
S.E.Fischer
02/24/2017 20:56
Another equation is given by
x^3*y^3 + x^6 + y^6 + 5*x^2*y^2 + x*y + 6 = 0.
Upper bound Nmax = 102

Submitted by Everett Howe
Date 07/16/2016
Reference Momonari Kudo, Shushi Harashita
Superspecial curves of genus $4$ in small characteristic
arxiv.org/abs/1607.01114
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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