Submitted by 
Gerrit Oomens 
Date 
01011900 
Reference 
Michael Zieve Private communication, 1999

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Explicit curves 
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Unique example

Everett Howe
11032021 21:25

The Magma programs associated to the paper "New methods for bounding the number of points on curves over finite fields" by Howe and Lauter (Geometry and Arithmetic (C. Faber, G. Farkas, and R. de Jong, eds.), European Mathematical Society, 2012, pp. 173–212) show that there is only one possible isogeny class of abelian fourfolds over F_128 that could contain the Jacobian of a genus4 curve with 215 points; its real Weil polynomial is (x^2 + 43*x + 461)^2. Arguing as in Section 5 of the cited paper, we find that any such curve can actually be defined over F_2, and that its real Weil polynomial over F_2 is (x^2 + x  1)^2. The LMFDB entry for this isogeny class (https://www.lmfdb.org/Variety/Abelian/Fq/4/2/c_h_k_v) lists one Jacobian in this isogeny class, and because the database includes a complete listing of genus4 curves over F_2, we find that there is exactly one Jacobian in the isogeny class.
Therefore there is exactly one genus4 curve over F_128 with 215 points. It can be defined over F_2, and one model for it in P^3 is given by the pair of equations
x^2 + x*y + y^2 + z*t = 0
x^2*y + x*y^2 + y^2*z + y*z^2 + x^2*t + x*t^2 = 0.


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