Entry details for q = 25 = 32, g = 4
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Lower bound Nmin = 71

Submitted by Gerrit Oomens
Date 01/01/1900
Reference M. Zieve
Private communication, 1999
Comments
Tags Explicit curves

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Explicit examples     
Everett Howe
04/16/2010 06:48
While searching for genus-4 double covers of the elliptic curve E: y^2 + x*y = x^3 + x having more than 72 points, we found a number of such double covers that have 71 points.

Let r in F_32 satisfy r^5 + r^2 + 1 = 0. The double cover of E defined by z^2 + z = (r^27*x^2 + r^13*x*y + r^14*x + r^25)/(x + r^30*y + r^27) has 71 rational points, and has real Weil polynomial (x + 7) * (x + 9) * (x + 11)^2.

The double cover defined by z^2 + z = (r^26*x^2 + r^29*x*y + r^16*x + r^18)/(x + r^28*y + r^4) has 71 points and has real Weil polynomial (x + 11) * (x^3 + 27*x^2 + 239*x + 691).

The double cover defined by z^2 + z = (r^14*x^2 + r^24*x + r^18)/(x + r) has 71 points and has real Weil polynomial (x + 9)^3 * (x + 11).
Upper bound Nmax = 74

Submitted by Everett Howe
Date 04/14/2010
Reference E. W. Howe and K. E. Lauter
Improved upper bounds for the number of points on curves over finite fields
Ann. Inst. Fourier (Grenoble) 53 (2003) 1677–1737.
Comments
A real Weil polynomial we don't know how to eliminate: (x + 8) * (x + 11)^3
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