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

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Tags 
Explicit curves 
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Explicit examples

Everett Howe
04/16/2010 06:48

While searching for genus4 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).


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