manYPoints – Table of Curves with Many Points
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 Michael 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 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 = 72

Submitted by Everett Howe
Date 04-16-2010
Reference Everett W. Howe, Kristin E. Lauter
New methods for bounding the number of points on curves over finite fields
Geometry and Arithmetic (C. Faber, G. Farkas, and R. de Jong, eds.), European Mathematical Society, 2012, pp. 173–212
Comments
The Magma routines in IsogenyClasses.magma (found at the URL linked to above) show that a genus-4 curve over F_32 having more than 72 points must be a double cover of the elliptic curve y^2 + x*y = x^3 + x. The programs in 32-4.magma, found at the same URL, enumerate such double covers, and show that no such curve has more than 71 points.

Combined with the results from IsogenyClasses.magma, these results show that if there is a genus-4 curve over F_32 with 72 points, its real Weil polynomial must be (x + 11)^2 * (x^2 + 17*x + 71).
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