manYPoints

Submitted by	Everett Howe
Date	04-20-2020
Reference	Everett W. Howe The maximum number of points on a curve of genus eight over the field of four elements J. Number Theory 220 (2021) 320–329
Comments	An example is given by y^2 + (x^3 + x + 1)y = x^6 + x^5 + x^4 + x^2 z^3 = (x+1)y + x^2 This is a degree-3 Kummer extension of the genus-2 curve defined by the first equation, ramified at 4 points.
Tags	Explicit curves

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Submitted by	Everett Howe
Date	04-14-2010
Reference	Jean-Pierre Serre Rational points on curves over finite fields. With contributions by Everett Howe, Joseph Oesterlé and Christophe Ritzenthaler. Edited by Alp Bassa, Elisa Lorenzo García, Christophe Ritzenthaler and René Schoof Documents Mathématiques 18, Société Mathématique de France, Paris, 2020
Comments	The Oesterlé bound. Here is a real Weil polynomial that we don't know how to eliminate: x * (x + 2)^4 * (x + 3) * (x + 4)^2
Tags	Oesterlé bound

Restrictions on curve	Everett Howe 05-04-2010 00:36
In fact, the real Weil polynomial listed above is the only possibility for a curve with 24 points. Also, if there is a curve with this real Weil polynomial, it must have a degree-3 map to the unique elliptic curve with 8 points. All this can be deduced from the Magma package IsogenyClasses.magma, found at http://alumnus.caltech.edu/~however/Magma/IsogenyClasses.magma