manYPoints - Table of Curves with Many Points http://www.manypoints.org manYPoints aims at providing an open access up-to-date source for information on curves over finite fields with many points Lower bound 35 for q=2, g=48 http://www.manypoints.org/Details.aspx?e1=7312 Let F be the hyperelliptic genus 2 field given by y^2 + x*y + x^5 + x. Using the notation of Magma, let D be the divisor (x^4 + x + 1, y + x^3 + x^2) + (x^4 + x + 1, y + x^3 + x^2 + x) + 2*(1/x, y/x^3). (The first two places in the support of D has degree 4, the last is rational.) Let S be the set { (x, y), (x + 1, y + x + 1), (x + 1, y + 1), (x^2 + x + 1)}. (The first three places in S are rational, the last has degree 4.) Then F^D_S, the largest abelian extension of F with conductor <=D such that all places in S split completely, has genus 48 and 35 rational places. This can easily be verified using Magma. Lower bound 36 for q=2, g=46 http://www.manypoints.org/Details.aspx?e1=7311 Let F be the hyperelliptic genus 2 field given by y^2 + x*y + x^5 + x^3 + x^2 + x. Using the notation of Magma, let D be the divisor (x^3 + x + 1) + (x^2 + x + 1, y + x + 1) + +(x^2 + x + 1, y + 1) (The first of the places in the support of D has degree 6, the other two has degree 2.) Let S be the set of rational places in F. Then F^D_S, the largest abelian extension of F with conductor <=D such that all places in S split completely, has genus 46 and 36 rational places. This can easily be verified using Magma. Lower bound 81 for q=5, g=46 http://www.manypoints.org/Details.aspx?e1=7310 Lower bound 96 for q=5, g=45 http://www.manypoints.org/Details.aspx?e1=7309 Lower bound 72 for q=5, g=40 http://www.manypoints.org/Details.aspx?e1=7308 Lower bound 80 for q=5, g=37 http://www.manypoints.org/Details.aspx?e1=7307 Lower bound 72 for q=5, g=35 http://www.manypoints.org/Details.aspx?e1=7306 Lower bound 60 for q=5, g=26 http://www.manypoints.org/Details.aspx?e1=7305 Lower bound 36 for q=5, g=12 http://www.manypoints.org/Details.aspx?e1=7304 Lower bound 31 for q=5, g=10 http://www.manypoints.org/Details.aspx?e1=7303 Lower bound 24 for q=5, g=8 http://www.manypoints.org/Details.aspx?e1=7302 Lower bound 72 for q=4, g=41 http://www.manypoints.org/Details.aspx?e1=7301 Lower bound 60 for q=3, g=46 http://www.manypoints.org/Details.aspx?e1=7300 Lower bound 48 for q=3, g=33 http://www.manypoints.org/Details.aspx?e1=7299 Lower bound 33 for q=3, g=22 http://www.manypoints.org/Details.aspx?e1=7298 Lower bound 28 for q=3, g=17 http://www.manypoints.org/Details.aspx?e1=7297 Lower bound 36 for q=2, g=45 http://www.manypoints.org/Details.aspx?e1=7296 Lower bound 18 for q=2, g=17 http://www.manypoints.org/Details.aspx?e1=7295 Upper bound 291 for q=73, g=13 http://www.manypoints.org/Details.aspx?e1=7292 A real Weil polynomial we don't know how to eliminate: (x + 13) * (x + 17)^12 Upper bound 375 for q=73, g=18 http://www.manypoints.org/Details.aspx?e1=7293 A real Weil polynomial we don't know how to eliminate: (x + 12) * (x + 17)^17