Entry details for q = 72 = 49, g = 3
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Lower bound Nmin = 92

Submitted by C. Ritzenthaler
Date 06/01/2009
Reference Jaap Top
Curves of genus 3 over small finite fields
Indag. Math. (N.S.) 14 (2003), no. 2, 275–283
This number of points in reached by the Fermat curve C: x^4+y^4+z^4=0.

Its Jacobian is isogenous to E^3 with trace(E)=-14. Its automorphism group (over F_q and geometric) has order 96.
Tags Explicit curves

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Another example     
Everett Howe
05/20/2010 23:45
This number of points is also reached by the hyperelliptic curve y^2 = x^7 - x, which is the reduction of the Klein quartic in characteristic 7. Its automorphism group is the direct product of PGL(2,F_7) with the group of order 2 containing the hyperelliptic involution, so the automorphism group has order 672.
Upper bound Nmax = 92

Submitted by Everett Howe
Date 06/10/2010
Reference Jean-Pierre Serre
Sur le nombre de points rationnels d'une courbe algébrique sur un corps fini
C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), 397–402. (= Œuvres III, No. 128, 658–663).
The Hasse-Weil-Serre bound
Tags Hasse-Weil-Serre bound

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