By David Kohel, Robert Rolland (ed.)
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Extra info for Arithmetic, Geometry, Cryptography and Coding Theory 2009
Of any genus, over any perfect ﬁeld)? 4. Extremal properties The curve C2 can be found in many places in the existing literature. It enjoys some remarkable properties concerning the number #C2 (F2m ) of F2m -rational points for various values of m. First, it has no F2 -rational points. However, over F4 and F8 it has 14 and 24 points, respectively; in both cases, this is the maximal number of rational points possible on a complete nonsingular genus 3 curve, and in each case C2 is the unique curve obtaining this bound (up to isomorphism).
255–281 (2009)  N. Elkies, The Klein quartic in number theory, pp. 51–102 in S. ), The eightfold way: the beauty of Klein’s quartic curve, MSRI Publication Series 35, Cambridge University Press, 352 pp. (1999)  A. Enge, How to distinguish hyperelliptic curves in even characteristic, proceedings of Public– key Cryptography and Computational Number Theory (Warsaw 2000), de Gruyter, Berlin, pp. 49–58 (2001) ´ndez Encinas, J. Mun ˜oz Masqu´  J. Espinosa Garc´ıa, L. Herna e, A review on the isomorphism classes of hyperelliptic curves of genus 2 over ﬁnite ﬁelds admitting a Weierstrass point, Acta Appl.
Cette courbe a un unique point a` l’inﬁni. Ses points singuliers sont les points (0, 0), (0, 1) et le point a` l’inﬁni. La valuation en (0, 0) de x (resp. u) est v(0,0) (x) = 3 (resp. v(0,0) (u) = 7). La valuation en l’inﬁni de x (resp. u) est v∞ (x) = −6 (resp. v∞ (u) = −7). 8). 7 Soit C2 la courbe d’´equation aﬃne (v 4 +v)3 = γx7 . Elle a un unique point a` l’inﬁni. Ses points singuliers sont le point a` l’inﬁni et les points (0, 0), (0, 1), (0, ζ), (0, ζ 2). La valuation en l’inﬁni de x (resp.