By David Kohel, Robert Rolland (ed.)
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Extra resources for Arithmetic, Geometry, Cryptography and Coding Theory 2009
If m is odd, its number of points is maximal. If m is even, its number of points is minimal. Although the curve C3 is not isomorphic over F3 to the Klein quartic, over F27 it has the same characteristic polynomial of Frobenius, being (T 2 + 27)3 . It follows that C3 shares the extremal properties of the Klein quartic over ﬁelds of the form F36m : C3 has the maximal number of points possible if m is odd, and the minimal number of points possible if m is even. References  V. Batyrev, Variations of the mixed Hodge structure of aﬃne hypersurfaces in algebraic tori, Duke Math.
69(2), pp. 349–409 (1993)  P. Beelen, R. Pellikaan, The Newton polygon of plane curves with many rational points, Des. Codes Cryptogr. 21, pp. 41–67 (2000)  W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user Language, J. Symbolic Computation 24(3-4), pp. 235–265 (1997) 28 8 WOUTER CASTRYCK AND JOHN VOIGHT `s, Curves of genus two over ﬁelds of even characteristic,  G. Cardona, E. Nart, J. Pujola Math. Z. 250(2005), no. 1, pp. 177–201  W. Castryck, J. Voight, On nondegeneracy of curves, Algebra & Number Theory 3(3), pp.
The polynomials f that passed this test were then checked for nondegeneracy with respect to the edges of Δ(f ). Checking nondegeneracy with respect to the edges boils down to checking squarefreeness of a number of univariate polynomials of small degree, which can be done very eﬃciently. The nondegeneracy condition with respect to the vertices of Δ(f ) is automatic. The nondegeneracy condition with respect to Δ(f ) itself is also automatic if f deﬁnes a genus g curve (by Baker’s inequality), so we can disregard any polynomial for which this condition is not satisﬁed.