Counting points on the Fricke-Macbeath curve over finite fields

Abstract

The Fricke-Macbeath curve is a smooth projective algebraic curve of genus 7 with automorphism group PSL₂(픽₈). We recall two models of it (introduced, respectively, by Maxim Hendriks and by Bradley Brock) defined over ℚ, and we establish an explicit isomorphism defined over ℚ( −7 ) between these models. Moreover, we decompose up to isogeny over ℚ the jacobian of one of these models. As a consequence we obtain a simple formula for the number of points over 픽q on (the reduction of) this model, in terms of the elliptic curve with equation y² = x³ + x² − 114x − 127. Moreover, twists by elements of PSL₂(픽₈) of the curve over finite fields are described. The curve leads to a number of new records as maintained on manYPoints of curves of genus 7 with many rational points over finite fields.