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Jaap Top; Carlo Verschoor
Counting points on the Fricke–Macbeath curve over finite fields, to appear, online on 13 December 2017, 13 p.
Article: PDF

Abstract

The Fricke-Macbeath curve is a smooth projective algebraic curve of genus $7$ with automorphism group $\mathop {\mathrm{PSL}}_2({\mathbb{F}}_8)$. We recall two models of it (introduced, respectively, by Maxim Hendriks and by Bradley Brock) defined over ${\mathbb{Q}}$, and we establish an explicit isomorphism defined over ${\mathbb{Q}}(\sqrt{-7})$ between these models. Moreover, we decompose up to isogeny over ${\mathbb{Q}}$ the jacobian of one of these models. As a consequence we obtain a simple formula for the number of points over ${\mathbb{F}}_q$ on (the reduction of) this model, in terms of the elliptic curve with equation $y^2=x^3 + x^2 - 114x - 127$. Moreover, twists by elements of $\mathop {\mathrm{PSL}}_2({\mathbb{F}}_8)$ 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.

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