An Elementary Approach to Hessian Curves with Torsion Group Z/6Z

Abstract

We investigate the geometry of Hessian curves H_c : x^3 + y^3 + z^3 + cxyz = 0. In particular, we present an elementary characterisation of Hessian curves with torsion group Z/6Z. As an application we show, for example, that a result of Mordell’s implies that the equation 7m^4 −26m^2e^2 − 49e^4 − n^2 = 0 does not have a solution in positive integers.