More configurations on elliptic curves (Q6581288)

A \((p_{\lambda}, l_{\pi})\) configuration is a set of \(p\) points and \(l\) lines in a projective plane such that each point is incident with \(\lambda\) lines and each line is incident with \(\pi\) points.\N\NLet \(r\ge s\ge1\) be integers. The authors construct \((3r_{s},sr_{3})\) configurations, where the points are on an elliptic curve. These configurations have a mirror symmetry and if \(3\) divides \(r\) also a rotational symmetry of order three. Moreover, they have the property that the points can be moved along the curve in such a way that incidences are preserved.\N\NIt is an open question whether the points of any such configuration can be placed on an elliptic curve.