Modular curves and Poncelet polygons (Q1318046)

A Poncelet-polygon is a polygon in the projective plane (over \(\mathbb{C})\) with its vertices on one smooth conic while its sides touch another smooth conic. It is well-known that Poncelet-\(n\)-gons are related to \(n\)- torsion elements in elliptic curves. We consider moduli of this situation and obtain a plane model for the modular curve parametrizing elliptic curves with a level-2-structure on an \(n\)-torsion point. Using this plane curve we compute numbers for Poncelet-configurations in a pencil of conics and for pairs of conics which are mutually related by Poncelet- polygons.