Regular polygons, line operators, and elliptic modular surfaces as realization spaces of matroids

For an integer $n\ge 7$, we investigate the matroid realization space of a specific deformation of the regular $n$-gon along with its lines of symmetry. It turns out that this particular realization space is birational to the elliptic modular surface ${\mathrm{\Xi }}_1(n)$ over the modular curve $X_1(n)$. In this way, we obtain a model of ${\mathrm{\Xi }}_1(n)$ defined over the rational numbers. Furthermore, a natural geometric operator acts on these matroid realizations. On the elliptic modular surface, this operator corresponds to the multiplication by $-2$ on the elliptic curves. This provides a new geometric approach to computing multiplication by $-2$ on elliptic curves.