  For a complex elliptic curve $E$ and a point $p$ of order $n$ on it, the
images of the points $p_k=kp$ under the Weierstrass embedding of $E$ into
$\mathbb{C}\mathbb{P}^2$ are collinear if and only if the sum of indices is
divisible by $n$. Thus, it provides a realization of a certain matroid. We
study this matroid in detail and prove that its realization space is isomorphic
(over $\mathbb{C}$) to the modular curve $X_1(n)$, provided $n\geq 10$, which
also provides an integral model of $X_1(n)$. In the process, we find a
connection to the classical Ceva and B\"or\"oczky examples of special point and
line configurations. We also discuss the situation for smaller values of $n$.
