  In 1983 Takeuchi showed that up to conjugation there are exactly 4 arithmetic
subgroups of $\textrm{PSL}_2 (\mathbb{R})$ with signature $(1; \infty)$.
Shinichi Mochizuki gave a purely geometric characterization of the
corresponding arithmetic $(1; \infty)$-curves, which also arise naturally in
the context of his recent work on inter-universal Teichm\"uller theory.
  Using Bely\u{\i} maps, we explicitly determine the canonical models of these
curves. We also study their arithmetic properties and modular interpretations.
