  Motivated by a demand for explicit genus 1 Belyi maps from theoretical
physics, we give an efficient method of explicitly computing genus one Belyi
maps by (1) composing covering maps from elliptic curves to the Riemann sphere
with simpler (univariate) genus zero Belyi maps, as well as by (2) composing
further with isogenies between elliptic curves. This gives many new explicit
dessins on the doubly periodic plane, including several which have been
realized in the physics literature as so-called brane-tilings in the context of
quiver gauge theories.
