  Hyperbolic geometry on the one-bordered torus is numerically uniformized
using Liouville theory. This geometry is relevant for the hyperbolic string
tadpole vertex describing the one-loop quantum corrections of closed string
field theory. We argue that the Lam\'e equation, upon fixing its accessory
parameter via Polyakov conjecture, provides the input for the characterization.
The explicit expressions for the Weil-Petersson metric as well as the local
coordinates and the associated vertex region for the tadpole vertex are given
in terms of classical torus conformal blocks. The relevance of this vertex for
vacuum shift computations in string theory is highlighted.
