  We describe a method to compute the norm on the cotangent space to the moduli
space of Riemann surfaces associated to the Finsler Teichm\"uller metric. Our
method involves computing the periods of abelian double covers and is easy to
implement for Riemann surfaces presented as algebraic curves using existing
tools for approximating period matrices of plane algebraic curves. We
illustrate our method by depicting the unit sphere in the cotangent space to
moduli space at a particular surface of genus zero with five punctures and by
corroborating the proof of a theorem of Royden's for our example.
