Computing Humbert Surfaces

Humbert surfaces are hypersurfaces in the Siegel modular threefold which 
parametrize principally polarized abelian surfaces with real multiplication.  
Using methods of Bernhard Runge, we produce equations of Humbert surfaces 
in terms of Rosenhain invariants of genus two curves.  Taking intersections 
of Humbert surfaces produce Shimura curves - an example will be provided.