  We study extremal properties of the determinant of the Laplacian in the
Bergman metric on the moduli space of compact genus two Riemann surfaces. By a
combination of analytical and numerical methods we identify four non-degenerate
critical points of this function and compute the signature of the Hessian at
these points. The curve with the maximal number of automorphisms (the Burnside
curve) turns out to be the point of the absolute maximum. Our results agree
with the mass formula for orbifold Euler characteristics of the moduli space. A
similar analysis is performed for the Bolza's strata of symmetric Riemann
surfaces of genus two.

Spectral Theory