  We provide a Hodge theoretical characterization of the set of algebraic
numbers which arises from the complete list, due to A. Beauville, of semistable
families of elliptic curves over $\mathbb{P}^1$ with four singular fibers. Our
technical innovation is the analysis of the periodicity of the uniformizing
Higgs bundle attached to $\mathbb{P}^1$ minus four points over the field of
complex numbers.
