  We study the Hodge structure of elliptic surfaces which are canonically
defined from bielliptic curves of genus three. We prove that the period map for
the second cohomology has one dimensional fibers, and the period map for the
total cohomology is of degree twelve, and moreover, by adding the information
of the Hodge structure of the canonical divisor, we prove a generic Torelli
theorem for these elliptic surfaces. Finally, we give explicit examples of the
pair of non-isomorphic elliptic surfaces which have the same Hodge structure on
themselves and the same Hodge structure on their canonical divisors.
