  The aim of this paper is to give an explicit extension of the classical
elliptic integrals to the Hilbert modular case for $\mathbb{Q}(\sqrt{5})$. We
study a family of Kummer surfaces corresponding to the Humbert surface of
invariant $5$ with two complex parameters. Our Kummer surface is given by a
double covering of the weighted projective space $\mathbb{P}(1:1:2)$ branched
along a parabola and a quintic curve. The period mapping for our family is
given by double integrals of an algebraic function on chambers coming from an
arrangement of a parabola and a quintic curve in $\mathbb{C}^2$.
