  A rational perfect cuboid is a rectangular parallelepiped whose edges and
face diagonals are given by rational numbers and whose space diagonal is equal
to unity. It is described by a system of four equations with respect to six
variables. The cuboid factor equations were derived from these four equations
by symmetrization procedure. They constitute a system of eight polynomial
equations, which has been solved parametrically. In the present paper its
parametric solution is expressed through intersections or rational and elliptic
curves arranged into parametric families.
