The relationship between face cuboids and elliptic curves

A rational face cuboid is a cuboid that all of edges, two of three face diagonals and space diagonal have rational lengths. \[ E1,s: y^2=x(x-(2s)^2)(x+(s^2-1)^2) \] for a rational number s ≠ 0, ± 1, and define à consisting of all pairs of a rational number s and a non-torsion rational point (α, β ) ∈ E1,s(Q). We construct a surjective map from à to the set F of equivalence classes of rational face cuboids, and prove that this map is a $32:1$-map. In this way, we show that the set F has infinite elements. Also, we prove that there are infinitely many s ∈ Q \ 0,± 1 \ with rank E1,s (Q)>0. In this proof, we construct pairs of s and (α, β) ∈ E1,s (Q) which are not parametric solutions.