We recall the analytical result of the 4-loop QED contribution
to the electron $g$-$2$ and slope, which contain elliptic constants. We describe the relations between four elliptic constants which have ${}_4F_3$ hypergeometric expressions. In particular we consider a quadratic relation found by Broahurst and Mellit, and, by introducing a free parameter, we generalize it to a new quadratic relation between values of the hypergeometric ${}_4F_3$.