On the Euler-Poincar\'{e} characteristics of a simply connected rationally elliptic CW-complex

For a simply connected rationally elliptic CW-complex $X$, we show that the cohomology and the homotopy Euler-Poincar\'e characteristics are related to two new numerical invariants namely $\eta_{X}$ and $\rho_{X}$ which we define using the Whitehead exact sequences of the Quillen and the Sullivan models of $X$.