  The complete $p$-elliptic integrals are generalizations of the complete
elliptic integrals by the generalized trigonometric function $\sin_p{\theta}$
and its half-period $\pi_p$. It is shown, only for $p=4$, that the generalized
$p$-elliptic integrals yield a computation formula of $\pi_p$ in terms of the
arithmetic-geometric mean. This is a $\pi_p$-version of the celebrated formula
of $\pi$, independently proved by Salamin and Brent in 1976.
