  Using the framework of non-relativistic effective field theory, the
finite-volume ground-state energy shift is calculated up-to-and-including
$O(L^{-6})$ for the system of three pions in the channel with the total isospin
$I=1$. The relativistic corrections are included perturbatively, up to the same
order in the inverse of the box size $L$. The obtained explicit expression,
together with the known result for the system with maximal isospin $I=3$, can
be used for the extraction of two independent effective three-body couplings
from the measured ground-state spectrum of three pions.
