  We present the three-pion spectrum with maximal isospin in a finite volume
determined from lattice QCD, including excited states in addition to the ground
states across various irreducible representations at zero and nonzero total
momentum. The required correlation functions, from which the spectrum is
extracted, are computed using a newly implemented algorithm which speeds up the
computation by more than an order of magnitude. On a subset of the data we
extract a nonzero value of the three-pion threshold scattering amplitude using
the $1/L$ expansion of the three-particle quantization condition, which
consistently describes all states at zero total momentum. The finite-volume
spectrum is publicly available to facilitate further explorations within the
available three-particle finite-volume approaches.
