  A precision measurement of the matrix elements for $\eta\to\pi^+\pi^-\pi^0$
and $\eta\to\pi^0\pi^0\pi^0$ decays is performed using a sample of
$(10087\pm44)\times10^6$ $J/\psi$ decays collected with the BESIII detector.
The decay $J/\psi \to \gamma \eta$ is used to select clean samples of 631,686
$\eta\to\pi^+\pi^-\pi^0$ decays and 272,322 $\eta\to\pi^0\pi^0\pi^0$ decays.
The matrix elements for both channels are in reasonable agreement with previous
measurements. The non-zero $gX^2Y$ term for the decay mode
$\eta\to\pi^+\pi^-\pi^0$ is confirmed, as reported by the KLOE Collaboration,
while the other higher-order terms are found to be insignificant. Dalitz plot
asymmetries in the $\eta\to\pi^+\pi^-\pi^0$ decay are also explored and are
found to be consistent with charge conjugation invariance. In addition, a cusp
effect is investigated in the $\eta\to\pi^0\pi^0\pi^0$ decay, and no obvious
structure around the $\pi^+\pi^-$ mass threshold is observed.
