  In this letter, we perform partial wave decomposition on coupled-channel
scattering amplitudes, $J/\psi J/\psi$-$J/\psi \psi(2S)$-$J/\psi \psi(3770)$,
to study the resonance appears in these processes. Effective Lagrangians are
used to describe the interactions of four charmed vector mesons, and the
scattering amplitudes are calculated up to the next-to-leading order. Partial
wave projections are performed, and unitarization is implemented by Pad\'e
approximation. Then we fit the amplitudes to the $J/\psi J/\psi$ invariant mass
spectra measured by LHCb and determine the unknown couplings. The pole
parameters of the $X(6900)$ are extracted as $M=6861.0^{+6.3}_{-8.8}$~MeV and
$\Gamma=129.0^{+5.6}_{-3.4}$~MeV. Our analysis implies that its quantum number
prefers to be $0^{++}$. The pole counting rule and phase shifts show that it is
a normal Breit-Wigner resonance and, hence, should be a compact tetraquark.
