We report a precise measurement of the J/ψ elliptic flow in Pb-Pb collisions at sNN=5.02 TeV with the ALICE detector at the LHC. The J/ψ mesons are reconstructed at midrapidity (|y|<0.9) in the dielectron decay channel and at forward rapidity (2.5<y<4.0) in the dimuon channel, both down to zero transverse momentum. At forward rapidity, the elliptic flow v2 of the J/ψ is studied as a function of the transverse momentum and centrality. A positive v2 is observed in the transverse momentum range 2<pT<8 GeV/c in the three centrality classes studied and confirms with higher statistics our earlier results at sNN=2.76 TeV in semicentral collisions. At midrapidity, the J/ψ v2 is investigated as a function of the transverse momentum in semicentral collisions and found to be in agreement with the measurements at forward rapidity. These results are compared to transport model calculations. The comparison supports the idea that at low pT the elliptic flow of the J/ψ originates from the thermalization of charm quarks in the deconfined medium but suggests that additional mechanisms might be missing in the models.