We report a direct lattice calculation of both the I=0 and I=2 $\pi\pi$ scattering phase shifts using G-parity boundary conditions on an ensemble of $32^3 \times 64$ gauge configurations at physical quark mass. This extends an earlier calculation of the RBC/UKQCD Collaboration \cite{prl} by including additional operators and more statistics. We apply both multistate fitting and the generalized eigenvalue treatment to this set of operators in order to understand and reduce excited state contamination and to study $\pi\pi$ scattering at energies in the energy region around the kaon mass. These results are compared with predictions from dispersion theory.