We calculate ab initio interaction potentials for electron-methane scattering and use them to perform converged scattering calculations for the electronically and vibrationally elastic rotational-state-to-rotational-state cross sections at 10 eV impact energy. The effective potential has static, local exchange, and polarization terms calculated from extended-basis-set Hartree-Fock wave functions for both unperturbed and polarized methane molecules. The polarization potential includes nonadiabatic effects in the semiclassical local kinetic energy approximation, and for comparison we also perform calculations based on the adiabatic polarization potentials. Five to 12 terms are retained in the angular expansion of the various parts of the interaction potential and the coupled channels calculations involved 41 total angular momenta, with 1-33 coupled channels for each. The resulting rotationally summed integral cross sections are in excellent agreement with recent experiments for scattering angles 40° and larger, but are larger than the experiment at small scattering angles. The rotationally inelastic cross sections for the full potential are smaller than those for the adiabatic potential by about a factor of 2.