  We study two-loop corrections to the scattering amplitude of four massive
leptons in quantum electrodynamics. These amplitudes involve previously unknown
elliptic Feynman integrals, which we compute analytically using the
differential equation method. In doing so, we uncover the details of the
elliptic geometry underlying this scattering amplitude and show how to exploit
its properties to obtain compact, easy-to-evaluate series expansions that
describe the scattering of four massive leptons in QED in the kinematical
regions relevant for Bhabha and M{\o}ller scattering processes.
