Recent work has demonstrated that breaking the up–down symmetry of tokamak flux surfaces removes a constraint that limits intrinsic momentum transport, and hence toroidal rotation, to be small. We show, through MHD analysis, that ellipticity is most effective at introducing up–down asymmetry throughout the plasma. We detail an extension to GS2, a local δf gyrokinetic code that self-consistently calculates momentum transport, to permit up–down asymmetric configurations. Tokamaks with tilted elliptical poloidal cross-sections were simulated to determine nonlinear momentum transport. The results, which are consistent with the experiment in magnitude, suggest that a toroidal velocity gradient, (∂uζi/∂ρ)/vthi, of 5% of the temperature gradient, (∂Ti/∂ρ)/Ti, is sustainable. Here vthi is the ion thermal speed, uζi is the ion toroidal mean flow, ρ is the minor radial coordinate normalized to the tokamak minor radius, and Ti is the ion temperature. Though other known core intrinsic momentum transport mechanisms scale poorly to larger machines, these results indicate that up–down asymmetry may be a feasible method to generate the current experimentally measured rotation levels in reactor-sized devices.