The electron energy budget of the nightside Venus ionosphere is analyzed by comparing two-dimensional solutions of the heat conduction equation with an empirical model of the electron temperature based on Pioneer Venus measurements. The theoretical and empirical models are confined to a narrow latitude band at low latitudes. The energy calculation includes solar EUV heating at the terminator, electron cooling to ions and neutrals, and heat conduction within the ionospheric plasma. Heating at the ionopause by the shocked solar wind is included through an upper boundary condition. Several simple, uniform magnetic field models are found to be inconsistent with the empirical temperature model. An optimum magnetic field for the equatorial region is derived by solving for the heat flux directions which force energy conservation while constrained by the observed temperature within the range of 80¿ to 170¿ SZA (solar zenith angle) and 160 to 700 km. The resulting heat flux vectors suggest a magnetic field that connects the lower nightside ionosphere (less than 300 km) to the dayside ionosphere and connects the upper ionosphere (greater than 300 km) to the ionosheath. Thus the lower ionosphere is heated through conduction of heat from the dayside, and the upper ionosphere is heated by the solar wind in the ionosheath, heat flowing downward and from the nightside to the dayside. The suggested magnetic field appears to be consistent with the magnetic field directions observed by the Pioneer Venus magnetometer.