The deuteron form factors are calculated using two model wave functions obtained from the 2007 covariant spectator theory high-precision fits to np scattering data. Included in the calculation is a new class of isoscalar np interaction currents which are automatically generated by the nuclear force model used in these fits. If the nuclear model WJC2 is used, a precision fit (?2/datum 1) to the Sick global analysis (GA) of all ed elastic scattering data can be obtained by adjusting the unknown off-shell nucleon form factors F3(Q2 ) (discussed before) and F4(Q2 ) (introduced in this paper) and predicting the high-Q2 behavior of the neutron charge form factor GEn(Q2 ) well beyond the region where it has been measured directly. Relativistic corrections, isoscalar interaction currents, and off-shell effects are defined and discussed, and their sizes are displayed. A rationale for extending ed elastic scattering measurements to higher Q2 is presented.