The motion of a microorganism with a flagellum is investigated with a view to applying its propulsion mechanism to the hydrodynamic driving of micromachines in the future. Using the boundary element method for the Stokes flow, the speed of advance and the power consumption of the microorganism are numerically determined. Optimizations are performed on six nondimensional parameters that describe the shape of the flagellum and two nondimensional parameters that represent the shape of the cell body. The optimum shape agrees well with the shape of the actual microorganism except for the cross-sectional shape of the flagellum. Low power consumption and high speed of advance are predicted when the cross section of the flagellum is elliptic with its major axis in the radial direction of the flagellar spiral.