  We investigate equilibrium sequences of relativistic stars containing purely
toroidal magnetic fields with four kinds of realistic equations of state (EOSs)
of SLy (Douchin et al.), FPS (Pandharipande et al.), Shen (Shen et al.), and LS
(Lattimer & Swesty). We numerically construct thousands of equilibrium
configurations. Particularly we pay attention to the equilibrium sequences of
constant baryon mass and/or constant magnetic flux, which model evolutions of
an isolated neutron star. Important properties obtained in this study are
summarized as follows ; (1) The dependence of the mass-shedding angular
velocity on the EOSs is determined from that of the non-magnetized case. The
stars with Shen(FPS) EOS reach the mass-shedding limit at the smallest(largest)
angular velocity, while the stars with SLy or Lattimer-Swesty EOSs take the
moderate values. (2) For the supramassive sequences, the equilibrium
configurations are found to be generally oblate for the realistic EOSs in sharp
contrast to the polytropic stars. For FPS(LS) EOS, the parameter region which
permits the prolately deformed stars is widest(narrowest). For SLy and Shen
EOS, it is in medium. Furthermore, the angular velocities $\Omega_{\rm up}$,
above which the stars start to spin up as they lose angular momentum, are found
to depend sharply on the realistic EOSs. Our analysis indicates that the
hierarchy of this spin up angular velocity is $\Omega_{\rm up,SLy} >
\Omega_{\rm up,FPS} > \Omega_{\rm up,LS}>\Omega_{\rm up,Shen}$ and this
relation holds even if the sequences have strong magnetic fields. Our results
suggest the EOSs within the relativistic stars containing purely toroidal
magnetic fields can be constrained by observing the angular velocity, the
gravitational wave, and the signature of the spin up.
