  We perform a series of two-dimensional magnetohydrodynamic core-collapse
simulations of rapidly rotating and strongly magnetized massive stars. To study
the properties of magnetic explosions for a longer time stretch of postbounce
evolution, we develop a new code under the framework of special relativity
including a realistic equation of state with a multiflavor neutrino leakage
scheme. Our results show the generation of the magnetically-dominated jets in
the two ways. One is launched just after the core-bounce in a prompt way and
another is launched at $ \sim 100 $ ms after the stall of the prompt shock. We
find that the shock-revival occurs when the magnetic pressure becomes strong,
due to the field wrapping, enough to overwhelm the ram pressure of the
accreting matter. The critical toroidal magnetic fields for the magnetic
shock-revival are found to be universal of $\sim 10^{15}\mathrm{G}$ behind the
jets. We point out that the time difference before the shock-revival has a
strong correlation with the explosions energies. Our results suggest that the
magnetically dominated jets are accompanied by the formation of the magnetars.
Since the jets are mildly relativistic, we speculate that they might be the
origin of some observed X-ray flashes.
