  Stars with $\sim 8-10~{M}_{\odot}$ evolve to form a strongly degenerate ONeMg
core. When the core mass becomes close to the Chandrasekhar mass, the core
undergoes electron captures on $^{24}$Mg and $^{20}$Ne, which induce the
electron-capture supernova (ECSN). In order to clarify whether the ECSN leads
to a collapse or thermonuclear explosion, we calculate the evolution of an
$8.4~M_\odot$ star from the main sequence until the oxygen ignition in the
ONeMg core. We apply the latest electron-capture rate on $^{20}$Ne including
the second forbidden transition, and investigate how the location of the oxygen
ignition (center or off-center) and the $Y_e$ distribution depend on the input
physics and the treatment of the semiconvection and convection. The central
density when the oxygen deflagration is initiated, $\rho_{\rm c,def}$, can be
significantly higher than that of the oxygen ignition thanks to the convection,
and we estimate $\log_{10}(\rho_{\rm c,def}/\mathrm{g~cm^{-3}})>10.10$. We
perform two-dimensional simulations of the flame propagation to examine how the
final fate of the ONeMg core depends on the $Y_e$ distribution and $\rho_{\rm
c,def}$. We find that the deflagration starting from $\log_{10}(\rho_{\rm
c,def}/\mathrm{g~cm^{-3}})>10.01 (< 10.01)$ leads to a collapse (thermonuclear
explosion). Since our estimate of $\rho_{\rm c,def}$ exceeds this critical
value, the ONeMg core is likely to collapse, although further studies of the
convection and semiconvection before the deflagration are important.
