First-principles based on density functional theory is used to study the phase stability, elastic, magnetic, and electronic properties of cubic (c)-Fe$_4$C. Our results show that c-Fe$_{4}$C has a ferromagnetic (FM) ground state structure compared wi
th antiferromagnetic (AFM) and nonmagnetic (NM)states. To study the phase stability of c-Fe$_4$C, BCC Fe$_4$C, FCC Fe$_4$C, and BCC Fe$_{16}$C, where C is considered at tetrahedral and octahedral interstitial sites, are also considered. Although, the formation energy of c-Fe$_4$C is smaller than BCC Fe$_4$C, but the shear moduli of c-Fe$_4$C is negative in the FM and AFM states indicating that c-Fe$_4$C is dynamically not stable in the magnetic (FM/AFM) states. However, NM state has positive shear moduli which illustrates that instability in c-Fe$_4$C is due to magnetism and can lead to soft phonon modes. The calculated formation energy also shows that c-Fe$_4$C has higher formation energy compared with the FCC Fe$_4$C indicating no possibility of c-Fe$_4$C in low carbon steels at low temperature. The magnetic moment of Fe in c-Fe$_4$C is also sensitive to lattice deformation. The electronic structure reveals the itinerant nature of electrons responsible for metallic behavior of c-Fe$_4$C.
