The intrinsic structure of \(^{12}\hbox {C}\) remains a longstanding challenge in self-consistent mean-field approaches, as different density functional theory calculations predict either spherical or deformed ground-state configurations. In this work, we investigate the interplay between the spin–orbit interaction, pairing correlations, and quadrupole deformation in \(^{12}\hbox {C}\) within the Skyrme–Hartree–Fock–Bogoliubov framework. By systematically varying the strengths of the spin–orbit interaction and the pairing interaction, we analyze the evolution of potential energy curves, single-particle shell structure, and occupation probabilities. For the normal spin–orbit strength, the spherical configuration remains remarkably robust against variations in the pairing strength. When the spin–orbit interaction is reduced, the shell structure around the Fermi surface becomes softened, allowing pairing correlations to play a decisive role in stabilizing an oblate intrinsic shape. These results demonstrate that, in \(^{12}\hbox {C}\) , the spin–orbit interaction controls the onset of deformation, whereas pairing correlations determine whether deformation is realized. It means that Gamow–Teller transition strengths might be highly sensitive to these structural changes and thus provide a valuable probe of the underlying intrinsic configuration.
