[en] Superconductivity has been predicted or measured for most alkali metals under high pressure, but the computed critical temperature (T_c) of sodium (Na) at the face-centered cubic (fcc) phase is vanishingly low. Here we report a thorough, first-principles investigation of superconductivity in Na under pressures up to 260 GPa, where the metal-to-insulator transition occurs. Linear-response calculations and density functional perturbation theory were employed to evaluate phonon distributions and the electron-phonon coupling for bcc, fcc, cI16, and tI19 Na. Our results indicate that the maximum electron-phonon coupling parameter, λ, is 0.5 for the cI16 phase, corresponding to a theoretical peak in the critical temperature at T_c ≈ 1.2 K. When pressure decreases or increases from 130 GPa, T_c drops quickly. This is mainly due to the lack of p-d hybridization in Na even at 260 GPa. Since current methods based on the Eliashberg and McMillian formalisms tend to overestimate the Tc (especially the peak values) of alkali metals, we conclude that under high pressure—before the metal-to-insulator transition at 260 GPa—superconductivity in Na is very weak, if it is measurable at all.