This paper reports low-energy electron scattering with C ₄ ^-anions with the ab initio R-matrix method in the single state close-coupling (CC) model and the fixed-nuclei approximation. We predicts the elastic integral scattering cross sections (ICS) of the four conformers for C ₄ ^-ions in the energy range of 0< E ≤12eV and discusses effect of configuration changes on the position and width of the resonances. Additionally, we compare and analyze the theoretical results and experimental data. The results indicate that the experimentally observed resonant peak at 8.8 eV is mainly from the $\Sigma_u^{+}$ and $\Sigma_u^{-}$ resonances of the conformer A and the A ₂resonance of the conformer C. The scattering cross-section reveals that the conformer A has five resonant states, and the conformer B has three resonances, while C and D each have four resonances. Finally, we used the Boltzmann distribution to calculate the population of different conformers at different temperatures, and simulated the low-energy electron elastic integrated scattering cross-section at room temperature, which is in good agreement with available experimental results. We also found a shape resonance at 3.3 eV with a width of 0.20 eV in our total cross sections, which is not detected in the available experimental results. This provides a new chance for the measurement.