  The $^{12}$C+$^{12}$C fusion reaction plays a significant role in the later
phases of stellar evolution. For a better understanding of the evolution
involved, one must understand the corresponding fusion-fission dynamics and
reaction characteristics. In the present analysis, we have studied the fusion
cross-section along with the S-factor for this reaction using the well-known
M3Y and recently developed R3Y nucleon-nucleon (NN) potential along with the
relativistic mean-field densities in double folding approach. The density
distributions and the microscopic R3Y NN potential are calculated using the
NL3$^*$ parameter set. The $\ell$- summed Wong model is employed to investigate
the fusion cross-section, with $\ell_{max}$-values from the sharp cut-off
model. The calculated results are also then compared with the experimental
data. It is found that the R3Y interaction gives a reasonable agreement with
the data.
