Cmatrix = ( {
    {C1 + CJ + CJ, -CJ, 0, -CJ},
    {-CJ, C2 + CJ + CJ, -CJ, 0},
    {0, -CJ, C1 + CJ + CJ, -CJ},
    {-CJ, 0, -CJ, C2 + CJ + CJ}
   } );

InvCmat = Inverse[Cmatrix];

[Omega]1 = Sqrt[1/L InvCmat[[1, 1]]];
[Omega]2 = Sqrt[1/L InvCmat[[2, 2]]];

J = 1/2 InvCmat[[1, 2]]/Sqrt[InvCmat[[1, 1]] InvCmat[[2, 2]]]
    Sqrt[[Omega]1*[Omega]2];

{{20, 20, 2, 7.70438, 7.50004, 0.156103}, {20, 20, 3, 7.60618,
  7.30572, 0.212588}, {20, 20, 4, 7.46063, 7.18521, 0.273479}, {20,
  20, 6, 7.25941, 6.79769, 0.354176}, {20, 20, 10, 6.93511, 6.59471,
  0.494317}, {13, 13, 3, 8.8022, 8.20026, 0.297579}, {16, 16, 3,
  8.22736, 7.78623, 0.252157}, {20, 20, 3, 7.58361, 7.359,
  0.212081}, {24, 24, 3, 7.08458, 6.90351, 0.178109}}