Fundamental matrix, drawing geometric relationship between two images, plays an important role in 3-dimensional computer vision. Degenerate configurations of the space points and the two camera optical centers affect stability of computation for fundamental matrix. In order to robustly estimate fundamental matrix, it is necessary to study these degenerate configurations. We analyze all the possible degenerate configurations caused by twisted cubic and give the corresponding degenerate rank for each case. Relationships with general degeneracies, the previous ruled quadric degeneracy and the homography degeneracy, are also reported.