Let F(x; y) = ax3 + bx2y + cxy2 + dy3 ? Z[x,y] be an irreducible cubic form. In this paper, we investigate arithmetic properties of the common indices of algebraic integers in cubic fields. For each integer k such that v2(k)??0 (mod 3) and 2v2(-2b3 - 27a2d + 9abc) = 3v2(b2 - 3ac), we prove that the cubic Thue equation F(x,y) = k has no solution (x,y) ? Z2. As application, we construct parametrized families of twisted elliptic curves E : ax3 + bx2 + cx + d = ey2 without integer points (x,y).