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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) E Z2. As application, we construct parametrized families of twisted elliptic curves E : ax3 + bx2 + cx + d = ey2 without integer points (x, y). [ABSTRACT FROM AUTHOR]