We consider the characteristic time operator T introduced in [E. A. Galapon, Proc. R. Soc. Lond. A, 458:2671 (2002)] which is bounded and self-adjoint. For a semibounded discrete Hamiltonian H with some growth condition, T satisfies the canonical relation [T,H]|ψ⟩=iℏ|ψ⟩ for |ψ⟩ in a dense subspace of the Hilbert space. While T is not covariant, we show that it still satisfies the canonical relation in a set of times of total measure zero called the time invariant set T. In the neighborhood of each time t in T, T is still canonically conjugate to H and its expectation value gives the parametric time. Its two-dimensional projection saturates the time-energy uncertainty relation in the neighborhood of T, and is proportional to the Pauli matrix σy. Thus, one can construct a quantum clock that tells the time in the neighborhood of T by measuring a compatible observable.