In the irreducible representation $V_j$, the normalized basis $|j,m\rangle$ may be chosen so that
\begin{align*}
J_+|j,m\rangle = \hbar\sqrt{j(j+1)-m(m+1)}\,|j,m+1\rangle.
\end{align*}
\begin{align*}
J_-|j,m\rangle = \hbar\sqrt{j(j+1)-m(m-1)}\,|j,m-1\rangle.
\end{align*}