To shed light on the nature of the controversial and not yet fully understood exotic states, we are carrying out a systematic study of their electromagnetic properties. The magnetic moment of a hadron state is as fundamental a dynamical quantity as its mass and contains valuable information on the deep underlying structure. In this study, we use the QCD light-cone sum rule to extract the magnetic moments of the $$\mathrm {P_{c}(4312)}$$ $P_c\left(4312\right)$ , $$\mathrm {P_{c}(4380)}$$ $P_c\left(4380\right)$ , and $$\mathrm {P_{c}(4440)}$$ $P_c\left(4440\right)$ pentaquarks by considering them as the molecular picture with spin-parity $$\mathrm {J^P= \frac{1}{2}^-}$$ $J^P={\frac{1}{2}}^{-}$ , $$\mathrm {J^P= \frac{3}{2}^-}$$ $J^P={\frac{3}{2}}^{-}$ , and $$\mathrm {J^P= \frac{3}{2}^-}$$ $J^P={\frac{3}{2}}^{-}$ , respectively. We define the isospin of the interpolating currents of these states, which is the key to solving the puzzle of the hidden-charm pentaquark states, to make these analyses more precise and reliable. We have compared our results with other theoretical predictions that could be a useful complementary tool for the interpretation of the hidden-charm pentaquark sector, and we observe that they are not in mutual agreement with each other. We have also calculated higher multipole moments for spin-3/2 $$\bar{D}^{*} \Sigma _c$$ ${\overline{D}}^{\ast }{\Sigma }_c$ and $$\bar{D} \Sigma _c^{*}$$ $\overline{D}\Sigma _c^{\ast }$ pentaquarks, indicating a non-spherical charge distribution.