  In order to explore the possible physical quantities for judging different
structures of the newly observed resonance $d^*(2380)$, we study its
electromagnetic form factors. In addition to the electric charge monopole $C0$,
we calculate its electric quadrupole $E2$, magnetic dipole $M1$, and six-pole
$M3$ form factors on the base of the realistic coupled $\Delta\Delta+CC$
channel $d^*$ wave function with both the $S$- and $D$-partial waves. The
results show that the magnetic dipole moment and electric quadrupole
deformation of $d^*$ are 7.602 and $2.53\times 10^{-2}~\rm{fm}^2$,
respectively. The calculated magnetic dipole moment in the naive constituent
quark model is also compared with the result of $D_{12}\pi$ picture. By
comparing with partial results where the $d^*$ state is considered with a
single $\Delta\Delta$ and with a $D_{12}\pi$ structures, we find that in
addition to the charge distribution of $d^*(2380)$, the magnetic dipole moment
and magnetic radius can be used to discriminate different structures of $d^*$.
Moreover, a quite small electric quadrupole deformation indicates that $d^*$ is
more inclined to an slightly oblate shape due to our compact hexaquark
dominated structure of $d^*(2380)$.
