  We studied firstly the ground state properties of C-isotopes using a deformed
Hartree-Fock (HF)+ BCS model with Skyrme interactions. Shallow deformation
minima are found in several neutron$-$rich C-isotopes. It is shown also that
the deformation minima appear in both the oblate and the prolate sides in
$^{17}$C and $^{19}$C having almost the same binding energies. Secondly, we
carried out shell model calculations to study electromagnetic moments and
electric dipole transitions of the C-isotopes. We point out the clear
configuration dependence of the quadrupole and magnetic moments in the odd
C-isotopes, which will be useful to find out the deformations and the
spin-parities of the ground states of these nuclei. We studied electric dipole
states of C-isotopes focusing on the interplay between low energy Pigmy
strength and giant dipole resonances.
  Reasonable agreement is obtained with available experimental data for the
photoreaction cross sections both in the low energy region below $\hbar \omega
$=14 MeV and in the high energy giant resonance region (14 MeV $<\hbar \omega
\leq $30 MeV).
  The calculated transition strength below dipole giant resonance ($\hbar
\omega \leq $14 MeV) in heavier C-isotopes than $^{15}$C is found to exhaust
about $12\sim16%$ of the classical Thomas-Reiche-Kuhn sum rule value and
$50\sim80%$ of the cluster sum rule value.
