  We present a systematic study of the $^{10-14}\text{C}$ isotopes within the
\textit{ab initio} no-core shell model theory. We apply four different
realistic nucleon-nucleon (NN) interactions: (i) the charge-dependent Bonn 2000
(CDB2K) potential (ii) the inside non-local outside Yukawa (INOY) potential
(iii) the next-to-next-to-next-to-leading order (N$^3$LO) potential, and (iv)
the optimized next-to-next-to-leading order (N$^2$LO$_{opt}$) potential. We
report the low-lying energy spectra of both positive and negative parity states
for the $^{10-14}\text{C}$ isotopes and investigate the level structures. We
also calculate electromagnetic properties such as transition strengths,
quadrupole and magnetic moments. The dependence of point-proton radii on the
harmonic oscillator frequency and basis space is shown. We present calculations
of the translation invariant one-body density matrix in the no-core shell model
and discuss isotopic trends in the density distribution. The maximum basis
space reached is $10 \hbar \Omega$ for $^{10}\text{C}$ and $8 \hbar \Omega$ for
$^{11-14}\text{C}$, with a maximum M-scheme dimension of $1.3 \times 10^{9}$
for $^{10}\text{C}$. We found that while the INOY interaction gives the best
description of the ground state energies, the N$^3$LO interaction best
reproduces the point-proton radii.
