  By measuring magnetizations of pristine and Ta-doped CsV$_{3}$Sb$_{5}$ single
crystals, we have carried out systematic studies on the lower critical field,
critical current density, and equilibrium magnetization of this kagome system.
The lower critical field has been investigated in the two typical samples, and
the temperature dependent lower critical field obtained in Ta-doped sample can
be fitted by using the model with two $s$-wave superconducting gaps yielding
the larger gap of $2\Delta_{s1}/k_\mathrm{B}T_\mathrm{c}=7.9\;(\pm1.8)$. This
indicates a strong-coupling feature of the V-based superconductors. The
measured magnetization hysteresis loops allow us to calculate the critical
current density, which shows a very weak bulk vortex pinning. The magnetization
hysteresis loops measured in these two kinds of samples can be well described
by a recently proposed generalized phenomenological model, which leads to the
determination of many fundamental parameters for these superconductors. Our
systematic results and detailed analysis conclude that this V-based kagome
system has features of strong-coupling superconductivity, relatively large
Ginzburg-Landau parameter and weak vortex coupling.
