$P_c\left(4312\right)$ observed by the LHCb collaboration is confirmed as a pentaquark and its structure, production, and decay behaviors attract great attention from theorists and experimentalists. Since its mass is very close to sum of ${\Sigma }_c$ and $\overline{D}$ masses, it is naturally tempted to be considered as a molecular state composed of ${\Sigma }_c$ and $\overline{D}$. Moreover, $P_c\left(4312\right)$ is observed in the channel with $J/\psi p$ final state, requiring that isospin conservation $P_c\left(4312\right)$ is an isospin-$1/2$ eigenstate. In the literature, several groups used various models to estimate its spectrum. We systematically study the pentaquarks within the framework of the Bethe-Salpeter equation; thus $P_c\left(4312\right)$ is an excellent target because of the available data. We calculate the spectrum of $P_c\left(4312\right)$ in terms of the Bethe-Salpter equations and further study its decay modes. Some predictions on other possible pentaquark states that can be tested in future experiments are made.