The smallest field of definition of a subgroup of the group $\mathrm{PSL}_2$

As previously proved by the author, for each semisimple algebraic group of adjoint type that is dense in the Zariski topology there exists a smallest field of definition which is an invariant of the class of commensurable subgroups. In the present paper an algorithm is given for finding the smallest field of definition of a dense finitely generated subgroup of the group $\mathrm{PSL}_2(\mathbb{C})$. A criterion for arithmeticity of a lattice in $\mathrm{PSL}_2(\mathbb{R})$ or $\mathrm{PSL}_2(\mathbb{C})$ in terms of this field is presented.