We consider a generalization of the abelian Higgs model with a Chern-Simons term by modifying two terms of the usual Lagrangian. We multiply a dielectric function with the Maxwell kinetic energy term and incorporate nonminimal interaction by considering generalized covariant derivative. We show that for a particular choice of the dielectric function this model admits both topological as well as nontopological charged vortices satisfying Bogomol'nyi bound for which the magnetic flux, charge and angular momentum are not quantized. However the energy for the topolgical vortices is quantized and in each sector these topological vortex solutions are infinitely degenerate. In the nonrelativistic limit, this model admits static self-dual soliton solutions with nonzero finite energy configuration. For the whole class of dielectric function for which the nontopological vortices exists in the relativistic theory, the charge density satisfies the same Liouville equation in the nonrelativistic limit.
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