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Relativistic momentum is the momentum of an object, taking relativity into account. It is not relativistically-invariant.
The term mass (i.e., with no qualifying adjective) has come to be used two ways in light of relativity: to mean the object's rest mass, which is the mass as we normally think of it (and is a recommended usage for the term mass), or to mean a relativistic mass, the mass equivalent of the object's total energy, including kinetic energy and relativistic effects. (Effectively, this potential terminology-confusion only arises due to relativity, and the difference is only for a moving object, and significant only if at relativistic speeds.)
Formulas for momentum:
(non-relativistic) momentum = MrestV relativistic momentum = γMrestV or given Mrel = γMrest, relativistic momentum = MrelV
(The above inclusion of the Lorentz factor accounts for special relativity, but does not fully account for general relativity.)