  We consider the onset of Boussinesq convection in a horizontal layer of
electrically conducting incompressible fluid with rigid electrically insulating
horizontal boundaries. The fluid is heated from below and rotates about a
vertical axis; vertical magnetic field is imposed. The instability of the
trivial steady state (fluid at rest) can be monotonic or oscillatory depending
on parameters of the problem (the kinematic and magnetic Prandtl, Taylor and
Chandrasekhar numbers). If the Taylor number is sufficiently large, convective
rolls, emerging in the monotonic instability, experience the K\"uppers-Lortz
instability. We study how the critical Rayleigh number, the type of instability
of the trivial steady state and the critical Taylor number for the
K\"uppers-Lortz instability depend on the Prandtl numbers. We consider Prandtl
numbers not exceeding one, which is typical for the inner core of the Earth.
