  We study the onset of magnetoconvection between two infinite horizontal
planes subject to a vertical magnetic field aligned with background rotation.
In order to gain insight into the convection taking place in the Earth's
tangent cylinder (TC), we target regimes of asymptotically strong rotation. The
critical Rayleigh number $Ra_{c}$ and critical wavenumber are computed
numerically by solving the linear stability problem in a systematic way. A
parametric study is conducted, varying the Ekman number, $E$ (ratio of viscous
to Coriolis forces) and the Elsasser number, $\Lambda$ (ratio of the Lorentz
force to the Coriolis force). $E$ is varied from $10^{-9}$ to $10^{-2}$ and
$\Lambda$ from $10^{-3}$ to $1$. Apply to arbitrary thermal and magnetic
Prandtl numbers, our results verify and confirm previous experimental and
theoretical results showing the existence of two distinct unstable modes at low
values of $E$ one being controlled by the magnetic field, the other being
controlled by viscosity (often called the viscous mode). Asymptotic scalings
for the onset of these modes have been numerically confirmed and precisely
quantified. We show that with no-slip boundary conditions, the asymptotic
behaviour is reached for $E<10^{-6}$ and establish a map in the $(E,\Lambda)$
plane. We distinguish regions where convection sets in either in the magnetic
mode or in the viscous mode. Our analysis gives the regime in which the
transition between magnetic and viscous modes may be observed. We also show
that within the asymptotic regime, the role played by the kinematic boundary
conditions is minimal.
