  In this study, we conducted a linear instability analysis of penetrative
magneto-convection in rapidly rotating Boussinesq flows within tilted f-planes,
under the influence of a uniform background magnetic field. We integrated wave
theory and convection theory to elucidate the penetration dynamics in rotating
magneto-convection. Our findings suggest that efficient penetration in rapidly
rotating flows with weakly stratified stable layers at low latitudes can be
attributed to the resonance of wave transmission near the interface between
unstable and stable layers. In the context of strongly stratified flows, we
derived the scaling relationships of penetrative distances $\Delta$ with the
stability parameter $\delta$. Our calculation shows that, for both
rotation-dominated and magnetism-dominated flows, $\Delta$ obeys a scaling of
$\Delta\sim O(\delta^{-1/2})$. In rotation-dominated flows, we noted a general
decrease in penetrative distance with increased rotational effect, and a minor
decrease in penetrative distance with increased latitude. When a background
magnetic field is introduced, we observed a significant shift in penetrative
distance as the Elsasser number $\Lambda$ approaches one. The penetrative
distance tends to decrease when $\Lambda \ll 1$ and increase when $\Lambda \gg
1$ with the rotational effect, indicating a transition from rotation-dominated
to magnetism-dominated flow. We have further investigated the impact of the
background magnetic field when it is not aligned with the rotational axis. This
presents a notable contrast to the case where the magnetic field is parallel to
the rotational axis.
