  In this work we study the complex entanglement between spin interactions,
electron correlation and Janh-Teller structural instabilities in the 5d$^1$
$J_{eff}=\frac{3}{2}$ spin-orbit coupled double perovskite $\rm Ba_2NaOsO_6$
using first principles approaches. By combining non-collinear magnetic
calculations with multipolar pseudospin Hamiltonian analysis and many-body
techniques we elucidate the origin of the observed quadrupolar canted
antifferomagnetic. We show that the non-collinear magnetic order originates
from Jahn-Teller distortions due to the cooperation of Heisenberg exchange,
quadrupolar spin-spin terms and both dipolar and multipolar
Dzyaloshinskii-Moriya interactions. We find a strong competition between
ferromagnetic and antiferromagnetic canted and collinear quadrupolar magnetic
phases: the transition from one magnetic order to another can be controlled by
the strength of the electronic correlation ($U$) and by the degree of
Jahn-Teller distortions.
