The modulus of the Kor\'anyi ellipsoidal ring

The Kor'anyi ellipsoidal ring $\mathcal{E}$ of radii $B$ and $A$, $0<B<A$, is defined as the image of the Kor'anyi spherical ring of the same radii and centred at the origin via a linear contact map $L$ in the Heisenberg group. If $K\ge 1$ is the maximal distortion of $L$ then we prove that the modulus of $\mathcal{E}$ is equal to $$\mathrm{m}\mathrm{o}\mathrm{d}(\mathcal{E})=\left(\frac{3}{8}\right(K^2+\frac{1}{K^2})+\frac{1}{4})\frac{{\pi }^2}{(\log (A\mathrm{/}B){)}^3}\mathrm{.}$$