\(\displaystyle \frac{\sigma(nz)}{\sigma^{n^2}(z)} = \frac{(-1)^{n-1}}{\left(1!2!3!\cdots(n-1)!\right)^2}\begin{vmatrix}

\wp'(z) & \wp''(z) & \cdots & \wp^{(n-1)}(z)\\

\wp''(z) & \wp'''(z) & \cdots & \wp^{(n)}(z)\\

. & . & \ldots & .\\

. & . & \ldots & .\\

\wp^{(n-1)}(z) & \wp^{(n)}(z) & \cdots & \wp^{(2n-3)}(z)

\end{vmatrix}\)

Do you guys have any ideas? Our king of elliptic functions, perhaps?