In Silverman's AEC Chapter VI we define the Weierstrass $\wp$ and $\sigma$ functions, particularly for $\Lambda = \mathbb{Z} + \tau \mathbb{Z}\subset \mathbb{C}$ a lattice, $$\wp(z) = \wp(z,\Lambda) = \frac{1}{z^2} + \sum_{\omega \in \Lambda, \omega\neq 0} \left( \frac{1}{(z-\omega)^2}- \frac{1}{\omega^2} \right)$$$$\sigma(z) = \sigma(z,\Lambda) = z\prod_{\omega \in \Lambda, \omega\neq 0}\left(1-\frac{z}{\omega}\right)e^{(z/\omega) + \frac12 (z/\omega)^2}.$$
In Exercise 6.3 of the same chapter we show for all $z,a\in \mathbb{C}\setminus\Lambda$, $$\wp(z)-\wp(a) = -\frac{\sigma(z+a)\sigma(z-a)}{\sigma(z)^2\sigma(a)^2}$$ and $$\wp'(z) = \frac{\sigma(2z)}{\sigma(z)^4}.$$
We now are required to show that for all $n\in\mathbb{Z}$, $\sigma(nz)/\sigma(z)^{n^2}$ is in $C(\Lambda)$, but I am not sure how to approach this, I tried induction but I wasn't getting anywhere.