Constructing $k$-morphisms between algebraic curves

Question

Let $k$ be a finite field containing a sixth root of unity $\xi_6$. Let $C: t^2 = s^6 - 1$ be an hyperelliptic curve over $k$. Let $E': \eta^2 = \xi^3 + 1$ be an elliptic curve over $k$. I am trying the determine the group of $k$-morphisms $\text{Mor}_k(C,E')$. Is there a systematic way of approaching this problem? I have found some morphisms such as $(s,t) \mapsto (-s^2,it)$ (only possible if $k$ contains $i = \sqrt{-1}$) or $(s,t)\mapsto (-\xi_6^2s^2,it)$, but I fail to construct more. How does one find more morphisms (if there are any). How does one determine there are no more. I am aware of this being a hard problem in general, but I was wondering whether there are some techniques one can use in specific cases.