Rank of $E: y^2 = x^3 + t^{p+1} + 1$ over $k(t)$ with $k = \mathbb{F}_{p^2}$ and $p \equiv -1 \mod 6$

Question

This is part of of my ongoing masters thesis and I was hoping someone has a good idea. I have the elliptic curve in my title and we can define the algebraic curve $C:s^6 = t^{p+1} + 1$ over $k$. Then the function field $k(C)$ is a degree 6 Galois extension of $k(t)$ and using an very old result (1966) from Tate and Shafarevich I concluded that the Mordell-Weil rank of $E(k(C))$ equals $10(p-1)$. From another reference I know that the rank of $E(k(t))$ must be $2(p-1)$ and I am wondering if anyone knows a way to get this from $r(E(k(C))) = 10(p-1)$. Maybe decompose using the Galois group? Any suggestion would be helpful.