Let $E$ be an elliptic curve over an algebraically closed field $k$ of characteristic $p$. Is there any nice computation for the group $H^1(E,\alpha_p)$ and $H^1(E,\mathbb{G}_a)$? The cohomology is taken in the flat topology.

When these groups are trivial? and is there any way to describe them? I am not asking for the description of $H^1$ using flat torsors on $E$, I looking for a description using which I could (at least) determine the groups are trivial or not.