Let $E$ be a smooth elliptic curve.
Let $I_2$ be the unique (up to isomorphism) non trivial self-extension of $\mathscr O_E$ and let $L$ and $L'$ be degree zero line bundles on E.
I am trying to compute the rank of
- $L\otimes L^{-1}\otimes I_2$
- $I_2^{\otimes 2}$
- $I_2^{\otimes 3}$
For $L\otimes L^{-1}\otimes I_2$, we have the exact sequence $0\to L\otimes L^{-1} \to L\otimes L^{-1}\otimes I_2 \to L\otimes L^{-1} \to 0$, so I think that $h^0(E,L\otimes L^{-1}\otimes I_2) =1$ since $L\otimes L^{-1}\otimes I_2$ can be seen as a point in $\operatorname{Ext}^1(L\otimes L^{-1},L\otimes L^{-1}) = H^1(\operatorname{Hom}(L\otimes L^{-1},L\otimes L^{-1})) $ and $\operatorname{Hom}(L\otimes L^{-1},L\otimes L^{-1})\cong \mathbb{C}$ by stability of $L\otimes L^{-1}$, but I am missing here something to prove it formally.
For the other two bundles, any help is much appreciated.