I'm trying to prove the following problem in the *Deformation theory* book by Hartshorne.

Any normalized vector bundle $\mathcal E$ of rank 2 degree 1 on an elliptic curve $\mathcal C$ can be written as a non-split extension

$0 \to \mathcal{O_C} \to \mathcal{E} \to \mathcal{O_C(p)} \to 0$

by a uniquely determined point p. (up to isomorphism)

It is easy to see that such data gives unique non-split extension. But the converse direction is not easy to show. I thought the proof of the classification of vector bundles on $\mathbb{P^1}$ may helps me, but I failed. How can I do this? I appreciate any helps or reference.