Throughout "curve" means smooth projective curve over an algebraically closed field.

**Motivation and Background**

I read somewhere that Atiyah has classified vector bundles on elliptic curves. My understanding is that the story is roughly: every vector bundles breaks up as a direct some of indecomposable vector bundles. The indecomposable vector bundles are further divided by their degree and rank. The set $Ind(d,r)$ of isomorphism classes of indecomposable vector bundles of a fixed degree $d$ and rank $r$ has the structure of variety isomorphic to the Jacobian of the curve; namely the curve itself. In fact $Ind(d,r) \cong Ind(0, gcd(d,r) )$ so its enough to consider the degree $0$ vector bundles.

Now for $V,V'$ vector bundles on any curve $C$ its true that $\deg V \otimes V' = \deg V\cdot rk(V') + \deg V'\cdot rk(V)$. So in the case of an elliptic curve the set $Ind(0,r)$ is a torsor for $Pic^0(C)$. Additionally, there is a unique isomorphism class $V_r \in Ind(0,r)$ characterized by the fact that $h^0(V_r) \ne 0$ (in fact $h^0(V_r)=1$).

**The Question**

Since curves of low genus are usually "simple enough" to easily describe explicitly, I would like to see how explicit I can be with a description of the $V_r$, or at least $V_2$. So my question is simply, given an explicit elliptic curve in $\mathbb{P}^2$, say $zy^2 - x(x-z)(x+z)$can you reasonably show how to construct $V_2$; i.e. give cocyles for it $\phi_{ij} \colon U_{ij} \to GL_2(k)$, or produce a graded module for it?

**Initial Thoughts**

$V_2$ should correspond to the unique nontrivial extension in $Ext^1(\mathcal{O}, \mathcal{O}) \cong H^1(\mathcal{O}) \cong k$.On a curve $C$, $0 \to \mathcal{O}_C \to K(C) \to K(C)/\mathcal{O}_C \to 0$ is a flasque resolution of the structure sheaf and an element in $H^1(\mathcal{O}_C)$ corresponds to a map $\alpha \colon \mathcal{O}_C \to K(C)/\mathcal{O}_C$. Then the desired extension should be the pullback of $0 \to \mathcal{O}_C \to K(C) \to K(C)/\mathcal{O}_C \to 0$ via $\alpha$.

The trouble with this is that I can't seem to pin down what $\alpha$ is. I now that, via Serre duality, it corresponds to a global section of $H^0(\omega_C)$ which I can explicitly describe as a differential on the curve. Using the example I mentioned above, on the affine patch where $z \ne 0$, it is: $\frac{dx}{2y} = -\frac{dy}{3x^2-1}$ but I guess I don't understand Serre duality well enough to determine $\alpha$ from this. Also I'm not even sure if this will lead to a reasonable way of getting at the cocylces of $V_2$. And this is so close to "just using the definitions" I have to imagine there might be a better way to go about this...