(Conjectural) description of rank of elliptic curves?

Question

The L-function of an elliptic curve E/$\mathbb{Q}$ is defined purely in terms of the étale cohomology of the curve. If the Birch-Swinnerton Dyer conjecture were true, then the rank of the ellitic curve can be computed as the order of vanishing of the L-function $\text{L}(E,s)$ at $s=1$ (the point of symmetry of the functional equation). So, conjeturaly, we should be able to determine the rank of an elliptic curve $E$ purely with the étale cohomology. However, I have no idea how to do this, any a literature search has yielded no result. My own attempt was to try to compute the derivative of the L-function and was not useful. Can anyone point me to a good source or provide a solution?