Let $E/\mathbb{Q}$ be an elliptic curve. The weak Birch and Swinnerton-Dyer conjecture predicts that
$$\text{ord}\_{s=1}L(E, s)=\text{rank} E(\mathbb{Q}).$$
Thanks to the work of Gross-Zagier and Kolyvagin, we know that this conjecture is true if $\text{ord}\_{s=1}L(E, s)\le 1$.


What is known in the case $\text{rank} E(\mathbb{Q})\le 1$? Is it known that if $\text{rank} E(\mathbb{Q})=1$, then $L'(E, 1)\neq 0$?


Thank you!



BSD conjecture for rank 1 elliptic curves
The following theorem is due to Chris Skinner, in [this 2014 paper](https://arxiv.org/abs/1405.7294).



> 
> Let E/Q be an elliptic curve such that rank E(Q) = 1 **and** the
>  Tate-Shafarevich group Sha(E / Q) is finite, and some other technical
>  assumptions hold. Then $ord\_{s = 1} L(E, s) = 1$, and in particular
>  $L'(E, 1) \ne 0$.
> 
> 
> 


This is, as far as I know, the best one can do at the moment; if you don't know that Sha (or at least its p-primary part for some p) is finite, then you're stuck.


