Birch and Swinnerton-Dyer conjecture for elliptic curves over number fields

Let $E/K$ be an elliptic curve. Write $L(E/K,s)$ for its Hasse–Weil $L$-function, $r=\operatorname{ord}_{s=1}L(E/K,s)$, and let $(r_1,r_2)$ be the signature of $K$. Let $\operatorname{Reg}_{E/K}$ be the regulator, $\operatorname{Sha}_{E/K}$ the Tate–Shafarevich group, $\operatorname{C}_{E/K}$ the product of Tamagawa numbers and other local factors, and $E(K)_{\operatorname{tors}}$ the torsion subgroup. Denote by $\Delta_K$ the discriminant of $K$ and by $\Omega_+(E),\Omega_-(E)$ the periods of $E$.

Birch and Swinnerton-Dyer conjecture. One has

$$\operatorname{ord}_{s=1}L(E/K,s)=\operatorname{rk}(E/K),$$

and the leading term at $s=1$ satisfies

$$\lim_{s\to1}\frac{L(E/K,s)}{(s-1)^r}\cdot\frac{\sqrt{|\Delta_K|}}{\Omega_+(E)^{r_1+r_2}|\Omega_-(E)|^{r_2}}=\frac{\operatorname{Reg}_{E/K}|\Sha_{E/K}|C_{E/K}}{|E(K)_{\operatorname{tors}}|^2}.$$

This conjecture relates the analytic behavior of the Hasse–Weil $L$-function to the arithmetic rank and the size of the Tate–Shafarevich group, assuming its finiteness. Its general status is not specified in the supplied text.