Birch and Swinnerton-Dyer invariants of an elliptic curve/$\mathbb{Q}$ (reviewed)

For an elliptic $E$ curve over $\mathbb{Q}$ the Birch/Swinnerton-Dyer invariants are

• rank, $r$
• regulator, \( \text{Reg} \)
• real period, \( \Omega \)
• Tamagawa numbers, \( \prod_p c_p \)
• torsion order, \( \#E_{\rm tor} \)
• analytic order of Sha, Ш\(_{an}\)

The Birch and Swinnerton-Dyer Conjecture asserts that the above invariants combine to give the order of vanishing and the first non-zero Taylor series coefficient of $L(E,s)$ at the central point $s=1.$