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, \( \#E_{tor} \)
- 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 $s=1.$

**Knowl status:**

- Review status: reviewed
- Last edited by John Cremona on 2019-02-08 12:15:40

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