On the algebraic and analytic ranks of the twin-prime elliptic curve $y^2=x(x-2)(x-p)$

Abstract: Let $p\ge 7$ and suppose that $p$ and $p-2$ are prime. We study $E_p:y<sup>2=x(x-2)(x-p)$ using the classical $2$-Selmer calculation of Qiu-Zhang and the Cassels-Tate pairing. These give $2<sup>\infty$-Selmer corank one for $p\equiv 3,5\pmod{8}$ and corank zero for $p\equiv 7\pmod{8}$. Assuming the low-corank Birch-Swinnerton-Dyer statement announced in the October 2026 OpenAI mathematics release, we deduce equality of the analytic and algebraic ranks in these cases, finiteness of the full Tate-Shafarevich group, and the exact BSD leading-term formula. The $2$-primary Tate-Shafarevich group is trivial, and the remaining factor has odd square order. We identify precisely the obstruction left in the class $p\equiv 1\pmod{8}$. An appendix gives a local descent proof of the known Selmer dimensions in the coordinates used here.