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

Let $p\ge 7$ and suppose that $p$ and $p-2$ are prime. We study $E_p:y^2=x(x-2)(x-p)$ using the classical $2$-Selmer calculation of Qiu-Zhang and the Cassels-Tate pairing. These give $2^\infty$-Selmer corank one for $p\equiv 3,5\pmod{8}$ and corank zero for $p\equiv 7\pmod{8}$. Assuming the low-cora