In 'Rational Points on Elliptic Curves' by Silverman and Tate, it is stated that the rank of the elliptic curve $E_p: y^2 = x^3 + px (p \equiv 1 \mod 8)$ is believed to be either $0$ or $2$. I am surprised that such a fundamental elliptic curve still has open problems associated with it.
My question is:
Is this problem still unresolved? Additionally, if we assume the validity of the parity conjecture, does this problem remain open?