Cube sums of form $3p$ and $3p^2$ II

Let $p\equiv 2,5\mathrm{m}\mathrm{o}\mathrm{d}9$ be a prime. We prove that both $3p$ and $3p^2$ are cube sums. We also establish some explicit Gross-Zagier formulae and investigate the 3 part full BSD conjecture of the related elliptic curves.