  Let $p\equiv 4,7\mod 9$ be a rational prime number such that $3\mod p$ is not
a cubic residue. In this paper we prove the 3-part of the product of the full
BSD conjectures for $E_p$ and $E_{3p^3}$ is true using an explicit Gross-Zagier
formula, where $E_p: x^3+y^3=p$ and $E_{3p^2}: x^3+y^3=3p^2$ are the elliptic
curves related to the Sylvester conjecture and cube sum problems.
