An explicit Gross--Zagier formula related to the Sylvester conjecture

Let p\equiv 4,7\mod 9 be a rational prime number such that 3\mod p is not a cube. In this paper, we prove the 3-part of \vert{\rm III}(E_p)\vert\cdot \vert{\rm III}(E_{3p^2})\vert is as predicted by the Birch and Swinnerton-Dyer conjecture, 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.