Explicit mock Heegner points and BSD formula on certain Mordell curves

For a natural number $a$, let $E_{2a}$ be the Mordell elliptic curve $X^3 +Y^3=2a$.

We give an explicit construction of (mock) Heegner point on the Mordell curve $E_{2p}$ for a prime $p\equiv 4 \mod 9$ and $E_{2p^2}$ for a prime $p\equiv 7 \mod 9$, under the assumption that $2$ is not a cube modulo $p$.

We also verify the explicit Gross-Zagier formula for these curves and go on to show that the BSD formula holds for these curves up to a $2$-adic unit.

Using a result of Burungale-Flach, we show that the full BSD formula holds for the rank zero curve $E_{2p}$ for $p\equiv 7 \mod 9$ and $E_{2p^2}$ for $p\equiv 4 \mod 9$, whenever $2$ is not a cube modulo $p$.