On the integer solutions to $u^2+163v^2=w^3$ and others

Question

It seems the solution of,

$$u^2+dv^2 = w^3\tag1$$

involves the class number $h(d)$. Assume $\gcd(u,v)=1$.

Q: For which $\color{red}{prime}\; d$ is the complete solution of $(1)$ in the integers given by,

$$(p^3 - 3 d p q^2)^2 + d(3 p^2 q - d q^3)^2 = (p^2+dq^2)^3\tag2$$

Equating terms between $(1)$ and $(2)$ and eliminating $q$, then the question is equivalent to solving,

$$u + 3 w p - 4 p^3 = 0\tag3$$

Using Mathematica to test various $d$, it seems it depends $\text{mod}\, 8$ and on the class number $h(d)$,

1. If $d=8n+1,8n+5$ with $h(4d)\neq3m$, then $(2)$ is complete with integer $p,q$. Ex. $d=5$.
2. If $d=8n+1,8n+5$ with $h(4d)=3m$, then $(2)$ is not complete. Ex. $d=29$.
3. If $d=8n+3$, then $(2)$ is complete with integer and half-integer $p,q$. Ex. $d=11,163$.
4. If $d=8n+7$, then $(2)$ is not complete. Ex. $d=7,23$.

Q: Is $\text{#}1,2,3,4$ true?