The primes $p$ of the form $p = -(4a^3 + 27b^2)$ The current question is motivated by this question.
It is known that the number of imaginary quadratic fields of class number 3 is finite.
Assuming the answer to this question is affirmative, I came up with the following question.
Let $f(X) = X^3 + aX + b$ be an irreducible polynomial in $\mathbb{Z}[X]$.
Let $p = -(4a^3 + 27b^2)$ be the discriminant of $f(X)$.
We consider the following conditions.
(1) $p = -(4a^3 + 27b^2)$ is a prime number.
(2) The class number of $\mathbb{Q}(\sqrt{p})$ is 3.
(3) $f(X) \equiv (X - s)^2(X - t)$ (mod $p$), where $s$ and $t$ are distinct rational integers mod $p$.
Question
Are there infinitely many primes $p$ satisfying (1), (2), (3)?
If this is too difficult, is there any such $p$?
I hope that someone would search for such primes using a computer.

For (229, -4,-1) the polynomial factors as $(x-200)^2(x-58)$
For (1373, -8,-5) the polynomial factors as $(x-860)(x-943)^2$
For (2713, -13,-15) the polynomial factors as $(x-520)^2(x-1673)$
