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.
 A: 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 $(x-520)^2(x-1673)$
A: Here's some code I wrote using Sage:
def quadClassNumber(p):
    K.<alpha> = NumberField([x^2-p]);
    return K.class_number();  

U=1000;
for p in Primes():
    if p < U and quadClassNumber(p)==3:
        J = EllipticCurve([0,-2^4*3^3*p]);
        L = J.integral_points();
        if len(L) > 0:
            for (a,b,1) in L:
                A = a/(-2^2*3);
                B = b/(2^2*3^3);
                if A.is_integral() and B.is_integral():
                    E = EllipticCurve([A,B]);
                    if E.has_multiplicative_reduction(p):
                        print "p=",p;
                        print E;
                        print E.discriminant().factor();
                        R.<t> = PolynomialRing(IntegerModRing(p),1);
                        print R(E(0,1,0).division_points(2,true)/4).factor();
                        print "    ";
                        break;
    elif p > U:
        break;

The integral points of the elliptic curve J correspond (roughly) to pairs of integers $(A,B)$ satisfying $p = -(4A^3 + 27B^2)$.  J is an integral model of the elliptic curve associated with this equation, but the use of J and Sage's integral_points() function comes at the cost of some powers of 2 and 3 that appear in the definitions of $A$ and $B$.
I searched for primes up to 10,000.  For instance, with p=8581, a = -16 and b=17 work, and the associated polynomial factors mod 8581 as $(x + 6166)^2(x + 4830)$.
