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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.

  • 0
    Is it known whether there are infinitely many primes satisfying (1)?2013-09-10

2 Answers 2

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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 $(x-520)^2(x-1673)$

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    I wrote a Maple program.2012-10-05
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Here's some code I wrote using Sage:

def quadClassNumber(p):     K. = 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. = 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 $(x + 6166)^2(x + 4830)$.