Given the [*j-function*][1] $j(\tau)$, I was looking at,

$$F(\tau) = \sqrt{d\big(j(\tau)-1728\big)}$$

which appears in Ramanujan-type pi formulas. Let $C_d$ be the prime factors of the **constant term** of the minimal polynomial for $F(\sqrt{-d})$. 


Let $d = 23,47,71,199,167$ which are the smallest *d* such that its [*class number*][2] is $h(-d) = 3,5,7,9,11$, respectively. Then,

$$C_{43} = 3, 7, 19, 43, \color{blue}{163}.$$

$$C_{23} = 3, 7, 11, 19, 23, 43, \color{red}{67}, 83.$$

$$C_{47} = 3, 11, 19, 31, 43, 47, \color{red}{67}, 107, 139, \color{blue}{163}, 179.$$

$$C_{71} = 5, 7, 11, 23, 47, 59, \color{red}{67}, 71, \color{blue}{163}, 283.$$

$$C_{199} = 3, 11, 19, \color{red}{67}, 71, 83, \color{blue}{163}, 199, 571, 787.$$

$$C_{167} = 23, 43, \color{red}{67}, 103, 131, 139, 151, \color{blue}{163}, 167, 227,307,\dots 659.$$

and so on.

*Does anybody know the reason for this "numerology"?*

P.S. For general $d \equiv -1 \,\text{mod}\,4$, I noticed that the prime factors $p_i$ of $C_d$ tend have $h(-p_i)\leq h(-d)$, though with exceptions.
 


  [1]: http://mathworld.wolfram.com/j-Function.html
  [2]: http://mathworld.wolfram.com/ClassNumber.html