TITLE: There's something strange about $\sqrt{\big(j(\tau)-1728\big)d}$
QUESTION [52 upvotes]: Given discriminant $d$ and j-function $j(\tau)$, I was looking at,
$$F(\tau) = \sqrt{\big(j(\tau)-1728\big)d}$$
which appears in Ramanujan-type pi formulas. Let $C_d$ be the odd prime factors of the constant term of the minimal polynomial for $F(\sqrt{-d})$. Then for prime $d>3$,
$$\begin{aligned}
C_{5} &= 5, 11, 19.\\
C_{7} &= 3, 7, 19.\\
C_{11} &=7, 11, 19, 43.\\
C_{13} &=3, 13, 43.\\
C_{17} &=17, 19, 43, 59, \color{red}{67}.\\
C_{19} &=3, 19, \color{red}{67}.\\
C_{23} &=3, 7, 11, 19, 23, 43, \color{red}{67}, 83.\\
C_{29} &=7, 23, 29, \color{red}{67}, 107.\\
C_{31} &=3, 11, 23, 31, 43.\\
C_{37} &=3, 7, 11, 37, \color{red}{67}, 139.\\
C_{41} &=23, 31, 41, 43, 83, 139,  \color{blue}{163}.\\
C_{43} &=3, 7, 19, 43, \color{blue}{163}.\\
C_{47} &=3, 11, 19, 31, 43, 47, \color{red}{67}, 107, 139,  \color{blue}{163}, 179.\\
C_{53} &=7, 11, 43, 53, 131,  \color{blue}{163}, 211.\\
C_{59} &=3, 5, 11, 23, 31, 43, 47, 59, \color{red}{67}, 211, 227.\\
C_{61} &=3, 19, 47, 61,  \color{blue}{163}.\\
C_{67} &=3, 7, 11, 31, 43, \color{red}{67}.\\
C_{71} &=5, 7, 11, 23, 47, 59, \color{red}{67}, 71, \color{blue}{163}, 283.\\
\vdots\\
C_{163} &=3, 7, 11, 19, 59, \color{red}{67}, 127, \color{blue}{163}, 211, 571, 643.\\
C_{167} &=3, 43, \color{red}{67}, 103, 131, 139, 151, \color{blue}{163}, 167, 227, 307,\dots 659.\\
\end{aligned}$$
and so on.  Notice that the d with $C_d$ divisible by $163$ are the first few primes of Euler's prime-generating polynomial,
$$P_1(n) = n^2+n+41 = 41, 43, 47, 53, 61, 71, 83, 97,\dots$$
and the lesser known,
$$P_2(n) = 4n^2+163 = 163, 167, 179, 199,\dots$$
Similarly, the d with $C_d$ divisible by $67$ intersect with,
$$Q_1(n) = n^2+n+17 = 17, 19, 23, 29, 37, 47, 59, 73, 89,\dots$$
and,
$$Q_2(n) = 4n^2+67 = 67, 71, 83, 103,\dots$$
Q: Does anybody know the reason for this "numerology"?

REPLY [65 votes]: "Numerology" such as you've observed is explained in the paper

Gross, B.H., and Zagier, D.: On singular moduli, J. reine angew. Math. 355 (1985), 191$-$220.  MR772491 (86j:11041)

which gives more generally the factorizations of the constant terms
of the minimal polynomials of $j(\tau) - j(\tau')$ where $\tau,\tau'$
are quadratic imaginaries not equivalent under ${\rm PSL}_2({\bf Z})$.
Your $j(\tau) - 1728$ is the special case $\tau' = i$.
Before seeking patterns in the appearance of factors such as $67$ and $163$,
one might wonder why all the constant terms, which are roughly exponential
in $\sqrt d$, factor into such small prime factors in the first place.
The reason is that these are the primes $p$ for which the elliptic curve
$E: y^2 = x^3 - x$, which has $j$-invariant $1728$, is also the reduction
mod $p$ of a curve of invariant $j(\sqrt{-d}\,)$, and thus has an action of
${\bf Z}[\sqrt{-d}\,]$.  Since $E$ already has an action of ${\bf Z}[i]$
[with $i$ acting by $(x,y) \mapsto (-x,iy)$], this makes $E$ supersingular.
The condition that the endomorphism ring of a supersingular curve $E \bmod p$
accommodate both ${\bf Z}[i]$ and ${\bf Z}[\sqrt{-d}\,]$ comes down to
the representability of $4d$ by the quadratic form $a^2+pb^2$.
In particular $p < 4d$, which explains why all the prime factors
are small.  Your $Q_1$ and $Q_2$ are obtained by setting $b=1$ and $b=2$,
but eventually higher $b$ arise too, e.g. you'll find $p=67$ among
the factors of $C_{151}$ (for which the minimal polynomial has degree $7$)
because $151 = \frac14 (1^2 + 67 \cdot 3^2)$, even though $151$ is not
represented by either $n^2+n+17$ or $n^2+67$.