Difference of j-invariant values and the abc conjecture I learned of the following example in a recent seminar: if $j(\tau)$ denotes the usual $j$-invariant, and $\alpha = (-1+i\sqrt{163})/2$, then
\begin{align*}
\frac{j(i)}{1728} &= 1 \\
\frac{-j(\alpha)}{1728} &= 151931373056000 = 2^{12}5^323^329^3 \\
\frac{j(i)-j(\alpha)}{1728} &= 151931373056001 = 3^37^211^219^2127^2163.
\end{align*}
A computation revealts that $(a,b,c) = (1, 151931373056000, 151931373056001)$ is a reasonably high-quality abc triple: if $R$ denotes the radical of $abc$ (the product of the distinct primes dividing $abc$), then
$$
q(a,b,c) = \frac{\log c}{\log R} = 1.20362\dots.
$$
My question is: is this a freak coincidence (including the fact that $163$ divides $c$), or is this example part of a larger theory? Are there families of high-quality abc triples that come from differences of $j$-invariants?
 A: There is a beautiful theory by Gross and Zagier (On singular moduli, J. Reine Angew. Math. 355 (1985), 191–220) that explains completely the factorizations of 
$$
  j(\tau_1)-j(\tau_2)
$$
for $\tau_1,\tau_2$ lying in (possibly two different) imaginary quadratic fields. There are recent extensions by Kristin Lauter and Bianca Viray, and there they state the result. So these numbers are always highly factorizable, this is not at all an isolated phenomenon! And the appearance of 163 here is predictable as well.
However, generally $j(\tau)$ would lie in some abelian extension of an imaginary quadratic field, this is part of the CM theory for elliptic curves. These are almost never rational numbers, in fact there are exactly 13 imaginary quadratic irrationalities $\tau$, including $i$ and $\alpha$ from your example, for which $j(\tau)$ is in ${\mathbb Q}$ - Noam Elkies has a complete list in his answer. So you won't be able to construct many ABC examples like that, unfortunately.
A: As T.Dokchitser explained, there is indeed good reason for this,
but $j(\alpha) = -640320^3$ is the last such example for the
classical $j$-invariant.  (One of the motivations for my
"Shimura curve computations"
(LNCS 1423 [=ANTS-3 proceedings, 1998]) was to find exotic ABC
triples coming from CM points on Shimura curves, but no spectacular
example turned up.)  Still the connection can be used in
the opposite direction.  If we somehow knew the ABC conjecture
(with effective constants) but didn't yet know that ${\bf Q}(\alpha)$
is the last quadratic imaginary field of class number one, then we could
use the the fact that $j(\alpha)$ is a cube and $j(\alpha)-12^3$ is almost
a square to solve the $h=1$ problem.  In 2000 Granville and Stark
(Invent. Math. 139 #3, 509$-$523) used this idea to prove that
an ABC conjecture over arbitrary number fields (with effective constants)
would imply a strong enough lower bound on $h(-D)$ to banish the Siegel zero
for imaginary quadratic characters!  Unfortunately this approach has yet
to yield an unconditional proof.
P.S. (In response to the OP's comment on T.Dokchitser's accepted answer)
Here's the list of $13$ discriminants $-D$ of imaginary quadratic orders
of class number 1 and the corresponding integers $j = j(\alpha_D)$ with
$\alpha_D = (D + \sqrt{-D})/2$:
$$
\begin{array}{c|cccccccc}
-D & -3 & -4 & -7 & -8 & -11 & -12 & -16 & -19 & 
\\ \hline 
 j & 0 & 12^3 & -15^3 & 20^3 & -32^3 & 2\cdot 30^3 & 66^3 & -96^3 &
\end{array}
$$ $$
\begin{array}{c|ccccc}
-D & -27 & -28 & -43 & -67 & -163 \\ 
\hline
j & -3 \cdot 160^3 & 255^3 & -960^3 & -5280^3 & -640320^3
\end{array}
$$
