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?

, $$x^2-163\cdot640320y^2=1$$ as $\displaystyle x=2u+1,\; y = \sqrt{\frac{4\cdot640320^2v}{163\cdot12^3}},$ and where, $$u = 151931373056000\\ v = 151931373056001$$ $\endgroup$ – Tito Piezas III Mar 7 '16 at 17:46Pell equation