In this MO comment, William Stein remarked that a "...spectacular example in which Heegner points fail in practice" is the Mordell curve, $$y^2 = x^3+N,\quad N = 7823$$
but doing a $4$-descent succeeds.
The article $4$-descent on the Elliptic curve $y^2 = x^3+7823$ pointed out that until 2002, the Mordell-Weil generators of all Mordell curves have been found for all $N<10000$ except that one. It would turn out to be the large,
$$x=\frac{2263582143321421502100209233517777}{143560497706190989485475151904721}$$
However, I had asked about the number $7824$ before so I knew that curve was in fact,
$$y^2=x^3+(48\times\color{blue}{163})-1$$
Q: Is it coincidence that the largest Heegner number $d=163$ appears here, and that it needs a $4$-descent, unlike smaller numbers?
