What's the time complexity of finding a generator point on an elliptic curve of rank 1 over the rationals?

Question

$E: y^2 = x^3+(4N^2+12N-3)x^2+32(N+3)x$, and we're looking only at cases where the torsion group is isomorphic to Z/6Z (which means that N is not 2).

Answer 1

This elliptic curve underlies finding an integer solution of \begin{equation*} N=\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} \end{equation*}

I spent several weeks finding generators for $N \in [3,999]$ several years ago.

The time to find a generator on a rank $1$ curve is essentially related to the $\mathbf{height}$ of the generator. Some $N$ have small heights, some are very large, so it is not possible to be dogmatic about the time.