[curves] How do you find a generator of an EC group?

[Ron Garret]

SLSIA.  I’m working on an introductory survey of ECC (the one suggested in the “Climbing the elliptic learning curve” thread a few weeks back) and I’m making good progress but I’m stuck on this issue.  I have, of course, found Shoof’s algorithm for counting curve points, but that only gets you so far.  With that as a baseline, I can kind of imagine an algorithm for finding a base point: compute the number of curve points, factor the result, pick the largest prime factor q, and then find a generator by brute-force search for a point P such that qP=-P.  Is that anywhere close to being the right answer?  Is there a reference I can cite?

Thanks,
rg