The Lagrange points are more like "regions" than points in reality. There is some complicated math you can do to work it out, but typically numerical stuff is a bit more reliable once you start to include the rest of the planets, the analytic theory gets pretty complicated the more bodies you include.

The trojans that you see there sort of do a little dance when you plot them in the rotating frame of Jupiter (more technical term is librate). The size of the region has to do with the mass of the parent body, since jupiter is so big, the capturing region is pretty large. Technically there are other trojans around some other planets, but most of them are not long term stable like jupiter is, mainly because jupiter is just big.

Yeah I knew they were regions more than points, I just sort of envisioned them as something more in the same of a tapered tail than something so tall compared the diameter of Jupiter.

I'd like to take the time to tell you that I really appreciate your contributions to this site. I'm always excited when there's a post about asteroids or something similarly sized in the solar system because I know that you'll post about it more often than not.

I've taken a course on the geology of the solar system and a course on exoplanet detection but I definitely consider myself a layman in this field. I've always had a particular fascination with these small tumbling rocks in our solar system so it's always a pleasure to see you talk about them and your work.