I've been using modular polynomials to compute isogeny vulcanoes with prime degree $l$ over finite fields $\mathbb{F}\_p$, excluding cases containing the $j$-invariants $0$ and $1728$ or $j$-invariants with double roots on the polynomial, building the graph by at each step using the polynomial to find the neighbors of a vertex. This seemed like the obvious thing to do, and I have not previously questioned myself in this regard. But, now that I think about it, I have not seen any paper explicitly talking about doing this, I just assumed it was like that, but maybe there is a better way to compute the isogeny volcano graph than this.



Better way to compute elliptic curves over finite fields?
The paper [Isogeny volcanoes by Andrew V. Sutherland](https://arxiv.org/abs/1208.5370) contains various improvements for computing isogeny volcanoes. The published version is in *ANTS X—Proceedings of the Tenth Algorithmic Number Theory Symposium*, 507–530,
Open Book Ser., 1, Math. Sci. Publ., Berkeley, CA, 2013.


