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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.

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The paper Isogeny volcanoes by Andrew V. Sutherland 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.

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  • $\begingroup$ I've skimmed the paper you linked, and in an algorithm to find a path to the floor of a volcano he does use finding the roots of the polynomial $\Phi_l(j,X)$ to get the neighbors of a vertex. He does not give an algorithm to construct a full volcano, but if he builds paths on the graph like this, then his approach might be similar. $\endgroup$
    – José
    Commented Jun 6, 2021 at 18:14

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