  The classical $\mathrm{AGM}$ produces wonderful interdependent infinite
sequences of arithmetic and geometric means with common limit. For finite
fields $\mathbb{F}_q,$ with $q\equiv 3\pmod 4,$ we introduce a finite field
analogue $\mathrm{AGM}_{\mathbb{F}_q}$ that spawns directed finite graphs
instead of infinite sequences. The compilation of these graphs reminds one of a
$\mathit{jellyfish~swarm},$ as the 3D renderings of the connected components
resemble $\mathit{jellyfish}$ (i.e. tentacles connected to a bell head). These
swarms turn out to be more than the stuff of child's play; they are taxonomical
devices in number theory. Each jellyfish is an isogeny graph of elliptic curves
with isomorphic groups of $\mathbb{F}_q$-points, which can be used to prove
that each swarm has at least $(1/2-\varepsilon)\sqrt{q}$ jellyfish.
Additionally, this interpretation gives a description of the
$\mathit{class~numbers}$ of Gauss, Hurwitz, and Kronecker which is akin to
counting types of spots on jellyfish.
