The classical AGM produces wonderful infinite sequences of arithmetic and geometric means with common limit. For finite fields Fq, with q ? 3 (mod 4), we introduce a finite field analogue AGMFq that spawns directed finite graphs instead of infinite sequences. The compilation of these graphs reminds one of a jellyfish swarm, as the 3D renderings of the connected components resemble 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 Fq -points. We will describe this theory in this lecture, whose players include Gaussian hypergeometric functions and Gauss?class numbers of binary quadratic forms.