Dynamical structure of AGM over finite fields with order congruent to 5 mod 8

Motivated by classical works of Gauss and Euler on the AGM, Ono and his collaborators investigated the union of AGM sequences over finite fields $\mathbb{F}_q$, where $q \equiv 3 \bmod 4$, which they refer to as swarms of jellyfish. A recent preprint extends some of their results to all finite fields with odd characteristic. For $q \equiv 5 \bmod 8$, we reveal finer details about the structure of the connected components, which turn out to be variants of jellyfish with longer and branched tentacles. Moreover, we determine the total population of these swarms in terms of the celebrated base "congruent number" elliptic curve.