  One of the most celebrated applications of Gauss' $_2F_1$ hypergeometric
functions is in connection with the rapid convergence of sequences and special
values that arise in the theory of arithmetic and geometric means. This theory
was the inspiration for a recent paper \cite{jelly1} in which a finite field
analogue of AGM$_\mathbb{R}$ was defined and then studied using finite field
hypergeometric functions. Instead of convergent sequences, one gets directed
graphs that combine to form disjoint unions of graphs that individually
resemble "jellyfish". Echoing the connection of hypergeometric functions to
periods of elliptic curves, these graphs organize elliptic curves over finite
fields. Here we use such "jellyfish swarms" to prove new identities for Gauss'
class numbers of positive definite binary quadratic forms. Moreover, we prove
that the sizes of jellyfish are in part dictated by the order of the prime
above 2 in certain class groups.
