  Let $p$ and $q$ be distinct primes, and let $X_{p,q}$ be the $(q+1)$-regular
graph whose nodes are supersingular elliptic curves over
$\overline{\mathbb{F}}_p$ and whose edges are $q$-isogenies. For fixed $p$, we
compute the distribution of the $\ell$-Sylow subgroup of the sandpile group
(i.e.\ Jacobian) of $X_{p,q}$ as $q \to \infty$. We find that the distribution
disagrees with the Cohen-Lenstra heuristic in this context. Our proof is via
Galois representations attached to modular curves. As a corollary of our
result, we give an upper bound on the probability that the Jacobian is cyclic,
which we conjecture to be sharp.
