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For a prime $p$ and a rational elliptic curve $E_{/\mathbb{Q}}$, set$K=\mathbb{Q}(E[p])$ to denote the torsion field generated by$E[p]:=\operatorname{ker}\{E\xrightarrow{p} E\}$. The class group$\operatorname{Cl}_K$ is a module over $\operatorname{Gal}(K/\mathbb{Q})$.Given a fixed odd prime number $p$, we study the average non-vanishing ofcertain Galois stable quotients of the mod-$p$ class group$\operatorname{Cl}_K/p\operatorname{Cl}_K$. Here, $E$ varies over rationalelliptic curves, ordered according to \emph{height}. Our results areconditional and rely on predictions made by Delaunay and Poonen-Rains for thestatistical variation of the $p$-primary parts of Tate-Shafarevich groups ofelliptic curves. We also prove results in the case when the elliptic curve$E_{/\mathbb{Q}}$ is fixed and the prime $p$ is allowed to vary.

Class group statistics for torsion fields generated by elliptic curves