More precisely, suppose E is an elliptic curve over $\mathbb{Q}$ of conductor $p$. Let $L=\mathbb{Q}(E[p])$, the field obtained by adding all p-torsion points of $E$. Are there commands in SAGE that allow us to test whether the p-part of $Cl(L)$ is non-trivial?
For cyclotomic fields $\mathbb{Q}(\zeta_p)$, one can use Kummer's criterion to see whether the p-part of $Cl(\mathbb{Q}(\zeta_p))$. However, I do not aware of any analogous criterion for $L=\mathbb{Q}(E[p])$.