I am a new user. I am very sorry if this question is inappropriate. I would like to test some of my speculations about the arithmetics of number fields obtained by adding p-torsion points of elliptic curves.
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 $L=\mathbb{Q}(E[p])$.
Thank you for reading.
