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Thank you. I have a follow-up naive question. In this algorithm, they show how to find primes $l$ such that $E/K$ has an $l$-isogeny. It seems to me that in this context the $l$-isogenies are thought to be defined over $K$, is that right?
@ChrisWuthrich, thanks for your answer! Indeed, $A_5$ is not solvable. For the other two cases do you know when it happens that $\bar{\rho}_{E,p}(G_l)$ is isomorphic to $A_4/S_4$, for example do you know if inertia looks in a certain way or any condition on $p$? Thank you!
Thank you @Ariyan! I just wanted to ask why $L/K$ being unramified implies that $B'$ is finite flat etale over $B$ and do you have a good reference for these notions?
@WillSawin, thanks for your reply. I am a bit confused why this is true. My reasoning was: if $E$ is supersingular, then $\pi^2=[-p]$ and $E[-p]=E[p]=0$. This implies $[-p]$ is injective, which implies $\pi$ is injective. Would you mind pointing out where this argument goes wrong?
