I was watching Mazur's 2019 Einstein lecture at the AMS joint sectional meeting, and he proposed the following question following a discussion, which I paraphrase here:
Given an elliptic curve $E/\mathbb Q$, consider all extension fields $\mathbb Q'$ of $\mathbb Q$ for which the Galois group $\mathrm{Gal}(\mathbb Q' / \mathbb Q) = M$, is the Monster group. Of these extensions $\mathbb Q'$, how many possess the property that $\mathrm{rank}\, E(\mathbb Q') > \mathrm{rank}\, E(\mathbb Q)$?
I'd like to be pointed to any useful papers that attempt to address this question, or have results towards the answer of such a question, and perhaps not necessarily just in the case of the Monster, but also other sporadic simple groups.