Do we have any examples of non-modular elliptic curves over number fields $K \neq \mathbb{Q}$?
In particular, I came across a paper by Freitas, Le Hung, and Siksek, "Elliptic curves over real quadratic fields are modular", which shows that there are at most finitely many non-modular elliptic curves over a fixed totally real number field. Is there any known example of $E/K$ where $E$ is a non-modular elliptic curve and $K$ is a totally real number field?
It is a widely believed conjecture that all elliptic curves, over any number field $K$, are modular (in the sense that there exists an automorphic representation [*] $\pi$ of $\operatorname{GL}_2 / K$ whose $L$-function is the same as that of $E$). No counterexamples are known, and it would be extremely big and disturbing news for number theory if somebody stumbled across one.
([*] If you are surprised not to see the word "cuspidal" here, then there's a reason for that: you need to take non-cuspidal $\pi$ if $E$ has CM, and the field it has CM by is a subfield of $K$. In all other cases $\pi$ will be cuspidal. In particular, if you are only looking at totally-real $K$, then you can validly write "cuspidal" there.)
