In arithmetic geometry one often encounters continuous representations $\rho:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to \operatorname{GL}_n(\mathbb{Q}_l)$ for some $n\geq 1$ and some prime number $l$ such that no Tate twist of $\rho$ factors through $\operatorname{GL}_n(K)$ for any number field $K$. An example is the first $l$-adic cohomology of an elliptic curve over $\mathbb{Q}$: if $\rho \otimes \mathbb{Q}_l(i)$ were to factor then $\wedge^2 (\rho\otimes \mathbb{Q}_l(i))\approx \wedge^2 \rho \otimes \mathbb{Q}_l(2i)\approx \mathbb{Q}_l(2i-1)$ would factor as well which is a contradiction since it is infinitely ramified.

This determinant trick is the only argument I know for proving that a representation does not factor. Are there explicit examples of representations unramified outside a finite set of primes and de Rham at $l$ with a trivial determinant and trivial Hodge-Tate weights that do not factor? Do such representations arise naturally in number theory or arithmetic geometry?