Existence of non-elliptic mod l Galois representations for every l >5

For $\mathrm{\ell }=3$ and 5 it is known that every odd, irreducible, 2-dimensional representation of $\text{Gal}(\bar{\text{Q}}\mathrm{/}\text{Q})$ with values in ${\text{F}}_{\mathrm{\ell }}$ and determinant equal to the cyclotomic character must “come from” the $\mathrm{\ell }$-torsion points of an elliptic curve defined over $\text{Q}$. We prove, by giving concrete counter-examples, that this result is false for every prime $\mathrm{\ell }>5$.