Elliptic \(\operatorname{mod} \ell\) Galois representations which are not minimally elliptic (Q2381102)

The author extends a result of \textit{F. Calegari} [Pac. J. Math. 225, No.~1, 1--11 (2006; Zbl 1119.11028)] by showing that for every prime number \(\ell>7\) there exists an irreducible representation of \(\text{Gal}(\overline{\mathbb Q} / {\mathbb Q})\) on \({\mathbb F}_\ell^2\) which comes from an elliptic curve, but whose (prime-to-\(\ell\))-conductor differs from that of any elliptic curve serving the purpose. More specifically, the action of the Galois-group on the \(\ell\)-torsion of the elliptic curve defined by \(Y^2 = X(X-3^\ell)(X-3^\ell-1)\) defines the sought after representation.