Generating Frobenius classes (awaiting review)

The image of a mod-$\ell$ Galois representation $\rho$ is generated by the images under $\rho$ of the Frobenius automorphisms of a finite set of unramified primes of the base field. We may canonicalize this set by choosing the minimal sequence of increasing unramified primes $p_1,\ldots,p_n$ such that the cardinalities of the subgroups generated by the images of Frobenius conjugacy classes $$H_k:=\langle \rho(\mathrm{Frob}_{p_1}).\ldots\rho(\mathrm{Frob}_{p_k})\rangle$$ are strictly increasing for every $1\le k\le n$.