Let $E/\mathbb{Q}_p$ be an unramified extension, and let $F_n$ be its cyclotomic extension by $p^n$th roots of unity. We study the Galois module structure of eigenspaces $D^r$ of the unit groups $U_i$ of $F_n$ for the $r$th powers of the Teichm\"{u}ller character, where $2 \leq r \leq p-2$. We use generators of $D^r$ and Sen's explicit reciprocity law to compute certain norm residue symbols. This allows us to determine the conductors of elements in Kummer extensions of $F_n$ and, in particular, to find elements in $U_i \cap D^r$ achieving minimal conductor for a large range of values of $i$.