  An abelian surface $A_{/{\mathbb Q}}$ of prime conductor $N$ is favorable if
its 2-division field $F$ is an ${\mathcal S}_5$-extension with ramification
index 5 over ${\mathbb Q}_2$. Let $A$ be favorable and let $B$ be any
semistable abelian variety of dimension $2d$ and conductor $N^d$ such that
$B[2]$ is filtered by copies of $A[2]$. We give a sufficient class field
theoretic criterion on $F$ to guarantee that $B$ is isogenous to $A^d$.
  As expected from our paramodular conjecture, we conclude that there is one
isogeny class of abelian surfaces for each conductor in $\{277,
349,461,797,971\}$. The general applicability of our criterion is discussed in
the data section.
