  An abelian threefold $A_{/{\mathbb Q}}$ of prime conductor $N$ is favorable
if its 2-division field $F$ is an ${\mathcal S}_7$-extension over ${\mathbb Q}$
with ramification index 7 over ${\mathbb Q}_2$. Let $A$ be favorable and let
$B$ be a semistable abelian variety of dimension $3d$ and conductor $N^d$ with
$B[2]$ filtered by copies of $A[2]$. We give a sufficient and computable class
field theoretic criterion on $F$ to guarantee that $B$ is isogenous to $A^d$.
