Certain Abelian varieties bad at only one prime

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.