  We say that an abelian variety $A_{/\mathbb{Q}}$ of dimension $g$ is prosaic
if it is semistable and its points of order $2$ generate a $2$-extension of
$\mathbb{Q}$. We focus on prosaic abelian varieties $A$ with good reduction
outside one prime $p$, whose ring of endomorphisms ${\rm End}_{\mathbb{Q}}(A)$
is $\mathbb{Z}$ or an order in a totally real field of degree $g$. We show that
$p \equiv 1 \bmod{8}$ and that $A$ has totally toroidal reduction at $p$.
  We construct indecomposable group schemes over $\mathbb{Z}[1/p]$ of exponent
2 whose Galois module is related to the 2-Sylow subgroup $H_2$ of the class
group of $\mathbb{Q}(\sqrt{-p})$. We find that $B[2]$ is such a group scheme
for some $B$ isogenous to $A$ and thereby prove that $2g+2 \le |H_2|$. Moreover
if $2g + 4 \le |H_2|$, then $p$ has the form $a^2+64b^2$, with $a \equiv \pm 1
\bmod{8}$.
  A few examples with real multiplication and one bad prime are given. In
contrast, under the Schinzel hypothesis, we exhibit infinite families of
prosaic abelian surfaces of conductor $pq$, for primes $p \ne q$.
