The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows from the paper "Number fields unramified away from $2$" by John W. Jones in the Journal of Number Theory in 2012.

The existence follows from the fact that $\mathbb{Q}(\zeta_{64})$ has class number $17$ and hence if $H$ is the Hilbert class field of $\mathbb{Q}(\zeta_{64})$, then $H/\mathbb{Q}$ is Galois with $G = Gal(H/\mathbb{Q})$ of order $32 \cdot 17$. If $P$ is a Sylow $2$-subgroup of $G$, then the fixed field of $P$ is a number field $K/\mathbb{Q}$ of degree $17$ ramified only at $2$. (Actually, one of the degree $16$ subfields of $\mathbb{Q}(\zeta_{64})$ also has class number $17$, and the Galois closure of $K/\mathbb{Q}$ has degree $272$. Harbater shows that this is the unique Galois extension of $\mathbb{Q}$ of degree $272$ ramified only at $2$.) My question is the following:

Is it possible to explicitly compute a degree $17$ polynomial $f(x)$ that has a root in $K$?

The motivation for this question is the following. If $C/\mathbb{Q}$ is a curve of genus $g$ with good reduction away from $2$ and $J(C)$ is the Jacobian of $C$, then $\mathbb{Q}(J(C)[2^{n}])/\mathbb{Q}$ is a Galois extension, ramified only at $2$, with Galois group contained in a subgroup of $GSp_{2g}(\mathbb{Z}/2^{n} \mathbb{Z})$. Knowing something about the number fields that can arise allows one to understand the arithmetic of $C$ (and in particular, allows one to find etale covers of degree $2^{n}$). For example, since there are no $A_{3}$ or $S_{3}$ extensions of $\mathbb{Q}$ ramified only at $2$, every elliptic curve $E/\mathbb{Q}$ with good reduction away from $2$ must have a rational $2$-torsion point. (Of course, there are other ways to see this.)

Here are three possible approaches to the question:

$\bullet$ Use existing software that will do computations with class field theory. As a test case, I tried using Magma to compute Hilbert class fields of cubic fields with class group of prime order $p$. For $p \leq 11$, the computation was doable, but for $p > 11$, it seemed quite difficult. Maybe there are more sensible ways to attempt the computation.

$\bullet$ I've seen hints in the literature at generalizations of explicit class field theory for imaginary quadratic fields to CM fields. Maybe there's a way to construct $K$ using CM abelian varieties.

$\bullet$ The rigidity method (sometimes) allows one to construct a Galois extension $L/\mathbb{Q}(t)$ with a given Galois group, and has the ability to construct specializations with a small number of ramified primes. This method seems promising, although $\mathbb{Z}/17\mathbb{Z} \rtimes (\mathbb{Z}/17/\mathbb{Z})^{\times}$ does not have a rationally rigid triple of conjugacy classes.