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A polynomial $p \in \mathbb{Z}[x]$ of degree $n$ can be encoded as a finite sequence $(a_0,a_1, \dots, a_n)$, i.e. $p(x)= \sum_{i=0}^n a_i x^i$.
Let $G(a_0,a_1, \dots, a_n)$ be the Galois group of the splitting extension of $\mathbb{Q}$ for $p$.

Question: Is there a paper, a website or a software, providing $G(a_0,a_1, \dots, a_n)$ for small $n$ and small $a_i$?

Remark: "providing" means like an atlas, it does not mean "computing" (see this post for that).

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  • $\begingroup$ How small do you want, on either side? $n\leq 10$ and $|a_i|\lt 10$, for instance, is already roughly $10^{13}$ polynomials... $\endgroup$
    – Steven Stadnicki
    Mar 5, 2020 at 16:31
  • $\begingroup$ Fair that! I suspect the computation is harder than I was initially giving it credit for, especially after reading the post you linked to. $\endgroup$
    – Steven Stadnicki
    Mar 5, 2020 at 17:12
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    $\begingroup$ Pari/GP's polgalois function can compute Galois groups of polynomials up to degree 11 (if the galdata package is installed) according to the doc. $\endgroup$
    – Gro-Tsen
    Mar 5, 2020 at 18:33
  • $\begingroup$ I don't know if polgalois function picks in a database or computes directly the group, but it is much quicker than expected ! $\endgroup$
    – Sebastien Palcoux
    Mar 5, 2020 at 21:17
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    $\begingroup$ maybe lmfdb.org/NumberField or the sources where the data comes from lmfdb.org/NumberField/Source $\endgroup$
    – Chris Wuthrich
    Mar 5, 2020 at 21:52

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