\\ Number fields downloaded from the LMFDB on 28 February 2024.
\\ Search link: https://www.lmfdb.org/NumberField/?galois_group=20T26
\\ Query "{'degree': 20, 'galois_label': '20T26'}" returned 2 fields, sorted by degree.
\\ Each entry in the following data list has the form:
\\ [Label, Polynomial, Discriminant, Galois group, Class group]
\\ For more details, see the definitions at the bottom of the file.
\\ To create a list of fields, type "fields = make_data()"
columns = ["label", "coeffs", "disc", "galois_label", "class_group"];
data = {[
["20.0.23383113568432629108428955078125.1", [1771561, 0, 0, 0, 0, -47916, 0, 0, 0, 0, 2486, 0, 0, 0, 0, -36, 0, 0, 0, 0, 1], 23383113568432629108428955078125, "20T26", [5]],
["20.0.23383113568432629108428955078125.2", [161051, 0, 0, 0, 0, -2904, 0, 0, 0, 0, 781, 0, 0, 0, 0, -54, 0, 0, 0, 0, 1], 23383113568432629108428955078125, "20T26", [5]]
]};
create_record(row) =
{
out = Map(["label",row[1];"coeffs",row[2];"disc",row[3];"galois_label",row[4];"class_group",row[5]]);
poly = Polrev(mapget(out, "coeffs"));
mapput(~out, "poly", poly);
field = nfinit(poly);
mapput(~out, "field", field);
return(out);
}
make_data() =
{
return(apply(create_record, data));
}
\\ Label --
\\ Each (global) number field has a unique label of the form d.r.D.i where
\\
\\ - \(d\) is the degree;
\\
- \(r\) is the real signature; the full signature is therefore \([r,(d-r)/2]\);
\\
- \(D\) is the absolute value of the discriminant;
\\
- \(i\) is the index, counting from 1. This is in case there is more than one
\\ field with the same signature and absolute value of the
\\ discriminant: for example 4.0.1008.1 and 4.0.1008.2.
\\

\\ The discriminant portion of the label can take the form \(a_1\) e \(\epsilon_1\) _ \(a_2\) e \(\epsilon_2\) _ \(\cdots\) _ \(a_k\) e \(\epsilon_k\) to mean the absolute value of the
\\ discriminant equals \(a_1^{\epsilon_1}a_2^{\epsilon_2}\cdots a_k^{\epsilon_k}\). The separators are the letter e and the underscore symbol.
\\Polynomial (coeffs) --
\\ A **defining polynomial** of a number field $K$ is an irreducible polynomial $f\in\Q[x]$ such that $K\cong \mathbb{Q}(a)$, where $a$ is a root of $f(x)$. Equivalently, it is a polynomial $f\in \Q[x]$ such that $K \cong \Q[x]/(f)$.
\\ A root \(a \in K\) of the defining polynomial is a generator of \(K\).
\\ {# original author: john.jones #}
\\Galois group (galois_label) --
\\ Let $K$ be a finite degree $n$ separable extension of a field $F$, and $K^{gal}$ be its
\\ Galois (or normal) closure.
\\ The **Galois group** for $K/F$ is the automorphism group $\Aut(K^{gal}/F)$.
\\ This automorphism group acts on the $n$ embeddings $K\hookrightarrow K^{gal}$ via composition. As a result, we get an injection $\Aut(K^{gal}/F)\hookrightarrow S_n$, which is well-defined up to the labelling of the $n$ embeddings, which corresponds to being well-defined up to conjugation in $S_n$.
\\ We use the notation $\Gal(K/F)$ for $\Aut(K/F)$ when $K=K^{gal}$.
\\ There is a naming convention for Galois groups up to degree $47$.
\\Class group (class_group) --
\\ The **ideal class group** of a number field $K$ with ring of integers $O_K$ is the group of equivalence classes of ideals, given by the quotient of the multiplicative group of all fractional ideals of $O_K$ by the subgroup of principal fractional ideals.
\\ Since $K$ is a number field, the ideal class group of $K$ is a finite abelian group, and so has the structure of a product of cyclic groups encoded by a finite list $[a_1,\dots,a_n]$, where the $a_i$ are positive integers with $a_i\mid a_{i+1}$ for $1\le i