# Number fields downloaded from the LMFDB on 06 June 2026.
# Search link: https://www.lmfdb.org/NumberField/?galois_group=20T6
# Query "{'degree': 20, 'galois_label': '20T6'}" returned 11 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.



"20.0.48883259296417236328125.1"	[1, 5, 10, -25, -35, 92, 65, -275, 50, 375, -261, -285, 535, -295, -5, 48, 45, -75, 40, -10, 1]	48883259296417236328125	"20T6"	[]
"20.4.439949333667755126953125.1"	[1, 0, 0, 0, 0, 3, 0, 0, 0, 0, -31, 0, 0, 0, 0, -3, 0, 0, 0, 0, 1]	439949333667755126953125	"20T6"	[]
"20.0.488281250000000000000000.1"	[1, 5, 5, -5, 0, 11, 45, -170, 220, -85, -194, 350, -205, -140, 420, -464, 325, -155, 50, -10, 1]	488281250000000000000000	"20T6"	[]
"20.4.7812500000000000000000000.1"	[1, 0, 0, 0, 0, 22, 0, 0, 0, 0, -6, 0, 0, 0, 0, -22, 0, 0, 0, 0, 1]	7812500000000000000000000	"20T6"	[]
"20.0.386587549251591827392578125.1"	[1, 1, 4, 10, 27, -64, 56, -31, 9, -17, 22, -8, -6, 11, -4, -1, 0, 1, 1, -2, 1]	386587549251591827392578125	"20T6"	[]
"20.0.20403517554797011816436767578125.1"	[1, -3, 13, -46, 174, 49, 309, 172, 837, -501, 704, -412, 538, -78, 123, -18, 30, -7, 6, -1, 1]	20403517554797011816436767578125	"20T6"	[29]
"20.0.623866452951915562152862548828125.1"	[28880, 0, 13000, 0, -25375, 0, 15700, 0, -950, 0, -3480, 0, 2275, 0, -800, 0, 170, 0, -20, 0, 1]	623866452951915562152862548828125	"20T6"	[5]
"20.0.2131588214553606414985382080078125.1"	[1, -45, 732, 3157, 9379, 15269, 21892, 15388, 24764, 5719, 13426, -772, 3653, -633, 406, -151, 94, -17, 13, -5, 1]	2131588214553606414985382080078125	"20T6"	[2, 58]
"20.20.19184293930982457734868438720703125.1"	[-101429, 185386, 616317, -651709, -1261123, 1017051, 1266895, -898678, -710001, 483926, 230822, -162403, -42360, 33772, 3737, -4198, -21, 284, -20, -8, 1]	19184293930982457734868438720703125	"20T6"	[]
"20.4.75487840807181783020496368408203125.1"	[1, 0, 0, 0, 0, 1918, 0, 0, 0, 0, -401, 0, 0, 0, 0, -18, 0, 0, 0, 0, 1]	75487840807181783020496368408203125	"20T6"	[5]
"20.20.1201292780904647323599596221923828125.1"	[-139, -27105, 344740, -1586360, 3317104, -2597056, -1533360, 4095435, -1892289, -1032083, 1183051, -135520, -215641, 75642, 11137, -8836, 431, 394, -49, -6, 1]	1201292780904647323599596221923828125	"20T6"	[]


# Label --
#    Each (global) number field has a unique label of the form d.r.D.i where
#    <ul>
#    <li>\(d\) is the degree;
#    <li>\(r\) is the real signature;  the full signature is therefore \([r,(d-r)/2]\);
#    <li>\(D\) is the absolute value of the discriminant;
#    <li>\(i\) 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 <a href="/NumberField/4.0.1008.1">4.0.1008.1</a> and <a href="/NumberField/4.0.1008.2">4.0.1008.2</a>.
#    </ul>
#    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 \(K\).




#Discriminant (disc) --
#    The **discriminant** of a number field $K$ is the square of the determinant of the matrix
#    \[
#    \left( \begin{array}{ccc}
#     \sigma_1(\beta_1) & \cdots & \sigma_1(\beta_n) \\
#    \vdots & & \vdots \\
#    \sigma_n(\beta_1) & \cdots & \sigma_n(\beta_n) \\
#    \end{array} \right)
#    \]
#    where $\sigma_1,..., \sigma_n$ are the embeddings of $K$ into the complex numbers $\mathbb{C}$, and $\{\beta_1, \ldots, \beta_n\}$ is an integral basis for the ring of integers of $K$.

#    The discriminant of $K$ is a non-zero integer divisible exactly by the primes which ramify in $K$.



#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 $1\leq i < n$.