# Number fields downloaded from the LMFDB on 04 October 2024.
# Search link: https://www.lmfdb.org/NumberField/?completions=17.12.9.3
# Query "{'local_algs': {'$contains': ['17.12.9.3']}}" returned 8 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.
"12.0.1772506678315561228125.1" [6561, 0, -3645, 729, 2997, 1647, 78, -216, -39, 18, -3, -3, 1] 1772506678315561228125 "12T12" [12]
"12.2.2954177797192602046875.1" [-1985, -6120, -2481, 5309, -1353, -1749, 3933, -2586, 402, 86, -12, -6, 1] -2954177797192602046875 "12T12" [4]
"12.2.7972425761140316000000.1" [17885, -72135, 105861, -70220, 24529, -13047, 3068, -987, 129, -10, 27, -1, 1] -7972425761140316000000 "12T12" [4]
"12.0.1335225603550355658203125.2" [7283051, -7454263, 5096688, -1873692, 492276, -128756, 35774, -5332, 963, -241, 23, -1, 1] 1335225603550355658203125 "12T1" [1492]
"12.12.65426054573967427251953125.2" [-492149, 4959038, -11454450, -1388646, 3755080, 184541, -357433, -9703, 13504, 181, -202, -1, 1] 65426054573967427251953125 "12T1" [4]
"12.0.21098872958222608828144613261.1" [493326523, 117963773, 102217297, 7971726, 8778862, -173877, 296307, -4273, 2281, 316, 76, -1, 1] 21098872958222608828144613261 "12T1" [13, 13, 52]
"20.20.6439150545149353253545643868160000000.1" [-2194, 24704, -57388, -124848, 296158, 279602, -558074, -353422, 524141, 262530, -271767, -114102, 80752, 28312, -13727, -3792, 1284, 246, -59, -6, 1] 6439150545149353253545643868160000000 "20T223" []
"24.0.4280568617115764108693321069663602825199127197265625.2" [113554323611, 30591970887, 376139557176, 251235790407, 248952353934, 164020859052, 61423569727, 5895919061, -1179057951, 1469803588, -262994586, -480510060, 30856912, 20246396, -5509477, -617183, 364189, 66684, -6137, -4159, 410, 118, -38, -1, 1] 4280568617115764108693321069663602825199127197265625 "24T2" [2, 14920]
# 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 #}
#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\le i