# Number fields downloaded from the LMFDB on 24 April 2024.
# Search link: https://www.lmfdb.org/NumberField/?galois_group=34T9
# Query "{'degree': 34, 'galois_label': '34T9'}" returned 7 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.
"34.0.770483087302237618924031213653875461627441440985549111296.1" [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] -770483087302237618924031213653875461627441440985549111296 "34T9" []
"34.0.3804212187770006845104225428814283599364285631706866468067.1" [3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] -3804212187770006845104225428814283599364285631706866468067 "34T9" []
"34.0.100988759218878889187610619236040764506432004552857893115789312.1" [2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] -100988759218878889187610619236040764506432004552857893115789312 "34T9" []
"34.2.100988759218878889187610619236040764506432004552857893115789312.1" [-2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 100988759218878889187610619236040764506432004552857893115789312 "34T9" []
"34.2.79666084014551601611187998196780089112851885147392749786376953125.1" [-5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 79666084014551601611187998196780089112851885147392749786376953125 "34T9" []
"34.2.65355867734067162277675143712675696467755092714334768555947173347328.1" [-3, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] 65355867734067162277675143712675696467755092714334768555947173347328 "34T9" []
"34.0.1368652901771550068097893439851509820974658000000000000000000000000000000000.1" [5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1] -1368652901771550068097893439851509820974658000000000000000000000000000000000 "34T9" NULL
# 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