# Number fields downloaded from the LMFDB on 15 July 2026.
# Search link: https://www.lmfdb.org/NumberField/?galois_group=16T50
# Query "{'degree': 16, 'galois_label': '16T50'}" returned 14 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.



"16.0.180143985094819840000.1"	[625, 0, 0, 0, 300, 0, 0, 0, 114, 0, 0, 0, -12, 0, 0, 0, 1]	180143985094819840000	"16T50"	[]
"16.0.1494186269970473680896.43"	[324, 0, 0, 0, 0, 0, 0, 0, 28, 0, 0, 0, 0, 0, 0, 0, 1]	1494186269970473680896	"16T50"	[]
"16.0.2123366400000000000000.1"	[2025, 0, 0, 0, -1350, 0, 0, 0, 415, 0, 0, 0, -30, 0, 0, 0, 1]	2123366400000000000000	"16T50"	[4]
"16.0.153177439332441840943104.128"	[324, 0, 0, 0, 1944, 0, 0, 0, 288, 0, 0, 0, 36, 0, 0, 0, 1]	153177439332441840943104	"16T50"	[2]
"16.0.153177439332441840943104.170"	[6561, 0, 0, 0, 972, 0, 0, 0, 450, 0, 0, 0, -12, 0, 0, 0, 1]	153177439332441840943104	"16T50"	[2]
"16.8.983761265267629401636864.1"	[-2351, 3322, 4391, -9798, 3215, 5144, -6833, 2946, 844, -1014, -41, 280, -145, 48, -4, -4, 1]	983761265267629401636864	"16T50"	[]
"16.8.1378596953991976568487936.3"	[36, 0, 0, 0, -504, 0, 0, 0, 336, 0, 0, 0, -36, 0, 0, 0, 1]	1378596953991976568487936	"16T50"	[]
"16.8.1378596953991976568487936.18"	[9, 0, 0, 0, -1404, 0, 0, 0, 1770, 0, 0, 0, -84, 0, 0, 0, 1]	1378596953991976568487936	"16T50"	[]
"16.0.2450839029319069455089664.150"	[324, 0, 2592, 0, 5832, 0, -8640, 0, 4770, 0, -1440, 0, 252, 0, -24, 0, 1]	2450839029319069455089664	"16T50"	[]
"16.0.9803356117276277820358656.279"	[26244, 0, 0, 0, 0, 0, 0, 0, 252, 0, 0, 0, 0, 0, 0, 0, 1]	9803356117276277820358656	"16T50"	[2]
"16.0.134588851614250885095358464.1"	[81, 0, 12312, 0, 36288, 0, 36792, 0, 17782, 0, 4520, 0, 608, 0, 40, 0, 1]	134588851614250885095358464	"16T50"	[2, 6, 6]
"16.16.134588851614250885095358464.1"	[81, 0, -12312, 0, 36288, 0, -36792, 0, 17782, 0, -4520, 0, 608, 0, -40, 0, 1]	134588851614250885095358464	"16T50"	[]
"16.0.57541523968884736000000000000.1"	[15625, 0, 81250, 0, 137000, 0, 103500, 0, 39975, 0, 8300, 0, 920, 0, 50, 0, 1]	57541523968884736000000000000	"16T50"	[2, 2, 10, 20]
"16.16.57541523968884736000000000000.1"	[15625, 0, -81250, 0, 137000, 0, -103500, 0, 39975, 0, -8300, 0, 920, 0, -50, 0, 1]	57541523968884736000000000000	"16T50"	[2]


# 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 $1\leq i < n$.