# Number fields downloaded from the LMFDB on 25 May 2026.
# Search link: https://www.lmfdb.org/NumberField/?galois_group=20T45
# Query "{'degree': 20, 'galois_label': '20T45'}" 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.



"20.0.45387298089433038375390625.1"	[25, -15, -2, 4, -84, 124, 53, -132, 88, -23, -30, 51, -24, -13, -4, 3, 11, -3, 2, -3, 1]	45387298089433038375390625	"20T45"	[]
"20.0.56477888187717354967269376.1"	[115, 102, -402, -970, -133, 1286, 1552, 686, 565, 594, 614, 482, 360, -18, 78, 26, -21, 12, -2, -2, 1]	56477888187717354967269376	"20T45"	[2]
"20.4.705364768808763611651745450561.1"	[-5791, 22901, -35670, 29044, -16447, 9203, -3783, 1852, -2447, 728, 674, -370, 224, -180, 153, -116, 43, -22, 7, -3, 1]	705364768808763611651745450561	"20T45"	[4]
"20.4.705364768808763611651745450561.2"	[51, 19, -186, -147, -686, 3137, -2813, 2361, -4986, 4072, -1083, 1536, -2250, 929, 337, -317, -6, 49, -4, -5, 1]	705364768808763611651745450561	"20T45"	[4]
"20.4.705364768808763611651745450561.3"	[1521, 0, 1390, 0, -6305, 0, -1089, 0, 1809, 0, 800, 0, -601, 0, -297, 0, 9, 0, 13, 0, 1]	705364768808763611651745450561	"20T45"	[2, 2]
"20.4.17287511078984605626766090564801.1"	[81, 567, 2745, 4197, 7697, 5475, 10393, 1900, 5801, -45, 4868, 2570, 1743, -843, -775, -97, 85, 30, -14, -2, 1]	17287511078984605626766090564801	"20T45"	[3]
"20.20.53384280905783887100407820071202644801.1"	[7067, -144650, 632356, -115897, -3759592, 5673768, 1117774, -6301742, 1308453, 2903891, -910055, -704949, 233926, 95007, -30388, -7034, 2099, 266, -73, -4, 1]	53384280905783887100407820071202644801	"20T45"	[]
"20.20.595720004130752289758344601308198076416.1"	[3655744, 0, -12777520, 0, 18445640, 0, -14361184, 0, 6629588, 0, -1886957, 0, 334817, 0, -36486, 0, 2322, 0, -77, 0, 1]	595720004130752289758344601308198076416	"20T45"	[4]


# 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 $1\leq i < n$.