# Number fields downloaded from the LMFDB on 07 June 2026.
# Search link: https://www.lmfdb.org/NumberField/?galois_group=20T28
# Query "{'degree': 20, 'galois_label': '20T28'}" returned 6 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.



"20.0.288467289442715712890625.1"	[1, 1, 5, 4, 23, 26, -12, 44, 19, -96, 68, 26, -34, 7, -6, -4, 12, -6, 3, -3, 1]	288467289442715712890625	"20T28"	[]
"20.0.591319123885120791015625.1"	[1, -3, 5, -9, 18, -29, 37, -24, 26, -13, 31, -13, 26, -24, 37, -29, 18, -9, 5, -3, 1]	591319123885120791015625	"20T28"	[]
"20.0.1083153147151571524051666259765625.1"	[12321, -86247, 229548, -227550, -39506, 182237, 5435, -219873, 349654, -360176, 279549, -175591, 91806, -41161, 15921, -5252, 1498, -354, 71, -10, 1]	1083153147151571524051666259765625	"20T28"	[16]
"20.0.75045768722673667967319488525390625.1"	[55238851, 70381245, -32489355, -43775185, 15953485, 7848131, 448205, 1129650, -1315060, 111430, 59367, -105110, 56490, -6250, -660, 1594, -460, 30, 5, -5, 1]	75045768722673667967319488525390625	"20T28"	[5]
"20.20.516853980088694624404163683037236690944.1"	[15313, 67764, -65073, -635688, -423490, 1666554, 2411685, -714960, -3200232, -1615050, 616108, 735960, 77348, -103968, -27524, 5760, 2380, -108, -83, 0, 1]	516853980088694624404163683037236690944	"20T28"	[2]
"20.20.17687695508088196514062500000000000000000000.1"	[916095316, 1007795640, -8806862800, -681612780, 10914530225, 784838688, -4729220200, -221247000, 1009834870, 25437000, -122060112, -1405800, 8908075, 37080, -400000, -372, 10790, 0, -160, 0, 1]	17687695508088196514062500000000000000000000	"20T28"	[]


# 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 $1\leq i < n$.