# Number fields downloaded from the LMFDB on 18 July 2026.
# Search link: https://www.lmfdb.org/NumberField/?galois_group=27T46
# Query "{'degree': 27, 'galois_label': '27T46'}" returned 4 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.



"27.27.2291640151035204503892377550163444370666522802608406528.1"	[881, -15038, -146172, 311093, 3996545, 4071159, -17146056, -21364000, 36423522, 39595704, -45471606, -37894050, 35322803, 21000324, -17537565, -6989285, 5630326, 1379370, -1166055, -148550, 152622, 5923, -12030, 327, 514, -39, -9, 1]	2291640151035204503892377550163444370666522802608406528	"27T46"	[]
"27.27.26103213595385376302149113032330483534623361298461380608.1"	[743, 63735, 583725, 590784, -5043156, -7959093, 16846006, 28955214, -29422410, -50817880, 29716908, 50276448, -18000857, -30155193, 6684192, 11392621, -1542348, -2762556, 219679, 430203, -18525, -42152, 831, 2472, -15, -78, 0, 1]	26103213595385376302149113032330483534623361298461380608	"27T46"	[]
"27.27.1670605670104664083337543234069150946215895123101528358912.1"	[-7561, 59427, 37389, -866850, -217893, 5218554, 2411455, -16024680, -11892138, 25541057, 26856084, -18957114, -29580077, 3861165, 16364808, 1998961, -4920498, -1289328, 833056, 303360, -78168, -36776, 3729, 2391, -69, -78, 0, 1]	1670605670104664083337543234069150946215895123101528358912	"27T46"	[2, 2]
"27.27.1670605670104664083337543234069150946215895123101528358912.2"	[2269, -19239, -245721, 1972746, 3200613, -30948720, -8018767, 97589034, 7729512, -140501779, -2975622, 114046686, -136513, -57051909, 528462, 18340951, -196776, -3856650, 35030, 531132, -3318, -47102, 159, 2571, -3, -78, 0, 1]	1670605670104664083337543234069150946215895123101528358912	"27T46"	[]


# 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 $1\leq i < n$.