# Number fields downloaded from the LMFDB on 07 September 2026. # Search link: https://www.lmfdb.org/NumberField/?completions=2.1.16.58n1.779 # Query "{'local_algs': {'$contains': ['2.1.16.58n1.779']}}" 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. "16.4.153177439332441840943104.2" [478, 1264, 2728, 4864, 6164, 5776, 3520, 944, -314, -336, -80, 24, 16, -8, -8, 0, 1] 153177439332441840943104 "16T333" [] "16.16.81030865406861733858902016.2" [-2, 32, -152, 80, 956, -1072, -2456, 1936, 2746, -1200, -1448, 264, 340, -16, -32, 0, 1] 81030865406861733858902016 "16T333" [] "16.8.81030865406861733858902016.3" [382, -1568, -152, 8128, -10756, 544, 7024, -4624, 982, 96, -56, -24, 40, -8, -8, 0, 1] 81030865406861733858902016 "16T333" [2] "16.0.1509221451981748002510926376155643641856.1" [281522905, 290739552, -11526000, -50965728, 40473724, -20351904, 4234848, -2430912, 883002, -146976, 31344, -4512, 1492, -288, 0, 0, 1] 1509221451981748002510926376155643641856 "16T1143" [2, 2, 2, 2, 30516] "20.4.4800653894273660658369343520768.1" [32, 0, -298, 0, 737, 0, 72, 0, -524, 0, -236, 0, 6, 0, 40, 0, 20, 0, 6, 0, 1] 4800653894273660658369343520768 "20T951" [] "20.4.4800653894273660658369343520768.2" [2, 0, -23, 0, 193, 0, -728, 0, 1100, 0, -270, 0, 210, 0, -120, 0, 10, 0, -7, 0, 1] 4800653894273660658369343520768 "20T951" [] # Label -- # Each (global) number field has a unique label of the form d.r.D.i where # # 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 $1\leq i < n$.