# Number fields downloaded from the LMFDB on 21 September 2026. # Search link: https://www.lmfdb.org/NumberField/?completions=149.1.10.9a1.2 # Query "{'local_algs': {'$contains': ['149.1.10.9a1.2']}}" 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. "15.3.114183003690700736771566380473600000000000000000.1" [31450063523, 18937195585, -11283996635, -5926881845, -1319451745, 35252577, 92089185, 13972715, -2845895, -867445, 4191, 18665, 485, -205, -5, 1] 114183003690700736771566380473600000000000000000 "15T11" [40] "20.0.39985533644639499021400991542095167230255126953125.1" [87368211275, -475118881950, 1008354191025, -1199779764500, 1005027627810, -563315121120, 232530371755, -82376920120, 26888672646, -6551196136, 1266915970, -236282030, 43472431, -5034680, 718724, -70190, 12221, -1060, 30, -6, 1] 39985533644639499021400991542095167230255126953125 "20T9" [5, 5, 50, 50] "24.4.52912165668287367688067162825230050544672851174254901707172393798828125.1" [-22732430452102984076, 10366847196876110809, -6789835303748244318, -6066224588639277287, 1767412848634061926, 168549865887958763, 107497115747347673, 6657109113531594, 2607823839824001, 137534406718492, 7929973713741, -2551487110059, -724791091972, -183509248078, -19081736696, -2729225003, -376351093, -27850734, -614796, -216487, -8391, 2424, 32, -2, 1] 52912165668287367688067162825230050544672851174254901707172393798828125 "24T1353" NULL "24.4.52912165668287367688067162825230050544672851174254901707172393798828125.4" [1175531490184802981207, 3802216454740899657001, 1122067748117692054268, 92249536788217761648, -10851373311045042219, -5702081496796538363, -632051043110538544, 66615430857564618, 1013019224328463, -322407516412444, 256040228284787, -36282628808304, 8787147531598, -1003459690667, 85657689791, -14383137583, 1051548701, -78745502, 11225948, -604234, 42792, -4684, 103, -7, 1] 52912165668287367688067162825230050544672851174254901707172393798828125 "24T1353" NULL "24.4.1322804141707184192201679070630751263616821279356372542679309844970703125.2" [-366698381038675868096, 69590173923840856824, 34739715477877596507, -14050215246264991187, -432792589826176049, 695526930216912573, -31510553075715397, -18797644677486931, 3845269392021726, -40156594860183, -6860181991789, 6400548586951, 685859047428, -58014463703, -1340980921, -1722788113, -93578148, -13274809, -1028271, -89837, 5764, 189, 32, -2, 1] 1322804141707184192201679070630751263616821279356372542679309844970703125 "24T1353" NULL "24.4.1322804141707184192201679070630751263616821279356372542679309844970703125.4" [7649352307379696165, 1126494081599767680, -6382631992225093239, 3951985230524279208, -723106578278422559, -380604735230379460, 66052933181887775, 317950721485879, 5021056994043387, 2969086642896399, 427374355794280, 102920474610045, 16255589971246, 1538479325013, 207290443976, 16276158725, 1732749360, 189095024, 1277447, 904519, 40695, -590, 271, -12, 1] 1322804141707184192201679070630751263616821279356372542679309844970703125 "24T1353" NULL # 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 $1\leq i < n$.