# Number fields downloaded from the LMFDB on 14 May 2026. # Search link: https://www.lmfdb.org/NumberField/?completions=5.1.15.17a1.4 # Query "{'local_algs': {'$contains': ['5.1.15.17a1.4']}}" returned 12 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. "15.5.2573571875000000000000.1" [-5, -25, 50, 45, 185, 85, 110, 95, 70, 30, -10, -20, -5, -5, 0, 1] -2573571875000000000000 "15T11" [] "15.3.2834497600000000000000000.1" [72, 40, -1225, 2675, -2065, 1247, -845, 785, -295, 175, -21, 15, -20, 10, -5, 1] 2834497600000000000000000 "15T64" [] "15.3.2834497600000000000000000.2" [89, -5, 835, -965, -925, -977, 800, 1900, -180, -540, -263, 145, 35, -5, -5, 1] 2834497600000000000000000 "15T64" [] "15.3.2938656153600000000000000000.1" [94480, -279300, 412200, -478980, 327600, -228150, 89040, -42150, 11700, -2480, 720, 0, -40, -15, 0, 1] 2938656153600000000000000000 "15T29" [] "18.6.5077997833420800000000000000000.1" [3100, 16080, 21600, -4840, -30390, -18540, 6025, 13260, 6210, -940, -2385, -900, 180, 240, 45, -20, -12, 0, 1] 5077997833420800000000000000000 "18T845" [] "18.6.126949945835520000000000000000000.1" [-200, 4800, 9000, 26360, 82800, 147600, 175670, 149820, 87660, 43140, 8490, -7200, -3230, 420, 360, -20, -30, 0, 1] 126949945835520000000000000000000 "18T227" [] "20.0.2434814898518097920000000000000000000000.1" [3070180, 8752080, 8424540, 1283240, -1904775, 399184, 1105960, -172960, -335920, 80240, 82648, -8680, -6670, 1040, -520, -48, 220, 0, -20, 0, 1] 2434814898518097920000000000000000000000 "20T1032" [] "20.0.21913334086662881280000000000000000000000.1" [234225072, 174841920, 11562000, -24257280, 10739860, 12388192, 2844720, -4294560, 1490240, 528960, -398016, 91520, 54950, -12280, -1120, 1920, 540, -80, -40, 0, 1] 21913334086662881280000000000000000000000 "20T1032" [4] "20.4.24264439127689134080000000000000000000000.1" [8792410, -26870825, 55965075, -4392785, -5760615, -16392250, 21264310, -11407390, 1784260, 970665, -388435, 136445, -130285, 74190, -12250, -2478, 1000, 35, -25, -5, 1] 24264439127689134080000000000000000000000 "20T559" [] "20.0.388231026043026145280000000000000000000000.1" [3773156, 18112880, 40912340, 62435320, 72196265, 71884872, 70268380, 57142880, 34578120, 15240040, 5121684, 1383720, 347650, 89480, 22860, 4208, 500, -40, 0, 0, 1] 388231026043026145280000000000000000000000 "20T811" [4] "20.4.50617612400283069972480000000000000000000000.1" [-39496248, -23055840, 14352120, -16312320, -19541415, 8363824, 8299980, -3997440, -4643510, -307920, 947856, 307840, -68265, -54960, -3520, 4320, 870, -160, -60, 0, 1] 50617612400283069972480000000000000000000000 "20T811" [2, 2] "20.4.1498476178648776262633390080000000000000000000000.1" [-122509757583, 310434398640, -318768052590, 169568098560, -50448894465, 10090345424, -2618033400, 647709600, 74473325, -92071680, 14821146, 2689760, -964695, 40560, -1000, 1920, 555, -80, -30, 0, 1] 1498476178648776262633390080000000000000000000000 "20T811" [2] # 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 $1\leq i < n$.