# Number fields downloaded from the LMFDB on 13 May 2026. # Search link: https://www.lmfdb.org/NumberField/?galois_group=18T23 # Query "{'degree': 18, 'galois_label': '18T23'}" returned 32 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. "18.0.5040742784941043712.1" [1, 0, 0, -5, 0, 0, 12, 0, 0, -9, 0, 0, 2, 0, 0, -1, 0, 0, 1] -5040742784941043712 "18T23" [] "18.0.5132738882980786176.1" [1, -7, 29, -79, 154, -224, 253, -232, 181, -108, 16, 64, -87, 51, -2, -20, 16, -6, 1] -5132738882980786176 "18T23" [] "18.0.1844362878529525198848.2" [1, 0, 0, 3, 0, 0, 12, 0, 0, -11, 0, 0, 6, 0, 0, -3, 0, 0, 1] -1844362878529525198848 "18T23" [] "18.0.16599265906765726789632.4" [1, 0, 0, 12, 0, 0, 285, 0, 0, 34, 0, 0, -3, 0, 0, -6, 0, 0, 1] -16599265906765726789632 "18T23" [] "18.18.106297170913362278088000000000.1" [-125, 750, 450, -6175, -570, 19710, 3176, -29310, -9144, 20685, 9132, -6570, -3614, 870, 612, -35, -42, 0, 1] 106297170913362278088000000000 "18T23" [] "18.0.450283905890997363000000000000.10" [117649, 0, 0, -29841, 0, 0, 366, 0, 0, -1141, 0, 0, 354, 0, 0, -21, 0, 0, 1] -450283905890997363000000000000 "18T23" [3, 3] "18.0.3238832304182321509797314520003.4" [46656, 0, 0, -12312, 0, 0, -639, 0, 0, -594, 0, 0, 267, 0, 0, -18, 0, 0, 1] -3238832304182321509797314520003 "18T23" [3, 3] "18.0.4052555153018976267000000000000.10" [34896, -15120, 170424, -35748, -150804, -4428, 48987, 14490, 945, -4056, -3240, 270, 177, 72, 36, -18, 9, 0, 1] -4052555153018976267000000000000 "18T23" [3, 3, 3] "18.18.116407003600838612509128000000000.1" [125, 1500, 4950, -2525, -34320, -18900, 84016, 52290, -104526, -39545, 65196, 7470, -16668, -120, 1692, -15, -72, 0, 1] 116407003600838612509128000000000 "18T23" [] "18.0.2361108749748912380642242285082187.6" [47523, 138402, 132030, 7146, -43209, -23787, 3276, 10323, -1323, -2565, -783, 54, 519, -117, 36, -36, 9, 0, 1] -2361108749748912380642242285082187 "18T23" [3, 3, 9, 9] "18.0.13266257117930788904129800273932288.5" [19476, -109404, 360072, -814854, 1123398, -836352, 283375, -19323, 17625, -78, -8355, 5337, -1676, 225, 69, -90, 39, -9, 1] -13266257117930788904129800273932288 "18T23" [3, 3] "18.0.50396986113842071567939653456703488.13" [8796, -56916, 174384, -315030, 408726, -339732, 160075, -43155, 6465, 12342, -10119, 3825, -1100, 225, 69, -90, 39, -9, 1] -50396986113842071567939653456703488 "18T23" [3, 3, 3, 9] "18.0.149893238403862212943036837746306920448.1" [66996, -520884, 1822716, -3847170, 5131062, -3954150, 1476721, -156159, 31989, 44796, -41133, 14733, -2756, 225, 69, -90, 39, -9, 1] -149893238403862212943036837746306920448 "18T23" [3, 3, 3, 9] "18.0.3238832304182321509797314520003000000000000.3" [385910388, 162804708, -287391240, 75997794, 91169694, -59070312, 13853359, 3557637, -912039, -66486, 62061, -207, -7292, 2889, -579, -18, 39, -9, 1] -3238832304182321509797314520003000000000000 "18T23" [3, 3, 3, 3, 3] "18.0.3631198257563192544205989811161007572909843.1" [174676879, 0, 0, -46126884, 0, 0, 46926843, 0, 0, -955510, 0, 0, 15423, 0, 0, -111, 0, 0, 1] -3631198257563192544205989811161007572909843 "18T23" [3, 3, 63] "18.0.5623949773935752894766481334356864599994368.1" [47870076, 9854892, -23111064, 11847282, 2088558, -1645560, 1661599, -1223703, 747021, -272778, 73689, -23931, 4408, -531, 177, -90, 39, -9, 1] -5623949773935752894766481334356864599994368 "18T23" [6, 6] "18.0.12303951687949724503891516957203000000000000.1" [1328391828, 35801892, 33435936, -85483590, 20343942, -236412, -4869773, 1417041, -379995, -168870, 190365, -55755, 2680, 3141, -615, -18, 39, -9, 1] -12303951687949724503891516957203000000000000 "18T23" [3, 3, 3, 3, 9, 9] "18.0.32680784318068732897853908300449068156188587.3" [174676879, 0, 0, -127849638, 0, 0, 44713203, 0, 0, 467768, 0, 0, -7815, 0, 0, -21, 0, 0, 1] -32680784318068732897853908300449068156188587 "18T23" [3, 27, 189] "18.0.872233768265190335916797362984151904157842243.1" [21867232251, 2120234994, -2708142030, 498275931, 104639490, -19419525, 4368925, -1186992, 2059689, -946308, 155823, -46008, 4363, 1845, -129, -99, 39, -9, 1] -872233768265190335916797362984151904157842243 "18T23" [3, 9, 9, 9, 27] "18.0.996620086723927684815212873533443000000000000.3" [2750600349, 8011432251, 7962223293, 2437342002, -546767253, -190141911, 129192813, 27041769, -2416230, 1015740, -127413, 44631, 29385, -3591, 1593, -171, 63, 0, 1] -996620086723927684815212873533443000000000000 "18T23" [3, 3, 3, 3, 9, 9] "18.0.7168555859200162677766007466667663950714918483.1" [451250391, 129533769, -288291726, 128113350, 67758066, -58489497, 21624805, 228735, -801240, 308622, 30651, -18045, -20, 3330, -876, 45, 39, -9, 1] -7168555859200162677766007466667663950714918483 "18T23" [3, 3, 3, 3, 18, 18] "18.0.8969580780515349163336915861800987000000000000.1" [40353607, 0, 0, -184875000, 0, 0, 313150767, 0, 0, 1705300, 0, 0, -28551, 0, 0, -60, 0, 0, 1] -8969580780515349163336915861800987000000000000 "18T23" [3, 3, 3, 3, 6, 18] "18.0.158167331320896150779524753864609204299658702848.1" [2314764156, 374601564, -952674696, 514036110, 146010546, -130560552, 48365515, 1561365, -2055975, 663882, 89241, -44235, 3940, 4365, -1191, 90, 39, -9, 1] -158167331320896150779524753864609204299658702848 "18T23" [3, 3, 3, 3, 9, 9] "18.0.5225877221356918592091419443200727020071175574107.1" [17373979, 0, 0, 33589614, 0, 0, 224141118, 0, 0, 74482, 0, 0, 3849, 0, 0, -96, 0, 0, 1] -5225877221356918592091419443200727020071175574107 "18T23" [3, 3, 3, 3, 9, 9, 27] "18.0.80996323443947526775687562993410084082675287079043.2" [678127017, 354328425, -587584620, 134872806, 206732358, -167179851, 61018867, -2349711, -205674, 408012, -41691, 513, -2378, 3870, -1020, 63, 39, -9, 1] -80996323443947526775687562993410084082675287079043 "18T23" [3, 3, 3, 3, 3, 3, 9, 9] "18.0.5171755899362511287780502990259247846315138714185728.1" [888131124, 404543268, -1123085628, 111900570, 1203660630, -884819106, 281242081, 7936227, -900645, -448638, -95403, 92007, -28406, 3843, -849, 18, 39, -9, 1] -5171755899362511287780502990259247846315138714185728 "18T23" [3, 3, 3, 3, 3, 3, 3, 3, 3] "18.0.12520069450520138163679370498940927487923000000000000.1" [51961428, -202915692, 468467820, -805226586, 1079608554, -810558162, 272888029, -4698873, 3378711, -812226, -233499, 131103, -30962, 2259, -489, -18, 39, -9, 1] -12520069450520138163679370498940927487923000000000000 "18T23" [3, 3, 3, 6, 6] "18.0.59046319790637747019476233422195951296270284280622347.1" [40353607, 0, 0, 51364578, 0, 0, 1020101190, 0, 0, 3025762, 0, 0, 2229, 0, 0, -60, 0, 0, 1] -59046319790637747019476233422195951296270284280622347 "18T23" [3, 3, 3, 3, 9, 9, 9, 18, 54] "18.0.1955476758561888850288369070881021250289363000000000000.1" [57871188, -374881932, 1157395500, -2002296426, 2444612454, -1846066752, 634588399, -21041973, 9141591, -832476, -868809, 294723, -46082, 2259, -489, -18, 39, -9, 1] -1955476758561888850288369070881021250289363000000000000 "18T23" [3, 3, 3, 3, 18, 18] "18.0.5548588090332456647311228183246490588513332762983149568.1" [256690812, -427567140, 983394720, -969139794, 831229938, -633662316, 227790487, -18695511, 7983021, -1046958, -382071, 110313, -17408, 3465, -795, 18, 39, -9, 1] -5548588090332456647311228183246490588513332762983149568 "18T23" [3, 3, 3, 3, 3, 3, 9, 18, 18] "18.0.20803075197746806021621278075959514606918210410888671875.1" [522764442927, 99972009, -14623618176, 14258110416, -81393138, 541076877, -143151569, -79679565, 36038982, -8088882, 1068837, -404667, 65374, 4122, -588, -63, 39, -9, 1] -20803075197746806021621278075959514606918210410888671875 "18T23" NULL "18.0.13985883359540449487500333263200539051545952247168701171875.1" [9167365520487, -1368271656, -132534531936, 134538470331, -4922138088, 6138070497, -1758315869, -160156980, 89645457, -25029222, 3126507, -1036062, 170809, -4383, 627, -63, 39, -9, 1] -13985883359540449487500333263200539051545952247168701171875 "18T23" 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 $1\leq i < n$.