# Number fields downloaded from the LMFDB on 26 August 2026. # Search link: https://www.lmfdb.org/NumberField/?completions=7.3.4.9a1.1 # Query "{'local_algs': {'$contains': ['7.3.4.9a1.1']}}" returned 44 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. "12.0.148231719413132544.2" [128, 0, 0, 25, 0, 0, 45, 0, 0, -13, 0, 0, 1] 148231719413132544 "12T83" [2] "16.8.368501202188674264398198472704.1" [41148, -297216, -128952, 321408, 170748, -83808, -30240, 42912, 14364, -9672, -4372, 816, 564, -24, -36, 0, 1] 368501202188674264398198472704 "16T1033" [] "18.4.2661084740845748882007.1" [-1, 0, 0, -8, 0, 0, -13, 0, 0, 7, 0, 0, 3, 0, 0, -5, 0, 0, 1] -2661084740845748882007 "18T228" [] "18.2.71849288002835219814189.1" [1, -3, 2, 6, -2, -12, -20, -49, 69, -6, -58, 64, 7, -59, 23, 13, -7, -2, 1] 71849288002835219814189 "18T228" [] "18.6.10899803098504187420700672.1" [4, 50, 147, 101, -287, -480, 500, 404, -141, -445, -94, 455, -18, -193, 36, 37, -10, -3, 1] 10899803098504187420700672 "18T228" [] "18.6.10899803098504187420700672.2" [-343, 0, 1372, 0, -245, 0, -294, 0, -42, 0, 98, 0, 16, 0, -18, 0, -1, 0, 1] 10899803098504187420700672 "18T734" [] "18.0.43599212394016749682802688.1" [2818, -6026, 9861, -11846, 7421, -3832, 5336, -3434, -373, 30, 682, 76, -188, -74, 51, -2, 8, -6, 1] -43599212394016749682802688 "18T734" [2] "18.0.294294683659613060358918144.1" [982, -1319, 1247, 825, 4781, 893, 3079, 1386, 1172, 25, 727, -95, 121, -12, 24, -11, 11, -3, 1] -294294683659613060358918144 "18T228" [2, 8] "18.6.239272477618363922259221151744.1" [-117649, 0, 84035, 0, 120050, 0, -3087, 0, -15435, 0, 1176, 0, 343, 0, 14, 0, -14, 0, 1] 239272477618363922259221151744 "18T154" [] "18.6.209808478676186554957580557615104.1" [-103161709, 0, 0, 0, 0, 0, 1982540, 0, 0, 0, 0, 0, -2838, 0, 0, 0, 0, 0, 1] 209808478676186554957580557615104 "18T152" [3] "18.0.527850880803275932384437883608576.1" [11663, -3756, 14211, -11939, 17733, -1023, 3744, 537, 2328, -259, 1209, 168, -243, -186, 81, 35, -12, -3, 1] -527850880803275932384437883608576 "18T61" [2, 2, 114] "18.0.4910338574526370951967483335540736.1" [24767, 517, 12414, -17749, 13119, 730, 10351, -6131, 2663, -2174, 913, 1069, -279, -286, 97, 43, -14, -3, 1] -4910338574526370951967483335540736 "18T111" [6, 120] "18.0.43543302538963085509483263938264576.1" [83777, 33276, 49471, -24459, 7728, 34451, 17291, -12478, -9313, 2712, 6177, 621, -1363, -263, 220, 45, -20, -3, 1] -43543302538963085509483263938264576 "18T61" [2, 6, 126] "18.0.465225639663371349906053850329513984.1" [77984, 13920, 7504, -53376, 41580, -9388, 20564, -9772, 7301, -7929, 1194, 2979, -643, -590, 175, 63, -20, -3, 1] -465225639663371349906053850329513984 "18T111" [2, 1462] "18.0.3922574212674389864293916110009401344.1" [563072, 566272, 263712, -49984, 198780, 274780, 41368, -121808, -62083, 21559, 26134, 439, -4647, -514, 511, 67, -32, -3, 1] -3922574212674389864293916110009401344 "18T111" [2, 2, 988] "18.18.7443610234613941598496861605272223744.1" [68, -9016, -25656, 88132, 167341, -341944, -236236, 451000, 108969, -240614, -7295, 56852, -4011, -6290, 765, 320, -48, -6, 1] 7443610234613941598496861605272223744 "18T111" [] "18.0.15038889554237188354035923909504139264.1" [216761, 29519, 4104, -146369, 109977, -29248, 20293, -1963, 17081, -21760, 2071, 6473, -1329, -1004, 283, 83, -26, -3, 1] -15038889554237188354035923909504139264 "18T111" [2, 2856] "18.0.20929939922429384638960473399085612544.1" [163271, -4442, -53431, 7021, 78341, -16649, 10102, 12881, -4812, -7319, 7201, 2464, -1587, -628, 267, 67, -24, -3, 1] -20929939922429384638960473399085612544 "18T111" [5474] "18.18.59744089436262355372588369743524134912.1" [-54784, -638976, 5651584, 6640768, -9529568, -5543552, 5927960, 1924168, -1864236, -352420, 332578, 36930, -35067, -2235, 2162, 73, -72, -1, 1] 59744089436262355372588369743524134912 "18T266" [] "18.18.125522374805580475657405315520300843008.1" [69632, -196608, -1185536, 3594752, -491264, -4784128, 2063696, 2106272, -1064328, -454832, 241096, 54528, -29118, -3708, 1955, 134, -69, -2, 1] 125522374805580475657405315520300843008 "18T111" [] "18.18.146283896473715117030063296049094066176.1" [8704, -169472, -352640, 2851968, -1262432, -5135136, 1974200, 2664376, -957532, -634428, 220562, 80294, -27403, -5595, 1888, 203, -68, -3, 1] 146283896473715117030063296049094066176 "18T266" [] "18.18.188923636599181292235646547360229044736.1" [-644608, -2155008, 2993664, 6782976, -5048064, -6826032, 3506056, 2998608, -1264992, -669732, 254538, 82398, -29525, -5664, 1959, 204, -69, -3, 1] 188923636599181292235646547360229044736 "18T176" [] "18.0.437339876307037388165317055614136000000.1" [488893, -172130, 260709, -554635, 407996, -43573, -91793, 27474, 72107, -56666, -1705, 13753, -1987, -1697, 406, 109, -32, -3, 1] -437339876307037388165317055614136000000 "18T111" [13266] "18.0.642619478370369858504658064222854270464.1" [454292, -36312, -193059, 72721, 149808, -53835, 29163, 35466, -22098, -17212, 14646, 4974, -2952, -954, 405, 83, -30, -3, 1] -642619478370369858504658064222854270464 "18T111" [2, 2, 9150] "18.18.4159781123407415509472868801127860417024.1" [8704, -322560, 2290944, 328448, -8648448, -212016, 6494872, 14976, -2147532, 7908, 382560, -1482, -39361, 93, 2343, -2, -75, 0, 1] 4159781123407415509472868801127860417024 "18T176" [] "18.18.4520796299502764426851080911862443074048.1" [-531968, 296960, 9115136, 5785472, -12660928, -5326352, 7106952, 1939664, -2100368, -367084, 360050, 39170, -36921, -2372, 2229, 76, -73, -1, 1] 4520796299502764426851080911862443074048 "18T176" [] "18.18.5660400357784239214306957196959196643328.1" [34816, 2643968, 6703616, -406528, -9539968, -1725056, 5704992, 1049792, -1830176, -248896, 334520, 30248, -35748, -2012, 2208, 70, -73, -1, 1] 5660400357784239214306957196959196643328 "18T266" [] "18.0.6440393968715739593928720279646940950016.1" [1115872, -454992, 718752, -1350036, 900714, -44640, -304797, 134329, 122102, -110821, 992, 22563, -3592, -2361, 574, 129, -38, -3, 1] -6440393968715739593928720279646940950016 "18T111" [2, 2, 12996] "18.18.16906055105504143395447398717583312551936.1" [-122368, 2195968, 6599296, -331904, -10732384, -1801888, 6564728, 1076264, -2053212, -254772, 363234, 30994, -37727, -2057, 2278, 71, -74, -1, 1] 16906055105504143395447398717583312551936 "18T266" [] "18.18.17350725915999986179625767734017065302528.1" [256, -768, -17040, 61864, 141639, -727497, 436815, 706758, -405264, -263916, 134430, 47652, -21664, -4350, 1776, 188, -69, -3, 1] 17350725915999986179625767734017065302528 "18T111" [] "18.18.21468546340507820716510118968858028286464.1" [55808, 541184, 1538560, 286976, -3834112, -2034704, 3144440, 1466992, -1273500, -395916, 269520, 52706, -31541, -3739, 2065, 136, -71, -2, 1] 21468546340507820716510118968858028286464 "18T176" [] "18.18.103046303499451833502859524474351055200256.1" [548864, -4116480, 4342272, 7678464, -8491328, -4646336, 5381200, 1414176, -1698656, -251072, 305204, 27284, -32620, -1790, 2051, 65, -70, -1, 1] 103046303499451833502859524474351055200256 "18T266" [] "18.18.1561382630189567606481071728056448250806272.1" [-1818112, 9728, 8024704, 2464128, -9561184, -2818016, 5471288, 1224552, -1736316, -259700, 318274, 30162, -34339, -1977, 2148, 69, -72, -1, 1] 1561382630189567606481071728056448250806272 "18T266" [] "18.0.1648740855991229336045752391589616883204096.1" [11239424, 0, 31962112, 0, 35179648, 0, 19502000, 0, 5957980, 0, 1025325, 0, 97513, 0, 4892, 0, 117, 0, 1] -1648740855991229336045752391589616883204096 "18T266" [2, 2, 2, 2, 31344] "18.18.5321114275027723901807412615657192712000000.1" [-2289152, -2179072, 7329536, 4312832, -8814848, -3171248, 5164904, 1155520, -1650416, -231380, 304042, 26882, -33065, -1814, 2091, 66, -71, -1, 1] 5321114275027723901807412615657192712000000 "18T266" [] "18.18.7491055727576623485514981994740130671235584.1" [-2848256, -965120, 9913088, 562816, -11375552, 74608, 6316408, -96672, -1925580, 22748, 341648, -2334, -35993, 111, 2211, -2, -73, 0, 1] 7491055727576623485514981994740130671235584 "18T266" [] "18.0.42157331015118325374988936657524102771769344.1" [54261112, -46377384, 40317820, -8546916, 6121006, -12013190, 15914937, -7223195, 5412147, -1463500, 854821, -152401, 74494, -8583, 3705, -252, 97, -3, 1] -42157331015118325374988936657524102771769344 "18T266" [2, 2, 2, 2, 574368] "18.14.2038562613295014888222190587826200156400622047232.1" [238335952, 648947488, 748672224, 529773984, 314846192, 159286048, 24770664, -34537200, -19959604, 637224, 2710648, 221696, -187423, -16914, 7463, 396, -149, -2, 1] 2038562613295014888222190587826200156400622047232 "18T950" [3] "36.0.21398978367532602050913228248132339152922471766474640072995579525693467262976.1" [3853712, 6957280, 9211792, -148537280, 365041592, -298086184, -98323744, 195622728, 175344416, -286151236, 24486230, 35139946, 91142993, -136657168, 89631925, -23798720, -2947235, 7596626, -9149200, 4919942, -1693615, 84728, 635387, -498860, 387102, -261006, 167673, -98968, 52417, -25700, 11152, -4264, 1401, -380, 81, -12, 1] 21398978367532602050913228248132339152922471766474640072995579525693467262976 "36T3379" NULL "36.0.17303778994656659820211322036533666992057745030749670455635346022689551189016576.1" [68719476736, 0, -12884901888, -2147483648, -2415919104, -1811939328, 1191182336, -25165824, 0, 137363456, 118751232, -18087936, -9338880, -15384576, -5554176, -208896, 1554432, 67584, 301568, 16896, 97152, -3264, -21696, -15024, -2280, -1104, 1812, 524, 0, -6, 71, -27, -9, -2, -3, 0, 1] 17303778994656659820211322036533666992057745030749670455635346022689551189016576 "36T1803" NULL "36.0.20437599181597888521607217689901451581497256981403444028932632651429364011106304.1" [3195392, -7037952, 14238336, 87904448, -152888384, -56244592, 692053512, -73724176, -417892300, 484355456, 662418, -115066148, 323548427, 24379819, -47139783, 87117071, 41426055, 11250407, 19134908, 3560178, 1910654, -14803, -338726, -65593, -109813, -11062, 6981, -1133, 4274, -1205, 731, -262, 94, -26, 13, 0, 1] 20437599181597888521607217689901451581497256981403444028932632651429364011106304 "36T1803" NULL "36.0.301047689812153559468949278808203982205593781388513760108410233287654372163190784.1" [3663748864, -11975800896, -36678744624, 84141618520, 183104592072, -110109059538, -162114898461, 208034846421, 66368918904, -201803107819, 128073851037, -7499756334, -21535096595, 6448161405, 5162377947, -2665629236, 30203313, 283125477, -87601572, -1082763, -10800621, -5812916, 7934277, -4344519, 2311886, -922797, 436041, -205392, 81279, -32463, 12677, -4506, 1473, -437, 102, -15, 1] 301047689812153559468949278808203982205593781388513760108410233287654372163190784 "36T807" NULL "36.0.2437915717857692035976377517369806384440911054782402404768751733575119787498074537984.1" [512, -5888, 27136, 1726400, -1101504, -609872, 894665816, 29250664, -298838636, 140749772, 306548006, -65564958, 200257783, 13773482, -124613413, 21684224, 7129517, 778766, 20361840, 155606, 2265243, 62334, -392723, 1428, -160184, 3864, -5077, 1084, 249, -84, 232, -32, 43, -6, 7, 0, 1] 2437915717857692035976377517369806384440911054782402404768751733575119787498074537984 "36T3379" NULL "36.0.1777240558318257494226779210162588854257424158936371353076420013776262325086096338190336.1" [3305531244544, -17686593536, 14589905272832, 9038185693184, 46988599885824, 25083388919808, 62930183213056, 19494359330816, 51321732841472, 8845110654976, 29950092829952, 1574606120192, 13075210905536, -299570401728, 4386525309664, -316069561248, 1132339328928, -98727072288, 223256681352, -19495741960, 34479543144, -2720369448, 4190952558, -286872742, 406458419, -23276523, 31232683, -1478352, 1899311, -71975, 88762, -2649, 3105, -66, 73, -1, 1] 1777240558318257494226779210162588854257424158936371353076420013776262325086096338190336 "36T3161" 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 $1\leq i < n$.