# Number fields downloaded from the LMFDB on 29 April 2026. # Search link: https://www.lmfdb.org/NumberField/?completions=17.3.4.9a1.1 # Query "{'local_algs': {'$contains': ['17.3.4.9a1.1']}}" returned 97 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. "12.0.33498139941871322753.1" [66403, 13806, 20040, 2872, 5679, -657, 1182, -268, 176, -23, 19, -1, 1] 33498139941871322753 "12T1" [26] "12.2.75170642536024371968.1" [-324, -432, 2324, -209, 1813, -4946, 3992, -1592, 338, -26, 0, -3, 1] -75170642536024371968 "12T12" [3, 6] "12.2.467122465288747467571.1" [-9728, 13568, -7264, -7136, -2184, -188, -1048, 265, -176, 39, -1, -4, 1] -467122465288747467571 "12T12" [2] "12.0.498370286073963148713.1" [371293, -86021, 184028, -40752, 41776, -7104, 5157, -674, 402, -37, 23, -1, 1] 498370286073963148713 "12T1" [370] "12.2.1442802841295277948928.1" [-79768, 116324, -88244, 37604, -9082, -297, 2859, -1447, -75, 97, -19, -1, 1] -1442802841295277948928 "12T12" [2] "12.2.1517924819161600000000.1" [-4096, -15872, -26752, -18992, -6656, 714, 1618, 587, 61, -24, -12, -1, 1] -1517924819161600000000 "12T12" [2] "12.2.4462955285120067627776.1" [-7056, 16464, -36752, 36900, -16648, -5384, 4992, -1566, -184, 157, -5, -1, 1] -4462955285120067627776 "12T12" [2] "12.12.10681804828402845265625.1" [77771, 249627, -276332, -180968, 146564, 36484, -27143, -2986, 2204, 99, -79, -1, 1] 10681804828402845265625 "12T1" [2] "12.0.12143398553292800000000.1" [251392, 28416, -95024, -41216, 22106, 8456, -3219, -458, 297, 20, -17, -2, 1] 12143398553292800000000 "12T12" [8] "12.0.38771164618966039808393.1" [40000, -12800, 72020, -33962, 23045, -11248, 4393, -1638, 440, -92, 21, -4, 1] 38771164618966039808393 "12T12" [8] "12.2.50783145446897692693504.1" [-64, -920, -5488, -15948, -21156, -8534, 1196, 517, -193, 26, 40, -5, 1] -50783145446897692693504 "12T12" [2, 4] "12.0.70520379243635681201673.1" [664747, -4499, 289996, -8594, 53766, -3360, 5035, -512, 360, -41, 15, -1, 1] 70520379243635681201673 "12T1" [3, 390] "12.12.137208381201904937996288.1" [15407168, 0, -7703584, 0, 1416100, 0, -123403, 0, 5474, 0, -119, 0, 1] 137208381201904937996288 "12T1" [2] "12.0.150051495494708906688512.1" [2160512, -537600, 298800, 145872, -20020, 25192, 100, 760, 509, -60, 22, -4, 1] 150051495494708906688512 "12T12" [2, 8] "12.2.167274716644927613730816.1" [-800, -1680, -14436, -17218, 5328, -12351, 3807, -651, 159, 79, -27, -3, 1] -167274716644927613730816 "12T12" [6] "12.0.212528495470874611248389.1" [1041353, -903163, 754326, -65001, -23581, 24387, -1624, 2430, 248, -27, 27, -1, 1] 212528495470874611248389 "12T1" [2, 1220] "12.0.212528495470874611248389.2" [1041353, 1229045, 244258, -469, 13105, -22907, 8100, -1990, 248, -27, 27, -1, 1] 212528495470874611248389 "12T1" [10, 20] "12.0.1211102005991051069543417.1" [259280239, -32202285, 53983724, -5261792, 4830676, -341188, 239825, -11010, 7168, -173, 125, -1, 1] 1211102005991051069543417 "12T1" [10642] "12.0.1338197733159420909846528.2" [284120, -464784, 319308, -72264, -47034, 27012, 41, -1782, 369, 120, -21, -6, 1] 1338197733159420909846528 "12T12" [24] "12.0.1468238569084753753837833.1" [1547713, -482611, 412674, -92274, 58454, -12084, 6437, -1004, 336, -5, 7, -1, 1] 1468238569084753753837833 "12T1" [13, 130] "12.12.1511496468699324442765625.1" [-167539, 465381, -228504, -234762, 154066, 41588, -31141, -3096, 2570, 95, -87, -1, 1] 1511496468699324442765625 "12T1" [2] "12.12.2041324691758953057128448.1" [147185389, 29826220, -43917151, -7179172, 5116627, 642746, -302031, -26980, 9577, 534, -155, -4, 1] 2041324691758953057128448 "12T1" [2] "12.12.8781336396921916031762432.1" [986058752, 0, -246514688, 0, 22657600, 0, -987224, 0, 21896, 0, -238, 0, 1] 8781336396921916031762432 "12T1" [2] "12.0.8781336396921916031762432.1" [986058752, 0, 246514688, 0, 22657600, 0, 987224, 0, 21896, 0, 238, 0, 1] 8781336396921916031762432 "12T1" [2, 2, 12010] "12.0.11380867074944436567483713.1" [33557129, -2044686, 8687928, -604359, 956959, -70408, 59792, -3963, 2468, -109, 66, -1, 1] 11380867074944436567483713 "12T1" [4562] "12.0.11917944092174430123082737.1" [56906617, 743219, 11284568, 1606014, 2070122, 82612, 102178, -4081, 6181, -163, 146, -1, 1] 11917944092174430123082737 "12T1" [2, 46, 92] "12.12.31469448068517527302765625.1" [5099, 12973, -146082, -77178, 182146, 36264, -37763, -3860, 2954, 131, -95, -1, 1] 31469448068517527302765625 "12T1" [2] "12.0.43752672577138054208000000.1" [6257111461, -566133668, 853330065, -63981204, 49180355, -2957446, 1527577, -69412, 26849, -826, 253, -4, 1] 43752672577138054208000000 "12T1" [2, 10, 1460] "12.0.73732909482806151463725033.1" [4182016, -1035392, 718264, -78020, 83358, -6579, 5845, -1295, 477, 7, -9, -1, 1] 73732909482806151463725033 "12T1" [225010] "12.0.97197978069926901998134097.1" [95825344, 4204256, 28242128, -921120, 3271044, -58354, 154111, -9533, 5655, -224, 85, -1, 1] 97197978069926901998134097 "12T1" [7, 70] "12.0.101202590329414889537679233.1" [15763541053, -1125206796, 1783472978, -95184737, 85217641, -3147004, 2215412, -50229, 33382, -377, 278, -1, 1] 101202590329414889537679233 "12T1" [132130] "12.12.130644780272572995656220672.1" [11836891513, 1082254876, -1611480844, -120290644, 88102348, 5128808, -2489370, -105316, 38443, 1044, -308, -4, 1] 130644780272572995656220672 "12T1" [2] "12.0.130644780272572995656220672.1" [17965056817, -1376257244, 2073287516, -131522884, 100710568, -5111584, 2626542, -100420, 38647, -996, 304, -4, 1] 130644780272572995656220672 "12T1" [2, 2, 16516] "12.0.264854403201433049715185433.1" [30932992, 18345984, 7602432, 1341344, 28872, -111708, -16102, 3741, 1869, 34, -48, -3, 1] 264854403201433049715185433 "12T1" [351030] "12.0.264854403201433049715185433.2" [33083497, 24144189, 10367943, 1833927, -107166, -147954, -15703, 5442, 1932, 6, -48, -3, 1] 264854403201433049715185433 "12T1" [13, 390] "12.12.288851473381931750202052608.1" [95601337, 33477028, -35256455, -8155824, 4849639, 709102, -314151, -28700, 10261, 550, -163, -4, 1] 288851473381931750202052608 "12T1" [2] "12.12.530034123439596105135457713.1" [-284423, -7301295, -10131165, 4006970, 5120249, -448394, -548281, 19649, 20291, -211, -271, -1, 1] 530034123439596105135457713 "12T1" [2] "12.12.570239809681692130490394953.1" [1541224, -1588472, -10637934, -956419, 3379968, 341359, -330007, -23653, 13076, 452, -213, -1, 1] 570239809681692130490394953 "12T1" [10] "12.12.870516717448702407673401344.1" [1596725, 0, -17053023, 0, 6033742, 0, -495924, 0, 16133, 0, -221, 0, 1] 870516717448702407673401344 "12T1" [2, 4] "12.12.870516717448702407673401344.2" [1596725, 0, -13987311, 0, 4651166, 0, -465868, 0, 16133, 0, -221, 0, 1] 870516717448702407673401344 "12T1" [2, 4] "12.0.1441985084096691048449342913.1" [410355856, 114799632, 46784916, 3439756, 1130217, -76773, 49195, 5016, 2328, -68, 3, -3, 1] 1441985084096691048449342913 "12T1" [13, 78] "12.0.1441985084096691048449342913.2" [423141391, 146661336, 53209677, 5014525, 860661, -93741, 50665, 6717, 2391, -96, 3, -3, 1] 1441985084096691048449342913 "12T1" [7278] "12.12.6013905178971151375719763968.1" [66417457, 8822928, -31811167, -4598492, 4865283, 627546, -326743, -31020, 10873, 606, -171, -4, 1] 6013905178971151375719763968 "12T1" [3, 6] "12.12.8025462320079492591473139857.3" [565118, -1517303, 1041834, 553946, -1025690, 403933, 11800, -39624, 5649, 748, -152, -4, 1] 8025462320079492591473139857 "12T45" [4] "12.12.48815899001546465784146890752.1" [38738336832, 0, -6456389472, 0, 395612100, 0, -11056851, 0, 139230, 0, -663, 0, 1] 48815899001546465784146890752 "12T1" [2, 2, 2] "12.12.1084843635513069771633399533568.1" [-2799287, 120750, 13714353, -16932020, 6009759, 521640, -555249, 8820, 18711, -140, -237, 0, 1] 1084843635513069771633399533568 "12T1" [6] "12.12.1084843635513069771633399533568.2" [-2065904, 2837856, 6677316, -13496224, 5918472, 417312, -557895, 7056, 18711, -112, -237, 0, 1] 1084843635513069771633399533568 "12T1" [6] "12.0.2171019769608585646634834792448.1" [82748755008, 0, 13791459168, 0, 845064900, 0, 23276349, 0, 273258, 0, 969, 0, 1] 2171019769608585646634834792448 "12T1" [2, 2, 2, 2, 97330] "12.12.5906370904460046534448508571648.1" [5123870857, 1995879270, -1652173719, -210307020, 154628859, 2963520, -4933565, 8820, 70119, -140, -441, 0, 1] 5906370904460046534448508571648 "12T1" [6] "12.12.5906370904460046534448508571648.2" [5225843728, 1604482656, -1702933260, -168196224, 154267680, 2370816, -4936211, 7056, 70119, -112, -441, 0, 1] 5906370904460046534448508571648 "12T1" [6] "15.3.3721413221257087109673.1" [-32, -240, -720, -1096, -906, -459, -216, -81, 0, 4, 12, 9, -6, -9, 0, 1] 3721413221257087109673 "15T41" [] "15.3.63264024761370480864441.1" [-55, 234, -288, -42, 255, -189, 137, -39, -36, -12, -12, 9, -5, -9, 0, 1] 63264024761370480864441 "15T42" [] "24.0.596343433841506725052569942119702789888841969.1" [1356903451, -31006530, -284979984, -139206728, 42723459, -20691777, 14643616, -1485088, 7823776, 796117, 2279730, -70397, 112180, 48190, -8155, 18083, 815, 1225, 448, 31, 105, -6, 18, -1, 1] 596343433841506725052569942119702789888841969 "24T2" [4810] "24.0.18826139872047260357517724966740901740301778944.2" [4409358409, 0, -2470826604, 0, 1076505210, 0, -394483712, 0, 114173281, 0, -24745369, 0, 4098030, 0, -575644, 0, 73608, 0, -7987, 0, 667, 0, -37, 0, 1] 18826139872047260357517724966740901740301778944 "24T2" [6, 156] "24.0.45168361387113570146383270207431468565055095321.1" [2961952501, 737103728, 924764015, 391776155, 329134451, 132846090, 115491882, 21471726, 34231607, -2996153, 7793062, -2185170, 1384735, -460447, 218475, -60493, 33197, -6824, 4432, -598, 458, -35, 31, -1, 1] 45168361387113570146383270207431468565055095321 "24T2" [10, 10, 610] "24.0.273956391495408902956372548458470580812744140625.2" [850632889, -988191228, 86972994, 1722816436, 561017523, -519218053, -651229652, -33176804, 377953978, 12905237, -93575986, -4733083, 7242076, 1339676, 460447, -178209, 22387, 33449, 2618, -521, 189, -18, -4, -1, 1] 273956391495408902956372548458470580812744140625 "24T2" [3224] "24.0.4167006497184784711228478664743753253547570888704.3" [137858491849, 0, -129257003967, 0, 57877521936, 0, -16322511586, 0, 3246857816, 0, -484149588, 0, 56626545, 0, -5374504, 0, 418294, 0, -26089, 0, 1259, 0, -45, 0, 1] 4167006497184784711228478664743753253547570888704 "24T2" [6, 2220] "24.0.77111868915905578424392601463770733528276086554624.2" [1856734153, 39169164, 157852396, -390505800, -26008830, 739028606, 233196872, 73723056, 103546863, -14564572, -16200181, -13556958, -322594, -2605414, 998016, -404696, 237802, -49130, 26781, -3514, 1729, -128, 63, -2, 1] 77111868915905578424392601463770733528276086554624 "24T2" [2, 1092] "24.0.77111868915905578424392601463770733528276086554624.4" [19791178681, -7614240180, 29856914972, -8843141656, 15305714146, -2402892610, 4110338488, -162285200, 610341731, 19543804, 65471451, 3707282, 6549646, 50458, 660432, -49304, 66214, -11778, 4533, -890, 317, -40, 15, -2, 1] 77111868915905578424392601463770733528276086554624 "24T2" [2, 2, 156130] "24.0.142037391384195401373751902380518266779180147411169.4" [169912819987, 20395438165, 156338774857, 20062528483, 68533171903, 8771160168, 18922982895, 2254325691, 3685268791, 377755884, 536780849, 43642412, 60888891, 3487011, 5543948, 174056, 410835, 2789, 24633, -360, 1184, -42, 44, -2, 1] 142037391384195401373751902380518266779180147411169 "24T2" [2, 2, 138, 8970] "24.24.10004982599740668091659577274049751561767717703778304.3" [98867482624, 0, -395469930496, 0, 568488025088, 0, -384383687040, 0, 139892997824, 0, -29826685436, 0, 3951668377, 0, -336913888, 0, 18728356, 0, -672214, 0, 14960, 0, -187, 0, 1] 10004982599740668091659577274049751561767717703778304 "24T2" NULL "24.0.10004982599740668091659577274049751561767717703778304.3" [629407744, 0, 1809547264, 0, 2579096576, 0, 1376719680, 0, 347269664, 0, -17551996, 0, -10521623, 0, -376124, 0, 738430, 0, -82838, 0, 4109, 0, -98, 0, 1] 10004982599740668091659577274049751561767717703778304 "24T2" [2, 2, 63640] "24.0.10004982599740668091659577274049751561767717703778304.4" [1858531648, -3687198336, -502038528, 2539405568, 3081470496, 1679318976, -3626388512, -797354336, 2817178564, 523577912, -502129800, -62415568, -31860071, -1270042, 4336652, -2101754, 1478744, -272110, 135466, -12472, 5862, -252, 123, -2, 1] 10004982599740668091659577274049751561767717703778304 "24T2" [2, 2, 4472] "24.0.17068058612468878177191848610790413326530850360131584.2" [1625711704, 2804605200, -152138896, -2882336368, -234996586, 989751948, -40021843, -165822990, 180176517, -224899680, 266056448, -153039072, 108366172, -41048024, 21602802, -5853788, 2404074, -476208, 157048, -22336, 5958, -564, 121, -6, 1] 17068058612468878177191848610790413326530850360131584 "24T2" [2, 15540] "24.0.17068058612468878177191848610790413326530850360131584.5" [798245323912, -186840932752, 562015256064, -114435063728, 191354221458, -33663487068, 42068824333, -6635200718, 6661711781, -935833696, 802289552, -104908640, 77084916, -10008808, 6135074, -813084, 404602, -52560, 21784, -2816, 978, -124, 41, -6, 1] 17068058612468878177191848610790413326530850360131584 "24T2" [2, 2, 3055460] "24.0.280936172010381001312620985877345789072364522011190369.1" [16045571869, 2315599625, 33638920739, 19228394124, 20006322122, 14793075042, 7989262601, 4963688792, 2053631779, 1062615210, 353025164, 145062297, 51344161, 11572317, 6619255, 386166, 690737, -24192, 51360, -3132, 2452, -128, 72, -2, 1] 280936172010381001312620985877345789072364522011190369 "24T2" [26, 1820] "24.24.315850215079549249226312095595604924531818850527739904.2" [12420656704, 0, -93127391360, 0, 165921786080, 0, -131915916048, 0, 55501649748, 0, -13367607068, 0, 1974791377, 0, -186983864, 0, 11560002, 0, -463512, 0, 11593, 0, -164, 0, 1] 315850215079549249226312095595604924531818850527739904 "24T2" [2, 2] "24.0.315850215079549249226312095595604924531818850527739904.3" [12420656704, 0, 93127391360, 0, 165921786080, 0, 131915916048, 0, 55501649748, 0, 13367607068, 0, 1974791377, 0, 186983864, 0, 11560002, 0, 463512, 0, 11593, 0, 164, 0, 1] 315850215079549249226312095595604924531818850527739904 "24T2" [2, 2, 2, 504420] "24.0.315850215079549249226312095595604924531818850527739904.6" [237380825780224, 0, 0, 0, 15709025235456, 0, 0, 0, 272726132560, 0, 0, 0, 1527476265, 0, 0, 0, 3426962, 0, 0, 0, 3213, 0, 0, 0, 1] 315850215079549249226312095595604924531818850527739904 "24T2" NULL "24.24.757799355357663982877023743056442553313139386101006336.1" [16708604563456, 0, -29240057986048, 0, 19780039225856, 0, -6941442337024, 0, 1446510221520, 0, -192924489732, 0, 17179547489, 0, -1042533687, 0, 43213014, 0, -1202240, 0, 21437, 0, -221, 0, 1] 757799355357663982877023743056442553313139386101006336 "24T2" [2, 2, 2] "24.0.757799355357663982877023743056442553313139386101006336.1" [66576900625, 0, -23420373467, 0, 14537693712, 0, 20887965463, 0, 1408202799, 0, -562463503, 0, 102591126, 0, -11031082, 0, 968704, 0, -51843, 0, 2291, 0, -69, 0, 1] 757799355357663982877023743056442553313139386101006336 "24T2" [2, 12, 14640] "24.0.757799355357663982877023743056442553313139386101006336.2" [66576900625, 0, 93778965725, 0, 39534765916, 0, 7225840051, 0, 3859970611, 0, -833929295, 0, 58249686, 0, -4390474, 0, 533776, 0, -41235, 0, 2291, 0, -69, 0, 1] 757799355357663982877023743056442553313139386101006336 "24T2" [2, 60, 240] "24.0.757799355357663982877023743056442553313139386101006336.3" [1084416070609, 0, -755335881587, 0, 402482427764, 0, -867904805, 0, 6182254755, 0, -179378135, 0, -90079690, 0, 5288230, 0, 106640, 0, -14275, 0, 1171, 0, -53, 0, 1] 757799355357663982877023743056442553313139386101006336 "24T2" [2, 4, 453840] "24.0.757799355357663982877023743056442553313139386101006336.4" [1084416070609, 0, 1001834009877, 0, 88108676904, 0, -79579168449, 0, 9515344047, 0, 70542809, 0, -1354378, 0, -15770, 0, 371840, 0, -24883, 0, 1171, 0, -53, 0, 1] 757799355357663982877023743056442553313139386101006336 "24T2" [2, 20, 7440] "24.24.40980408728537776503437628514507782397000571714675933184.7" [404961208827904, 0, -809922417655808, 0, 582131737690112, 0, -196804447764480, 0, 35812607442944, 0, -3817815735808, 0, 252906776128, 0, -10781244416, 0, 299653696, 0, -5377712, 0, 59840, 0, -374, 0, 1] 40980408728537776503437628514507782397000571714675933184 "24T2" NULL "24.0.40980408728537776503437628514507782397000571714675933184.10" [1124842074616, 621767604432, -1404395011728, 613010426192, 419109200022, -466134237876, 542125913645, -441041017806, 313671208293, -154516331872, 78159685056, -27934746912, 11339678908, -2741111256, 957074034, -158289308, 48053610, -5500656, 1451576, -113088, 25830, -1268, 249, -6, 1] 40980408728537776503437628514507782397000571714675933184 "24T2" [2, 2, 1771560] "24.0.40980408728537776503437628514507782397000571714675933184.11" [1380670378459, -2880413731128, 2410499761518, -637954066624, -147095468850, -126725318322, -496380712532, 305178734740, 208786087003, 38912623880, 66094441773, 260510210, 8569856752, -262308442, 648930938, -22611236, 31289000, -966370, 968977, -24190, 18735, -336, 207, -2, 1] 40980408728537776503437628514507782397000571714675933184 "24T2" [2, 2, 124488] "24.0.40980408728537776503437628514507782397000571714675933184.16" [124011187398337, -108888605526312, 108532753978098, -19055388773680, 568850791956, 12106484873202, -2361424051754, -2005243681268, 581850731011, 199682100860, -74054427759, -14018153014, 6252259204, 712151282, -358122508, -28207316, 14924456, 829886, -443945, -18142, 9453, 252, -129, -2, 1] 40980408728537776503437628514507782397000571714675933184 "24T2" NULL "24.0.40980408728537776503437628514507782397000571714675933184.17" [404961208827904, 0, 809922417655808, 0, 582131737690112, 0, 196804447764480, 0, 35812607442944, 0, 3817815735808, 0, 252906776128, 0, 10781244416, 0, 299653696, 0, 5377712, 0, 59840, 0, 374, 0, 1] 40980408728537776503437628514507782397000571714675933184 "24T2" NULL "24.0.40980408728537776503437628514507782397000571714675933184.19" [1792860550916968, -256498899381072, 33400156881600, 13745817871376, 22901231302098, -2813263586076, -1218361998643, -198246239118, 316679779845, 36722444192, -23201012592, -5421341088, 1984547668, 418069272, -103857054, -24834140, 5059386, 1025136, -171304, -31104, 4914, 580, -87, -6, 1] 40980408728537776503437628514507782397000571714675933184 "24T2" [2, 2, 10, 888740] "24.0.69910768076672525013777811909797532985470363075098968064.1" [55088497105399801, -17779428581368908, -2975968299981864, 1134909892293408, 596211000002129, -174059427260076, -13079489382932, 4565771441424, 2590841600655, -559021203264, -17103765162, 5249553216, 5555422037, -873922536, 9197772, -228888, 6866281, -699312, 35442, -2448, 4305, -204, 20, 0, 1] 69910768076672525013777811909797532985470363075098968064 "24T2" NULL "24.24.69910768076672525013777811909797532985470363075098968064.3" [19791178681, 118928047580, -8911158056, -566893758512, 74122868941, 671068081824, -59048575828, -337536344444, 26254299907, 91071085320, -7352405426, -14672513584, 1290447157, 1486723296, -142874544, -96539344, 10030509, 3982000, -441314, -100052, 11673, 1384, -168, -8, 1] 69910768076672525013777811909797532985470363075098968064 "24T2" [2, 2] "24.0.69910768076672525013777811909797532985470363075098968064.5" [13149858685993, -8436303982036, 13336474559176, -6780008973584, 6149105988181, -2722120658400, 1816036592588, -703968495068, 372662857315, -125173647000, 54817953718, -15859582480, 5812271149, -1407317280, 432776568, -85179088, 22047765, -3418496, 745342, -86612, 15873, -1256, 192, -8, 1] 69910768076672525013777811909797532985470363075098968064 "24T2" [2, 2, 4, 346836] "24.0.167855754152090732558080526395423876698114341743312622321664.2" [183482852093832256, 0, -14285685345414784, 0, 4057026789172448, 0, -222139872289776, 0, 25631546597300, 0, -805656766964, 0, 47771697841, 0, -989417212, 0, 35300482, 0, -459130, 0, 10577, 0, -70, 0, 1] 167855754152090732558080526395423876698114341743312622321664 "24T2" NULL "24.24.167855754152090732558080526395423876698114341743312622321664.2" [153664, 0, -40259968, 0, 2701950944, 0, -56037940560, 0, 162602891444, 0, -139604963708, 0, 43648167913, 0, -4578188188, 0, 216707890, 0, -5176234, 0, 64841, 0, -406, 0, 1] 167855754152090732558080526395423876698114341743312622321664 "24T2" [2, 2, 2] "24.0.167855754152090732558080526395423876698114341743312622321664.3" [279375876567076864, 0, -11301344203784192, 0, 536500759625216, 0, 19784037543168, 0, 2851116955088, 0, 92594721440, 0, 2365280329, 0, 104627248, 0, 7989694, 0, 320212, 0, 8789, 0, 136, 0, 1] 167855754152090732558080526395423876698114341743312622321664 "24T2" NULL "24.24.167855754152090732558080526395423876698114341743312622321664.4" [112957160281, 0, -545698857424, 0, 885611318228, 0, -687774981096, 0, 290317220270, 0, -70644063224, 0, 10296331486, 0, -916425832, 0, 49857157, 0, -1624000, 0, 30002, 0, -280, 0, 1] 167855754152090732558080526395423876698114341743312622321664 "24T2" [2, 2, 2] "24.0.167855754152090732558080526395423876698114341743312622321664.4" [279375876567076864, 0, 11301344203784192, 0, 536500759625216, 0, -19784037543168, 0, 2851116955088, 0, -92594721440, 0, 2365280329, 0, -104627248, 0, 7989694, 0, -320212, 0, 8789, 0, -136, 0, 1] 167855754152090732558080526395423876698114341743312622321664 "24T2" NULL "24.0.167855754152090732558080526395423876698114341743312622321664.11" [112957160281, 0, 545698857424, 0, 885611318228, 0, 687774981096, 0, 290317220270, 0, 70644063224, 0, 10296331486, 0, 916425832, 0, 49857157, 0, 1624000, 0, 30002, 0, 280, 0, 1] 167855754152090732558080526395423876698114341743312622321664 "24T2" [2, 2, 2, 2, 57503880] "36.0.14555723661975701129520414713591505717264293810334661879947459060497463940377.1" [88739165414377, 1197785699568, 16930660662, 3992608699248, 112871077641, 3593254105584, 289702503252, 1197349695360, 372475050696, 132056987296, 269011171638, -1517431968, 114129248142, -1478523456, 28224185886, -985682304, 3769723674, -456602832, 215269598, -146860560, 3996135, -32169456, 1937520, -4577472, 712530, -381456, 197316, -14128, 40455, 0, 5952, 0, 594, 0, 36, 0, 1] 14555723661975701129520414713591505717264293810334661879947459060497463940377 "36T1" NULL "36.0.18578968244125548124616820622906134052834598472886893781101495507970684317577737.1" [314475702946201, -272443564852044, 754616110153278, -473667528448616, 687799720872915, -457082453688360, 400600813897184, -233310397996224, 143219398499181, -72036142424760, 36830473035138, -15714568173240, 7435976619205, -3076563513732, 1559993144460, -670744519740, 297973081137, -109247726748, 39134525526, -12586522968, 4137101529, -1289435708, 397087680, -116457576, 34393824, -9357384, 2203584, -357316, 30030, 6912, -1072, -852, 777, -304, 78, -12, 1] 18578968244125548124616820622906134052834598472886893781101495507970684317577737 "36T3" NULL "36.0.2998349842185358863255204841514792544831190227347970050003854382739702385841842497.1" [2826300694492367, -586679398245228, 5253509890808868, -588086588594215, 4089367655099508, -527221390149000, 2080476220438645, -440122012765293, 757959816846441, -203260101629048, 204477541027983, -57999232619976, 40772365057554, -10931252013075, 6019321841520, -1399230501317, 675163203918, -126832652097, 60579704221, -8779986459, 4825880994, -606887805, 393700200, -56559951, 34342340, -5825274, 2926128, -564358, 208287, -38607, 11483, -2016, 558, -70, 24, -3, 1] 2998349842185358863255204841514792544831190227347970050003854382739702385841842497 "36T3" NULL "36.36.1000261713564783921567367061747723580144754632062761593433475437810167889837736142569472.1" [279783902667866112, 0, -1888541343008096256, 0, 4665808023902355456, 0, -5593106510134272000, 0, 3768710043816437760, 0, -1572953428060768512, 0, 436300597312728192, 0, -84528781438983696, 0, 11839562342924616, 0, -1226991733923327, 0, 95452554903345, 0, -5613466120956, 0, 249566849946, 0, -8324380854, 0, 204748119, 0, -3598050, 0, 42687, 0, -306, 0, 1] 1000261713564783921567367061747723580144754632062761593433475437810167889837736142569472 "36T1" NULL "36.36.1276736976031068373012854033710083107755464584613452023949707297014799943531783333643026432.1" [-54760480865012411, 799436524344590466, 798249314697436251, -5747262287938234880, -2396568579255331908, 10579589510484652182, 3412076707125906269, -8988435735842252748, -2427475735553660616, 4446580168463488472, 997940535494413266, -1433575501980734520, -260509181878012121, 320851626786690816, 45562912767195228, -51787651952164156, -5474391245015079, 6169887991972554, 450788704258077, -549740319318900, -24239384955576, 36825369190058, 687272217531, -1851102909252, 6406521661, 69165715854, -1561023087, -1884717672, 73451997, 36227178, -1919207, -463380, 30060, 3524, -264, -12, 1] 1276736976031068373012854033710083107755464584613452023949707297014799943531783333643026432 "36T3" NULL "36.36.206045032226446039803277454337390353284393213875429378828064627965012618047422191247967649792.1" [64978125597648069101, 114426950525433569346, -415778434695684669792, -262602075824947071046, 699393186533525699931, 193132746535654425018, -518855667187095697877, -73641857796913941000, 219306587679199150137, 17286393399958718852, -59873147190767332926, -2712327235603464990, 11329809839163057516, 296583297120354300, -1551478555245821412, -22966457995790158, 157996881375724086, 1249015358949066, -12169710913578236, -45422387005302, 715595082013542, 917785278920, -32213571117606, 1654964172, 1106390745734, -728913978, -28689888816, 23084500, 550743459, -374364, -7567610, 3252, 70209, -12, -393, 0, 1] 206045032226446039803277454337390353284393213875429378828064627965012618047422191247967649792 "36T3" NULL "36.36.262212606640726716335355871034795250193466558267460575149024985169308651313623503357731667968.1" [73343671380965094064128, 0, -247534890910757192466432, 0, 305778394654464767164416, 0, -183274914124079824896000, 0, 61746545357888516259840, 0, -12885634482673815650304, 0, 1787087246592934674432, 0, -173114944387038609408, 0, 12123711839154806784, 0, -628219767768743424, 0, 24435854055256320, 0, -718523663482368, 0, 15972278396544, 0, -266380187328, 0, 3275969904, 0, -28784400, 0, 170748, 0, -612, 0, 1] 262212606640726716335355871034795250193466558267460575149024985169308651313623503357731667968 "36T1" NULL "36.0.262212606640726716335355871034795250193466558267460575149024985169308651313623503357731667968.1" [73343671380965094064128, 0, 247534890910757192466432, 0, 305778394654464767164416, 0, 183274914124079824896000, 0, 61746545357888516259840, 0, 12885634482673815650304, 0, 1787087246592934674432, 0, 173114944387038609408, 0, 12123711839154806784, 0, 628219767768743424, 0, 24435854055256320, 0, 718523663482368, 0, 15972278396544, 0, 266380187328, 0, 3275969904, 0, 28784400, 0, 170748, 0, 612, 0, 1] 262212606640726716335355871034795250193466558267460575149024985169308651313623503357731667968 "36T1" 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 $1\leq i < n$.