# Number fields downloaded from the LMFDB on 06 June 2026. # Search link: https://www.lmfdb.org/NumberField/?completions=71.1.5.4a1.4 # Query "{'local_algs': {'$contains': ['71.1.5.4a1.4']}}" returned 313 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. "5.1.609880344.1" [-1065, 213, -71, 0, 0, 1] 609880344 "5T5" [5] "5.5.2032934480.1" [-1, 25, 34, -28, -1, 1] 2032934480 "5T5" [5] "5.1.3176460125.2" [496, -472, 176, -28, -1, 1] 3176460125 "5T3" [5] "5.1.7725151024.1" [-600, 400, -12, 30, -2, 1] 7725151024 "5T5" [5] "5.3.8716206583.1" [-387, -1304, 59, 30, -2, 1] -8716206583 "5T5" [5] "5.1.10977846192.2" [-3551, 451, 176, -28, -1, 1] 10977846192 "5T5" [5] "5.1.13036192353.1" [-143, 380, -179, 43, -1, 1] 13036192353 "5T5" [5] "5.1.14637128256.1" [-498, 238, 34, -28, -1, 1] 14637128256 "5T4" [15] "5.1.16466769288.2" [-245, -26, -83, -41, -2, 1] 16466769288 "5T5" [15] "5.1.20329344800.1" [-3125, -1537, -250, -28, -1, 1] 20329344800 "5T5" [5] "5.1.21955692384.1" [-6564, 3240, -438, 30, -2, 1] 21955692384 "5T5" [5] "5.1.24420625441.1" [-1097, 258, -83, 30, -2, 1] 24420625441 "5T4" [15] "5.3.25208387552.1" [-640, 96, 34, -28, -1, 1] -25208387552 "5T5" [5] "5.3.28638964487.1" [-5113, 2652, -37, -99, -1, 1] -28638964487 "5T5" [10] "5.1.31129309225.1" [1136, 852, -71, 142, 0, 1] 31129309225 "5T2" [5] "5.3.35576353400.1" [-214, -46, -321, -99, -1, 1] -35576353400 "5T5" [5] "5.1.43911384768.2" [-3834, -1917, 0, 71, 0, 1] 43911384768 "5T5" [20] "5.1.43911384768.3" [-1, 25, -37, 43, -1, 1] 43911384768 "5T5" [15] "5.1.44826205284.1" [-2485, 1278, -71, 71, 0, 1] 44826205284 "5T4" [10] "5.1.49400307864.2" [1104, -1304, 343, 30, -2, 1] 49400307864 "5T5" [5] "5.1.51458654025.1" [-14303, 2459, -367, 101, -2, 1] 51458654025 "5T4" [5] "5.1.52043122688.3" [-4150, 1465, 272, 30, -2, 1] 52043122688 "5T5" [5] "5.1.65867077152.1" [-8876, -3880, -179, 43, -1, 1] 65867077152 "5T5" [5] "5.1.76870335025.1" [-10509, -2957, -889, -28, -1, 1] 76870335025 "5T2" [5] "5.1.76870335025.2" [-4008, -736, -12, 101, -2, 1] 76870335025 "5T4" [10] "5.1.79691031616.1" [-782, -614, -250, -28, -1, 1] 79691031616 "5T4" [5] "5.1.79691031616.2" [1348, -756, 176, -28, -1, 1] 79691031616 "5T4" [15] "5.1.79691031616.3" [-4900, 3220, -392, -28, -1, 1] 79691031616 "5T4" [5] "5.1.85484894884.1" [212, -472, 602, -99, -1, 1] 85484894884 "5T4" [5] "5.1.100858961889.2" [9411, -1162, -367, 30, -2, 1] 100858961889 "5T4" [5] "5.1.117097026048.1" [-1705, 2581, -392, -28, -1, 1] 117097026048 "5T5" [10] "5.1.123500769660.1" [17820, -1182, 318, -99, -1, 1] 123500769660 "5T5" [5] "5.1.156129368064.1" [-8552, 8068, -296, 172, -2, 1] 156129368064 "5T5" [5] "5.5.164667692880.1" [-14485, 4285, 602, -170, -1, 1] 164667692880 "5T5" [5] "5.1.196788057664.5" [-569, -259, -108, -28, -1, 1] 196788057664 "5T4" [5] "5.1.197601231456.2" [252, -594, 414, -41, -2, 1] 197601231456 "5T5" [15] "5.1.203293448000.1" [-7385, 2865, -250, 114, -1, 1] 203293448000 "5T5" [5] "5.3.214068000744.1" [-640, -1466, -250, 43, -1, 1] -214068000744 "5T5" [5] "5.1.239073094848.1" [-40187, 1303, 176, -28, -1, 1] 239073094848 "5T5" [10] "5.1.239073094848.2" [-711, 1587, 176, -28, -1, 1] 239073094848 "5T5" [5] "5.1.257293270125.1" [-5751, 0, 0, 0, 0, 1] 257293270125 "5T3" [5, 5] "5.1.268957231704.1" [5324, 4498, 1028, 43, -1, 1] 268957231704 "5T5" [10] "5.1.274446154800.4" [34221, 13799, 1099, -99, -1, 1] 274446154800 "5T5" [5] "5.1.274852741696.2" [-30704, -2440, -1148, 30, -2, 1] 274852741696 "5T4" [5] "5.1.301916181961.1" [638, 25, 34, 43, -1, 1] 301916181961 "5T4" [15] "5.1.324481754689.1" [4725, 1607, -83, 101, -2, 1] 324481754689 "5T4" [15] "5.1.341736286088.1" [-4008, 1678, 272, -41, -2, 1] 341736286088 "5T5" [5] "5.3.360210578175.1" [4725, -3150, -225, 30, -2, 1] -360210578175 "5T5" [10] "5.3.364301858816.1" [-2415, -1537, 1170, -170, -1, 1] -364301858816 "5T5" [5] "5.1.365928206400.1" [-2131, 4711, -392, -28, -1, 1] 365928206400 "5T4" [15] "5.5.372052421521.4" [10933, 9397, 531, -312, -1, 1] 372052421521 "5T1" [5] "5.5.375889585352.1" [-8836, 14316, 343, -254, -2, 1] 375889585352 "5T5" [5] "5.5.397057515625.1" [-1704, -9940, -3550, -355, 0, 1] 397057515625 "5T2" [5] "5.1.411669232200.1" [-1208, 1232, -179, 43, -1, 1] 411669232200 "5T5" [5] "5.1.422875783521.1" [47925, 6319, -213, -71, 0, 1] 422875783521 "5T4" [5] "5.1.457410258000.1" [-32945, 3291, -1031, -99, -1, 1] 457410258000 "5T3" [5] "5.1.504294809445.1" [-13207, 664, 176, 43, -1, 1] 504294809445 "5T5" [5] "5.1.537914463408.1" [1814, 1181, 1124, -41, -2, 1] 537914463408 "5T5" [5] "5.3.559870155792.1" [107919, 8829, -1386, -170, -1, 1] -559870155792 "5T5" [5] "5.1.594862040529.1" [-1807, 755, 59, -41, -2, 1] 594862040529 "5T4" [5] "5.1.594862040529.2" [4188, 4356, 957, 43, -1, 1] 594862040529 "5T4" [5, 5] "5.1.658670771520.1" [18570, -3079, 272, 30, -2, 1] 658670771520 "5T5" [10] "5.1.702582156288.2" [-13806, 1465, -296, 30, -2, 1] 702582156288 "5T5" [5] "5.1.813173792000.1" [-1704, -355, 0, 0, 0, 1] 813173792000 "5T5" [5] "5.1.1026657324081.1" [26624, -11264, 1312, -28, -1, 1] 1026657324081 "5T4" [15] "5.1.1053873234432.2" [-569, 877, 176, -28, -1, 1] 1053873234432 "5T5" [5] "5.1.1067925894025.1" [54354, -2795, 769, 243, -2, 1] 1067925894025 "5T4" [15] "5.1.1146702105125.1" [35468, 6861, 201, 172, -2, 1] 1146702105125 "5T5" [5] "5.1.1210307542668.1" [-16188, 639, -284, 0, 0, 1] 1210307542668 "5T5" [5] "5.1.1245172369000.1" [20518, 8119, 1809, 114, -1, 1] 1245172369000 "5T5" [5] "5.1.1270584050000.1" [-1136, 0, 0, 0, 0, 1] 1270584050000 "5T3" [5, 5] "5.1.1301078067200.2" [-48494, 4214, 176, 256, -1, 1] 1301078067200 "5T3" [5] "5.1.1301078067200.4" [-4900, 3220, -108, -28, -1, 1] 1301078067200 "5T5" [10] "5.3.1657324423139.2" [38024, 3311, -438, -112, -2, 1] -1657324423139 "5T5" [10] "5.1.1771092518976.1" [-102130, 19073, 1124, -254, -2, 1] 1771092518976 "5T4" [15] "5.3.1801052890875.2" [181, -239, 130, -41, -2, 1] -1801052890875 "5T5" [5] "5.1.1829641032000.1" [-13096, 6932, -1432, 172, -2, 1] 1829641032000 "5T3" [5] "5.1.1985287578125.6" [23998, 1065, -355, 0, 0, 1] 1985287578125 "5T5" [5] "5.1.2058346161000.1" [-17640, 5796, -438, -41, -2, 1] 2058346161000 "5T5" [10] "5.5.2470015393200.1" [31950, 7881, -710, -213, 0, 1] 2470015393200 "5T5" [5] "5.1.2521474047225.1" [-87504, -14439, -1290, -41, -2, 1] 2521474047225 "5T4" [15] "5.1.2521474047225.2" [-14800, 2956, 343, -112, -2, 1] 2521474047225 "5T4" [5] "5.1.2634683086080.1" [40580, 5654, 1124, 172, -2, 1] 2634683086080 "5T5" [5] "5.1.2744461548000.1" [11328, -4712, 1408, 30, -2, 1] 2744461548000 "5T5" [5] "5.3.2830225971375.1" [12890, -14155, 3680, -254, -2, 1] -2830225971375 "5T5" [5] "5.1.2868877138176.1" [-1704, 1633, -1704, -142, 0, 1] 2868877138176 "5T4" [15] "5.1.3087519241500.1" [-99006, -6558, -1077, 172, -2, 1] 3087519241500 "5T5" [5] "5.1.3172597549488.1" [86904, -7668, -2414, 213, 0, 1] 3172597549488 "5T5" [5] "5.1.3227486780448.2" [-102596, 9752, -1031, 43, -1, 1] 3227486780448 "5T5" [5] "5.1.3227486780448.3" [-26768, 13160, -1883, 43, -1, 1] 3227486780448 "5T5" [5] "5.1.3227486780448.5" [44982, 17937, 1692, 30, -2, 1] 3227486780448 "5T5" [5] "5.1.3705023089800.1" [212, 238, -37, 43, -1, 1] 3705023089800 "5T5" [5] "5.1.3760115614208.1" [15548, -756, 1028, 256, -1, 1] 3760115614208 "5T5" [5] "5.1.3825169517568.1" [16866, 8281, 3112, 30, -2, 1] 3825169517568 "5T5" [5] "5.3.3853224013392.1" [5040, -3312, -108, 256, -1, 1] -3853224013392 "5T5" [5] "5.1.4116692322000.6" [-21230, 8900, 460, -170, -1, 1] 4116692322000 "5T5" [55] "5.1.4742429554944.1" [-63222, 24753, -1716, 172, -2, 1] 4742429554944 "5T4" [5] "5.3.4870504427184.1" [141, 25, 34, -28, -1, 1] -4870504427184 "5T5" [10] "5.1.5082336200000.4" [-96986, 25205, 0, -355, 0, 1] 5082336200000 "5T3" [5] "5.1.5082336200000.5" [24140, 4970, -1420, 0, 0, 1] 5082336200000 "5T5" [10] "5.3.5796302789376.1" [-13633, 5137, 602, -170, -1, 1] -5796302789376 "5T5" [5] "5.1.6352920250000.2" [84632, 52540, -710, -355, 0, 1] 6352920250000 "5T4" [15] "5.3.7729598068175.1" [7849, -7978, 2118, -183, -2, 1] -7729598068175 "5T5" [5] "5.1.8233384644000.1" [36746, 15026, 1550, -41, -2, 1] 8233384644000 "5T5" [5] "5.3.8658674537216.1" [33622, -12167, 556, 30, -2, 1] -8658674537216 "5T5" [5] "5.1.9682460341344.4" [-23888, 14032, -2852, 172, -2, 1] 9682460341344 "5T5" [5] "5.1.9682460341344.7" [48350, -6578, -463, 43, -1, 1] 9682460341344 "5T5" [5] "5.1.9799786072521.1" [-44304, 10437, -497, 213, 0, 1] 9799786072521 "5T4" [5] "5.5.9926437890625.3" [69651, 71710, -3195, -710, 0, 1] 9926437890625 "5T1" [5] "5.3.10958634961164.1" [86904, 5112, -3550, -426, 0, 1] -10958634961164 "5T5" [5] "5.1.11155118078656.1" [-32590, -7714, -463, 43, -1, 1] 11155118078656 "5T5" [20] "5.1.11601270961973.1" [65461, -19074, 673, 114, -1, 1] 11601270961973 "5T5" [5] "5.1.12646478813184.2" [13347, -827, -250, 114, -1, 1] 12646478813184 "5T5" [5] "5.5.16141728476329.1" [-72, 96, 176, -99, -1, 1] 16141728476329 "5T2" [5] "5.1.17802635946489.1" [147222, -11457, 1053, -41, -2, 1] 17802635946489 "5T4" [5] "5.1.18230568654529.3" [226458, 32421, -8248, 314, -2, 1] 18230568654529 "5T2" [5, 5] "5.3.22592407463136.1" [1704, 426, -1420, -213, 0, 1] -22592407463136 "5T5" [5] "5.5.23468221048801.1" [-112039, 14509, 6779, -880, -1, 1] 23468221048801 "5T1" [5] "5.3.24656928662619.2" [47711, 1516, -1954, -170, -1, 1] -24656928662619 "5T5" [5] "5.1.24700153932000.1" [-31840, -10960, -2710, -254, -2, 1] 24700153932000 "5T5" [5] "5.1.24700153932000.2" [-16362, 8991, -864, 172, -2, 1] 24700153932000 "5T5" [5] "5.1.25729327012500.3" [-2556, 1065, -710, 0, 0, 1] 25729327012500 "5T5" [5] "5.1.26181680345981.2" [-151972, -51075, -5621, -254, -2, 1] 26181680345981 "5T3" [5] "5.5.27530811901952.1" [6358, -8049, 2544, -254, -2, 1] 27530811901952 "5T3" [5] "5.1.31764601250000.3" [724768, -52185, -2130, 0, 0, 1] 31764601250000 "5T5" [5] "5.3.33197006884608.1" [-10398, -9327, -2568, -254, -2, 1] -33197006884608 "5T5" [5] "5.1.40658689600000.1" [-141148, 25205, 0, 0, 0, 1] 40658689600000 "5T5" [5, 15] "5.3.45184814926272.2" [9016, 1232, -818, -170, -1, 1] -45184814926272 "5T5" [5] "5.1.49632189453125.3" [-8875, 0, 0, 0, 0, 1] 49632189453125 "5T3" [5, 5] "5.1.50201004520629.1" [-73911, 8946, -1065, 213, 0, 1] 50201004520629 "5T5" [10] "5.1.51639788487168.1" [17008, 5512, -296, 172, -2, 1] 51639788487168 "5T5" [10] "5.1.66785074128125.1" [-197735, 25205, 0, -355, 0, 1] 66785074128125 "5T3" [5, 5] "5.5.71729247018528.1" [11896, -9824, 2686, -254, -2, 1] 71729247018528 "5T5" [5] "5.5.71807337114241.3" [-4829, -36469, -16651, -1164, -1, 1] 71807337114241 "5T1" [5] "5.3.72354984251472.1" [142709, 7267, -4297, -383, -1, 1] -72354984251472 "5T5" [5] "5.1.83711794842225.1" [-1390212, 196431, -2781, -467, -2, 1] 83711794842225 "5T4" [5, 45] "5.1.95245419909376.1" [-338986, 23049, 1124, 314, -2, 1] 95245419909376 "5T4" [5, 15] "5.5.97279091328125.1" [-60492, 25205, 0, -355, 0, 1] 97279091328125 "5T3" [5] "5.1.100183824348192.1" [-45796, 23668, -1741, -383, -1, 1] 100183824348192 "5T5" [10] "5.1.102917308050000.2" [-3834, 0, 0, 0, 0, 1] 102917308050000 "5T3" [5, 5] "5.1.102917308050000.4" [-2556, 0, 0, 0, 0, 1] 102917308050000 "5T3" [5, 5] "5.1.102917308050000.5" [-1704, 0, 0, 0, 0, 1] 102917308050000 "5T3" [5, 5] "5.1.110788423769664.1" [2808, -1020, -12, 30, -2, 1] 110788423769664 "5T4" [5] "5.3.111567444262400.1" [12890, 329, -864, 30, -2, 1] -111567444262400 "5T5" [10] "5.3.121351856267916.1" [17749, -4519, 460, -170, -1, 1] -121351856267916 "5T5" [10] "5.1.122026892162000.1" [-3976, 0, 0, 0, 0, 1] 122026892162000 "5T3" [5, 5] "5.1.127058405000000.2" [-90738, 25205, 0, -355, 0, 1] 127058405000000 "5T3" [5] "5.1.160808293828125.3" [311832, 6390, -3195, 0, 0, 1] 160808293828125 "5T5" [5, 25] "5.3.168852285213632.1" [149526, 28045, -1136, -355, 0, 1] -168852285213632 "5T5" [5] "5.1.190667019003125.4" [-170471, 0, 0, 0, 0, 1] 190667019003125 "5T3" [5, 5] "5.1.206559153948672.3" [78525, -20565, 2874, -170, -1, 1] 206559153948672 "5T5" [5] "5.1.265301126100125.1" [-1207, 0, 0, 0, 0, 1] 265301126100125 "5T3" [5, 5] "5.1.289454928890625.1" [-1397493, -172530, -5325, 0, 0, 1] 289454928890625 "5T4" [5] "5.1.289454928890625.3" [-850722, -224715, -24140, -1065, 0, 1] 289454928890625 "5T4" [5] "5.1.289454928890625.5" [-8946, 9585, -7455, 1065, 0, 1] 289454928890625 "5T4" [5] "5.1.301188188123712.1" [5364, 187, 130, 30, -2, 1] 301188188123712 "5T5" [5] "5.1.308751924150000.1" [-86904, 17040, -710, 0, 0, 1] 308751924150000 "5T5" [5] "5.1.317646012500000.2" [-606056, -45440, -9230, -710, 0, 1] 317646012500000 "5T5" [5] "5.5.351846094668721.4" [565727, 123139, -23041, -1732, -1, 1] 351846094668721 "5T1" [5] "5.1.411669232200000.2" [-2556, 6390, 0, 0, 0, 1] 411669232200000 "5T5" [5] "5.1.486395456640625.2" [-1691291, 226135, -355, -710, 0, 1] 486395456640625 "5T4" [15] "5.1.486395456640625.5" [-14413, 29110, -2485, 710, 0, 1] 486395456640625 "5T2" [5] "5.3.514524332454912.1" [1104, -168, 272, -112, -2, 1] -514524332454912 "5T5" [5] "5.5.586705526220025.1" [193900, 49725, -2025, -880, -1, 1] 586705526220025 "5T2" [5, 5] "5.1.778312142128125.1" [-19028, -14910, -4615, -355, 0, 1] 778312142128125 "5T5" [10] "5.1.790404925824000.1" [22546, -13871, 3112, -254, -2, 1] 790404925824000 "5T5" [5] "5.1.794115031250000.6" [-710, 0, 0, 0, 0, 1] 794115031250000 "5T3" [5, 10] "5.1.794115031250000.10" [-177500, 0, 0, 0, 0, 1] 794115031250000 "5T3" [5, 5] "5.1.794115031250000.12" [-101530, 30175, -3550, 0, 0, 1] 794115031250000 "5T5" [5, 30] "5.1.794115031250000.18" [-14200, 0, 0, 0, 0, 1] 794115031250000 "5T3" [5, 20] "5.1.804041469140625.2" [-2050551, 291810, -5680, 1065, 0, 1] 804041469140625 "5T4" [5, 5] "5.1.804041469140625.5" [3139407, 307785, -6035, -1065, 0, 1] 804041469140625 "5T4" [5, 10] "5.3.862946533460736.1" [30072, -1020, 840, -396, -2, 1] -862946533460736 "5T5" [5] "5.3.951270395934375.1" [-22720, 12780, -710, -355, 0, 1] -951270395934375 "5T5" [5] "5.1.1029173080500000.3" [-2556, 3195, 0, 0, 0, 1] 1029173080500000 "5T5" [10, 10] "5.1.1097784619200000.1" [-2556, 1065, 0, 0, 0, 1] 1097784619200000 "5T5" [5] "5.1.1162663817253125.2" [-1039511, 0, 0, 0, 0, 1] 1162663817253125 "5T3" [5, 5] "5.1.1162663817253125.4" [-367070, 121055, 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"20.0.547802134397995945234404562265039130889892578125.1" [349801250000, 0, 6084822500, 0, -9104080799, 0, -47621500, 0, 35735106, 0, 6635510, 0, -2289, 0, -22990, 0, 11, 0, 5, 0, 1] 547802134397995945234404562265039130889892578125 "20T5" [5, 5, 5, 5] "20.4.3594711530218651723346797822128029972198486328125.1" [4953851095, 10820557800, 28654320680, 33987849415, 38978227086, 23941961614, 10634445100, 4204843245, 1199740810, 181625352, -5087262, -13529625, -5871830, -1338440, -20812, 56177, 8350, -470, -160, -1, 1] 3594711530218651723346797822128029972198486328125 "20T9" [5, 5] "20.12.5041588277632109128867132309980221276160000000000.1" [140496435439, -1204913308336, -852156581038, 1060735253720, 671853804877, 124835981336, -126778202726, -49202516284, 6894970312, 3211938640, 30073560, -73156640, -5336018, 2566648, 148422, -81148, 1049, 1112, -122, -8, 1] 5041588277632109128867132309980221276160000000000 "20T391" [5] "20.0.13664178114277213081586252812156928000000000000000.2" [426134955776, -1568774760448, 2659395772928, -2746911907328, 1923006566528, -956752937728, 342378907648, -85514174848, 12927757648, -245233088, -413171456, 102138784, -8557944, -1157456, 401816, -37072, -2201, 856, -56, -4, 1] 13664178114277213081586252812156928000000000000000 "20T9" [5, 5, 10] "20.4.330073376720491401772996684343968086112091064453125.1" [15785671355, -28987740445, 60325038285, -32324253005, 24980468221, 17651326864, 1653368750, 1596672870, 869759700, 27708432, -36728292, -949290, -1620095, -1578385, -179777, 58197, 11610, -435, -185, -1, 1] 330073376720491401772996684343968086112091064453125 "20T9" [5, 5] "20.0.787945364716753081303307267876029728000000000000000.2" [1279140753061, -4343904353218, 6780185537153, -6420934576988, 4105885932783, -1861359444628, 606011140638, -137443804408, 18753847113, -262825158, -507972001, 109019364, -6445179, -1590196, 406686, -28912, -2661, 766, -31, -4, 1] 787945364716753081303307267876029728000000000000000 "20T9" [10, 10, 10, 10] "20.4.527183737490700757793047555945274642280045440673828125.1" [2045479300595, -4939653256540, 4679432139900, -1741143920465, -180807195734, 327048348754, -54056993680, -22771586355, 8070665670, 735412692, -613843842, 1469655, 29040910, -1770820, -1042832, 58617, 24690, -330, -260, -1, 1] 527183737490700757793047555945274642280045440673828125 "20T9" [5] "20.0.806856053469955155254586642305054441472000000000000000.1" [29309515676416, -72462369691648, 82571066444288, -58244056790528, 28332224519808, -9944297421568, 2556425983488, -467716755328, 54866661648, -2019244608, -497997376, 81707424, 6209256, -2977456, 497736, -14992, -741, 496, 44, -4, 1] 806856053469955155254586642305054441472000000000000000 "20T9" [10, 10, 10, 10] "20.0.5353818246888671111864679787071686816515909454345703125.1" [24993820599205, 68990735516805, 85253194568735, 60827928058195, 30430900709421, 10853334912664, 2949097256650, 578477384370, 85150675300, 6806057532, 278801808, -69099090, 7144555, 857065, 561173, -4303, 3760, -785, 65, -1, 1] 5353818246888671111864679787071686816515909454345703125 "20T9" [10, 10, 10, 10] "20.4.10430390854185059522817398813761529702769345733642578125.1" [14631866202805, -32691163445070, 27425622259310, -8494652133605, -1430155574004, 1557486672964, -181652526950, -106716259755, 27567864400, 3831090882, -1837108092, -55742265, 75072580, -1258610, -2045002, 54197, 36160, -260, -310, -1, 1] 10430390854185059522817398813761529702769345733642578125 "20T9" [2, 2, 2, 2, 40, 40] "20.4.103592834128629528858754748616438010978669821807861328125.1" [55102573827055, -128664365473440, 107324971439120, -31957997240745, -5429016401114, 5367512128534, -535800166540, -340329665555, 75592068130, 11586448352, -4411410982, -193839225, 154727250, -46400, -3486612, 46017, 49830, -190, -360, -1, 1] 103592834128629528858754748616438010978669821807861328125 "20T9" [2, 2, 2, 2, 10] "20.4.1514905022769963177176946292275359011786457584259033203125.1" [106144180014505, -518589009883695, 530310779287835, -175085380964205, -20306596438979, 24147239445064, -2434299307150, -1325178218630, 270468294100, 39591261332, -12944264392, -631055490, 369390255, 3436165, -6656727, 26697, 74460, -85, -435, -1, 1] 1514905022769963177176946292275359011786457584259033203125 "20T9" NULL "20.0.165200865944802404921885709756070546171554577350616455078125.1" [844524446299985350656, -24770180624970240, -49539634732370880, -49537455200971080, -24764368612266815, 19805246992138942, 5376016926290, 2034300335855, -145226801645, -436055442460, 174492070329, -62222280, 7671515, 3409165, -2556880, 681763, 70, -35, 15, -5, 1] 165200865944802404921885709756070546171554577350616455078125 "20T5" NULL "20.0.9257010277357974581704544938058391067398630232569433624267578125.1" [20612429021424952783602001, -2268195365946653420588532, 235971919264358965180432, -15935280267138936344262, 678043906921564338362, -238194120512777773244, 18323624464641278256, -1482478778428908936, 45583860055261646, 3643069204444274, 256870001118486, -4596320535916, 1191080273576, 3111933494, -1081232634, -297244, 996192, 4928, -878, -2, 1] 9257010277357974581704544938058391067398630232569433624267578125 "20T5" NULL "20.0.9257010277357974581704544938058391067398630232569433624267578125.2" [3984719442689747148713041, -766455670574990083687322, 89322541982902977975362, -8492325485397424362042, 803839026763828463482, -4284768172529240644, -5179711802191291354, 505782040879996194, -14209539002664794, -1971370537266966, 335585102950166, -23982643599326, 612397645526, 35269027714, 178003486, -47222564, 578002, -17082, -878, -2, 1] 9257010277357974581704544938058391067398630232569433624267578125 "20T5" NULL "20.0.811381968494063713161816447547608958640076735492156328155517578125.1" [366802150886848102410000, -113231540710208423313000, 22222015091454257442900, -3478040613241072944570, 472127518524660145981, -55703907552530229286, 5892381441580246600, -567450398473762865, 50088481955872055, -4043033729429338, 299467429985573, -20178230984450, 1216239703405, -63770503025, 2837923358, -94068163, 1544600, 24685, 1165, -1, 1] 811381968494063713161816447547608958640076735492156328155517578125 "20T5" NULL "24.4.1941796499846363571121266433601267635822296142578125.2" [22510900, -123126400, 338239525, -76326125, -341349450, 240607480, 45242120, -176073970, 97414985, -309820, -38550914, 26201869, -8927744, -207581, 1610411, -1006306, 367091, -108451, 27186, -6501, 1526, -281, 56, -6, 1] 1941796499846363571121266433601267635822296142578125 "24T1353" [] "24.4.1941796499846363571121266433601267635822296142578125.4" [42458350172, -98967528776, 113849162501, -70856028046, 17385063266, 11301458793, -12436243534, 4202740599, 114562276, -585209216, 248864042, -36143236, -21630444, 11112529, -886349, -65007, -140549, 2469, 26821, -6236, -488, 309, -19, -6, 1] 1941796499846363571121266433601267635822296142578125 "24T1353" [] "24.4.48544912496159089278031660840031690895557403564453125.2" [283087976, 5255357659, -3117642654, -2030571896, -3162417234, -3094008901, -2196318159, -1886984496, -1103889354, -504463441, -196715769, -75574446, -25257224, 1769524, 996271, -698281, -9054, 46949, -3874, -1671, 261, 174, -19, -6, 1] 48544912496159089278031660840031690895557403564453125 "24T1353" [2] "24.4.48544912496159089278031660840031690895557403564453125.3" [329094400, 2472876800, 5625659600, 7589871000, 3591219300, -2709913130, -5779907435, -5189632570, -2787489045, -749879320, 147904819, 273343518, 161308416, 57547132, 13217099, 1824426, 9347, -37656, 8568, 3416, 736, -8, -36, -2, 1] 48544912496159089278031660840031690895557403564453125 "24T1353" [10] # 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 $1\leq i < n$.