Note: Search results may be incomplete due to uncomputed quantities: Class number (201181 objects)
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Label | Polynomial | Discriminant | Galois group | Class group |
---|---|---|---|---|
19.1.272...391.1 | $x^{19} - x^{18} + 2 x^{17} - 5 x^{16} + 8 x^{15} - 14 x^{14} + 13 x^{13} - 10 x^{12} - x^{11} + 9 x^{10} - 18 x^{9} + 25 x^{8} - 10 x^{7} - 4 x^{6} + 38 x^{5} - 42 x^{4} + 37 x^{3} - 16 x^{2} + 4 x + 1$ | $-\,311^{9}$ | $D_{19}$ (as 19T2) | trivial |
19.1.990...639.1 | $x^{19} - 2 x^{18} + 2 x^{17} - 2 x^{16} - 3 x^{15} + 14 x^{14} - 7 x^{13} - 22 x^{12} + 30 x^{11} - 9 x^{10} + 5 x^{9} - 2 x^{8} - 51 x^{7} + 90 x^{6} - 19 x^{5} - 91 x^{4} + 113 x^{3} - 59 x^{2} + 14 x - 1$ | $-\,359^{9}$ | $D_{19}$ (as 19T2) | trivial |
19.1.193...555.1 | $x^{19} - x - 1$ | $-\,5\cdot 659\cdot 588489604729898953429$ | $S_{19}$ (as 19T8) | trivial |
19.1.201...403.1 | $x^{19} + x - 1$ | $-\,149\cdot 2063\cdot 6564253087266573169$ | $S_{19}$ (as 19T8) | trivial |
19.1.756...064.1 | $x^{19} - 2 x^{18} + 8 x^{17} - 12 x^{16} + 43 x^{15} - 74 x^{14} + 108 x^{13} - 18 x^{12} - 25 x^{11} + 338 x^{9} - 70 x^{8} - 264 x^{7} - 128 x^{6} + 120 x^{5} + 98 x^{4} + 43 x^{3} - 22 x^{2} - 31 x - 8$ | $-\,2^{18}\cdot 19^{16}$ | $F_{19}$ (as 19T6) | trivial |
19.3.146...972.1 | $x^{19} - 2 x^{18} + 7 x^{16} - 10 x^{15} - x^{14} + 19 x^{13} - 19 x^{12} - 9 x^{11} + 33 x^{10} - 19 x^{9} - 20 x^{8} + 32 x^{7} - 9 x^{6} - 17 x^{5} + 17 x^{4} - 3 x^{3} - 5 x^{2} + 4 x - 1$ | $2^{2}\cdot 139\cdot 126311\cdot 1021043\cdot 2043393285469$ | $S_{19}$ (as 19T8) | trivial |
19.1.467...879.1 | $x^{19} - 5 x^{18} + 10 x^{17} - 5 x^{16} + 9 x^{15} - 42 x^{14} + 51 x^{13} - 6 x^{12} - 23 x^{11} + 4 x^{10} - 29 x^{9} + 112 x^{8} - 63 x^{7} - 147 x^{6} + 183 x^{5} + 15 x^{4} - 77 x^{3} - 17 x^{2} + 37 x + 1$ | $-\,919^{9}$ | $D_{19}$ (as 19T2) | trivial |
19.1.714...419.1 | $x^{19} + 19 x^{15} - 19 x^{14} - 19 x^{13} + 171 x^{12} + 19 x^{11} - 266 x^{10} + 19 x^{9} + 475 x^{8} + 741 x^{7} + 437 x^{6} + 38 x^{5} - 38 x^{4} - 19 x^{3} + 1$ | $-\,19^{21}$ | $C_{19}:C_{6}$ (as 19T4) | trivial |
19.1.173...823.1 | $x^{19} + 9 x^{17} - 10 x^{16} + 8 x^{15} - 45 x^{14} + 39 x^{13} + 7 x^{12} + 42 x^{11} - 36 x^{10} - 60 x^{9} + 9 x^{8} + 55 x^{7} + 63 x^{6} + 15 x^{5} - 88 x^{4} - 7 x^{3} + 27 x^{2} + 51 x + 1$ | $-\,1063^{9}$ | $D_{19}$ (as 19T2) | trivial |
19.1.512...799.1 | $x^{19} - 3 x^{18} + 17 x^{17} - 47 x^{16} + 135 x^{15} - 310 x^{14} + 709 x^{13} - 1319 x^{12} + 2423 x^{11} - 3757 x^{10} + 5497 x^{9} - 6561 x^{8} + 7142 x^{7} - 5699 x^{6} + 4212 x^{5} - 1490 x^{4} + 482 x^{3} + 408 x^{2} - 210 x + 49$ | $-\,11^{9}\cdot 109^{9}$ | $D_{19}$ (as 19T2) | trivial |
19.3.787...824.1 | $x^{19} - 2 x^{18} + 7 x^{16} - 10 x^{15} - x^{14} + 19 x^{13} - 19 x^{12} - 9 x^{11} + 33 x^{10} - 19 x^{9} - 20 x^{8} + 33 x^{7} - 9 x^{6} - 18 x^{5} + 17 x^{4} - 3 x^{3} - 5 x^{2} + 4 x - 1$ | $2^{8}\cdot 317\cdot 25373\cdot 137363\cdot 27854432176313$ | $S_{19}$ (as 19T8) | trivial |
19.1.206...091.1 | $x^{19} + 2 x - 1$ | $-\,7\cdot 233\cdot 12649189464522827348824661$ | $S_{19}$ (as 19T8) | trivial |
19.1.223...888.1 | $x^{19} + 4 x - 4$ | $-\,2^{18}\cdot 1699471\cdot 109607929\cdot 458633653$ | $S_{19}$ (as 19T8) | trivial |
19.1.473...375.1 | $x^{19} - 6 x^{18} + 18 x^{17} - 22 x^{16} - 38 x^{15} + 219 x^{14} - 403 x^{13} + 380 x^{12} - 536 x^{11} + 1378 x^{10} - 1608 x^{9} + 691 x^{8} - 1073 x^{7} + 1881 x^{6} - 757 x^{5} + 716 x^{4} - 1464 x^{3} + 513 x^{2} - 10 x + 295$ | $-\,5^{9}\cdot 307^{9}$ | $D_{19}$ (as 19T2) | trivial |
19.1.495...343.1 | $x^{19} - 9 x^{18} + 24 x^{17} - 6 x^{16} - 42 x^{15} + 4 x^{14} + 34 x^{13} + 74 x^{12} + 46 x^{11} - 238 x^{10} - 49 x^{9} + 181 x^{8} + 195 x^{7} + 116 x^{6} - 193 x^{5} - 275 x^{4} + 106 x^{3} - 8 x^{2} + 121 x - 1$ | $-\,1543^{9}$ | $D_{19}$ (as 19T2) | trivial |
19.1.101...431.1 | $x^{19} - 5 x^{18} + 11 x^{17} - 10 x^{16} + 3 x^{15} + 3 x^{14} - 80 x^{13} + 481 x^{12} - 1180 x^{11} + 1220 x^{10} + 888 x^{9} - 5070 x^{8} + 8854 x^{7} - 9440 x^{6} + 6893 x^{5} - 3748 x^{4} + 2592 x^{3} - 2790 x^{2} + 2673 x - 1053$ | $-\,3^{9}\cdot 557^{9}$ | $D_{19}$ (as 19T2) | trivial |
19.7.105...804.1 | $x^{19} - 3 x^{18} - x^{17} + 15 x^{16} - 22 x^{15} - 6 x^{14} + 59 x^{13} - 70 x^{12} - 6 x^{11} + 106 x^{10} - 107 x^{9} + 92 x^{7} - 73 x^{6} - x^{5} + 35 x^{4} - 19 x^{3} - x^{2} + 4 x - 1$ | $2^{2}\cdot 131\cdot 149\cdot 397\cdot 20117\cdot 218819\cdot 773146117559$ | $S_{19}$ (as 19T8) | trivial |
19.1.197...516.1 | $x^{19} - 3 x^{18} + 2 x^{17} + 4 x^{16} - 8 x^{15} + 14 x^{13} - 9 x^{12} - 16 x^{11} + 21 x^{10} + 8 x^{9} - 25 x^{8} + 2 x^{7} + 16 x^{6} - 4 x^{5} - 8 x^{4} + 2 x^{3} + 3 x^{2} - x - 1$ | $-\,2^{2}\cdot 49323595893763542101396994379$ | $S_{19}$ (as 19T8) | trivial |
19.1.231...871.1 | $x^{19} - 6 x^{18} + 18 x^{17} - 39 x^{16} + 73 x^{15} - 200 x^{14} + 265 x^{13} + 305 x^{12} - 931 x^{11} + 1905 x^{10} - 5214 x^{9} + 10284 x^{8} - 13343 x^{7} + 12719 x^{6} - 8662 x^{5} + 4443 x^{4} - 1732 x^{3} + 614 x^{2} - 152 x + 39$ | $-\,1831^{9}$ | $D_{19}$ (as 19T2) | trivial |
19.1.239...368.1 | $x^{19} - 3 x^{18} + 2 x^{17} + 4 x^{16} - 8 x^{15} + 14 x^{13} - 9 x^{12} - 16 x^{11} + 21 x^{10} + 8 x^{9} - 25 x^{8} + 2 x^{7} + 17 x^{6} - 4 x^{5} - 9 x^{4} + 2 x^{3} + 3 x^{2} - x - 1$ | $-\,2^{7}\cdot 7\cdot 60259\cdot 4440286064038502416537$ | $S_{19}$ (as 19T8) | trivial |
19.1.436...528.1 | $x^{19} - 8 x - 8$ | $-\,2^{18}\cdot 17\cdot 8627\cdot 11343650175290580793$ | $S_{19}$ (as 19T8) | trivial |
19.1.477...752.1 | $x^{19} - 4 x - 4$ | $-\,2^{18}\cdot 593\cdot 10949399\cdot 280461391706269$ | $S_{19}$ (as 19T8) | trivial |
19.1.498...864.1 | $x^{19} - 2 x - 2$ | $-\,2^{18}\cdot 11\cdot 29\cdot 8255287577\cdot 721386928637$ | $S_{19}$ (as 19T8) | trivial |
19.1.518...552.1 | $x^{19} - x - 2$ | $-\,2^{19}\cdot 821\cdot 883013\cdot 15385297\cdot 88689509$ | $S_{19}$ (as 19T8) | trivial |
19.1.518...855.1 | $x^{19} - x - 4$ | $-\,5\cdot 103726168442653430174403870371$ | $S_{19}$ (as 19T8) | trivial |
19.1.518...976.1 | $x^{19} - 2$ | $-\,2^{18}\cdot 19^{19}$ | $F_{19}$ (as 19T6) | trivial |
19.1.539...088.1 | $x^{19} + 2 x - 2$ | $-\,2^{18}\cdot 7\cdot 47\cdot 1031\cdot 108139\cdot 56081694178207$ | $S_{19}$ (as 19T8) | trivial |
19.1.601...424.1 | $x^{19} + 8 x - 8$ | $-\,2^{18}\cdot 46553428447\cdot 49259334849493$ | $S_{19}$ (as 19T8) | trivial |
19.9.159...156.1 | $x^{19} - 3 x^{18} - 8 x^{17} + 30 x^{16} + 16 x^{15} - 115 x^{14} + 31 x^{13} + 208 x^{12} - 175 x^{11} - 165 x^{10} + 280 x^{9} + 5 x^{8} - 206 x^{7} + 82 x^{6} + 75 x^{5} - 54 x^{4} - 13 x^{3} + 13 x^{2} + x - 1$ | $-\,2^{2}\cdot 937\cdot 971\cdot 4676220259\cdot 93910895860973$ | $S_{19}$ (as 19T8) | trivial |
19.1.207...120.1 | $x^{19} - 4 x - 8$ | $-\,2^{20}\cdot 5\cdot 928238261\cdot 426273972622159$ | $S_{19}$ (as 19T8) | trivial |
19.1.386...375.1 | $x^{19} + 19 x^{17} + 152 x^{15} + 665 x^{13} + 1729 x^{11} + 2717 x^{9} + 2508 x^{7} + 1254 x^{5} + 285 x^{3} + 19 x - 1$ | $-\,5^{9}\cdot 19^{19}$ | $F_{19}$ (as 19T6) | trivial |
19.7.832...904.1 | $x^{19} - 3 x^{18} - x^{17} + 15 x^{16} - 22 x^{15} - 6 x^{14} + 59 x^{13} - 70 x^{12} - 6 x^{11} + 106 x^{10} - 107 x^{9} - x^{8} + 92 x^{7} - 72 x^{6} - x^{5} + 35 x^{4} - 19 x^{3} - x^{2} + 4 x - 1$ | $2^{10}\cdot 2746819\cdot 2958679616604904629509$ | $S_{19}$ (as 19T8) | trivial |
19.5.896...328.1 | $x^{19} - 2 x^{18} - 3 x^{17} + 9 x^{16} + x^{15} - 16 x^{14} + 5 x^{13} + 17 x^{12} - 10 x^{11} - 12 x^{10} + 10 x^{9} + x^{8} + 3 x^{6} - 9 x^{5} + 2 x^{4} + 6 x^{3} - 5 x^{2} - x + 1$ | $-\,2^{4}\cdot 3947\cdot 1431584761\cdot 99216821342546749$ | $S_{19}$ (as 19T8) | trivial |
19.5.239...200.1 | $x^{19} - x^{18} - 6 x^{17} + 7 x^{16} + 13 x^{15} - 21 x^{14} - 16 x^{13} + 41 x^{12} + 10 x^{11} - 51 x^{10} + 5 x^{9} + 43 x^{8} - 19 x^{7} - 24 x^{6} + 18 x^{5} + 5 x^{4} - 11 x^{3} + 2 x^{2} + 3 x - 1$ | $-\,2^{3}\cdot 5^{2}\cdot 430709\cdot 19005923\cdot 14610723851226353$ | $S_{19}$ (as 19T8) | trivial |
19.1.331...008.1 | $x^{19} + 4 x - 8$ | $-\,2^{24}\cdot 311\cdot 13913\cdot 49356007\cdot 9263974663$ | $S_{19}$ (as 19T8) | trivial |
19.3.457...029.1 | $x^{19} - 3 x - 1$ | $15451\cdot 1192721\cdot 1892107519\cdot 1311498030721$ | $S_{19}$ (as 19T8) | trivial |
19.1.457...987.1 | $x^{19} + 3 x - 1$ | $-\,577\cdot 70079\cdot 1130954781811714100492789$ | $S_{19}$ (as 19T8) | trivial |
19.1.462...984.1 | $x^{19} + 3 x - 2$ | $-\,2^{19}\cdot 613\cdot 7331\cdot 3070062337\cdot 6393898213$ | $S_{19}$ (as 19T8) | trivial |
19.1.629...707.1 | $x^{19} - 9 x - 9$ | $-\,3^{18}\cdot 125471\cdot 142573\cdot 90800058249161$ | $S_{19}$ (as 19T8) | trivial |
19.1.720...723.1 | $x^{19} - 3 x - 3$ | $-\,3^{18}\cdot 30011\cdot 461269\cdot 134390022739373$ | $S_{19}$ (as 19T8) | trivial |
19.1.766...619.1 | $x^{19} - 2 x - 3$ | $-\,3^{18}\cdot 109\cdot 2111\cdot 2503\cdot 3435034858449743$ | $S_{19}$ (as 19T8) | trivial |
19.1.766...307.1 | $x^{19} - x - 3$ | $-\,3^{18}\cdot 733\cdot 5827\cdot 8053\cdot 2874523\cdot 20009947$ | $S_{19}$ (as 19T8) | trivial |
19.1.766...731.1 | $x^{19} - 3$ | $-\,3^{18}\cdot 19^{19}$ | $F_{19}$ (as 19T6) | trivial |
19.1.766...155.1 | $x^{19} + x - 3$ | $-\,3^{18}\cdot 5\cdot 11\cdot 277\cdot 1202769413\cdot 107967633989$ | $S_{19}$ (as 19T8) | trivial |
19.13.801...664.1 | $x^{19} - 2 x^{18} - 18 x^{17} + 32 x^{16} + 132 x^{15} - 202 x^{14} - 502 x^{13} + 645 x^{12} + 1045 x^{11} - 1122 x^{10} - 1176 x^{9} + 1078 x^{8} + 700 x^{7} - 557 x^{6} - 220 x^{5} + 151 x^{4} + 34 x^{3} - 20 x^{2} - 2 x + 1$ | $-\,2^{3}\cdot 13^{2}\cdot 17\cdot 3229\cdot 34365587\cdot 85132367\cdot 3692009831$ | $S_{19}$ (as 19T8) | trivial |
19.1.812...739.1 | $x^{19} + 3 x - 3$ | $-\,3^{18}\cdot 581393\cdot 3605923841336588507$ | $S_{19}$ (as 19T8) | trivial |
19.1.903...755.1 | $x^{19} + 9 x - 9$ | $-\,3^{18}\cdot 5\cdot 7\cdot 11\cdot 17\cdot 293990087\cdot 1212233453773$ | $S_{19}$ (as 19T8) | trivial |
19.9.138...476.1 | $x^{19} - 2 x^{18} - 14 x^{17} + 23 x^{16} + 83 x^{15} - 104 x^{14} - 274 x^{13} + 234 x^{12} + 555 x^{11} - 274 x^{10} - 708 x^{9} + 162 x^{8} + 554 x^{7} - 50 x^{6} - 246 x^{5} + 17 x^{4} + 56 x^{3} - 7 x^{2} - 5 x + 1$ | $-\,2^{2}\cdot 34\!\cdots\!19$ | $S_{19}$ (as 19T8) | trivial |
19.1.171...279.1 | $x^{19} + 7 x - 4$ | $-\,17\!\cdots\!79$ | $S_{19}$ (as 19T8) | trivial |
19.13.610...484.1 | $x^{19} - 3 x^{18} - 12 x^{17} + 40 x^{16} + 52 x^{15} - 216 x^{14} - 88 x^{13} + 614 x^{12} + 8 x^{11} - 987 x^{10} + 143 x^{9} + 904 x^{8} - 163 x^{7} - 461 x^{6} + 72 x^{5} + 128 x^{4} - 14 x^{3} - 18 x^{2} + x + 1$ | $-\,2^{2}\cdot 90313\cdot 16881409121807\cdot 1000636759045331$ | $S_{19}$ (as 19T8) | trivial |