Number field search results

Note: Search results may be incomplete. Given $p$-adic completions contain an unramified field and completions are only searched for ramified primes.

Results (1-50 of 6244 matches)

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Label	Polynomial	Degree	Signature	Discriminant	Ram. prime count	Root discriminant	Galois root discriminant	CM field	Galois	Monogenic	Galois group	Class group	Unit group torsion	Unit group rank	Regulator
10.0.656097439.1	$x^{10} - 2 x^{9} + 4 x^{8} - 4 x^{7} + 5 x^{6} - 5 x^{5} + 4 x^{4} - 5 x^{3} + 4 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,17\cdot 71\cdot 617\cdot 881$	$4$	$7.61547215032$	$25614.399055999733$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.3089847497496137$
10.0.751885979.1	$x^{10} - x^{9} + 3 x^{8} - 2 x^{7} + 3 x^{6} - 2 x^{5} + 3 x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,13\cdot 17\cdot 883\cdot 3853$	$4$	$7.71996264022$	$27420.539363768905$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.368120673914704$
10.0.863184299.1	$x^{10} - 2 x^{9} + 2 x^{8} - 3 x^{6} + 5 x^{5} - 3 x^{4} + 3 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,17\cdot 421\cdot 120607$	$3$	$7.82727068912$	$29379.998281143584$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.4432818169992423$
10.2.912496573.1	$x^{10} - x^{9} - 2 x^{8} + 2 x^{7} + 2 x^{5} + x^{4} - 4 x^{3} + x^{2} + 2 x - 1$	$10$	[2,4]	$17\cdot 1721\cdot 31189$	$3$	$7.87087697424$	$30207.558209825565$			?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$1.7899907173124556$
10.2.1009570789.1	$x^{10} - x^{8} - 2 x^{7} - x^{6} + 2 x^{5} + 4 x^{4} + 2 x^{3} - 2 x^{2} - 3 x - 1$	$10$	[2,4]	$17\cdot 37\cdot 1605041$	$3$	$7.95085215006$	$31773.743704511748$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$1.9054776059277028$
10.0.1155332716.1	$x^{10} - 2 x^{9} + 4 x^{7} - 3 x^{6} - 2 x^{5} + 4 x^{4} - x^{3} - x^{2} + x + 1$	$10$	[0,5]	$-\,2^{2}\cdot 17\cdot 127\cdot 133781$	$4$	$8.05880611231$	$26978.02817591211$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.603632136790284$
10.2.1187270633.1	$x^{10} - 2 x^{9} + x^{8} + x^{6} - 5 x^{5} + 7 x^{4} - 3 x^{3} + 2 x - 1$	$10$	[2,4]	$17\cdot 61^{2}\cdot 137^{2}$	$3$	$8.08081143331$	$376.9204160031664$			?	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$5$	$2.11334506145$
10.2.1237253489.1	$x^{10} - x^{9} + 3 x^{8} - x^{7} + x^{6} + 2 x^{5} - 2 x^{4} + x^{3} - x^{2} - x - 1$	$10$	[2,4]	$17\cdot 72779617$	$2$	$8.11420300177$	$35174.61426938468$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.167549619823552$
10.0.1288089167.1	$x^{10} - 2 x^{9} + 3 x^{8} - 5 x^{7} + 7 x^{6} - 6 x^{5} + 7 x^{4} - 5 x^{3} + 5 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,17\cdot 75769951$	$2$	$8.14694140534$	$35889.95913901268$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.0203205913427573$
10.0.1324470799.1	$x^{10} - x^{7} - 3 x^{6} + x^{5} + 2 x^{2} + 1$	$10$	[0,5]	$-\,17\cdot 193\cdot 403679$	$3$	$8.16966481702$	$36393.279585659766$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.9915952537815476$
10.0.1392508879.1	$x^{10} - 2 x^{9} + 3 x^{8} - 5 x^{7} + 7 x^{6} - 7 x^{5} + 6 x^{4} - 5 x^{3} + 4 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,17\cdot 19\cdot 4311173$	$3$	$8.21069268344$	$37316.335283626126$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.0300146931169873$
10.2.1395650309.1	$x^{10} - x^{9} + x^{8} - 4 x^{7} + 2 x^{6} - 2 x^{5} + 3 x^{4} - 1$	$10$	[2,4]	$17\cdot 103\cdot 347\cdot 2297$	$4$	$8.2125430967$	$37358.403458927416$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.343092286753046$
10.0.1447655927.1	$x^{10} - x^{9} + x^{8} - x^{7} - x^{5} + x^{4} + x^{2} + x + 1$	$10$	[0,5]	$-\,17\cdot 101\cdot 843131$	$3$	$8.2426438523$	$38048.07389343119$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.422821751636724$
10.0.1452448023.1	$x^{10} - 2 x^{9} + 5 x^{8} - 6 x^{7} + 7 x^{6} - 6 x^{5} + 6 x^{4} - 5 x^{3} + 5 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,3\cdot 13^{2}\cdot 17\cdot 43\cdot 3919$	$5$	$8.24536831355$	$10570.088504832871$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.404056608025986$
10.0.1454840399.1	$x^{10} - 3 x^{9} + 6 x^{8} - 9 x^{7} + 11 x^{6} - 12 x^{5} + 13 x^{4} - 11 x^{3} + 8 x^{2} - 4 x + 1$	$10$	[0,5]	$-\,17\cdot 6367\cdot 13441$	$3$	$8.2467254303$	$38142.37012824452$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.9241772853029844$
10.2.1521575633.1	$x^{10} - x^{9} - 3 x^{8} + 3 x^{7} + 3 x^{6} - 2 x^{5} - x^{3} - 2 x^{2} + 1$	$10$	[2,4]	$17\cdot 149\cdot 600701$	$3$	$8.28379522224$	$39007.37921214395$			?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.4885517660538223$
10.0.1538775332.1	$x^{10} - 3 x^{9} + 3 x^{8} - 2 x^{7} + 5 x^{6} - 6 x^{5} + 5 x^{4} - 2 x^{3} + 3 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,2^{2}\cdot 17\cdot 67^{2}\cdot 71^{2}$	$4$	$8.29311182634$	$568.7495054942905$			✓	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$4$	$3.11760635261$
10.0.1542579575.1	$x^{10} - x^{9} - x^{8} - 2 x^{7} + 4 x^{6} + x^{5} - 4 x^{3} + x^{2} + x + 1$	$10$	[0,5]	$-\,5^{2}\cdot 17\cdot 53\cdot 68483$	$4$	$8.29515981659$	$22968.560987074317$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.181266338143171$
10.2.1556007637.1	$x^{10} - 3 x^{9} + 5 x^{8} - 6 x^{7} + 7 x^{6} - 7 x^{5} + 5 x^{4} - 3 x^{3} + x^{2} - 1$	$10$	[2,4]	$17\cdot 167\cdot 548083$	$3$	$8.30235257224$	$39446.26264933092$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.5434775259208036$
10.0.1588132027.1	$x^{10} - 2 x^{8} - x^{7} + 2 x^{6} - x^{5} - x^{3} + x^{2} + 2 x + 1$	$10$	[0,5]	$-\,17\cdot 93419531$	$2$	$8.31933591446$	$39851.37421720862$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.1936501110934286$
10.2.1628406149.1	$x^{10} - 3 x^{9} + 4 x^{8} - 5 x^{7} + 8 x^{6} - 7 x^{5} + x^{4} + 5 x^{3} - 7 x^{2} + 3 x - 1$	$10$	[2,4]	$17\cdot 37\cdot 1609^{2}$	$3$	$8.34019631505$	$1006.0124253705816$			?	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$5$	$2.59250226506$
10.2.1637352433.1	$x^{10} - x^{9} - 3 x^{8} + 2 x^{7} + 5 x^{6} - 2 x^{5} - 5 x^{4} + x^{3} + 3 x^{2} - x - 1$	$10$	[2,4]	$17\cdot 239\cdot 402991$	$3$	$8.34476703841$	$40464.21175557483$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.63894471863352$
10.0.1660970227.1	$x^{10} - 3 x^{9} + 6 x^{8} - 10 x^{7} + 14 x^{6} - 14 x^{5} + 11 x^{4} - 8 x^{3} + 6 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,7\cdot 17\cdot 13957733$	$3$	$8.35672642344$	$40755.002478223454$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.118125156610222$
10.2.1703798497.1	$x^{10} - 3 x^{8} - 3 x^{7} + 4 x^{6} + 6 x^{5} - 5 x^{3} - 2 x^{2} + 2 x + 1$	$10$	[2,4]	$17\cdot 881\cdot 113761$	$3$	$8.37802830108$	$41277.094095878405$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.647106937400503$
10.0.1754069639.1	$x^{10} - 2 x^{7} + 4 x^{4} + 4 x^{3} + 4 x^{2} + x + 1$	$10$	[0,5]	$-\,7\cdot 17\cdot 1877\cdot 7853$	$4$	$8.40242574483$	$41881.61457011895$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.2396111233732574$
10.0.1917649147.1	$x^{10} - x^{9} + 2 x^{6} - 3 x^{5} + 2 x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,17\cdot 8011\cdot 14081$	$3$	$8.47767796156$	$43790.97106710469$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.206101779294455$
10.0.1932906307.1	$x^{10} - 2 x^{9} + 2 x^{8} - x^{7} - x^{6} + 4 x^{5} - 3 x^{4} + x^{2} - x + 1$	$10$	[0,5]	$-\,17\cdot 37\cdot 3072983$	$3$	$8.48439892597$	$43964.830341990404$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.5688942204285796$
10.0.1949868703.1	$x^{10} - 2 x^{9} + x^{8} - x^{5} + 2 x^{4} - 3 x^{3} + 4 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,17\cdot 383\cdot 299473$	$3$	$8.49181524755$	$44157.31766083624$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.5198585493419774$
10.0.2005455439.1	$x^{10} - 2 x^{8} + 3 x^{6} - 3 x^{5} + 5 x^{3} - 2 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,13\cdot 17\cdot 9074459$	$3$	$8.51571859617$	$44782.31167548187$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.6124315429401905$
10.2.2027774909.1	$x^{10} - x^{9} - 2 x^{8} + 2 x^{7} + x^{6} - x^{3} - x^{2} + x + 1$	$10$	[2,4]	$17\cdot 47\cdot 83\cdot 30577$	$4$	$8.52514892726$	$45030.821766874295$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.173239685289735$
10.2.2070159173.1	$x^{10} - 2 x^{9} + 4 x^{7} - 2 x^{6} - 4 x^{5} + 5 x^{4} + x^{3} - 5 x^{2} + 4 x - 1$	$10$	[2,4]	$17\cdot 121774069$	$2$	$8.54280265449$	$45499.00189015139$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.2188522474818755$
10.0.2151551195.1	$x^{10} - 2 x^{8} - 2 x^{7} + 3 x^{6} + x^{5} + 2 x^{4} - 2 x^{3} - 2 x^{2} + x + 1$	$10$	[0,5]	$-\,5\cdot 17\cdot 47\cdot 538561$	$4$	$8.57581035389$	$46384.81642736123$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.550706982673931$
10.0.2182025123.1	$x^{10} - x^{8} - 2 x^{7} + 3 x^{6} + 2 x^{4} - 4 x^{3} + 2 x^{2} - x + 1$	$10$	[0,5]	$-\,17\cdot 128354419$	$2$	$8.58788014488$	$46712.15177017646$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.2827926545609993$
10.0.2247792088.1	$x^{10} - 2 x^{9} + 4 x^{8} - 4 x^{7} + 7 x^{6} - 7 x^{5} + 9 x^{4} - 9 x^{3} + 8 x^{2} - 4 x + 1$	$10$	[0,5]	$-\,2^{3}\cdot 17\cdot 29\cdot 569927$	$4$	$8.61341978984$	$47410.885754223156$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$4.059000349537326$
10.0.2249894699.1	$x^{10} - 3 x^{9} + 4 x^{8} - 2 x^{7} + x^{6} - 2 x^{5} + 3 x^{4} - 3 x^{3} + 4 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,13\cdot 17\cdot 389\cdot 26171$	$4$	$8.61422516024$	$47433.05491953897$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.1535170598014104$
10.0.2254246087.1	$x^{10} - x^{8} - x^{7} + 3 x^{6} + 2 x^{5} - x^{4} - x^{3} + 2 x^{2} + 3 x + 1$	$10$	[0,5]	$-\,17\cdot 137\cdot 967903$	$3$	$8.61588973828$	$47478.90149318958$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.9428110707686046$
10.0.2308391155.1	$x^{10} - x^{9} - x^{8} + 2 x^{7} - x^{5} - 2 x^{4} + 3 x^{3} + 2 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,5\cdot 7\cdot 17\cdot 29\cdot 133781$	$5$	$8.6363640346$	$48045.719424315$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.775623020663207$
10.0.2315517079.1	$x^{10} - 2 x^{9} + 3 x^{8} - 4 x^{7} + 5 x^{6} - 4 x^{5} + 3 x^{4} - 3 x^{3} + 3 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,17\cdot 136206887$	$2$	$8.6390263542$	$48119.820022522945$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.6797776964195252$
10.0.2321676791.1	$x^{10} - x^{9} + x^{8} - 2 x^{7} + 3 x^{6} - 3 x^{5} + 4 x^{4} - 3 x^{3} + 3 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,7^{3}\cdot 17\cdot 631^{2}$	$3$	$8.64132175165$	$274.02372160088623$			✓	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$4$	$3.05592051071$
10.2.2325220781.1	$x^{10} - 2 x^{8} - 3 x^{7} + x^{6} + 3 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} - 3 x - 1$	$10$	[2,4]	$13\cdot 17\cdot 10521361$	$3$	$8.64263992572$	$48220.54314293857$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.1811489164205655$
10.2.2332581781.1	$x^{10} - 3 x^{9} + 7 x^{7} - x^{6} - 14 x^{5} + 9 x^{4} + 7 x^{3} - 5 x^{2} - 3 x + 1$	$10$	[2,4]	$17\cdot 53\cdot 1609^{2}$	$3$	$8.64537205443$	$1204.0386206430423$			?	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$5$	$3.25925591671$
10.2.2335711141.1	$x^{10} - 3 x^{9} + 8 x^{8} - 14 x^{7} + 22 x^{6} - 28 x^{5} + 29 x^{4} - 24 x^{3} + 15 x^{2} - 6 x + 1$	$10$	[2,4]	$17\cdot 137394773$	$2$	$8.64653120617$	$48329.19553437653$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.252095859772139$
10.0.2365879052.1	$x^{10} - 2 x^{9} + 4 x^{8} - 2 x^{7} + x^{6} + 4 x^{5} - 4 x^{4} + 3 x^{3} + x^{2} - 3 x + 1$	$10$	[0,5]	$-\,2^{2}\cdot 17\cdot 34792339$	$3$	$8.65763463302$	$38605.83387986929$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$4.0976093512062235$
10.0.2375413111.1	$x^{10} - x^{9} - 2 x^{7} + 5 x^{4} - 3 x^{3} + 2 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,17\cdot 139730183$	$2$	$8.66111719044$	$48738.20996918126$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.689482207139167$
10.0.2386374847.1	$x^{10} - 2 x^{9} + x^{8} + 2 x^{7} - 2 x^{6} + x^{5} - 2 x^{3} + 5 x^{2} - 4 x + 1$	$10$	[0,5]	$-\,17\cdot 337\cdot 416543$	$3$	$8.66510573036$	$48850.535790306334$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.0465797541998247$
10.2.2435340797.1	$x^{10} - 3 x^{9} + 3 x^{8} - 5 x^{6} + 7 x^{5} - 3 x^{4} - x^{3} + 2 x^{2} - x - 1$	$10$	[2,4]	$17\cdot 143255341$	$2$	$8.68272356425$	$49349.17220177052$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.554087965007053$
10.2.2484151469.1	$x^{10} - x^{7} + x^{6} - 4 x^{4} - x^{3} - 2 x - 1$	$10$	[2,4]	$17\cdot 8831\cdot 16547$	$3$	$8.69997104681$	$49841.26271474269$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.4840220870101133$
10.0.2512217432.1	$x^{10} - x^{9} + 4 x^{8} - 2 x^{7} + 6 x^{6} + x^{5} + 3 x^{4} + 3 x^{3} + 3 x^{2} + x + 1$	$10$	[0,5]	$-\,2^{3}\cdot 17\cdot 31\cdot 595877$	$4$	$8.70975066282$	$50122.02541797368$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$4.5573031975744795$
10.2.2518433941.1	$x^{10} - x^{9} - 2 x^{8} + 2 x^{7} + 4 x^{6} - 2 x^{5} - 5 x^{4} + 2 x^{3} + 3 x^{2} - 1$	$10$	[2,4]	$17\cdot 148143173$	$2$	$8.71190350381$	$50184.000846883464$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.6237237477548736$
10.0.2537790107.1	$x^{10} - 4 x^{9} + 8 x^{8} - 10 x^{7} + 9 x^{6} - 8 x^{5} + 8 x^{4} - 6 x^{3} + 3 x^{2} - x + 1$	$10$	[0,5]	$-\,17\cdot 31\cdot 151\cdot 31891$	$4$	$8.71857624789$	$50376.48367045878$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.946583261145957$

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