LMFDB - 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.

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Degree	Signature	Class number	Class group structure	CM field

Ramified prime count	Ramified	Unramified primes	Galois group	Galois

Discriminant	Root discriminant	Galois root discriminant	Regulator	Intermediate field

$p$-adic completions	Minimal sibling	Monogenic	Index	Ines. $p$

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Results (1-50 of 509 matches)

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Label	Polynomial	Degree	Signature	Discriminant	Ram. prime count	Root discriminant	Galois root discriminant	Monogenic	Galois group	Class group	Unit group torsion	Unit group rank	Regulator
9.3.393718863.1	$x^{9} - 2 x^{8} + x^{6} + x^{5} - 3 x^{3} + x^{2} - x + 1$	$9$	[3,3]	$-\,3\cdot 1009\cdot 130069$	$3$	$9.01614092744$	$19842.350238819996$	?	$S_9$ (as 9T34)	trivial	$2$	$5$	$5.3041678109$
9.1.22588608000000.1	$x^{9} - 2 x^{8} + 3 x^{7} - 9 x^{6} - 10 x^{5} - 69 x^{4} + 168 x^{3} + 8 x^{2} - 64 x - 80$	$9$	[1,4]	$2^{12}\cdot 3\cdot 5^{6}\cdot 7^{6}$	$4$	$30.4624967972$	$290.20983261187956$	?	$S_9$ (as 9T34)	trivial	$2$	$4$	$12786.0880539$
9.1.421836492000000.1	$x^{9} - 4 x^{8} + 8 x^{7} - 18 x^{6} + 29 x^{5} - 34 x^{4} + 41 x^{3} - 22 x^{2} + 12 x - 6$	$9$	[1,4]	$2^{8}\cdot 3\cdot 5^{6}\cdot 7^{4}\cdot 11^{4}$	$5$	$42.171205542$	$368.8520196121334$	?	$S_9$ (as 9T34)	trivial	$2$	$4$	$30384.6509372$
9.3.218843499093450000.1	$x^{9} - 3 x^{8} + 16 x^{7} - 92 x^{6} - 146 x^{5} - 806 x^{4} - 4045 x^{3} - 4865 x^{2} - 14940 x - 14090$	$9$	[3,3]	$-\,2^{4}\cdot 3\cdot 5^{5}\cdot 7^{7}\cdot 11^{6}$	$5$	$84.4659199873$	$754.2705080154259$		$S_9$ (as 9T34)	$[3]$	$2$	$5$	$1212571.28672$
10.0.639041643.1	$x^{10} - 2 x^{9} + 5 x^{8} - 7 x^{7} + 11 x^{6} - 12 x^{5} + 12 x^{4} - 10 x^{3} + 7 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,3^{5}\cdot 541\cdot 4861$	$3$	$7.59543962557$	$12153.006870093279$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.4081174667226268$
10.0.822235563.1	$x^{10} - x^{9} + x^{8} - x^{7} - 2 x^{6} + 3 x^{5} - x^{4} + 2 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 199^{2}\cdot 769$	$3$	$7.78932143185$	$3405.440928711274$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.6494326103768342$
10.0.952269939.1	$x^{10} - x^{9} + x^{8} - 3 x^{7} + 2 x^{6} + x^{5} + x^{3} - 2 x + 1$	$10$	[0,5]	$-\,3^{4}\cdot 1559\cdot 7541$	$3$	$7.90452918852$	$14835.397966023034$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.8057735946924467$
10.0.1016022339.1	$x^{10} - x^{8} + 2 x^{6} - 2 x^{5} + 3 x^{3} - x^{2} - 2 x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 83\cdot 453379$	$3$	$7.95591850166$	$22100.96364298382$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.8833239101160306$
10.0.1063643859.1	$x^{10} - x^{9} + 3 x^{8} - 4 x^{7} + 4 x^{6} - 4 x^{5} + 3 x^{4} - x^{3} + x^{2} + 1$	$10$	[0,5]	$-\,3^{3}\cdot 39394217$	$2$	$7.99244439903$	$22612.97487876421$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.9451707468091348$
10.0.1094195979.1	$x^{10} - x^{9} + 3 x^{8} - 4 x^{7} + 5 x^{6} - 8 x^{5} + 9 x^{4} - 8 x^{3} + 7 x^{2} - 4 x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 61\cdot 664357$	$3$	$8.01511045078$	$22935.443317747635$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.985347814547003$
10.0.1144574523.1	$x^{10} - 2 x^{9} + x^{8} + 2 x^{7} - 3 x^{6} + 3 x^{4} - x^{2} - x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 29\cdot 1461781$	$3$	$8.05127030566$	$23457.494282080777$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.0544014492311957$
10.0.1190950587.1	$x^{10} - 2 x^{7} - x^{6} - x^{5} + 2 x^{4} + 2 x^{2} - x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 44109281$	$2$	$8.08331260013$	$23928.002927937145$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.090863273528543$
10.0.1208712699.1	$x^{10} - 2 x^{9} + 4 x^{8} - 7 x^{7} + 7 x^{6} - 9 x^{5} + 8 x^{4} - 5 x^{3} + 5 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,3^{4}\cdot 17\cdot 157\cdot 5591$	$4$	$8.09528808497$	$16714.011859339607$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.1152906020636064$
10.0.1218896127.1	$x^{10} - x^{9} + 3 x^{8} - 5 x^{7} + 7 x^{6} - 8 x^{5} + 9 x^{4} - 7 x^{3} + 5 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 45144301$	$2$	$8.10208266056$	$24207.10925475623$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.1789876756075377$
10.0.1291262607.1	$x^{10} - x^{9} + 2 x^{8} - 4 x^{7} + 3 x^{6} - 3 x^{5} + 2 x^{4} + x^{2} + x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 47824541$	$2$	$8.14894632954$	$24915.342930945008$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.185676213395155$
10.0.1332547227.1	$x^{10} - x^{8} - x^{7} - x^{5} + x^{4} + x^{3} + 1$	$10$	[0,5]	$-\,3^{3}\cdot 11^{2}\cdot 17\cdot 23993$	$4$	$8.17463294089$	$7631.405735620997$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.230642388730771$
10.0.1348219863.1	$x^{10} - 2 x^{9} + 3 x^{8} - 5 x^{7} + 5 x^{6} - 5 x^{5} + 4 x^{4} - x^{3} + 2 x^{2} + 1$	$10$	[0,5]	$-\,3^{3}\cdot 47\cdot 1062427$	$3$	$8.18419695259$	$25458.918005888514$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.317410312666733$
10.0.1380733155.1	$x^{10} - 3 x^{9} + 3 x^{8} + x^{7} - 2 x^{6} - 2 x^{5} + 4 x^{4} - x^{3} + x^{2} - 2 x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 5\cdot 661\cdot 15473$	$4$	$8.20372276255$	$25764.069292381733$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.458812203729755$
10.0.1422093888.1	$x^{10} - 2 x^{9} + 3 x^{7} - 4 x^{5} + x^{4} + 3 x^{3} - x^{2} - x + 1$	$10$	[0,5]	$-\,2^{6}\cdot 3^{3}\cdot 822971$	$3$	$8.22797241449$	$10993.506109919888$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.3835289544655973$
10.0.1422782631.1	$x^{10} - 2 x^{9} + 2 x^{8} - 2 x^{7} + 4 x^{6} - 4 x^{5} + 3 x^{4} + 2 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 52695653$	$2$	$8.22837082164$	$26153.4423368486$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.2959770718899377$
10.0.1469725479.1	$x^{10} - x^{9} + 3 x^{8} - 3 x^{7} + 5 x^{6} - 4 x^{5} + 3 x^{4} - 2 x^{3} + x^{2} - x + 1$	$10$	[0,5]	$-\,3^{5}\cdot 173\cdot 34961$	$3$	$8.2551244018$	$18430.506694653814$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.4192182088976586$
10.0.1472989239.1	$x^{10} - x^{9} + 3 x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 4 x^{4} - 5 x^{3} + 3 x^{2} - x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 113\cdot 482789$	$3$	$8.25695575461$	$26610.888103776786$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.3903947096230147$
10.0.1520103699.1	$x^{10} + x^{8} - x^{7} - x^{6} - x^{5} - x^{4} + x^{2} + x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 56300137$	$2$	$8.28299351967$	$27033.121084486047$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.443933079313193$
10.0.1534442859.1	$x^{10} - 3 x^{9} + 6 x^{8} - 7 x^{7} + 7 x^{6} - 6 x^{5} + 7 x^{4} - 5 x^{3} + 3 x^{2} - x + 1$	$10$	[0,5]	$-\,3^{4}\cdot 89\cdot 212851$	$3$	$8.29077390955$	$18831.916667672387$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.430978807752423$
10.0.1656859023.1	$x^{10} - 2 x^{9} + 5 x^{8} - 7 x^{7} + 9 x^{6} - 9 x^{5} + 8 x^{4} - 5 x^{3} + 4 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 53\cdot 1157833$	$3$	$8.35465567385$	$28222.947011145043$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.6287821024796365$
10.0.1667981403.1	$x^{10} - x^{9} - x^{8} + 3 x^{7} - 6 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} + x + 1$	$10$	[0,5]	$-\,3^{5}\cdot 11\cdot 173\cdot 3607$	$4$	$8.36024722606$	$19634.27030455711$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.598944921581374$
10.0.1688791059.1	$x^{10} - 2 x^{9} + 4 x^{8} - 7 x^{7} + 8 x^{6} - 9 x^{5} + 10 x^{4} - 7 x^{3} + 5 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 62547817$	$2$	$8.37061933192$	$28493.614367560458$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.5764351180740026$
10.0.1696079547.1	$x^{10} - 3 x^{9} + 5 x^{8} - 7 x^{7} + 9 x^{6} - 8 x^{5} + 6 x^{4} - 4 x^{3} + 3 x^{2} + 1$	$10$	[0,5]	$-\,3^{3}\cdot 23\cdot 2731207$	$3$	$8.37422492863$	$28555.03456991974$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.700681490216298$
10.0.1707327099.1	$x^{10} + 2 x^{8} - x^{7} + x^{6} - 3 x^{5} - x^{4} - x^{3} + x^{2} + x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 19\cdot 23\cdot 144701$	$4$	$8.37976179318$	$28649.559478329065$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.590168926544706$
10.0.1718974683.1	$x^{10} - x^{9} + x^{6} + x^{4} - x^{3} + x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 563\cdot 113083$	$3$	$8.38546108935$	$28747.118684174613$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.8290572317127083$
10.0.1719985347.1	$x^{10} - 3 x^{9} + 4 x^{8} - 4 x^{7} + 3 x^{6} - x^{5} + x^{4} - x^{3} + 3 x^{2} - x + 1$	$10$	[0,5]	$-\,3^{5}\cdot 7078129$	$2$	$8.38595397865$	$19937.997492730126$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.700512954089847$
10.0.1797311727.1	$x^{10} - x^{9} + 4 x^{8} - 2 x^{7} + 5 x^{6} - x^{5} + 3 x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 66567101$	$2$	$8.42291350226$	$29394.85253109358$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.8850181474188794$
10.0.1809722007.1	$x^{10} - x^{9} + x^{8} + 2 x^{7} - x^{6} + x^{5} + 2 x^{4} + x^{3} + x^{2} + x + 1$	$10$	[0,5]	$-\,3^{4}\cdot 113\cdot 163\cdot 1213$	$4$	$8.4287114575$	$20451.496742242838$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.7096659656713964$
10.0.1844569179.1	$x^{10} - x^{9} + 2 x^{7} + x^{6} - 4 x^{5} + 3 x^{4} + 2 x^{3} - 2 x^{2} + 1$	$10$	[0,5]	$-\,3^{4}\cdot 61\cdot 577\cdot 647$	$4$	$8.44480245455$	$20647.460187468932$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.956874086295188$
10.0.1873759299.1	$x^{10} - x^{9} + 3 x^{8} - 4 x^{7} + 8 x^{6} - 8 x^{5} + 10 x^{4} - 5 x^{3} + 5 x^{2} - x + 1$	$10$	[0,5]	$-\,3\cdot 624586433$	$2$	$8.45807204525$	$43286.941437343434$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.396560007426601$
10.0.1984965939.1	$x^{10} - 2 x^{9} + 4 x^{8} - 4 x^{7} + 6 x^{6} - 6 x^{5} + 6 x^{4} - 5 x^{3} + 5 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 11\cdot 6683387$	$3$	$8.5069779259$	$30891.294742465292$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.106921244894728$
10.0.1998632043.1	$x^{10} - 2 x^{9} + 4 x^{8} - 6 x^{7} + 8 x^{6} - 7 x^{5} + 6 x^{4} - 4 x^{3} + 3 x^{2} - x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 41\cdot 67\cdot 26947$	$4$	$8.5128167477$	$30997.45261156884$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.8757150585018842$
10.0.2024659968.1	$x^{10} - 4 x^{5} + x^{4} - 2 x^{3} + 2 x^{2} + 2 x + 1$	$10$	[0,5]	$-\,2^{10}\cdot 3\cdot 659069$	$3$	$8.52383844105$	$5157.714654401969$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.5606287790799636$
10.0.2084751027.1	$x^{10} - 2 x^{9} + 3 x^{8} - 3 x^{7} + 2 x^{5} + x^{4} - 3 x^{3} + 2 x^{2} - x + 1$	$10$	[0,5]	$-\,3^{4}\cdot 23\cdot 1119029$	$3$	$8.54880517317$	$21950.593224971653$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.1711520478642536$
10.0.2143445139.1	$x^{10} - 2 x^{8} - 2 x^{7} + x^{6} + 4 x^{4} - 2 x^{3} + 2 x^{2} + 1$	$10$	[0,5]	$-\,3^{3}\cdot 11\cdot 7216987$	$3$	$8.57257389184$	$32100.793503693192$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.0536921015635383$
10.0.2145719859.1	$x^{10} - 2 x^{8} - x^{7} + 4 x^{6} + x^{5} - 2 x^{4} - x^{3} + 3 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,3\cdot 715239953$	$2$	$8.57348321761$	$46321.91553681691$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.704050610128935$
10.0.2164482891.1	$x^{10} - 2 x^{9} + 3 x^{8} - 6 x^{7} + 8 x^{6} - 8 x^{5} + 9 x^{4} - 7 x^{3} + 4 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,3^{5}\cdot 17\cdot 103\cdot 5087$	$4$	$8.5809508758$	$22366.407963833517$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.1136863503872934$
10.0.2247263163.1	$x^{10} - 3 x^{9} + 5 x^{8} - 7 x^{7} + 8 x^{6} - 6 x^{5} + 4 x^{4} - 3 x^{3} + x^{2} + 1$	$10$	[0,5]	$-\,3^{3}\cdot 83231969$	$2$	$8.61321708712$	$32869.00430093382$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.097033680150176$
10.0.2257944363.1	$x^{10} - 2 x^{9} + 5 x^{8} - 7 x^{7} + 10 x^{6} - 11 x^{5} + 12 x^{4} - 9 x^{3} + 6 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 19\cdot 4401451$	$3$	$8.61730220336$	$32947.02458652265$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.040506089287545$
10.0.2338708167.1	$x^{10} + 2 x^{8} - 2 x^{7} + 4 x^{6} - 3 x^{5} + 4 x^{4} - 5 x^{3} + 4 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 809\cdot 107069$	$3$	$8.64764003027$	$33531.084329470454$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.493000157703164$
10.0.2363770839.1	$x^{10} - 3 x^{7} - x^{6} + 4 x^{4} - x + 1$	$10$	[0,5]	$-\,3\cdot 109\cdot 7228657$	$3$	$8.65686284965$	$48618.62646147051$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.9268946631680217$
10.0.2423566023.1	$x^{10} - x^{7} + 3 x^{6} - x^{5} + x^{4} + 3 x^{2} + 2 x + 1$	$10$	[0,5]	$-\,3\cdot 807855341$	$2$	$8.67851634069$	$49229.72702544673$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.9897880435312456$
10.0.2432941299.1	$x^{10} - x^{9} + 2 x^{8} - 2 x^{7} + 4 x^{6} - 3 x^{5} + x^{4} + 2 x^{3} - 2 x^{2} + 1$	$10$	[0,5]	$-\,3^{3}\cdot 73\cdot 709\cdot 1741$	$4$	$8.68186769139$	$34199.94416694223$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.3803063152022035$
10.0.2444516523.1	$x^{10} - x^{9} - x^{8} + x^{7} - 3 x^{6} + 3 x^{5} + 2 x^{4} - x^{3} + 4 x^{2} - x + 1$	$10$	[0,5]	$-\,3^{3}\cdot 5101\cdot 17749$	$3$	$8.68598945354$	$34281.20430063735$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.5256388256510767$
10.0.2466143739.1	$x^{10} - x^{9} + 2 x^{8} - 3 x^{7} + 2 x^{6} - 4 x^{5} + 2 x^{4} - 2 x^{3} + 3 x^{2} + 1$	$10$	[0,5]	$-\,3^{4}\cdot 30446219$	$2$	$8.69364372948$	$23874.17409115883$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.37471464471985$

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.