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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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
10.0.220168975.1	$x^{10} - x^{9} - x^{8} + 2 x^{7} - x^{6} - x^{5} + 4 x^{4} - x^{3} - 3 x^{2} + x + 1$	$10$	[0,5]	$-\,5^{2}\cdot 13^{2}\cdot 31\cdot 41^{2}$	$4$	$6.82771857501$	$287.42825191689144$			✓	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$4$	$0.586605631071$
10.0.416054627.1	$x^{10} - x^{9} + 3 x^{8} - 4 x^{7} + 4 x^{6} - 5 x^{5} + 3 x^{4} - 2 x^{3} + x^{2} + 1$	$10$	[0,5]	$-\,31\cdot 43\cdot 107\cdot 2917$	$4$	$7.27637437289$	$20397.417164925562$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$0.9052131093147093$
10.0.621349399.1	$x^{10} - x^{9} - x^{8} + 3 x^{6} - 2 x^{5} - 3 x^{4} + 4 x^{3} - x + 1$	$10$	[0,5]	$-\,11^{4}\cdot 31\cdot 37^{2}$	$3$	$7.57414453312$	$167.5110647172689$			✓	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$4$	$1.26310066768$
10.0.793643059.1	$x^{10} - x^{9} + x^{8} + 2 x^{6} - 2 x^{5} + 4 x^{4} - 3 x^{3} + 3 x^{2} - x + 1$	$10$	[0,5]	$-\,11\cdot 31\cdot 2327399$	$3$	$7.76180132702$	$28171.67121418252$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.4225159422481624$
10.0.825258688.1	$x^{10} - x^{9} - x^{8} + 3 x^{7} - 3 x^{5} + 2 x^{4} + 2 x^{3} - 2 x^{2} - x + 1$	$10$	[0,5]	$-\,2^{6}\cdot 31\cdot 415957$	$3$	$7.79218061517$	$10156.639995589092$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.3804645602988417$
10.0.839849179.1	$x^{10} - x^{9} + 2 x^{8} + 2 x^{6} + x^{5} + 2 x^{4} + 2 x^{3} + x^{2} + 2 x + 1$	$10$	[0,5]	$-\,13\cdot 31\cdot 167\cdot 12479$	$4$	$7.80584871907$	$28980.151466132815$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.4833457017737917$
10.0.941784619.1	$x^{10} - 3 x^{9} + 6 x^{8} - 7 x^{7} + 7 x^{6} - 7 x^{5} + 9 x^{4} - 9 x^{3} + 7 x^{2} - 4 x + 1$	$10$	[0,5]	$-\,31\cdot 30380149$	$2$	$7.89578218619$	$30688.509559768456$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.5942466964907376$
10.0.990648679.1	$x^{10} - 3 x^{9} + 6 x^{8} - 8 x^{7} + 11 x^{6} - 11 x^{5} + 10 x^{4} - 5 x^{3} + 3 x^{2} + 1$	$10$	[0,5]	$-\,31\cdot 5653^{2}$	$2$	$7.93582288475$	$418.6203530646832$			✓	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$4$	$1.75104776952$
10.0.1038167587.1	$x^{10} - 3 x^{9} + 5 x^{8} - 6 x^{7} + 8 x^{6} - 10 x^{5} + 10 x^{4} - 7 x^{3} + 5 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,31\cdot 101\cdot 331577$	$3$	$7.97309147099$	$32220.608110338326$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.5842124508228403$
10.2.1084561381.1	$x^{10} - x^{9} + x^{8} - 2 x^{7} + 3 x^{6} + x^{5} - 5 x^{4} + x^{3} + 2 x^{2} + x - 1$	$10$	[2,4]	$31\cdot 34985851$	$2$	$8.00802487684$	$32932.6795296101$			?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$1.9936401536045387$
10.2.1128353221.1	$x^{10} - 2 x^{9} + 3 x^{7} - 5 x^{6} + 4 x^{5} - 3 x^{3} + 4 x^{2} - 2 x + 1$	$10$	[2,4]	$31\cdot 37\cdot 811\cdot 1213$	$4$	$8.03978633846$	$33590.9693370108$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.048646469404427$
10.2.1221154573.1	$x^{10} - 2 x^{9} + 4 x^{8} - 4 x^{7} + 3 x^{6} - x^{5} - 2 x^{4} + 4 x^{3} - 4 x^{2} + 3 x - 1$	$10$	[2,4]	$31\cdot 39392083$	$2$	$8.10358261424$	$34945.02214908441$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.1812779024011295$
10.0.1285719575.1	$x^{10} - x^{9} - x^{8} + 3 x^{7} - 3 x^{5} + x^{4} + 3 x^{3} - x^{2} - 2 x + 1$	$10$	[0,5]	$-\,5^{2}\cdot 7^{2}\cdot 31\cdot 33857$	$4$	$8.14544143722$	$7925.635706673423$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.1524234976885492$
10.0.1292954944.1	$x^{10} - 2 x^{9} + 3 x^{8} - 3 x^{7} + 5 x^{6} - 7 x^{5} + 8 x^{4} - 6 x^{3} + 4 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,2^{6}\cdot 31\cdot 173\cdot 3767$	$4$	$8.15001370628$	$15118.343234884178$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.817008253132742$
10.2.1327825201.1	$x^{10} - x^{9} - x^{8} + x^{6} + 2 x^{5} - 3 x^{3} + x - 1$	$10$	[2,4]	$31\cdot 42833071$	$2$	$8.17173154116$	$36439.33590229108$			?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.315317590377933$
10.2.1331413141.1	$x^{10} - x^{9} - 2 x^{8} + 2 x^{7} + 2 x^{6} - 4 x^{5} - x^{4} + 4 x^{3} + x^{2} - 2 x - 1$	$10$	[2,4]	$31\cdot 42948811$	$2$	$8.17393695886$	$36488.53437725336$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.261711718036827$
10.2.1359444581.1	$x^{10} - x^{9} + x^{8} + x^{7} - x^{6} - x^{5} + 2 x^{4} - x^{3} - 2 x^{2} + 3 x - 1$	$10$	[2,4]	$11\cdot 31\cdot 3986641$	$3$	$8.1909853804$	$36870.64660403991$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.3149175474552326$
10.0.1400056379.1	$x^{10} - 3 x^{9} + 5 x^{8} - 3 x^{7} - 2 x^{6} + 6 x^{5} - 4 x^{4} - x^{3} + 5 x^{2} - 4 x + 1$	$10$	[0,5]	$-\,31\cdot 311\cdot 145219$	$3$	$8.21513212174$	$37417.32725623251$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$1.8457976841100077$
10.0.1441574803.1	$x^{10} - x^{9} + x^{8} - 2 x^{7} + 4 x^{6} - 3 x^{5} + x^{4} - 3 x^{3} + 6 x^{2} - 4 x + 1$	$10$	[0,5]	$-\,31\cdot 151\cdot 223\cdot 1381$	$4$	$8.23917482753$	$37968.07610348462$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.3324042821194357$
10.2.1457986637.1	$x^{10} - 3 x^{9} + 6 x^{8} - 10 x^{7} + 12 x^{6} - 13 x^{5} + 12 x^{4} - 10 x^{3} + 7 x^{2} - 4 x + 1$	$10$	[2,4]	$31\cdot 47031827$	$2$	$8.24850713401$	$38183.59120093342$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.4034817008530727$
10.0.1472062311.1	$x^{10} - 4 x^{9} + 6 x^{8} - 5 x^{7} + 4 x^{6} - x^{5} - 3 x^{4} + 3 x^{2} - x + 1$	$10$	[0,5]	$-\,3^{2}\cdot 31\cdot 2297^{2}$	$3$	$8.25643601071$	$462.1915187452059$			?	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$4$	$1.86680054887$
10.0.1602120083.1	$x^{10} - 3 x^{9} + 6 x^{8} - 10 x^{7} + 15 x^{6} - 18 x^{5} + 18 x^{4} - 15 x^{3} + 11 x^{2} - 5 x + 1$	$10$	[0,5]	$-\,31\cdot 51681293$	$2$	$8.32663459296$	$40026.49226449902$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.1780071191933126$
10.0.1641899407.1	$x^{10} - 2 x^{9} + 2 x^{7} - x^{6} + x^{4} - 3 x^{3} + 3 x^{2} + 1$	$10$	[0,5]	$-\,31\cdot 41\cdot 1291817$	$3$	$8.34708151297$	$40520.35793277251$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.2699806462826215$
10.0.1657358611.1	$x^{10} - x^{9} + 2 x^{8} - x^{6} + 2 x^{5} - x^{3} + 3 x^{2} - x + 1$	$10$	[0,5]	$-\,17\cdot 31\cdot 3144893$	$3$	$8.35490755523$	$40710.66949830228$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.3082725962419137$
10.2.1679598941.1	$x^{10} - 2 x^{9} + x^{7} + x^{6} - 3 x^{4} + x^{2} + x + 1$	$10$	[2,4]	$31\cdot 97\cdot 558563$	$3$	$8.36605199156$	$40982.91035297518$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.7053004070234525$
10.2.1762946657.1	$x^{10} - 2 x^{9} + 4 x^{8} - 5 x^{7} + 9 x^{6} - 14 x^{5} + 18 x^{4} - 18 x^{3} + 12 x^{2} - 5 x + 1$	$10$	[2,4]	$31\cdot 3499\cdot 16253$	$3$	$8.40666840203$	$41987.4583298394$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.905980938625884$
10.2.1988089613.1	$x^{10} - x^{9} + 2 x^{8} + x^{7} - 2 x^{6} + 6 x^{5} - 4 x^{4} + 5 x^{3} - 2 x^{2} + 2 x - 1$	$10$	[2,4]	$31\cdot 47\cdot 61\cdot 22369$	$4$	$8.50831569328$	$44587.99853099486$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.0006103955666914$
10.0.2009646052.1	$x^{10} - 2 x^{9} + 4 x^{8} - 3 x^{7} + 5 x^{5} - 6 x^{4} + 3 x^{3} + 2 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,2^{2}\cdot 31\cdot 1229\cdot 13187$	$4$	$8.51749637534$	$44829.07596638592$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.688717130195472$
10.0.2026040216.1	$x^{10} - x^{9} + 2 x^{8} - 3 x^{7} + 5 x^{6} - 5 x^{5} + 5 x^{4} - 4 x^{3} + 4 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,2^{3}\cdot 31\cdot 8169517$	$3$	$8.52441934863$	$45011.556471644035$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.7106688670028607$
10.2.2030367289.1	$x^{10} - 2 x^{9} - x^{8} + 6 x^{7} - 4 x^{6} - 4 x^{5} + 8 x^{4} - 2 x^{3} - 3 x^{2} + 3 x - 1$	$10$	[2,4]	$31\cdot 3463\cdot 18913$	$3$	$8.5262381863$	$45059.59707986746$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.011612278046071$
10.2.2101917893.1	$x^{10} - 3 x^{8} + 5 x^{6} - 2 x^{5} - 5 x^{4} + 2 x^{3} + 3 x^{2} - x - 1$	$10$	[2,4]	$17\cdot 31\cdot 59\cdot 67601$	$4$	$8.55581873028$	$45846.67810212644$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.020962239094592$
10.0.2165308987.1	$x^{10} - x^{9} - 2 x^{8} + 3 x^{7} - x^{6} - 2 x^{5} + 4 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 1$	$10$	[0,5]	$-\,23\cdot 31\cdot 3036899$	$3$	$8.58127831993$	$46532.88070816162$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.392147685378283$
10.0.2214793667.1	$x^{10} - 3 x^{9} + 5 x^{8} - 6 x^{7} + 6 x^{6} - 4 x^{5} + 3 x^{4} - 2 x^{3} + x^{2} - x + 1$	$10$	[0,5]	$-\,31\cdot 71\cdot 1006267$	$3$	$8.60069065152$	$47061.59439500536$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.0247129050836$
10.2.2253938297.1	$x^{10} - x^{9} + 3 x^{7} + x^{6} - 3 x^{5} + 3 x^{3} - x^{2} - 3 x - 1$	$10$	[2,4]	$13^{2}\cdot 31\cdot 109\cdot 3947$	$4$	$8.61577209151$	$13167.378972293613$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.22580340427767$
10.0.2260135259.1	$x^{10} - 2 x^{9} + 2 x^{8} - x^{7} - x^{6} + 2 x^{5} - 2 x^{3} + 3 x^{2} - 2 x + 1$	$10$	[0,5]	$-\,31\cdot 2693\cdot 27073$	$3$	$8.61813798013$	$47540.87987195862$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.424227285904649$
10.0.2266426523.1	$x^{10} - 3 x^{9} + 5 x^{8} - 6 x^{7} + 8 x^{6} - 10 x^{5} + 9 x^{4} - 6 x^{3} + 5 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,31\cdot 103\cdot 653\cdot 1087$	$4$	$8.62053390689$	$47607.00077719663$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.0787368837148703$
10.2.2271620201.1	$x^{10} - x^{9} - 2 x^{7} + x^{6} - 3 x^{5} + 2 x^{4} + 2 x^{3} + x^{2} - 3 x + 1$	$10$	[2,4]	$31\cdot 73278071$	$2$	$8.62250732955$	$47661.516981732755$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.1376050108773907$
10.2.2293196573.1	$x^{10} - 3 x^{9} + 6 x^{8} - 8 x^{7} + 10 x^{6} - 9 x^{5} + 8 x^{4} - 6 x^{3} + 4 x^{2} - 3 x + 1$	$10$	[2,4]	$31\cdot 73974083$	$2$	$8.63066238968$	$47887.33207227147$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.2081040902958082$
10.0.2328641167.1	$x^{10} - 4 x^{9} + 6 x^{8} - 3 x^{7} - 2 x^{6} + 3 x^{5} - x^{4} - 2 x^{3} + 3 x^{2} - x + 1$	$10$	[0,5]	$-\,31\cdot 5527\cdot 13591$	$3$	$8.64391041217$	$48255.99617664109$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.457826653595387$
10.0.2332083035.1	$x^{10} - 2 x^{9} + x^{8} + 2 x^{7} - x^{6} - 3 x^{5} + 4 x^{4} - 3 x^{2} + x + 1$	$10$	[0,5]	$-\,5\cdot 17\cdot 31\cdot 885041$	$4$	$8.64518718376$	$48291.645602526325$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.7850088877044157$
10.0.2388102887.1	$x^{10} - 2 x^{9} + 2 x^{8} - x^{5} + 2 x^{4} - x^{2} + 1$	$10$	[0,5]	$-\,31\cdot 739\cdot 104243$	$3$	$8.66573299025$	$48868.21960129098$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$4.131065330070257$
10.2.2396091277.1	$x^{10} - 2 x^{9} + 3 x^{8} - 5 x^{7} + 6 x^{6} - 5 x^{5} + 5 x^{4} - 5 x^{3} + 2 x^{2} - 2 x + 1$	$10$	[2,4]	$31\cdot 77293267$	$2$	$8.66862739114$	$48949.88536248068$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.2778022129966287$
10.2.2424721141.1	$x^{10} - 2 x^{9} + 4 x^{8} - 5 x^{7} + 3 x^{6} - 2 x^{5} - 2 x^{4} + 5 x^{3} - 6 x^{2} + 4 x - 1$	$10$	[2,4]	$31\cdot 311\cdot 251501$	$3$	$8.67892988672$	$49241.45754341559$			?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.410173748455915$
10.0.2435685376.1	$x^{10} - 2 x^{7} + 2 x^{5} + 2 x^{4} + 6 x^{3} + 4 x^{2} + 2 x + 2$	$10$	[0,5]	$-\,2^{10}\cdot 31\cdot 277^{2}$	$3$	$8.68284640923$	$209.23338267867777$			?	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$4$	$4.09919502276$
10.2.2498598481.1	$x^{10} - 2 x^{9} + 2 x^{8} - 2 x^{7} + 4 x^{6} - 7 x^{5} + 8 x^{4} - 3 x^{3} - 2 x^{2} + 3 x - 1$	$10$	[2,4]	$31\cdot 80599951$	$2$	$8.70501747254$	$49985.982845193714$			?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.441671975654899$
10.2.2502015053.1	$x^{10} - 3 x^{9} + 5 x^{8} - 4 x^{7} + x^{6} + 4 x^{5} - 7 x^{4} + 5 x^{3} - 2 x + 1$	$10$	[2,4]	$31\cdot 80710163$	$2$	$8.70620706079$	$50020.14647119698$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.3609153512122476$
10.0.2581671847.1	$x^{10} - 2 x^{9} + 4 x^{8} - 6 x^{7} + 8 x^{6} - 9 x^{5} + 9 x^{4} - 7 x^{3} + 6 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,31\cdot 83279737$	$2$	$8.73353578111$	$50810.15495941731$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$4.418339123812683$
10.2.2606448473.1	$x^{10} - x^{9} + 4 x^{8} - 4 x^{7} + 5 x^{6} - 9 x^{5} + 2 x^{4} - 7 x^{3} + 2 x^{2} - x + 1$	$10$	[2,4]	$31\cdot 563\cdot 149341$	$3$	$8.74188148456$	$51053.38845757449$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.543075609546572$
10.0.2616420367.1	$x^{10} - x^{9} + 2 x^{6} - 3 x^{5} + 2 x^{4} - x^{3} - x^{2} + 2 x + 1$	$10$	[0,5]	$-\,11\cdot 31\cdot 7672787$	$3$	$8.74522025756$	$51150.95665772049$			?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$4.428954805620562$
10.2.2618382233.1	$x^{10} - x^{9} - 2 x^{8} + 2 x^{7} + 2 x^{6} - x^{5} - 3 x^{4} + x^{3} + 2 x^{2} - x - 1$	$10$	[2,4]	$31\cdot 71\cdot 1189633$	$3$	$8.74587577775$	$51170.13028124904$			✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.8831391602420746$

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