LMFDB - Number field search results

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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

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Results (1-50 of 207005 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
8.2.4286875.1	$x^{8} - x^{7} + x^{5} - 2 x^{4} - x^{3} + 2 x^{2} + 2 x - 1$	$8$	[2,3]	$-\,5^{4}\cdot 19^{3}$	$2$	$6.74555367758$	$9.746794344808963$	?	$D_{8}$ (as 8T6)	trivial	$2$	$4$	$0.831403828897$
8.2.4296211.1	$x^{8} - x^{7} + 3 x^{5} - 4 x^{4} + 2 x^{3} + 2 x^{2} - 3 x + 1$	$8$	[2,3]	$-\,199\cdot 21589$	$2$	$6.74738824726$	$2072.7303249578804$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.83312070768$
8.2.4297507.1	$x^{8} - x^{7} - 2 x^{6} + x^{5} + x^{3} + 2 x^{2} - 1$	$8$	[2,3]	$-\,2011\cdot 2137$	$2$	$6.74764264181$	$2073.0429325028463$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.83318984365$
8.2.4364587.1	$x^{8} - x^{7} + 2 x^{6} - 3 x^{5} + 3 x^{4} - 4 x^{3} + 2 x^{2} - 2 x + 1$	$8$	[2,3]	$-\,29\cdot 150503$	$2$	$6.76071913838$	$2089.159400333062$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8412077944090119$
8.2.4386467.1	$x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} - 2 x^{3} + 2 x^{2} + x - 1$	$8$	[2,3]	$-\,41\cdot 83\cdot 1289$	$3$	$6.76494637444$	$2094.3894098280766$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8443442011032247$
8.2.4421387.1	$x^{8} - 2 x^{7} + 2 x^{6} + x^{5} - 2 x^{4} + x^{3} + x^{2} - 1$	$8$	[2,3]	$-\,1321\cdot 3347$	$2$	$6.77165488361$	$2102.7094426001895$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.848612129114202$
8.2.4423907.1	$x^{8} - x^{7} - 2 x^{6} + 3 x^{5} + 2 x^{4} - 2 x^{3} - 2 x^{2} + x + 1$	$8$	[2,3]	$-\,4423907$	$1$	$6.772137207270576$	$2103.308584112184$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8486556559484179$
8.2.4456891.1	$x^{8} - x^{7} - x^{6} + 2 x^{5} - 2 x^{3} + x - 1$	$8$	[2,3]	$-\,4456891$	$1$	$6.778428222351358$	$2111.1350027887843$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8533890882180042$
8.2.4461875.1	$x^{8} - x^{7} + x^{6} + 2 x^{5} - 2 x^{4} + 2 x^{2} - x - 1$	$8$	[2,3]	$-\,5^{4}\cdot 11^{2}\cdot 59$	$3$	$6.7793752718$	$56.96490147450446$	✓	$C_2 \wr C_2\wr C_2$ (as 8T35)	trivial	$2$	$4$	$0.853816779689$
8.2.4505651.1	$x^{8} - 2 x^{7} + 4 x^{6} - 5 x^{5} + 5 x^{4} - 3 x^{3} + x^{2} + x - 1$	$8$	[2,3]	$-\,4505651$	$1$	$6.787653961428009$	$2122.6518791360963$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8590899748258677$
8.2.4542739.1	$x^{8} - 3 x^{7} + 5 x^{6} - 6 x^{5} + 4 x^{4} - x^{3} - x^{2} + x - 1$	$8$	[2,3]	$-\,4542739$	$1$	$6.794612960385022$	$2131.370216550846$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8640907466005752$
8.2.4570091.1	$x^{8} - x^{6} - x^{5} + x^{4} - x^{3} - 2 x^{2} + x + 1$	$8$	[2,3]	$-\,1249\cdot 3659$	$2$	$6.799713367076283$	$2137.7771165395143$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8691071409989591$
8.2.4570723.1	$x^{8} - 2 x^{7} + 3 x^{6} - 3 x^{5} + 2 x^{3} - 4 x^{2} + 3 x - 1$	$8$	[2,3]	$-\,4570723$	$1$	$6.7998309019$	$2137.9249285229826$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8669605427357915$
8.2.4584491.1	$x^{8} - x^{7} + 2 x^{6} - 2 x^{5} + 4 x^{4} - 5 x^{3} + 4 x^{2} - 3 x + 1$	$8$	[2,3]	$-\,19\cdot 101\cdot 2389$	$3$	$6.80238785315$	$2141.1424520568453$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8688179378958599$
8.2.4596992.1	$x^{8} - 2 x^{7} + 3 x^{6} - 4 x^{5} + 4 x^{4} - 4 x^{3} + 2 x^{2} - 2 x + 1$	$8$	[2,3]	$-\,2^{8}\cdot 17957$	$2$	$6.804703687569884$	$491.52785359466174$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8711682249043199$
8.2.4603987.1	$x^{8} - x^{7} - 2 x^{6} + x^{5} + 2 x^{4} - x - 1$	$8$	[2,3]	$-\,4603987$	$1$	$6.8059971215378265$	$2145.690331804662$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8718212555217068$
8.2.4614499.1	$x^{8} - x^{7} + x^{5} + x^{4} - 2 x^{3} + 2 x - 1$	$8$	[2,3]	$-\,4614499$	$1$	$6.807937648205655$	$2148.1384964661847$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8729503709676886$
8.2.4616192.1	$x^{8} - 2 x^{7} + 3 x^{6} - 4 x^{5} + 3 x^{4} - 4 x^{3} + 3 x^{2} - 2 x + 1$	$8$	[2,3]	$-\,2^{12}\cdot 7^{2}\cdot 23$	$3$	$6.80824981614$	$35.888716889852724$	✓	$C_2 \wr C_2\wr C_2$ (as 8T35)	trivial	$2$	$4$	$0.873136825319$
8.2.4623371.1	$x^{8} + x^{6} - x^{3} - x^{2} - 1$	$8$	[2,3]	$-\,17\cdot 31^{2}\cdot 283$	$3$	$6.809572421507061$	$684.4769184408666$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.873839750719525$
8.2.4648192.1	$x^{8} - 2 x^{7} + 2 x^{6} - 2 x^{4} + 2 x^{3} - x^{2} + 1$	$8$	[2,3]	$-\,2^{8}\cdot 67\cdot 271$	$3$	$6.814131452560941$	$494.2575233501126$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8773171765831781$
8.2.4663051.1	$x^{8} - x^{6} - 2 x^{5} + x^{4} + x^{2} + 2 x - 1$	$8$	[2,3]	$-\,31\cdot 359\cdot 419$	$3$	$6.816850516631215$	$2159.409873090331$	?	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8785601974574553$
8.2.4690927.1	$x^{8} - x^{7} - x^{6} + 2 x^{5} + x^{4} - 2 x^{3} + 2 x - 1$	$8$	[2,3]	$-\,443\cdot 10589$	$2$	$6.8219311864100645$	$2165.854796610336$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8833361607728416$
8.2.4711123.1	$x^{8} - x^{7} - x^{5} + 2 x^{3} + x - 1$	$8$	[2,3]	$-\,43\cdot 331^{2}$	$2$	$6.82559562487$	$119.30213744941874$	✓	$C_2 \wr S_4$ (as 8T44)	trivial	$2$	$4$	$0.884280803625$
8.2.4725251.1	$x^{8} - 4 x^{6} - 2 x^{5} + 7 x^{4} + 5 x^{3} - 3 x^{2} - 4 x - 1$	$8$	[2,3]	$-\,59\cdot 283^{2}$	$2$	$6.82815090024$	$129.2168719633779$	✓	$C_2 \wr S_4$ (as 8T44)	trivial	$2$	$4$	$0.886962638902$
8.2.4761667.1	$x^{8} - x^{7} + x^{6} + x^{5} - 3 x^{4} + 4 x^{3} - 4 x^{2} + 3 x - 1$	$8$	[2,3]	$-\,23\cdot 207029$	$2$	$6.834706625519568$	$2182.1244235835866$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.892409675427044$
8.2.4775363.1	$x^{8} - 2 x^{7} - x^{6} + 4 x^{5} - 2 x^{4} - 3 x^{3} + 3 x^{2} - 1$	$8$	[2,3]	$-\,1931\cdot 2473$	$2$	$6.837160875217895$	$2185.2603963830034$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.892761210808111$
8.2.4785667.1	$x^{8} - x^{7} + 2 x^{6} - x^{5} + x^{3} - 2 x^{2} + 2 x - 1$	$8$	[2,3]	$-\,29\cdot 59\cdot 2797$	$3$	$6.839003240119339$	$2187.616739742133$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8954120433168544$
8.2.4809907.1	$x^{8} - 2 x^{7} + 3 x^{6} - 5 x^{5} + 5 x^{4} - 5 x^{3} + 4 x^{2} - 3 x + 1$	$8$	[2,3]	$-\,19\cdot 253153$	$2$	$6.843323725534318$	$2193.15001766865$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.8973516802448191$
8.2.4858379.1	$x^{8} + 3 x^{6} - x^{5} + 2 x^{4} - 3 x^{3} - 2 x + 1$	$8$	[2,3]	$-\,17^{2}\cdot 16811$	$2$	$6.851906434395191$	$534.5904974838218$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9045989163533711$
8.2.4931267.1	$x^{8} - 2 x^{7} + x^{6} + x^{5} - 3 x^{4} + 4 x^{3} - 4 x^{2} + 2 x - 1$	$8$	[2,3]	$-\,11\cdot 67\cdot 6691$	$3$	$6.864672373301241$	$2220.6456268391858$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9159002625287757$
8.2.4960000.1	$x^{8} - 2 x^{7} + 4 x^{5} - 8 x^{4} + 10 x^{3} - 7 x^{2} + 4 x - 1$	$8$	[2,3]	$-\,2^{8}\cdot 5^{4}\cdot 31$	$3$	$6.86965947012$	$35.21363372331802$	✓	$C_2 \wr C_2\wr C_2$ (as 8T35)	trivial	$2$	$4$	$0.915024622331$
8.2.5040467.1	$x^{8} - 3 x^{7} + 3 x^{6} - 4 x^{4} + 4 x^{3} - x + 1$	$8$	[2,3]	$-\,5040467$	$1$	$6.883492554051752$	$2245.098438821781$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9288588955969742$
8.2.5040547.1	$x^{8} - 3 x^{7} + 4 x^{6} - 4 x^{5} + 2 x^{4} - x^{3} + x^{2} + 1$	$8$	[2,3]	$-\,37\cdot 59\cdot 2309$	$3$	$6.883506210414857$	$2245.116255341803$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9268647190390614$
8.2.5103467.1	$x^{8} - x^{7} - x^{6} + x^{5} + x^{4} + x^{3} - 2 x^{2} + 1$	$8$	[2,3]	$-\,5103467$	$1$	$6.894188661972919$	$2259.0854344180966$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9341515370278446$
8.2.5107019.1	$x^{8} - 3 x^{7} + 5 x^{6} - 3 x^{5} - x^{4} + 4 x^{3} - 3 x^{2} + 2 x - 1$	$8$	[2,3]	$-\,5107019$	$1$	$6.894788271629269$	$2259.871456521366$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9337002743186916$
8.2.5118587.1	$x^{8} - x^{7} - x^{6} + x^{5} + x^{2} - x - 1$	$8$	[2,3]	$-\,29\cdot 176503$	$2$	$6.896738528281753$	$2262.429446413744$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9358964080809821$
8.2.5149367.1	$x^{8} - x^{6} - x^{5} - x^{3} + x^{2} + x - 1$	$8$	[2,3]	$-\,47\cdot 331^{2}$	$2$	$6.90190902798$	$124.72770341828635$	?	$C_2 \wr S_4$ (as 8T44)	trivial	$2$	$4$	$0.937944317141$
8.2.5155867.1	$x^{8} - x^{7} - 2 x^{6} + x^{5} + 3 x^{4} - x^{3} - 3 x^{2} + 1$	$8$	[2,3]	$-\,449\cdot 11483$	$2$	$6.902997454301547$	$2270.653430182598$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9405039275650298$
8.2.5165819.1	$x^{8} - 2 x^{6} - x^{5} + 2 x^{4} + 3 x^{3} - x - 1$	$8$	[2,3]	$-\,641\cdot 8059$	$2$	$6.904661594549214$	$2272.843813375657$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9414451308962976$
8.2.5204491.1	$x^{8} - x^{7} + x^{6} - 2 x^{4} + x^{3} - x^{2} - x + 1$	$8$	[2,3]	$-\,5204491$	$1$	$6.911101682109057$	$2281.335354567583$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9486003676975986$
8.2.5233147.1	$x^{8} - 2 x^{7} + 2 x^{5} + x^{4} - x^{3} - 2 x^{2} + x + 1$	$8$	[2,3]	$-\,5233147$	$1$	$6.915846841129293$	$2287.607265244627$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9515125685677428$
8.2.5272027.1	$x^{8} - x^{7} - x^{6} + 3 x^{5} - x^{4} - 3 x^{3} + 2 x^{2} - 1$	$8$	[2,3]	$-\,317\cdot 16631$	$2$	$6.922248776687624$	$2296.08950173986$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9542437305912765$
8.2.5286727.1	$x^{8} - x^{7} - x^{6} + 2 x^{5} - x^{4} - 2 x^{3} + 2 x^{2} - 1$	$8$	[2,3]	$-\,5286727$	$1$	$6.924658503091751$	$2299.2883681695953$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9546924661447783$
8.2.5293867.1	$x^{8} - x^{7} + 2 x^{5} - x^{4} - x^{3} + x^{2} + x - 1$	$8$	[2,3]	$-\,227\cdot 23321$	$2$	$6.925826826912146$	$2300.840498600457$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9584670756759093$
8.2.5344939.1	$x^{8} - 2 x^{7} + 4 x^{6} - 5 x^{5} + 4 x^{4} - x^{3} - x^{2} + 2 x - 1$	$8$	[2,3]	$-\,521\cdot 10259$	$2$	$6.934143805490464$	$2311.912411835708$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9628254972480232$
8.2.5346947.1	$x^{8} - x^{6} - x^{5} + x^{4} + 2 x^{3} - 2 x - 1$	$8$	[2,3]	$-\,839\cdot 6373$	$2$	$6.934469381534787$	$2312.3466435636333$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.9655694198061011$
8.2.5359051.1	$x^{8} - x^{7} + x^{6} - 2 x^{4} + 4 x^{3} - 4 x^{2} + 3 x - 1$	$8$	[2,3]	$-\,5359051$	$1$	$6.936429654555921$	$2314.9624187014356$	✓	$S_8$ (as 8T50)	trivial	$2$	$4$	$0.966607895416933$
8.2.5365963.1	$x^{8} - x^{7} + x^{6} - 2 x^{5} + 5 x^{4} - 5 x^{3} + 3 x^{2} - 1$	$8$	[2,3]	$-\,67\cdot 283^{2}$	$2$	$6.93754733307$	$137.69894698217558$	?	$C_2 \wr S_4$ (as 8T44)	trivial	$2$	$4$	$0.964152352733$
8.2.5365963.2	$x^{8} - 2 x^{5} + x^{4} - x^{3} - 2 x^{2} + 3 x + 1$	$8$	[2,3]	$-\,67\cdot 283^{2}$	$2$	$6.93754733307$	$137.69894698217558$	?	$C_2 \wr S_4$ (as 8T44)	trivial	$2$	$4$	$0.96720132041$
8.2.5369375.1	$x^{8} - 2 x^{7} + 2 x^{6} - 4 x^{3} + 5 x^{2} - 4 x + 1$	$8$	[2,3]	$-\,5^{4}\cdot 11^{2}\cdot 71$	$3$	$6.93809859314$	$62.489999199871974$	?	$C_2 \wr C_2\wr C_2$ (as 8T35)	trivial	$2$	$4$	$0.964343847115$

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