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Degree	Signature	Ramified prime count	Ramified	Unramified primes

Galois group	Field is	Intermediate field	Class group structure	Class number

Discriminant	Root discriminant	Galois root discriminant	CM field	Relative class number

Regulator	$p$ -adic completions	Monogenic	Index	Ines. $p$

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Results (1-50 of 748622 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
4.2.275.1	$x^{4} - x^{3} + 2 x - 1$	$4$	[2,1]	$-\,5^{2}\cdot 11$	$2$	$4.07223819929$	$7.416198487095663$	✓	$D_{4}$ (as 4T3)	trivial	$2$	$2$	$0.369184373575$
4.2.283.1	$x^{4} - x - 1$	$4$	[2,1]	$-\,283$	$1$	$4.10153676581$	$16.822603841260722$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$0.37819933246$
4.2.331.1	$x^{4} - x^{3} + x^{2} + x - 1$	$4$	[2,1]	$-\,331$	$1$	$4.26537283232$	$18.193405398660254$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$0.432203091185$
4.2.400.1	$x^{4} - x^{2} - 1$	$4$	[2,1]	$-\,2^{4}\cdot 5^{2}$	$2$	$4.472135955$	$6.324555320336759$	✓	$D_{4}$ (as 4T3)	trivial	$2$	$2$	$0.51069810943$
4.2.448.1	$x^{4} - 2 x^{3} + x^{2} - 2 x + 1$	$4$	[2,1]	$-\,2^{6}\cdot 7$	$2$	$4.60065326758$	$7.483314773547883$	✓	$D_{4}$ (as 4T3)	trivial	$2$	$2$	$0.557886846205$
4.2.475.1	$x^{4} - 2 x^{3} + 2 x^{2} - x - 1$	$4$	[2,1]	$-\,5^{2}\cdot 19$	$2$	$4.66845742379$	$9.746794344808963$	✓	$D_{4}$ (as 4T3)	trivial	$2$	$2$	$0.57693147741$
4.2.491.1	$x^{4} - x^{3} - x^{2} + 3 x - 1$	$4$	[2,1]	$-\,491$	$1$	$4.70728369723$	$22.15851980616034$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$0.63355773488$
4.2.507.1	$x^{4} - x^{3} - x^{2} - x + 1$	$4$	[2,1]	$-\,3\cdot 13^{2}$	$2$	$4.74517233601$	$6.244997998398398$	✓	$D_{4}$ (as 4T3)	trivial	$2$	$2$	$0.649395711926$
4.2.563.1	$x^{4} - x^{3} + x^{2} - x - 1$	$4$	[2,1]	$-\,563$	$1$	$4.87110059796$	$23.727621035409346$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$0.704267195983$
4.2.643.1	$x^{4} - x^{3} - 2 x + 1$	$4$	[2,1]	$-\,643$	$1$	$5.03561760524$	$25.357444666211933$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$0.720184467868$
4.2.688.1	$x^{4} - 2 x - 1$	$4$	[2,1]	$-\,2^{4}\cdot 43$	$2$	$5.12149920406$	$14.720951755301632$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$0.998743211853$
4.2.731.1	$x^{4} - x^{3} + 2 x^{2} - 1$	$4$	[2,1]	$-\,17\cdot 43$	$2$	$5.19971265256$	$27.03701166919155$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$0.866830073687$
4.2.751.1	$x^{4} - 2 x^{3} + x^{2} - x - 1$	$4$	[2,1]	$-\,751$	$1$	$5.23491921734$	$27.40437921208944$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.06794059626$
4.2.775.1	$x^{4} - x^{3} - 3 x - 1$	$4$	[2,1]	$-\,5^{2}\cdot 31$	$2$	$5.27625073458$	$12.449899597988733$		$D_{4}$ (as 4T3)	trivial	$2$	$2$	$0.867905309463$
4.2.848.1	$x^{4} - x^{2} - 2 x + 1$	$4$	[2,1]	$-\,2^{4}\cdot 53$	$2$	$5.39633575282$	$20.591260281974$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$0.992994938315$
4.2.976.1	$x^{4} - 2 x^{3} + 3 x^{2} - 1$	$4$	[2,1]	$-\,2^{4}\cdot 61$	$2$	$5.58936478534$	$22.090722034374522$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$0.990699215006$
4.2.1024.1	$x^{4} - 2 x^{2} - 1$	$4$	[2,1]	$-\,2^{10}$	$1$	$5.65685424949$	$6.727171322029716$	✓	$D_{4}$ (as 4T3)	trivial	$2$	$2$	$1.34724203432$
4.2.1099.1	$x^{4} - x^{3} + x^{2} - 3 x + 1$	$4$	[2,1]	$-\,7\cdot 157$	$2$	$5.75770517728$	$33.15116890850155$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.01244916602$
4.2.1107.1	$x^{4} - x^{3} - 2 x - 1$	$4$	[2,1]	$-\,3^{3}\cdot 41$	$2$	$5.76815477594$	$23.06924557594881$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.28804807411$
4.2.1156.1	$x^{4} - x^{3} - 2 x^{2} - x + 1$	$4$	[2,1]	$-\,2^{2}\cdot 17^{2}$	$2$	$5.83095189485$	$8.246211251235321$	✓	$D_{4}$ (as 4T3)	trivial	$2$	$2$	$1.53512616922$
4.2.1192.1	$x^{4} - x^{3} + 2 x^{2} + x - 1$	$4$	[2,1]	$-\,2^{3}\cdot 149$	$2$	$5.87582785684$	$34.52535300326414$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.5689723036$
4.2.1255.1	$x^{4} - x^{2} - 3 x - 1$	$4$	[2,1]	$-\,5\cdot 251$	$2$	$5.95197271229$	$35.4259791678367$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.58232994952$
4.2.1323.1	$x^{4} - x^{3} - 3 x^{2} - x + 1$	$4$	[2,1]	$-\,3^{3}\cdot 7^{2}$	$2$	$6.03100878452$	$6.03100878451909$	✓	$D_{4}$ (as 4T3)	trivial	$2$	$2$	$1.35144993247$
4.2.1328.1	$x^{4} - 3 x^{2} - 2 x + 1$	$4$	[2,1]	$-\,2^{4}\cdot 83$	$2$	$6.03669895858$	$18.2208671582886$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.3668896797$
4.2.1371.1	$x^{4} + 2 x^{2} - x - 1$	$4$	[2,1]	$-\,3\cdot 457$	$2$	$6.08498292217$	$37.027017163147235$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.34876542307$
4.2.1375.1	$x^{4} - x^{3} + x^{2} + 4 x - 4$	$4$	[2,1]	$-\,5^{3}\cdot 11$	$2$	$6.08941642815$	$11.08980336897316$		$D_{4}$ (as 4T3)	trivial	$2$	$2$	$1.33061932583$
4.2.1399.1	$x^{4} - x^{3} + x - 2$	$4$	[2,1]	$-\,1399$	$1$	$6.11581625121$	$37.403208418530085$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.70613352231$
4.2.1423.1	$x^{4} - x^{3} + x^{2} - 2 x - 1$	$4$	[2,1]	$-\,1423$	$1$	$6.14187855853$	$37.72267222772003$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.69769459199$
4.2.1424.1	$x^{4} + x^{2} - 2 x - 1$	$4$	[2,1]	$-\,2^{4}\cdot 89$	$2$	$6.14295731128$	$26.68332812825267$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.47325330706$
4.2.1456.1	$x^{4} - 2 x^{2} - 2 x + 1$	$4$	[2,1]	$-\,2^{4}\cdot 7\cdot 13$	$3$	$6.17718123878$	$21.415210999701447$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.76052319498$
4.2.1472.1	$x^{4} - 2 x^{3} + 2 x^{2} - 2$	$4$	[2,1]	$-\,2^{6}\cdot 23$	$2$	$6.19408202936$	$13.564659966250536$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.95632588969$
4.2.1472.2	$x^{4} - 2 x^{3} + 3 x^{2} - 2 x - 1$	$4$	[2,1]	$-\,2^{6}\cdot 23$	$2$	$6.19408202936$	$13.564659966250536$	✓	$D_{4}$ (as 4T3)	trivial	$2$	$2$	$1.25951480215$
4.2.1475.1	$x^{4} - x^{3} - x^{2} + 5 x - 5$	$4$	[2,1]	$-\,5^{2}\cdot 59$	$2$	$6.19723557236$	$17.175564037317667$		$D_{4}$ (as 4T3)	trivial	$2$	$2$	$1.23477697451$
4.2.1588.1	$x^{4} - x^{3} - 3 x^{2} + 2$	$4$	[2,1]	$-\,2^{2}\cdot 397$	$2$	$6.31266328029$	$39.84971769034255$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.93358038312$
4.2.1600.1	$x^{4} - 2 x^{2} - 4$	$4$	[2,1]	$-\,2^{6}\cdot 5^{2}$	$2$	$6.32455532034$	$8.94427190999916$		$D_{4}$ (as 4T3)	trivial	$2$	$2$	$1.41413886912$
4.2.1728.1	$x^{4} - 2 x^{3} - 2 x + 1$	$4$	[2,1]	$-\,2^{6}\cdot 3^{3}$	$2$	$6.44741959094$	$6.4474195909412515$	✓	$D_{4}$ (as 4T3)	trivial	$2$	$2$	$2.18995070591$
4.2.1732.1	$x^{4} - x^{3} + 3 x - 1$	$4$	[2,1]	$-\,2^{2}\cdot 433$	$2$	$6.45114750206$	$41.617304093369626$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$2.05155660923$
4.2.1775.1	$x^{4} - 2 x^{3} + 3 x^{2} - 2 x - 4$	$4$	[2,1]	$-\,5^{2}\cdot 71$	$2$	$6.49082035384$	$18.841443681416774$		$D_{4}$ (as 4T3)	trivial	$2$	$2$	$1.51169722096$
4.2.1791.1	$x^{4} - x^{3} - x^{2} - 2 x + 1$	$4$	[2,1]	$-\,3^{2}\cdot 199$	$2$	$6.50539836897$	$24.43358344574123$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$2.01913849925$
4.2.1792.1	$x^{4} - 2 x^{2} - 4 x - 2$	$4$	[2,1]	$-\,2^{8}\cdot 7$	$2$	$6.50630624679$	$10.583005244258363$	✓	$D_{4}$ (as 4T3)	trivial	$2$	$2$	$1.98905564548$
4.2.1823.1	$x^{4} - x^{3} + 3 x - 2$	$4$	[2,1]	$-\,1823$	$1$	$6.53426385145$	$42.69660408041839$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$2.28192620748$
4.2.1856.1	$x^{4} - 2 x^{3} + x^{2} - 2 x - 1$	$4$	[2,1]	$-\,2^{6}\cdot 29$	$2$	$6.56363606982$	$21.540659228538015$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.76145911782$
4.2.1879.1	$x^{4} - x^{3} - 2 x^{2} - 3 x + 1$	$4$	[2,1]	$-\,1879$	$1$	$6.58387679133$	$43.347433603386484$		$S_4$ (as 4T5)	trivial	$2$	$2$	$1.6589571085$
4.2.1927.1	$x^{4} - 2 x^{3} + x^{2} - 3 x + 1$	$4$	[2,1]	$-\,41\cdot 47$	$2$	$6.62552700812$	$43.89760813529594$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$2.07547726245$
4.2.1931.1	$x^{4} - 3 x + 1$	$4$	[2,1]	$-\,1931$	$1$	$6.62896259496$	$43.94314508543966$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.39084475468$
4.2.1963.1	$x^{4} - x^{3} + 2 x^{2} - 2 x - 1$	$4$	[2,1]	$-\,13\cdot 151$	$2$	$6.65625689361$	$44.30575583375144$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$1.47885191076$
4.2.1968.1	$x^{4} - 4 x^{2} - 2 x + 2$	$4$	[2,1]	$-\,2^{4}\cdot 3\cdot 41$	$3$	$6.66049142524$	$24.897412647668382$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$2.39920685452$
4.2.1975.1	$x^{4} - 2 x^{3} + x^{2} - 5$	$4$	[2,1]	$-\,5^{2}\cdot 79$	$2$	$6.66640623474$	$19.87460691435179$		$D_{4}$ (as 4T3)	trivial	$2$	$2$	$1.77633002997$
4.2.1984.1	$x^{4} - 2 x^{3} + x^{2} - 2$	$4$	[2,1]	$-\,2^{6}\cdot 31$	$2$	$6.67398793096$	$15.748015748023622$	✓	$D_{4}$ (as 4T3)	trivial	$2$	$2$	$3.00091979939$
4.2.1984.2	$x^{4} - 2 x^{3} + 2 x^{2} + 2 x - 1$	$4$	[2,1]	$-\,2^{6}\cdot 31$	$2$	$6.67398793096$	$15.748015748023622$	✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$2.16801442334$

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