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	Ramified prime count	Ramified	Unramified primes

Galois group	Field is	Intermediate field	Class group structure	Class number

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Results (1-50 of 124247 matches)

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Label	Polynomial	Degree	Signature	Discriminant	Ram. prime count	Root discriminant	Galois root discriminant	CM field	Monogenic	Galois group	Class group	Unit group torsion	Unit group rank	Regulator
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	$2$	$2$	$0.992994938315$
4.0.2597.1	$x^{4} + 4 x^{2} - x + 1$	$4$	[0,2]	$7^{2}\cdot 53$	$2$	$7.13868119648$	$26.640147687438905$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$1.33950218822$
4.0.3392.1	$x^{4} - 2 x^{3} - 6 x + 17$	$4$	[0,2]	$2^{6}\cdot 53$	$2$	$7.63157120875$	$20.591260281974$		✓	$D_{4}$ (as 4T3)	trivial	$4$	$1$	$4.87147534378$
4.0.4240.1	$x^{4} - 5 x^{2} - 6 x + 18$	$4$	[0,2]	$2^{4}\cdot 5\cdot 53$	$3$	$8.06940409103$	$32.55764119219941$			$D_{4}$ (as 4T3)	trivial	$4$	$1$	$8.6189189872$
4.0.5300.1	$x^{4} - x^{3} + 4 x + 4$	$4$	[0,2]	$2^{2}\cdot 5^{2}\cdot 53$	$3$	$8.53235599895$	$33.791276769558415$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$5.3586317518$
4.0.5724.1	$x^{4} - x^{3} - 3 x^{2} + 6$	$4$	[0,2]	$2^{2}\cdot 3^{3}\cdot 53$	$3$	$8.698110213$	$52.457718022675515$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$5.48337257369$
4.2.7155.1	$x^{4} - x^{3} - 3$	$4$	[2,1]	$-\,3^{3}\cdot 5\cdot 53$	$3$	$9.19713178547$	$58.64951172160915$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$4.71614770996$
4.0.7473.1	$x^{4} + x^{2} - 3 x + 3$	$4$	[0,2]	$3\cdot 47\cdot 53$	$3$	$9.29766181664$	$86.44651525654461$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$4.37359486372$
4.0.10865.1	$x^{4} - x^{3} + 4 x^{2} - 5 x + 4$	$4$	[0,2]	$5\cdot 41\cdot 53$	$3$	$10.2095695656$	$104.23531071570709$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$7.38941718743$
4.0.11024.1	$x^{4} - 2 x^{3} - 11 x^{2} + 12 x + 40$	$4$	[0,2]	$2^{4}\cdot 13\cdot 53$	$3$	$10.2467184009$	$52.49761899362675$			$D_{4}$ (as 4T3)	trivial	$4$	$1$	$18.8280785288$
4.0.11024.2	$x^{4} - 2 x^{3} - 7 x^{2} + 8 x + 41$	$4$	[0,2]	$2^{4}\cdot 13\cdot 53$	$3$	$10.2467184009$	$52.49761899362675$			$D_{4}$ (as 4T3)	trivial	$4$	$1$	$3.31738659771$
4.2.11448.1	$x^{4} - x^{3} - 3 x^{2} + 8 x - 8$	$4$	[2,1]	$-\,2^{3}\cdot 3^{3}\cdot 53$	$3$	$10.3438545524$	$46.93792312435236$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$8.19666709785$
4.2.13356.1	$x^{4} - x^{3} - 7 x^{2} + 7 x - 2$	$4$	[2,1]	$-\,2^{2}\cdot 3^{2}\cdot 7\cdot 53$	$4$	$10.7502633319$	$52.95832566335522$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$11.2441574258$
4.0.14416.1	$x^{4} - 2 x^{2} - 2 x + 5$	$4$	[0,2]	$2^{4}\cdot 17\cdot 53$	$3$	$10.9574927861$	$67.38512791277556$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$5.92373999934$
4.4.15317.1	$x^{4} - 2 x^{3} - 4 x^{2} + 5 x + 2$	$4$	[4,0]	$17^{2}\cdot 53$	$2$	$11.1248311501$	$30.01666203960727$		✓	$D_{4}$ (as 4T3)	trivial	$2$	$3$	$15.164099755$
4.4.15529.1	$x^{4} - x^{3} - 6 x^{2} - x + 2$	$4$	[4,0]	$53\cdot 293$	$2$	$11.1631271764$	$124.61540835707277$		✓	$S_4$ (as 4T5)	trivial	$2$	$3$	$9.77175030835$
4.0.15529.2	$x^{4} - x^{3} + 3 x^{2} - 4 x + 5$	$4$	[0,2]	$53\cdot 293$	$2$	$11.1631271764$	$124.61540835707277$		✓	$S_4$ (as 4T5)	$[2]$	$2$	$1$	$3.12361015112$
4.2.16271.1	$x^{4} - x^{3} - 2 x^{2} + 7 x - 6$	$4$	[2,1]	$-\,53\cdot 307$	$2$	$11.2941502568$	$127.55783002230793$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$8.91756509991$
4.0.17172.1	$x^{4} - x^{3} - 3 x^{2} + 3 x + 6$	$4$	[0,2]	$2^{2}\cdot 3^{4}\cdot 53$	$3$	$11.4473568131$	$50.00187286567662$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$14.301810469$
4.0.20776.1	$x^{4} - x^{3} + 11 x^{2} - x + 22$	$4$	[0,2]	$2^{3}\cdot 7^{2}\cdot 53$	$3$	$12.0057828555$	$54.47935388750494$			$D_{4}$ (as 4T3)	$[2]$	$2$	$1$	$8.19589383834$
4.0.21200.1	$x^{4} + 6 x^{2} - 6 x + 2$	$4$	[0,2]	$2^{4}\cdot 5^{2}\cdot 53$	$3$	$12.0665735727$	$47.78808189741242$		✓	$S_4$ (as 4T5)	$[2]$	$2$	$1$	$6.16433571337$
4.2.21412.1	$x^{4} - x^{3} - 3 x^{2} + 7 x + 4$	$4$	[2,1]	$-\,2^{2}\cdot 53\cdot 101$	$3$	$12.0966275379$	$146.32839779072276$			$S_4$ (as 4T5)	trivial	$2$	$2$	$21.8041516249$
4.4.22896.1	$x^{4} - 9 x^{2} - 4 x + 3$	$4$	[4,0]	$2^{4}\cdot 3^{3}\cdot 53$	$3$	$12.3009854302$	$74.18641627881125$		✓	$S_4$ (as 4T5)	trivial	$2$	$3$	$12.9810322765$
4.0.23373.1	$x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 28$	$4$	[0,2]	$3^{2}\cdot 7^{2}\cdot 53$	$3$	$12.3645585313$	$33.36165463522455$			$D_{4}$ (as 4T3)	$[2]$	$2$	$1$	$6.99760268097$
4.0.24645.1	$x^{4} + 2 x^{2} - 7 x + 5$	$4$	[0,2]	$3\cdot 5\cdot 31\cdot 53$	$4$	$12.5294557196$	$156.9872606296447$		✓	$S_4$ (as 4T5)	$[3]$	$2$	$1$	$2.03064876616$
4.0.25228.1	$x^{4} - x^{3} + 6 x^{2} - 9 x + 7$	$4$	[0,2]	$2^{2}\cdot 7\cdot 17\cdot 53$	$4$	$12.6029062478$	$158.8332458901473$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$10.7823880293$
4.2.26871.1	$x^{4} - 2 x^{3} + 5 x^{2} - 4 x - 9$	$4$	[2,1]	$-\,3\cdot 13^{2}\cdot 53$	$3$	$12.803271565$	$45.46427168667722$			$D_{4}$ (as 4T3)	$[2]$	$2$	$2$	$7.24847935197$
4.0.27136.1	$x^{4} - 18 x^{2} + 106$	$4$	[0,2]	$2^{9}\cdot 53$	$2$	$12.8347217444$	$48.97454646839284$			$D_{4}$ (as 4T3)	$[3]$	$4$	$1$	$6.71152624662$
4.2.27560.1	$x^{4} - 5 x^{2} - 8 x + 8$	$4$	[2,1]	$-\,2^{3}\cdot 5\cdot 13\cdot 53$	$4$	$12.8845662618$	$166.01204775557707$			$S_4$ (as 4T5)	trivial	$2$	$2$	$14.1905674233$
4.2.30263.1	$x^{4} - x^{3} - 4 x^{2} + 5 x - 4$	$4$	[2,1]	$-\,53\cdot 571$	$2$	$13.1894897425$	$173.9626396672573$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$11.6118907602$
4.0.32065.1	$x^{4} - x^{3} - 6 x^{2} - x + 23$	$4$	[0,2]	$5\cdot 11^{2}\cdot 53$	$3$	$13.3815928259$	$53.99073994677235$			$D_{4}$ (as 4T3)	$[2]$	$2$	$1$	$3.84795173568$
4.0.32701.1	$x^{4} - x^{3} + 5 x^{2} - 7 x + 5$	$4$	[0,2]	$53\cdot 617$	$2$	$13.4474599157$	$180.83417818543043$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$4.81989682968$
4.2.33072.1	$x^{4} - 2 x^{3} - 4 x^{2} + 2 x + 6$	$4$	[2,1]	$-\,2^{4}\cdot 3\cdot 13\cdot 53$	$4$	$13.4854398055$	$102.06383904464309$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$20.5394184925$
4.2.33867.1	$x^{4} - 2 x^{3} - 2 x^{2} - 9 x - 6$	$4$	[2,1]	$-\,3^{2}\cdot 53\cdot 71$	$3$	$13.5657616401$	$106.24970588194586$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$30.1372012979$
4.2.34291.1	$x^{4} - 8 x^{2} - 7 x + 25$	$4$	[2,1]	$-\,53\cdot 647$	$2$	$13.6080230917$	$185.17829246431668$			$S_4$ (as 4T5)	trivial	$2$	$2$	$8.93363668602$
4.2.35563.1	$x^{4} - 2 x^{3} - 2 x^{2} - 5 x - 3$	$4$	[2,1]	$-\,11\cdot 53\cdot 61$	$3$	$13.7324996759$	$188.58154734755996$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$11.0503192173$
4.0.38160.2	$x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 148$	$4$	[0,2]	$2^{4}\cdot 3^{2}\cdot 5\cdot 53$	$4$	$13.9766178725$	$56.39148871948674$			$D_{4}$ (as 4T3)	$[2]$	$4$	$1$	$13.698312275$
4.2.38531.1	$x^{4} - 2 x^{3} + 2 x^{2} + x - 6$	$4$	[2,1]	$-\,53\cdot 727$	$2$	$14.010465664$	$196.29314812290315$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$27.2183240733$
4.2.38796.1	$x^{4} - x^{3} + x^{2} + 9 x + 2$	$4$	[2,1]	$-\,2^{2}\cdot 3\cdot 53\cdot 61$	$4$	$14.0344933044$	$156.3328133369983$			$S_4$ (as 4T5)	trivial	$2$	$2$	$23.3830905933$
4.2.40492.1	$x^{4} - 8 x^{2} - 6 x + 8$	$4$	[2,1]	$-\,2^{2}\cdot 53\cdot 191$	$3$	$14.1854235338$	$159.7133731913881$			$S_4$ (as 4T5)	trivial	$2$	$2$	$24.2212565596$
4.2.41287.1	$x^{4} - x^{3} + 4 x^{2} - 9 x - 4$	$4$	[2,1]	$-\,19\cdot 41\cdot 53$	$3$	$14.2545440966$	$203.1920274026518$			$S_4$ (as 4T5)	trivial	$2$	$2$	$19.8560850828$
4.2.42188.1	$x^{4} - x^{3} - 6 x - 2$	$4$	[2,1]	$-\,2^{2}\cdot 53\cdot 199$	$3$	$14.331684347$	$163.02384680310564$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$23.4787923394$
4.0.43248.1	$x^{4} + 7 x^{2} - 6 x + 13$	$4$	[0,2]	$2^{4}\cdot 3\cdot 17\cdot 53$	$4$	$14.420871503$	$147.0510115572144$			$S_4$ (as 4T5)	trivial	$2$	$1$	$16.3132665507$
4.0.44573.1	$x^{4} - 2 x^{3} + 8 x^{2} - 7 x + 5$	$4$	[0,2]	$29^{2}\cdot 53$	$2$	$14.5300786918$	$39.20459156782532$	✓	✓	$D_{4}$ (as 4T3)	trivial	$2$	$1$	$3.29446229274$
4.2.45315.1	$x^{4} - 2 x^{3} - 2 x^{2} - 9 x + 15$	$4$	[2,1]	$-\,3^{2}\cdot 5\cdot 19\cdot 53$	$4$	$14.5901748343$	$122.90240030202827$			$S_4$ (as 4T5)	trivial	$2$	$2$	$15.6939522675$
4.2.45527.1	$x^{4} - x^{3} - 2 x^{2} - 7 x + 5$	$4$	[2,1]	$-\,53\cdot 859$	$2$	$14.6072095099$	$213.3705696669529$			$S_4$ (as 4T5)	trivial	$2$	$2$	$15.2112105882$
4.2.45580.1	$x^{4} - x^{3} - 6 x^{2} + 4 x - 2$	$4$	[2,1]	$-\,2^{2}\cdot 5\cdot 43\cdot 53$	$4$	$14.6114588804$	$169.45087998326878$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$26.377892763$
4.0.46004.1	$x^{4} - x^{3} + 3 x^{2} + 7 x + 4$	$4$	[0,2]	$2^{2}\cdot 7\cdot 31\cdot 53$	$4$	$14.6453211211$	$170.23719919477608$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$12.4422882564$
4.0.48336.1	$x^{4} + 7 x^{2} - 6 x + 6$	$4$	[0,2]	$2^{4}\cdot 3\cdot 19\cdot 53$	$4$	$14.8274912692$	$109.92724866929036$			$S_4$ (as 4T5)	$[6]$	$2$	$1$	$3.96546283565$
4.0.48389.1	$x^{4} - x^{3} - 5 x^{2} - x + 15$	$4$	[0,2]	$11\cdot 53\cdot 83$	$3$	$14.8315541525$	$219.97499857938402$		✓	$S_4$ (as 4T5)	$[4]$	$2$	$1$	$2.61844619346$

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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.