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

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 115272 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
4.0.1593.1	$x^{4} - 2 x^{3} + 3 x^{2} - x + 2$	$4$	[0,2]	$3^{3}\cdot 59$	$2$	$6.31762645991$	$27.67371536012528$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$2.08381549833$
4.2.3776.2	$x^{4} - 2 x^{3} - 3 x^{2} + 2 x + 3$	$4$	[2,1]	$-\,2^{6}\cdot 59$	$2$	$7.83895184211$	$21.72556098240043$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$2.8249567292$
4.2.7375.1	$x^{4} - 2 x^{3} - x^{2} + 2 x - 19$	$4$	[2,1]	$-\,5^{3}\cdot 59$	$2$	$9.26702866007$	$25.683458749990002$			$D_{4}$ (as 4T3)	trivial	$2$	$2$	$3.67083710185$
4.0.11328.2	$x^{4} - 2 x^{3} + 9 x^{2} - 6 x + 9$	$4$	[0,2]	$2^{6}\cdot 3\cdot 59$	$3$	$10.3166408082$	$37.62977544445356$			$D_{4}$ (as 4T3)	trivial	$2$	$1$	$6.0890810539$
4.2.13275.1	$x^{4} - x^{3} + 2 x^{2} - 14 x - 29$	$4$	[2,1]	$-\,3^{2}\cdot 5^{2}\cdot 59$	$3$	$10.7339268778$	$29.748949561287034$			$D_{4}$ (as 4T3)	trivial	$2$	$2$	$2.80856200422$
4.0.14160.1	$x^{4} - 2 x^{3} + 4 x^{2} + 6 x + 2$	$4$	[0,2]	$2^{4}\cdot 3\cdot 5\cdot 59$	$4$	$10.9085195258$	$66.78413371922893$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$6.26608652331$
4.2.15104.1	$x^{4} - 2 x^{2} - 4 x + 2$	$4$	[2,1]	$-\,2^{8}\cdot 59$	$2$	$11.0859520099$	$30.72458299147443$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$9.25799179456$
4.2.15576.1	$x^{4} - x^{3} + 7 x^{2} - 4$	$4$	[2,1]	$-\,2^{3}\cdot 3\cdot 11\cdot 59$	$4$	$11.171564174$	$124.80384609458156$			$S_4$ (as 4T5)	trivial	$2$	$2$	$10.5505342279$
4.2.16048.1	$x^{4} - 4 x^{2} - 12 x - 4$	$4$	[2,1]	$-\,2^{4}\cdot 17\cdot 59$	$3$	$11.2552521507$	$71.09713954990194$			$S_4$ (as 4T5)	trivial	$2$	$2$	$8.58996393544$
4.2.16756.1	$x^{4} + 3 x^{2} - 8 x - 4$	$4$	[2,1]	$-\,2^{2}\cdot 59\cdot 71$	$3$	$11.3773884966$	$129.44496900227526$			$S_4$ (as 4T5)	trivial	$2$	$2$	$19.3759573618$
4.2.17051.1	$x^{4} - 2 x^{3} + x - 4$	$4$	[2,1]	$-\,17^{2}\cdot 59$	$2$	$11.4271377743$	$31.670175244226233$		✓	$D_{4}$ (as 4T3)	trivial	$2$	$2$	$13.3907143433$
4.2.17051.2	$x^{4} - x^{3} + 4 x^{2} - 4 x - 1$	$4$	[2,1]	$-\,17^{2}\cdot 59$	$2$	$11.4271377743$	$31.670175244226233$		✓	$S_4$ (as 4T5)	$[2]$	$2$	$2$	$2.8636187974$
4.0.20532.1	$x^{4} - x^{3} + 2 x^{2} - 4 x + 10$	$4$	[0,2]	$2^{2}\cdot 3\cdot 29\cdot 59$	$4$	$11.9703765983$	$113.72928162184428$			$S_4$ (as 4T5)	trivial	$2$	$1$	$15.2887836011$
4.0.21417.1	$x^{4} - 2 x^{3} + x^{2} + 11$	$4$	[0,2]	$3\cdot 11^{2}\cdot 59$	$3$	$12.0973336588$	$44.12482294582042$			$D_{4}$ (as 4T3)	trivial	$2$	$1$	$7.3320336956$
4.2.22892.1	$x^{4} - x^{3} + 4 x^{2} - 4 x - 4$	$4$	[2,1]	$-\,2^{2}\cdot 59\cdot 97$	$3$	$12.3004481403$	$120.08770268889616$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$17.0740354578$
4.2.23600.1	$x^{4} + 3 x^{2} - 59$	$4$	[2,1]	$-\,2^{4}\cdot 5^{2}\cdot 59$	$3$	$12.3944711447$	$34.351128074635334$			$D_{4}$ (as 4T3)	trivial	$2$	$2$	$7.23230946522$
4.0.23600.1	$x^{4} + 16 x^{2} + 59$	$4$	[0,2]	$2^{4}\cdot 5^{2}\cdot 59$	$3$	$12.3944711447$	$48.579831205964474$	✓		$D_{4}$ (as 4T3)	$[2]$	$2$	$1$	$0.962423650119$
4.2.25488.1	$x^{4} - 3 x^{2} - 4 x + 3$	$4$	[2,1]	$-\,2^{4}\cdot 3^{3}\cdot 59$	$3$	$12.6352529198$	$78.27308716708362$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$11.6963470405$
4.4.25488.1	$x^{4} - 9 x^{2} - 6 x + 6$	$4$	[4,0]	$2^{4}\cdot 3^{3}\cdot 59$	$3$	$12.6352529198$	$35.01845187552935$			$D_{4}$ (as 4T3)	trivial	$2$	$3$	$23.0109465771$
4.2.26019.3	$x^{4} - x^{3} + 2 x^{2} - 14 x + 7$	$4$	[2,1]	$-\,3^{2}\cdot 7^{2}\cdot 59$	$3$	$12.700553559$	$35.199431813596085$			$D_{4}$ (as 4T3)	trivial	$2$	$2$	$8.34469057272$
4.0.26845.1	$x^{4} - x^{3} + 2 x^{2} + 5$	$4$	[0,2]	$5\cdot 7\cdot 13\cdot 59$	$4$	$12.8001733745$	$163.84443841644426$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$5.99663778213$
4.4.28025.1	$x^{4} - x^{3} - 17 x^{2} + 8 x + 64$	$4$	[4,0]	$5^{2}\cdot 19\cdot 59$	$3$	$12.9385737401$	$74.86654793697917$			$D_{4}$ (as 4T3)	trivial	$2$	$3$	$11.6867896386$
4.0.28733.1	$x^{4} - x^{3} + 3 x^{2} + 5 x + 3$	$4$	[0,2]	$59\cdot 487$	$2$	$13.0195280982$	$169.50811190028637$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$6.68159829077$
4.2.30267.1	$x^{4} - x^{3} + 6 x - 3$	$4$	[2,1]	$-\,3^{3}\cdot 19\cdot 59$	$3$	$13.1899255498$	$120.6269286470954$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$8.68986031241$
4.2.31860.1	$x^{4} - x^{3} - 9 x^{2} + 8 x - 2$	$4$	[2,1]	$-\,2^{2}\cdot 3^{3}\cdot 5\cdot 59$	$4$	$13.3601533437$	$78.30363886048863$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$16.2446685255$
4.2.32627.1	$x^{4} - x^{3} + 5 x^{2} + 3 x - 1$	$4$	[2,1]	$-\,7\cdot 59\cdot 79$	$3$	$13.4398457938$	$180.6294549623621$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$6.99288370976$
4.2.32863.2	$x^{4} - x^{3} - 6 x^{2} + 7 x + 4$	$4$	[2,1]	$-\,59\cdot 557$	$2$	$13.4640836665$	$181.28154897837783$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$12.4823944525$
4.2.32863.3	$x^{4} + 9 x^{2} - 9 x + 2$	$4$	[2,1]	$-\,59\cdot 557$	$2$	$13.4640836665$	$181.28154897837783$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$17.8239961758$
4.0.33217.1	$x^{4} - x^{3} + 7 x^{2} + 6 x + 5$	$4$	[0,2]	$59\cdot 563$	$2$	$13.5001968661$	$182.25531542317222$			$S_4$ (as 4T5)	trivial	$2$	$1$	$9.57677214844$
4.2.33984.2	$x^{4} - 2 x^{3} + 4 x^{2} + 12 x - 18$	$4$	[2,1]	$-\,2^{6}\cdot 3^{2}\cdot 59$	$3$	$13.5774628686$	$37.62977544445356$			$D_{4}$ (as 4T3)	trivial	$2$	$2$	$16.522239197$
4.2.34515.1	$x^{4} - x^{3} - x^{2} + x - 5$	$4$	[2,1]	$-\,3^{2}\cdot 5\cdot 13\cdot 59$	$4$	$13.6301918719$	$107.26136303441235$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$9.13244302114$
4.2.36639.1	$x^{4} + 3 x^{2} - 3 x - 9$	$4$	[2,1]	$-\,3^{3}\cdot 23\cdot 59$	$3$	$13.8352146963$	$132.7184764912722$			$S_4$ (as 4T5)	trivial	$2$	$2$	$19.467521549$
4.0.36816.1	$x^{4} - 2 x^{3} - 4 x^{2} + 2 x + 27$	$4$	[0,2]	$2^{4}\cdot 3\cdot 13\cdot 59$	$4$	$13.8518937075$	$107.68617990825199$			$S_4$ (as 4T5)	trivial	$2$	$1$	$19.129495043$
4.2.37111.1	$x^{4} - x^{3} + 4 x^{2} + 3 x - 2$	$4$	[2,1]	$-\,17\cdot 37\cdot 59$	$3$	$13.8795589016$	$192.6421553035576$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$12.3124803457$
4.4.37229.1	$x^{4} - x^{3} - 6 x^{2} + 2 x + 3$	$4$	[4,0]	$59\cdot 631$	$2$	$13.8905788062$	$192.9481795716145$		✓	$S_4$ (as 4T5)	trivial	$2$	$3$	$13.9596253836$
4.0.37229.2	$x^{4} - x^{3} + 6 x^{2} - 6 x + 5$	$4$	[0,2]	$59\cdot 631$	$2$	$13.8905788062$	$192.9481795716145$		✓	$S_4$ (as 4T5)	$[2]$	$2$	$1$	$3.1966550931$
4.2.37996.1	$x^{4} - x^{3} - 5 x^{2} - 5 x - 4$	$4$	[2,1]	$-\,2^{2}\cdot 7\cdot 23\cdot 59$	$4$	$13.9615768036$	$154.71257255279158$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$20.3394415274$
4.2.41300.1	$x^{4} - x^{3} + 10 x^{2} - x - 1$	$4$	[2,1]	$-\,2^{2}\cdot 5^{2}\cdot 7\cdot 59$	$4$	$14.255666043$	$118.84612454577355$			$S_4$ (as 4T5)	trivial	$2$	$2$	$20.9228858961$
4.0.42421.1	$x^{4} - x^{3} + 3 x^{2} - 3 x + 7$	$4$	[0,2]	$59\cdot 719$	$2$	$14.3514316016$	$205.96358901514608$		✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$7.74076573795$
4.2.42775.1	$x^{4} - x^{3} - 23 x - 101$	$4$	[2,1]	$-\,5^{2}\cdot 29\cdot 59$	$3$	$14.3812787609$	$92.49324299644812$			$D_{4}$ (as 4T3)	$[2]$	$2$	$2$	$3.40559589451$
4.2.42775.2	$x^{4} - 2 x^{3} - 7 x^{2} + 8 x - 109$	$4$	[2,1]	$-\,5^{2}\cdot 29\cdot 59$	$3$	$14.3812787609$	$92.49324299644812$			$D_{4}$ (as 4T3)	$[2]$	$2$	$2$	$4.27177902501$
4.0.44368.1	$x^{4} - 2 x^{3} - 6 x + 19$	$4$	[0,2]	$2^{4}\cdot 47\cdot 59$	$3$	$14.5133431273$	$118.21609148264857$			$S_4$ (as 4T5)	trivial	$2$	$1$	$13.4357953065$
4.2.45076.1	$x^{4} - x^{3} - 8 x^{2} + 11 x + 3$	$4$	[2,1]	$-\,2^{2}\cdot 59\cdot 191$	$3$	$14.5708988228$	$212.31109250343$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$23.6129237616$
4.0.45312.2	$x^{4} + 8 x^{2} - 16 x + 41$	$4$	[0,2]	$2^{8}\cdot 3\cdot 59$	$3$	$14.5899333491$	$53.21653878260028$			$D_{4}$ (as 4T3)	$[2]$	$2$	$1$	$5.68133941153$
4.2.46256.1	$x^{4} - 3 x^{2} - 6 x + 1$	$4$	[2,1]	$-\,2^{4}\cdot 7^{2}\cdot 59$	$3$	$14.6653360323$	$56.21532098938853$		✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$9.66250412405$
4.2.46256.2	$x^{4} - 2 x^{3} + 2 x^{2} + 12 x - 4$	$4$	[2,1]	$-\,2^{4}\cdot 7^{2}\cdot 59$	$3$	$14.6653360323$	$56.21532098938853$			$S_4$ (as 4T5)	trivial	$2$	$2$	$11.9116666458$
4.2.47200.1	$x^{4} - 2 x^{3} - 3 x^{2} - 16 x + 4$	$4$	[2,1]	$-\,2^{5}\cdot 5^{2}\cdot 59$	$3$	$14.739593272$	$68.70225614927067$		✓	$S_4$ (as 4T5)	$[2]$	$2$	$2$	$12.8550977515$
4.2.47200.2	$x^{4} - 2 x^{3} - 18 x - 1$	$4$	[2,1]	$-\,2^{5}\cdot 5^{2}\cdot 59$	$3$	$14.739593272$	$89.83922566626457$			$S_4$ (as 4T5)	trivial	$2$	$2$	$25.6787674667$
4.0.48321.1	$x^{4} - x^{3} - x^{2} - 2 x + 7$	$4$	[0,2]	$3^{2}\cdot 7\cdot 13\cdot 59$	$4$	$14.82634079$	$126.91335627111907$		✓	$S_4$ (as 4T5)	$[2]$	$2$	$1$	$6.00732187396$
4.2.49619.1	$x^{4} - x^{3} - 2 x^{2} + 12 x - 1$	$4$	[2,1]	$-\,29^{2}\cdot 59$	$2$	$14.9249196543$	$41.36423575989287$			$D_{4}$ (as 4T3)	trivial	$2$	$2$	$5.42447041305$

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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.