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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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.