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	Class number	Class group structure	CM field

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Results (1-50 of 63550 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
3.1.411.1	$x^{3} - x^{2} + 5 x - 2$	$3$	[1,1]	$-\,3\cdot 137$	$2$	$7.43499374162$	$20.273134932713294$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$4.02853094339$
3.1.959.1	$x^{3} - x^{2} + 6 x + 1$	$3$	[1,1]	$-\,7\cdot 137$	$2$	$9.86142181275$	$30.967725134404045$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.82256798591$
3.1.1096.1	$x^{3} - x^{2} + x - 13$	$3$	[1,1]	$-\,2^{3}\cdot 137$	$2$	$10.310273471$	$33.1058907144937$		$S_3$ (as 3T2)	trivial	$2$	$1$	$8.77878491855$
3.1.2740.1	$x^{3} - x^{2} - 10$	$3$	[1,1]	$-\,2^{2}\cdot 5\cdot 137$	$3$	$13.9931939707$	$52.3450093132096$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$17.2509650403$
3.3.3973.1	$x^{3} - 10 x - 1$	$3$	[3,0]	$29\cdot 137$	$2$	$15.8382133311$	$63.0317380372777$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$13.1506676099$
3.1.4795.1	$x^{3} - x^{2} - 5 x + 42$	$3$	[1,1]	$-\,5\cdot 7\cdot 137$	$3$	$16.8627940998$	$69.24593850905626$		$S_3$ (as 3T2)	trivial	$2$	$1$	$18.425873936$
3.1.5343.1	$x^{3} - x^{2} + 6 x - 15$	$3$	[1,1]	$-\,3\cdot 13\cdot 137$	$3$	$17.4821586876$	$73.09582751429797$	✓	$S_3$ (as 3T2)	$[4]$	$2$	$1$	$2.61291676651$
3.1.6028.1	$x^{3} - x^{2} + 11 x + 3$	$3$	[1,1]	$-\,2^{2}\cdot 11\cdot 137$	$3$	$18.1994283925$	$61.62306422422653$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$14.8196608733$
3.1.6987.1	$x^{3} - x^{2} + x + 48$	$3$	[1,1]	$-\,3\cdot 17\cdot 137$	$3$	$19.1174625345$	$83.58827668997608$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$12.0834642347$
3.1.7535.1	$x^{3} - x^{2} - 16 x - 25$	$3$	[1,1]	$-\,5\cdot 11\cdot 137$	$3$	$19.6047399337$	$86.80437776978762$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$6.25014291497$
3.3.8220.1	$x^{3} - x^{2} - 20 x + 12$	$3$	[3,0]	$2^{2}\cdot 3\cdot 5\cdot 137$	$4$	$20.1816779915$	$90.66421565314509$		$S_3$ (as 3T2)	trivial	$2$	$2$	$52.459522391$
3.3.8905.1	$x^{3} - x^{2} - 20 x + 37$	$3$	[3,0]	$5\cdot 13\cdot 137$	$3$	$20.7273910614$	$94.36630754670864$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$10.9545554016$
3.1.10412.1	$x^{3} - x^{2} - 9 x + 63$	$3$	[1,1]	$-\,2^{2}\cdot 19\cdot 137$	$3$	$21.8362497642$	$80.98857318012652$		$S_3$ (as 3T2)	trivial	$2$	$1$	$37.7186399497$
3.3.11097.1	$x^{3} - 21 x - 31$	$3$	[3,0]	$3^{4}\cdot 137$	$2$	$22.3049812248$	$50.643295250437625$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$16.5359455291$
3.1.11371.1	$x^{3} - 26 x - 55$	$3$	[1,1]	$-\,83\cdot 137$	$2$	$22.4870707612$	$106.63489110042735$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$31.9309447915$
3.1.11919.1	$x^{3} + 3 x - 63$	$3$	[1,1]	$-\,3\cdot 29\cdot 137$	$3$	$22.8426563712$	$109.17417276993676$		$S_3$ (as 3T2)	trivial	$2$	$1$	$17.8125456632$
3.1.12056.1	$x^{3} - x^{2} - 11 x - 41$	$3$	[1,1]	$-\,2^{3}\cdot 11\cdot 137$	$3$	$22.9298429277$	$109.79981785048643$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$21.9396505529$
3.1.13700.1	$x^{3} + 5 x - 90$	$3$	[1,1]	$-\,2^{2}\cdot 5^{2}\cdot 137$	$3$	$23.9280251071$	$68.44950031880546$		$S_3$ (as 3T2)	trivial	$2$	$1$	$68.3128388079$
3.3.14385.1	$x^{3} - 33 x - 23$	$3$	[3,0]	$3\cdot 5\cdot 7\cdot 137$	$4$	$24.3203575447$	$119.93748371547571$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$31.9568604859$
3.1.14796.1	$x^{3} + 6 x - 70$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}\cdot 137$	$3$	$24.5498084308$	$66.94041665484356$		$S_3$ (as 3T2)	$[6]$	$2$	$1$	$5.94286138756$
3.1.14796.2	$x^{3} + 12 x - 44$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}\cdot 137$	$3$	$24.5498084308$	$66.94041665484356$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$4.90412092671$
3.1.14796.3	$x^{3} + 24 x - 12$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}\cdot 137$	$3$	$24.5498084308$	$66.94041665484356$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$4.59481465016$
3.1.16303.1	$x^{3} - x^{2} - 8 x + 29$	$3$	[1,1]	$-\,7\cdot 17\cdot 137$	$3$	$25.3564923648$	$127.68320171424274$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$16.8018676213$
3.1.16851.1	$x^{3} - x^{2} + 3 x + 24$	$3$	[1,1]	$-\,3\cdot 41\cdot 137$	$3$	$25.637473602$	$129.81140165640304$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$32.5996285413$
3.1.17399.1	$x^{3} - x^{2} - 6 x - 125$	$3$	[1,1]	$-\,127\cdot 137$	$2$	$25.9124272637$	$131.90526903804866$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$5.00286382$
3.1.17399.2	$x^{3} - x^{2} - 24 x - 81$	$3$	[1,1]	$-\,127\cdot 137$	$2$	$25.9124272637$	$131.90526903804866$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$11.2473439679$
3.1.17399.3	$x^{3} - x^{2} + 127$	$3$	[1,1]	$-\,127\cdot 137$	$2$	$25.9124272637$	$131.90526903804866$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$4.11195150151$
3.1.17399.4	$x^{3} - x^{2} - 34 x + 104$	$3$	[1,1]	$-\,127\cdot 137$	$2$	$25.9124272637$	$131.90526903804866$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$24.8532014411$
3.1.18084.1	$x^{3} - x^{2} - 16 x + 112$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 11\cdot 137$	$4$	$26.2481177789$	$134.47676379211393$		$S_3$ (as 3T2)	trivial	$2$	$1$	$61.958041134$
3.1.18495.1	$x^{3} + 33 x - 29$	$3$	[1,1]	$-\,3^{3}\cdot 5\cdot 137$	$3$	$26.4454794585$	$94.29458415489817$		$S_3$ (as 3T2)	trivial	$2$	$1$	$14.3227792634$
3.1.19043.1	$x^{3} - x^{2} + 17 x - 10$	$3$	[1,1]	$-\,137\cdot 139$	$2$	$26.704131368$	$137.99637676402958$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$25.3106221787$
3.1.20276.1	$x^{3} - x^{2} + 4 x - 56$	$3$	[1,1]	$-\,2^{2}\cdot 37\cdot 137$	$3$	$27.2684693689$	$142.3938200906205$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$10.6113018798$
3.1.20276.2	$x^{3} - x^{2} + 36 x - 18$	$3$	[1,1]	$-\,2^{2}\cdot 37\cdot 137$	$3$	$27.2684693689$	$142.3938200906205$		$S_3$ (as 3T2)	$[15]$	$2$	$1$	$4.96289387073$
3.1.20276.3	$x^{3} - x^{2} - 11 x - 53$	$3$	[1,1]	$-\,2^{2}\cdot 37\cdot 137$	$3$	$27.2684693689$	$142.3938200906205$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$7.95330586803$
3.1.20276.4	$x^{3} + 26 x - 20$	$3$	[1,1]	$-\,2^{2}\cdot 37\cdot 137$	$3$	$27.2684693689$	$142.3938200906205$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$14.7755919298$
3.3.20276.1	$x^{3} - x^{2} - 37 x + 81$	$3$	[3,0]	$2^{2}\cdot 37\cdot 137$	$3$	$27.2684693689$	$113.01804990281076$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$36.6094480691$
3.1.20824.1	$x^{3} - x^{2} + 5 x - 57$	$3$	[1,1]	$-\,2^{3}\cdot 19\cdot 137$	$3$	$27.5119507287$	$144.30523206037958$		$S_3$ (as 3T2)	trivial	$2$	$1$	$41.2384806677$
3.1.22468.1	$x^{3} - x^{2} + 17 x - 57$	$3$	[1,1]	$-\,2^{2}\cdot 41\cdot 137$	$3$	$28.2176908931$	$149.89329538041386$		$S_3$ (as 3T2)	trivial	$2$	$1$	$45.1743704898$
3.1.23564.1	$x^{3} - x^{2} - 21 x - 41$	$3$	[1,1]	$-\,2^{2}\cdot 43\cdot 137$	$3$	$28.6692504131$	$121.8375550060875$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$16.1355294945$
3.1.24523.1	$x^{3} - x^{2} - 25 x + 66$	$3$	[1,1]	$-\,137\cdot 179$	$2$	$29.0530144016$	$156.59821199490113$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$17.9326319476$
3.3.25893.1	$x^{3} - 24 x - 33$	$3$	[3,0]	$3^{3}\cdot 7\cdot 137$	$3$	$29.5842654382$	$111.5708565949157$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$2$	$29.7681523156$
3.1.26715.1	$x^{3} - x^{2} + 25 x - 90$	$3$	[1,1]	$-\,3\cdot 5\cdot 13\cdot 137$	$4$	$29.8940708519$	$163.44723919356974$		$S_3$ (as 3T2)	$[2]$	$2$	$1$	$26.9129375039$
3.1.27263.1	$x^{3} - x^{2} + 20 x + 116$	$3$	[1,1]	$-\,137\cdot 199$	$2$	$30.0970928345$	$165.1151113617406$		$S_3$ (as 3T2)	$[2]$	$2$	$1$	$33.0951068354$
3.1.28907.1	$x^{3} - x^{2} + 13 x - 32$	$3$	[1,1]	$-\,137\cdot 211$	$2$	$30.6902910798$	$170.02058698875263$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$12.607221798$
3.1.29044.1	$x^{3} - x^{2} + 13 x + 59$	$3$	[1,1]	$-\,2^{2}\cdot 53\cdot 137$	$3$	$30.7386985592$	$170.42300314218147$		$S_3$ (as 3T2)	trivial	$2$	$1$	$48.7361682115$
3.1.29592.1	$x^{3} - 21 x - 106$	$3$	[1,1]	$-\,2^{3}\cdot 3^{3}\cdot 137$	$3$	$30.9308204129$	$119.27426277916068$		$S_3$ (as 3T2)	$[2]$	$2$	$1$	$31.9948136767$
3.3.29592.1	$x^{3} - 51 x - 46$	$3$	[3,0]	$2^{3}\cdot 3^{3}\cdot 137$	$3$	$30.9308204129$	$119.27426277916068$		$S_3$ (as 3T2)	trivial	$2$	$2$	$80.6101190453$
3.1.32195.1	$x^{3} - x^{2} + 15 x + 22$	$3$	[1,1]	$-\,5\cdot 47\cdot 137$	$3$	$31.8123786572$	$179.42965195307045$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$17.2528665334$
3.1.32332.1	$x^{3} - x^{2} - 9 x - 67$	$3$	[1,1]	$-\,2^{2}\cdot 59\cdot 137$	$3$	$31.8574387438$	$142.71609472165542$		$S_3$ (as 3T2)	trivial	$2$	$1$	$28.1281587056$
3.1.32743.1	$x^{3} - 29 x - 92$	$3$	[1,1]	$-\,137\cdot 239$	$2$	$31.9918599087$	$180.95026941123908$		$S_3$ (as 3T2)	trivial	$2$	$1$	$110.247034591$

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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.