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	Narrow class group	Narrow class number

CM field	Relative class number	Regulator	$p$-adic completions	Monogenic

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Results (1-50 of at least 1000)

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Label	Polynomial	Degree	Signature	Discriminant	Ram. prime count	Root discriminant	Galois root discriminant	Monogenic	Galois group	Class group	Narrow class group	Unit group torsion	Unit group rank	Regulator
3.1.87.1	$x^{3} - x^{2} + 2 x + 1$	$3$	[1,1]	$-\,3\cdot 29$	$2$	$4.43104762169$	$9.327379053088816$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$0.934844845546$
3.1.204.1	$x^{3} - x^{2} + x - 3$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 17$	$3$	$5.88676531688$	$11.336311000024526$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$2.35464367875$
3.1.231.1	$x^{3} - x^{2} + 3$	$3$	[1,1]	$-\,3\cdot 7\cdot 11$	$3$	$6.13579243966$	$15.198684153570664$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$1.74549013515$
3.1.255.1	$x^{3} - x^{2} - 3$	$3$	[1,1]	$-\,3\cdot 5\cdot 17$	$3$	$6.34132570538$	$15.968719422671311$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$1.99294483445$
3.1.300.1	$x^{3} - x^{2} - 3 x - 3$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 5^{2}$	$3$	$6.69432950082$	$8.039467687661697$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$3.14854957566$
3.3.321.1	$x^{3} - x^{2} - 4 x + 1$	$3$	[3,0]	$3\cdot 107$	$2$	$6.84702127757$	$17.916472867168917$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$2$	$2.56925968013$
3.1.327.1	$x^{3} - x^{2} - 2 x - 3$	$3$	[1,1]	$-\,3\cdot 109$	$2$	$6.88941877452$	$18.083141320025124$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$2.1985916392$
3.1.339.1	$x^{3} - x^{2} - x + 4$	$3$	[1,1]	$-\,3\cdot 113$	$2$	$6.97268264894$	$18.411952639521967$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$3.54626923722$
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	trivial	$2$	$1$	$4.02853094339$
3.1.492.1	$x^{3} - x^{2} + 3 x + 3$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 41$	$3$	$7.89444677297$	$17.60512931716603$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$4.42072409068$
3.1.516.1	$x^{3} - x^{2} + x - 9$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 43$	$3$	$8.02077931412$	$22.715633383201094$		$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$6.38507778986$
3.1.519.1	$x^{3} - x^{2} - 4 x + 7$	$3$	[1,1]	$-\,3\cdot 173$	$2$	$8.03629343269$	$22.781571499789035$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$2.68057999706$
3.1.543.1	$x^{3} - x^{2} + 2 x - 5$	$3$	[1,1]	$-\,3\cdot 181$	$2$	$8.15830510742$	$23.302360395462088$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$3.0129067369$
3.3.564.1	$x^{3} - x^{2} - 5 x + 3$	$3$	[3,0]	$2^{2}\cdot 3\cdot 47$	$3$	$8.26214922567$	$18.849343120394256$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$2$	$5.40274790998$
3.1.588.1	$x^{3} - x^{2} + 5 x + 1$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 7^{2}$	$3$	$8.37771872824$	$10.061112020813587$	✓	$S_3$ (as 3T2)	$[3]$	$[3]$	$2$	$1$	$1.65400419943$
3.1.687.1	$x^{3} - x^{2} + 4 x + 3$	$3$	[1,1]	$-\,3\cdot 229$	$2$	$8.82373071436$	$26.210684844162312$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$3.45484825256$
3.1.696.1	$x^{3} - x^{2} - 2 x + 6$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 29$	$3$	$8.86209524339$	$26.38181191654584$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$7.80993441396$
3.1.744.1	$x^{3} - x^{2} - 6 x - 6$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 31$	$3$	$9.06130979217$	$27.27636339397171$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$8.29429929653$
3.1.759.1	$x^{3} - x^{2} + 6 x - 3$	$3$	[1,1]	$-\,3\cdot 11\cdot 23$	$3$	$9.12180096782$	$27.54995462791182$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$3.13746079313$
3.1.771.1	$x^{3} - x^{2} + 3 x - 6$	$3$	[1,1]	$-\,3\cdot 257$	$2$	$9.16962255455$	$27.76688675382964$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$6.13970801431$
3.1.780.1	$x^{3} - x^{2} - x - 5$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 5\cdot 13$	$4$	$9.20516408252$	$22.16684933541527$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$6.15920712154$
3.1.804.1	$x^{3} - x^{2} + 4 x - 6$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 67$	$3$	$9.29862391479$	$28.35489375751565$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$8.57148683343$
3.1.843.1	$x^{3} - x^{2} - 2 x + 12$	$3$	[1,1]	$-\,3\cdot 281$	$2$	$9.44660721967$	$29.03446228191595$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$6.73586181539$
3.1.867.1	$x^{3} - x^{2} + 6 x - 12$	$3$	[1,1]	$-\,3\cdot 17^{2}$	$2$	$9.53541719596$	$11.451434893652847$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$6.87929864473$
3.1.876.1	$x^{3} - x^{2} - x - 11$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 73$	$3$	$9.56829820512$	$23.491390334630143$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$5.94796000997$
3.1.888.1	$x^{3} - x^{2} + 9 x + 3$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 37$	$3$	$9.61179106741$	$29.79932885150268$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$9.6414091891$
3.1.948.1	$x^{3} - x^{2} + 6$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 79$	$3$	$9.82357229868$	$30.789608636681304$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$10.0586517309$
3.1.984.1	$x^{3} - x^{2} - 12$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 41$	$3$	$9.94637966654$	$31.368774282716245$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$9.97919841126$
3.3.993.1	$x^{3} - x^{2} - 6 x + 3$	$3$	[3,0]	$3\cdot 331$	$2$	$9.9766120095$	$31.51190251317746$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$2$	$5.5546439553$
3.1.996.1	$x^{3} - x^{2} - 6$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 83$	$3$	$9.98664884928$	$31.559467676119$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$10.0286655667$
3.1.1011.1	$x^{3} - x^{2} - 5 x - 6$	$3$	[1,1]	$-\,3\cdot 337$	$2$	$10.0365330379$	$31.796226191169293$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$7.0950891288$
3.1.1059.1	$x^{3} - x^{2} + x + 6$	$3$	[1,1]	$-\,3\cdot 353$	$2$	$10.1929208858$	$32.54228019054596$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$7.47876163689$
3.1.1068.1	$x^{3} - x^{2} - x + 13$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 89$	$3$	$10.2217146006$	$25.93834691424768$		$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$7.16938023975$
3.1.1083.1	$x^{3} - x^{2} - 6 x - 12$	$3$	[1,1]	$-\,3\cdot 19^{2}$	$2$	$10.2693467638$	$12.332838034173273$		$S_3$ (as 3T2)	$[3]$	$[3]$	$2$	$1$	$2.62907295987$
3.3.1101.1	$x^{3} - x^{2} - 9 x + 12$	$3$	[3,0]	$3\cdot 367$	$2$	$10.3259283287$	$33.18132004607411$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$2$	$9.18395840633$
3.1.1164.1	$x^{3} - x^{2} - x + 7$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 97$	$3$	$10.5192391032$	$27.07903342142803$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$6.88237750613$
3.1.1176.1	$x^{3} - x^{2} - 2 x - 6$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 7^{2}$	$3$	$10.5552641758$	$17.926863604818365$	✓	$S_3$ (as 3T2)	$[3]$	$[3]$	$2$	$1$	$3.71059897095$
3.1.1191.1	$x^{3} - x^{2} - 10 x - 11$	$3$	[1,1]	$-\,3\cdot 397$	$2$	$10.5999525333$	$34.51086785347479$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$4.14539858913$
3.1.1203.1	$x^{3} - x^{2} - 3 x - 6$	$3$	[1,1]	$-\,3\cdot 401$	$2$	$10.6354338106$	$34.68429039204925$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$8.4385821355$
3.1.1236.1	$x^{3} - x^{2} + 4 x + 12$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 103$	$3$	$10.7318062545$	$35.156791662493895$		$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$11.525634303$
3.3.1257.1	$x^{3} - x^{2} - 8 x + 9$	$3$	[3,0]	$3\cdot 419$	$2$	$10.7922440886$	$35.45419580247167$	✓	$S_3$ (as 3T2)	trivial	$[2]$	$2$	$2$	$6.19689047505$
3.1.1272.1	$x^{3} - x^{2} - 3 x + 15$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 53$	$3$	$10.83500303$	$35.66510900025401$		$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$12.1661720923$
3.1.1347.1	$x^{3} - x^{2} - 7 x - 8$	$3$	[1,1]	$-\,3\cdot 449$	$2$	$11.0439016875$	$36.701498607005135$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$9.03336267791$
3.1.1356.1	$x^{3} - x^{2} + x + 21$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 113$	$3$	$11.068443772$	$29.22715298876584$		$S_3$ (as 3T2)	$[3]$	$[3]$	$2$	$1$	$2.72140555263$
3.1.1356.2	$x^{3} - x^{2} + 11 x + 1$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 113$	$3$	$11.068443772$	$29.22715298876584$		$S_3$ (as 3T2)	$[3]$	$[3]$	$2$	$1$	$2.40678503563$
3.1.1356.3	$x^{3} - x^{2} + 3 x - 15$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 113$	$3$	$11.068443772$	$29.22715298876584$		$S_3$ (as 3T2)	$[3]$	$[3]$	$2$	$1$	$3.12599092174$
3.1.1371.1	$x^{3} - x^{2} + 3 x + 6$	$3$	[1,1]	$-\,3\cdot 457$	$2$	$11.109107046$	$37.027017163147235$	✓	$S_3$ (as 3T2)	$[4]$	$[4]$	$2$	$1$	$2.13833439909$
3.1.1383.1	$x^{3} - x^{2} + 6 x + 3$	$3$	[1,1]	$-\,3\cdot 461$	$2$	$11.1414246303$	$37.1887079635741$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$5.01156761866$
3.1.1407.1	$x^{3} - x^{2} - 8 x - 9$	$3$	[1,1]	$-\,3\cdot 7\cdot 67$	$3$	$11.2055032479$	$37.5099986670221$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$5.37065976135$
3.1.1419.1	$x^{3} - x^{2} + 10 x - 12$	$3$	[1,1]	$-\,3\cdot 11\cdot 43$	$3$	$11.2372695503$	$37.66961640367472$		$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$8.61938596484$

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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.