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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 1179936 matches)

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Label	Polynomial	Degree	Signature	Discriminant	Ram. prime count	Root discriminant	Galois root discriminant	Galois	Monogenic	Galois group	Class group	Unit group torsion	Unit group rank	Regulator
3.1.23.1	$x^{3} - x^{2} + 1$	$3$	[1,1]	$-\,23$	$1$	$2.84386697985$	$4.795831523312719$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$0.281199574323$
3.1.31.1	$x^{3} + x - 1$	$3$	[1,1]	$-\,31$	$1$	$3.14138065239$	$5.5677643628300215$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$0.38224508584$
3.1.44.1	$x^{3} - x^{2} + x + 1$	$3$	[1,1]	$-\,2^{2}\cdot 11$	$2$	$3.53034833533$	$5.264813681193971$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$0.609377863436$
3.3.49.1	$x^{3} - x^{2} - 2 x + 1$	$3$	[3,0]	$7^{2}$	$1$	$3.65930571002$	$3.6593057100229713$	✓	✓	$C_3$ (as 3T1)	trivial	$2$	$2$	$0.525454682123$
3.1.59.1	$x^{3} + 2 x - 1$	$3$	[1,1]	$-\,59$	$1$	$3.89299641587$	$7.681145747868608$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$0.790985720639$
3.1.76.1	$x^{3} - 2 x - 2$	$3$	[1,1]	$-\,2^{2}\cdot 19$	$2$	$4.23582358425$	$6.919320768399539$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.01859181978$
3.3.81.1	$x^{3} - 3 x - 1$	$3$	[3,0]	$3^{4}$	$1$	$4.32674871092$	$4.326748710922225$	✓	✓	$C_3$ (as 3T1)	trivial	$2$	$2$	$0.849287450646$
3.1.83.1	$x^{3} - x^{2} + x - 2$	$3$	[1,1]	$-\,83$	$1$	$4.36207067145$	$9.1104335791443$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.04069259944$
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	$2$	$1$	$0.934844845546$
3.1.104.1	$x^{3} - x - 2$	$3$	[1,1]	$-\,2^{3}\cdot 13$	$2$	$4.70266937544$	$10.198039027185569$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.57608295323$
3.1.107.1	$x^{3} - x^{2} + 3 x - 2$	$3$	[1,1]	$-\,107$	$1$	$4.74745939852$	$10.344080432788601$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.25605672147$
3.1.108.1	$x^{3} - 2$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}$	$2$	$4.7622031559$	$5.71910575798162$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.34737734833$
3.1.116.1	$x^{3} - x^{2} - 2$	$3$	[1,1]	$-\,2^{2}\cdot 29$	$2$	$4.87699896107$	$10.770329614269007$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.71752979598$
3.1.135.1	$x^{3} + 3 x - 1$	$3$	[1,1]	$-\,3^{3}\cdot 5$	$2$	$5.12992784003$	$8.05613000539548$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.13262826357$
3.1.139.1	$x^{3} - x^{2} + x + 2$	$3$	[1,1]	$-\,139$	$1$	$5.18010146738$	$11.789826122551595$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.66355473537$
3.1.140.1	$x^{3} + 2 x - 2$	$3$	[1,1]	$-\,2^{2}\cdot 5\cdot 7$	$3$	$5.19249410185$	$9.391191271220128$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.47367088284$
3.3.148.1	$x^{3} - x^{2} - 3 x + 1$	$3$	[3,0]	$2^{2}\cdot 37$	$2$	$5.28957247269$	$9.65578363946814$		✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$1.66233652077$
3.1.152.1	$x^{3} - x^{2} - 2 x - 2$	$3$	[1,1]	$-\,2^{3}\cdot 19$	$2$	$5.33680329744$	$12.328828005937952$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.13064558596$
3.3.169.1	$x^{3} - x^{2} - 4 x - 1$	$3$	[3,0]	$13^{2}$	$1$	$5.52877481368$	$5.528774813678872$	✓	✓	$C_3$ (as 3T1)	trivial	$2$	$2$	$1.36504986759$
3.1.172.1	$x^{3} - x^{2} - x + 3$	$3$	[1,1]	$-\,2^{2}\cdot 43$	$2$	$5.56129776652$	$10.409284811693793$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.88195290962$
3.1.175.1	$x^{3} - x^{2} + 2 x - 3$	$3$	[1,1]	$-\,5^{2}\cdot 7$	$2$	$5.59344471041$	$7.736223764452809$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.28850651272$
3.1.199.1	$x^{3} - x^{2} + 4 x - 1$	$3$	[1,1]	$-\,199$	$1$	$5.83827246081$	$14.106735979665885$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.33665777698$
3.1.200.1	$x^{3} - x^{2} + 2 x + 2$	$3$	[1,1]	$-\,2^{3}\cdot 5^{2}$	$2$	$5.84803547643$	$8.270371084000274$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.60439940759$
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	$2$	$1$	$2.35464367875$
3.1.211.1	$x^{3} - 2 x - 3$	$3$	[1,1]	$-\,211$	$1$	$5.95334181314$	$14.52583904633395$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.2376328728$
3.1.212.1	$x^{3} - x^{2} + 4 x - 2$	$3$	[1,1]	$-\,2^{2}\cdot 53$	$2$	$5.96273195774$	$14.560219778561036$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.71275620908$
3.1.216.1	$x^{3} + 3 x - 2$	$3$	[1,1]	$-\,2^{3}\cdot 3^{3}$	$2$	$6.0$	$10.190287977389717$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$3.02357707871$
3.3.229.1	$x^{3} - 4 x - 1$	$3$	[3,0]	$229$	$1$	$6.11803317264$	$15.132745950421556$		✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$2.35545459084$
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	$2$	$1$	$1.74549013515$
3.1.239.1	$x^{3} - x - 3$	$3$	[1,1]	$-\,239$	$1$	$6.2058217949$	$15.459624833740307$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.09654202363$
3.1.243.1	$x^{3} - 3$	$3$	[1,1]	$-\,3^{5}$	$1$	$6.24025146916$	$7.494148598900439$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.52468140471$
3.1.244.1	$x^{3} + x - 6$	$3$	[1,1]	$-\,2^{2}\cdot 61$	$2$	$6.24879976953$	$15.620499351813308$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$3.30267551049$
3.1.247.1	$x^{3} + x - 3$	$3$	[1,1]	$-\,13\cdot 19$	$2$	$6.27430535741$	$15.716233645501712$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.54453228949$
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	$2$	$1$	$1.99294483445$
3.3.257.1	$x^{3} - x^{2} - 4 x + 3$	$3$	[3,0]	$257$	$1$	$6.35786117973$	$16.0312195418814$		✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$1.97459387078$
3.1.268.1	$x^{3} - x^{2} - 3 x + 5$	$3$	[1,1]	$-\,2^{2}\cdot 67$	$2$	$6.44730572727$	$12.993437600801144$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.52123333635$
3.1.283.1	$x^{3} + 4 x - 1$	$3$	[1,1]	$-\,283$	$1$	$6.56541442735$	$16.822603841260722$		✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$1.40134232731$
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	$2$	$1$	$3.14854957566$
3.1.307.1	$x^{3} - x^{2} + 3 x + 2$	$3$	[1,1]	$-\,307$	$1$	$6.74599671171$	$17.52141546793523$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.9581482587$
3.3.316.1	$x^{3} - x^{2} - 4 x + 2$	$3$	[3,0]	$2^{2}\cdot 79$	$2$	$6.81128460769$	$17.776388834631177$		✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$3.91345849211$
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	$2$	$2$	$2.56925968013$
3.1.324.1	$x^{3} - 3 x - 4$	$3$	[1,1]	$-\,2^{2}\cdot 3^{4}$	$2$	$6.86828545532$	$8.65349742184445$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$4.04764172645$
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	$2$	$1$	$2.1985916392$
3.1.331.1	$x^{3} - x^{2} + 3 x - 4$	$3$	[1,1]	$-\,331$	$1$	$6.91739641661$	$18.193405398660254$		✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$1.50285283332$
3.1.335.1	$x^{3} - x^{2} + 4 x + 1$	$3$	[1,1]	$-\,5\cdot 67$	$2$	$6.94514955803$	$18.303005217723125$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.45573302761$
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	$2$	$1$	$3.54626923722$
3.1.351.1	$x^{3} + 3 x - 3$	$3$	[1,1]	$-\,3^{3}\cdot 13$	$2$	$7.05400406316$	$12.99011931146003$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.70227535834$
3.1.356.1	$x^{3} - x^{2} + x + 7$	$3$	[1,1]	$-\,2^{2}\cdot 89$	$2$	$7.0873410615$	$18.867962264113206$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$3.75521003542$
3.3.361.1	$x^{3} - x^{2} - 6 x + 7$	$3$	[3,0]	$19^{2}$	$1$	$7.1203673589$	$7.120367358901993$	✓	✓	$C_3$ (as 3T1)	trivial	$2$	$2$	$1.95215669651$
3.1.364.1	$x^{3} + 4 x - 2$	$3$	[1,1]	$-\,2^{2}\cdot 7\cdot 13$	$3$	$7.14003698192$	$15.142840918429636$		✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.93617392275$

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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.