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
3.1.104.1	$x^{3} - x - 2$	$3$	[1,1]	$-\,2^{3}\cdot 13$	$2$	$4.70266937544$	$10.198039027185569$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$1.57608295323$
3.1.247.1	$x^{3} + x - 3$	$3$	[1,1]	$-\,13\cdot 19$	$2$	$6.27430535741$	$15.716233645501712$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$1.54453228949$
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	trivial	$2$	$1$	$1.70227535834$
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	trivial	$2$	$1$	$2.93617392275$
3.1.728.1	$x^{3} - x^{2} + 6 x - 2$	$3$	[1,1]	$-\,2^{3}\cdot 7\cdot 13$	$3$	$8.99588289055$	$26.981475126464083$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$6.05215007604$
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	$2$	$1$	$6.15920712154$
3.1.1144.1	$x^{3} - x^{2} + 6 x + 2$	$3$	[1,1]	$-\,2^{3}\cdot 11\cdot 13$	$3$	$10.4586430635$	$33.823069050575526$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$8.79557852648$
3.1.1196.1	$x^{3} - x^{2} + 5 x - 7$	$3$	[1,1]	$-\,2^{2}\cdot 13\cdot 23$	$3$	$10.6147652308$	$27.44873016802661$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$4.31445449716$
3.1.1235.1	$x^{3} - x^{2} - 5 x + 10$	$3$	[1,1]	$-\,5\cdot 13\cdot 19$	$3$	$10.7289112433$	$35.14256678161116$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$5.79310657438$
3.1.1300.1	$x^{3} - x^{2} - 3 x - 13$	$3$	[1,1]	$-\,2^{2}\cdot 5^{2}\cdot 13$	$3$	$10.9139288306$	$21.085391770985456$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$10.1398630277$
3.3.1300.1	$x^{3} - 10 x - 10$	$3$	[3,0]	$2^{2}\cdot 5^{2}\cdot 13$	$3$	$10.9139288306$	$16.735486539211966$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$2$	$6.55000546582$
3.1.1547.1	$x^{3} - x^{2} - x + 8$	$3$	[1,1]	$-\,7\cdot 13\cdot 17$	$3$	$11.5654740341$	$39.331920878594275$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$6.46158046594$
3.3.1573.1	$x^{3} - x^{2} - 7 x + 2$	$3$	[3,0]	$11^{2}\cdot 13$	$2$	$11.6299069738$	$17.833371889561775$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$2$	$8.44494130232$
3.1.1599.1	$x^{3} + 3 x - 23$	$3$	[1,1]	$-\,3\cdot 13\cdot 41$	$3$	$11.6936337636$	$39.98749804626441$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$4.46392196445$
3.1.1612.1	$x^{3} - x^{2} - 3 x - 7$	$3$	[1,1]	$-\,2^{2}\cdot 13\cdot 31$	$3$	$11.7252383329$	$31.866853723191245$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$7.12686306677$
3.1.1755.1	$x^{3} + 3 x - 16$	$3$	[1,1]	$-\,3^{3}\cdot 5\cdot 13$	$3$	$12.0621772758$	$29.046789816257387$		$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$9.01614859838$
3.1.1807.1	$x^{3} - x^{2} + 8 x - 3$	$3$	[1,1]	$-\,13\cdot 139$	$2$	$12.1801522662$	$42.50882261366456$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$4.60366780573$
3.3.1937.1	$x^{3} - x^{2} - 8 x - 1$	$3$	[3,0]	$13\cdot 149$	$2$	$12.4655048946$	$44.01136216933077$	✓	$S_3$ (as 3T2)	trivial	$[2]$	$2$	$2$	$6.542242836$
3.1.1963.1	$x^{3} - x^{2} + 5 x + 6$	$3$	[1,1]	$-\,13\cdot 151$	$2$	$12.5210312617$	$44.30575583375144$	✓	$S_3$ (as 3T2)	$[4]$	$[4]$	$2$	$1$	$2.26113105698$
3.1.2132.1	$x^{3} - x^{2} - 3 x + 19$	$3$	[1,1]	$-\,2^{2}\cdot 13\cdot 41$	$3$	$12.8705093209$	$46.17358552246078$		$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$10.7300504916$
3.1.2184.1	$x^{3} - x^{2} + 13 x - 9$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 7\cdot 13$	$4$	$12.9743082334$	$46.73328578219169$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$16.947113779$
3.1.2327.1	$x^{3} - x^{2} + 8 x - 7$	$3$	[1,1]	$-\,13\cdot 179$	$2$	$13.2515128212$	$48.23898838076935$	✓	$S_3$ (as 3T2)	$[2]$	$[2]$	$2$	$1$	$2.17308027352$
3.1.2444.1	$x^{3} - 4 x - 10$	$3$	[1,1]	$-\,2^{2}\cdot 13\cdot 47$	$3$	$13.4699836071$	$39.23803668599558$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$8.27970067148$
3.1.2548.1	$x^{3} - x^{2} + 12 x + 8$	$3$	[1,1]	$-\,2^{2}\cdot 7^{2}\cdot 13$	$3$	$13.6583993889$	$26.38762874017197$		$S_3$ (as 3T2)	$[3]$	$[3]$	$2$	$1$	$5.23033576204$
3.1.2795.1	$x^{3} - x^{2} + 10 x - 20$	$3$	[1,1]	$-\,5\cdot 13\cdot 43$	$3$	$14.086202824$	$52.86775955154521$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$9.64588296043$
3.1.2808.1	$x^{3} + 3 x - 10$	$3$	[1,1]	$-\,2^{3}\cdot 3^{3}\cdot 13$	$3$	$14.1080081263$	$36.74160581422285$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$16.4533466421$
3.3.2808.1	$x^{3} - 9 x - 2$	$3$	[3,0]	$2^{3}\cdot 3^{3}\cdot 13$	$3$	$14.1080081263$	$36.74160581422285$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$2$	$20.3128560863$
3.1.2860.1	$x^{3} - x^{2} - x + 21$	$3$	[1,1]	$-\,2^{2}\cdot 5\cdot 11\cdot 13$	$4$	$14.1945624886$	$42.4462848945543$		$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$11.5965585919$
3.1.2951.1	$x^{3} - x^{2} + 6 x - 11$	$3$	[1,1]	$-\,13\cdot 227$	$2$	$14.3435417932$	$54.323107422164284$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$5.70715444811$
3.1.2964.1	$x^{3} - x^{2} + 5 x + 19$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 13\cdot 19$	$4$	$14.3645734095$	$54.4426303552648$		$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$18.7942487797$
3.1.3055.1	$x^{3} - x^{2} + 10 x - 33$	$3$	[1,1]	$-\,5\cdot 13\cdot 47$	$3$	$14.5100999787$	$55.27205442174191$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$7.17246698248$
3.1.3107.1	$x^{3} - x^{2} - 13 x + 26$	$3$	[1,1]	$-\,13\cdot 239$	$2$	$14.5919640522$	$55.74047003748713$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$7.89134138651$
3.1.3159.1	$x^{3} - 9 x - 24$	$3$	[1,1]	$-\,3^{5}\cdot 13$	$2$	$14.6729197395$	$27.020537039282146$		$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$26.1590930965$
3.1.3159.2	$x^{3} - 9 x - 15$	$3$	[1,1]	$-\,3^{5}\cdot 13$	$2$	$14.6729197395$	$27.020537039282146$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$7.6916561974$
3.1.3159.3	$x^{3} + 9 x - 3$	$3$	[1,1]	$-\,3^{5}\cdot 13$	$2$	$14.6729197395$	$27.020537039282146$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$4.43039304741$
3.1.3263.1	$x^{3} - x - 11$	$3$	[1,1]	$-\,13\cdot 251$	$2$	$14.8322040409$	$57.12267500739089$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$4.8455487924$
3.1.3471.1	$x^{3} - x^{2} + 6 x - 24$	$3$	[1,1]	$-\,3\cdot 13\cdot 89$	$3$	$15.1408946583$	$58.915193286621744$		$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$33.3613959484$
3.1.3523.1	$x^{3} + 4 x - 11$	$3$	[1,1]	$-\,13\cdot 271$	$2$	$15.2161301528$	$59.3548650070068$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$13.3960783617$
3.1.3575.1	$x^{3} - x^{2} - 8 x - 33$	$3$	[1,1]	$-\,5^{2}\cdot 11\cdot 13$	$3$	$15.2906289177$	$34.966166531003054$		$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$11.1512809626$
3.1.3692.1	$x^{3} - x^{2} - 3 x + 13$	$3$	[1,1]	$-\,2^{2}\cdot 13\cdot 71$	$3$	$15.4556484481$	$48.22669652905884$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$10.3622725367$
3.1.3783.1	$x^{3} - 9 x - 37$	$3$	[1,1]	$-\,3\cdot 13\cdot 97$	$3$	$15.5816022429$	$61.506097258727124$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$7.52129628774$
3.1.3796.1	$x^{3} - x^{2} + 8 x + 6$	$3$	[1,1]	$-\,2^{2}\cdot 13\cdot 73$	$3$	$15.5994301786$	$61.61168720299745$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$17.9412342226$
3.1.3835.1	$x^{3} - 17 x - 36$	$3$	[1,1]	$-\,5\cdot 13\cdot 59$	$3$	$15.6526709674$	$61.927376821564145$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$14.5558656986$
3.1.3991.1	$x^{3} - x^{2} + 10 x - 7$	$3$	[1,1]	$-\,13\cdot 307$	$2$	$15.8620960715$	$63.17436188834834$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$6.20459041894$
3.1.4108.1	$x^{3} + 10 x - 2$	$3$	[1,1]	$-\,2^{2}\cdot 13\cdot 79$	$3$	$16.015609766$	$50.871188667890095$	✓	$S_3$ (as 3T2)	$[2]$	$[2]$	$2$	$1$	$5.53334267878$
3.1.4147.1	$x^{3} - x^{2} - x - 12$	$3$	[1,1]	$-\,11\cdot 13\cdot 29$	$3$	$16.0661325289$	$64.39720490828775$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$14.0836917164$
3.1.4212.1	$x^{3} + 9 x - 36$	$3$	[1,1]	$-\,2^{2}\cdot 3^{4}\cdot 13$	$3$	$16.1496378363$	$31.200628666555602$		$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$19.0652515517$
3.3.4212.1	$x^{3} - 12 x - 10$	$3$	[3,0]	$2^{2}\cdot 3^{4}\cdot 13$	$3$	$16.1496378363$	$24.76395538367976$	✓	$S_3$ (as 3T2)	$[3]$	$[6]$	$2$	$2$	$5.45742079518$
3.1.4472.1	$x^{3} + 14 x - 16$	$3$	[1,1]	$-\,2^{3}\cdot 13\cdot 43$	$3$	$16.4753227686$	$66.87301398920195$		$S_3$ (as 3T2)	$[5]$	$[5]$	$2$	$1$	$2.84363127265$
3.1.4511.1	$x^{3} - x^{2} + 13$	$3$	[1,1]	$-\,13\cdot 347$	$2$	$16.5230775944$	$67.16397844082793$	✓	$S_3$ (as 3T2)	trivial	trivial	$2$	$1$	$2.79909436604$

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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.