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

Ramified prime count	Ramified	Unramified primes	Galois group	Galois

Discriminant	Root discriminant	Galois root discriminant	Regulator	Intermediate field

$p$-adic completions	Minimal sibling	Monogenic	Index	Ines. $p$

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Results (1-50 of 1267589 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
3.1.140.1	$x^{3} + 2 x - 2$	$3$	[1,1]	$-\,2^{2}\cdot 5\cdot 7$	$3$	$5.19249410185$	$9.391191271220128$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.47367088284$
3.1.175.1	$x^{3} - x^{2} + 2 x - 3$	$3$	[1,1]	$-\,5^{2}\cdot 7$	$2$	$5.59344471041$	$7.736223764452809$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.28850651272$
3.1.231.1	$x^{3} - x^{2} + 3$	$3$	[1,1]	$-\,3\cdot 7\cdot 11$	$3$	$6.13579243966$	$15.198684153570664$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.74549013515$
3.1.364.1	$x^{3} + 4 x - 2$	$3$	[1,1]	$-\,2^{2}\cdot 7\cdot 13$	$3$	$7.14003698192$	$15.142840918429636$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.93617392275$
3.3.469.1	$x^{3} - x^{2} - 5 x + 4$	$3$	[3,0]	$7\cdot 67$	$2$	$7.76946201167$	$21.656407827707714$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$3.85282661562$
3.1.567.1	$x^{3} - 3 x - 5$	$3$	[1,1]	$-\,3^{4}\cdot 7$	$2$	$8.27677252914$	$11.447501074569505$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.46402039258$
3.1.679.1	$x^{3} + x - 5$	$3$	[1,1]	$-\,7\cdot 97$	$2$	$8.78934661221$	$26.057628441590765$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$3.44325590957$
3.1.707.1	$x^{3} + 2 x - 5$	$3$	[1,1]	$-\,7\cdot 101$	$2$	$8.90853870595$	$26.589471600616662$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$4.18689198434$
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	$2$	$1$	$6.05215007604$
3.1.756.1	$x^{3} - 6 x - 12$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}\cdot 7$	$3$	$9.10976691563$	$19.06428314197696$		$S_3$ (as 3T2)	trivial	$2$	$1$	$7.10745968178$
3.3.756.1	$x^{3} - 6 x - 2$	$3$	[3,0]	$2^{2}\cdot 3^{3}\cdot 7$	$3$	$9.10976691563$	$15.13133155729692$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$5.69171840461$
3.1.812.1	$x^{3} - x^{2} - 7 x - 7$	$3$	[1,1]	$-\,2^{2}\cdot 7\cdot 29$	$3$	$9.32936339099$	$22.616983579985163$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$3.84421980906$
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	$2$	$1$	$1.82256798591$
3.1.1036.1	$x^{3} + 4 x - 12$	$3$	[1,1]	$-\,2^{2}\cdot 7\cdot 37$	$3$	$10.1185881266$	$25.546802223478856$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$5.23153744514$
3.1.1099.1	$x^{3} - x^{2} - x - 6$	$3$	[1,1]	$-\,7\cdot 157$	$2$	$10.3196720846$	$33.15116890850155$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$3.18157943984$
3.1.1267.1	$x^{3} - x^{2} + 7 x - 4$	$3$	[1,1]	$-\,7\cdot 181$	$2$	$10.8207875806$	$35.59494346111537$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$7.61033921916$
3.1.1295.1	$x^{3} - x^{2} + 7$	$3$	[1,1]	$-\,5\cdot 7\cdot 37$	$3$	$10.899918637$	$35.98610843089316$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$3.52724980954$
3.1.1316.1	$x^{3} - x^{2} + 9 x + 7$	$3$	[1,1]	$-\,2^{2}\cdot 7\cdot 47$	$3$	$10.9585214764$	$36.27671429443411$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$7.74628402629$
3.1.1351.1	$x^{3} - x^{2} - 7$	$3$	[1,1]	$-\,7\cdot 193$	$2$	$11.0548227346$	$36.75595189897821$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$3.5340812844$
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	$2$	$1$	$5.37065976135$
3.1.1484.1	$x^{3} + 8 x - 12$	$3$	[1,1]	$-\,2^{2}\cdot 7\cdot 53$	$3$	$11.4062958965$	$30.57550357757$		$S_3$ (as 3T2)	trivial	$2$	$1$	$10.3829156899$
3.1.1491.1	$x^{3} - x^{2} + 6 x + 12$	$3$	[1,1]	$-\,3\cdot 7\cdot 71$	$3$	$11.4242021989$	$38.61346915261564$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$9.40968165124$
3.1.1512.1	$x^{3} - 6 x - 16$	$3$	[1,1]	$-\,2^{3}\cdot 3^{3}\cdot 7$	$3$	$11.4775870966$	$26.960967776304578$		$S_3$ (as 3T2)	trivial	$2$	$1$	$11.7719996659$
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	$2$	$1$	$6.46158046594$
3.1.1603.1	$x^{3} - 5 x - 16$	$3$	[1,1]	$-\,7\cdot 229$	$2$	$11.7033764332$	$40.0374824383352$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$8.71144109499$
3.1.1687.1	$x^{3} - 5 x - 9$	$3$	[1,1]	$-\,7\cdot 241$	$2$	$11.904331921$	$41.0731055558257$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$4.91389648033$
3.1.1708.1	$x^{3} - x^{2} - 5 x + 11$	$3$	[1,1]	$-\,2^{2}\cdot 7\cdot 61$	$3$	$11.953523934$	$32.80202092265386$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$7.88795891425$
3.3.1708.1	$x^{3} - x^{2} - 8 x - 2$	$3$	[3,0]	$2^{2}\cdot 7\cdot 61$	$3$	$11.953523934$	$41.32795663954365$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$12.8730037743$
3.1.1736.1	$x^{3} + 2 x - 16$	$3$	[1,1]	$-\,2^{3}\cdot 7\cdot 31$	$3$	$12.0184900138$	$41.66533331199932$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$9.00614272602$
3.1.1743.1	$x^{3} - x^{2} - 8 x + 15$	$3$	[1,1]	$-\,3\cdot 7\cdot 83$	$3$	$12.0346222346$	$41.7492514902962$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$6.14192640156$
3.1.1932.1	$x^{3} - x^{2} + 7 x + 3$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 7\cdot 23$	$4$	$12.4547698696$	$34.88672719242158$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$10.4888543724$
3.1.1967.1	$x^{3} - x^{2} + 2 x + 25$	$3$	[1,1]	$-\,7\cdot 281$	$2$	$12.5295301825$	$44.35087372307337$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$8.59674432309$
3.1.1988.1	$x^{3} - x^{2} + 9 x - 17$	$3$	[1,1]	$-\,2^{2}\cdot 7\cdot 71$	$3$	$12.5739615124$	$44.58699361921591$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$10.662399127$
3.1.2023.1	$x^{3} - x^{2} + 6 x + 5$	$3$	[1,1]	$-\,7\cdot 17^{2}$	$2$	$12.647323508$	$17.492355738674252$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$5.67671865574$
3.1.2051.1	$x^{3} - x^{2} - 7 x + 14$	$3$	[1,1]	$-\,7\cdot 293$	$2$	$12.7054061758$	$45.28796749689701$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$3.24302461654$
3.3.2177.1	$x^{3} - x^{2} - 8 x + 5$	$3$	[3,0]	$7\cdot 311$	$2$	$12.9604319574$	$46.65833258915282$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$7.51804024088$
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	$2$	$1$	$16.947113779$
3.1.2191.1	$x^{3} + x - 9$	$3$	[1,1]	$-\,7\cdot 313$	$2$	$12.9881548909$	$46.8081189538738$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$2.52792006055$
3.1.2219.1	$x^{3} + 2 x - 9$	$3$	[1,1]	$-\,7\cdot 317$	$2$	$13.0432484662$	$47.10626285325551$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$7.50166037388$
3.3.2233.1	$x^{3} - x^{2} - 8 x + 1$	$3$	[3,0]	$7\cdot 11\cdot 29$	$3$	$13.0706215758$	$47.25462940284264$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$5.52254734341$
3.3.2296.1	$x^{3} - x^{2} - 14 x - 14$	$3$	[3,0]	$2^{3}\cdot 7\cdot 41$	$3$	$13.1924045682$	$47.916594202843754$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$14.2655509459$
3.1.2380.1	$x^{3} - x^{2} - x - 9$	$3$	[1,1]	$-\,2^{2}\cdot 5\cdot 7\cdot 17$	$4$	$13.3513644937$	$38.72087356161918$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$10.6870010325$
3.1.2387.1	$x^{3} - x^{2} + 6 x - 20$	$3$	[1,1]	$-\,7\cdot 11\cdot 31$	$3$	$13.3644412547$	$48.85693400122443$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$8.45829604356$
3.1.2408.1	$x^{3} - x^{2} + 2 x - 10$	$3$	[1,1]	$-\,2^{3}\cdot 7\cdot 43$	$3$	$13.4035187908$	$49.07137658554119$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$12.1110859192$
3.3.2429.1	$x^{3} - x^{2} - 14 x - 4$	$3$	[3,0]	$7\cdot 347$	$2$	$13.4423697872$	$49.28488612140643$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$13.278665959$
3.1.2443.1	$x^{3} + 4 x - 9$	$3$	[1,1]	$-\,7\cdot 349$	$2$	$13.4681462065$	$49.42671342502959$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$5.6771289362$
3.1.2555.1	$x^{3} - x^{2} + 14 x - 4$	$3$	[1,1]	$-\,5\cdot 7\cdot 73$	$3$	$13.6708956441$	$50.547007824400445$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$9.76411086644$
3.1.2604.1	$x^{3} - x^{2} + 9 x - 3$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 7\cdot 31$	$4$	$13.7577366928$	$40.50206423088982$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$11.6838574054$
3.1.2660.1	$x^{3} - x^{2} + 10$	$3$	[1,1]	$-\,2^{2}\cdot 5\cdot 7\cdot 19$	$4$	$13.8556598224$	$51.57518783291051$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$14.3468532588$
3.1.2723.1	$x^{3} + 8 x - 5$	$3$	[1,1]	$-\,7\cdot 389$	$2$	$13.9641942171$	$52.182372502598994$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$8.55101376792$

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