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	CM field	Relative class number

Regulator	$p$ -adic completions	Monogenic	Index	Ines. $p$

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Results (1-50 of 186409 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
3.1.244.1	$x^{3} + x - 6$	$3$	[1,1]	$-\,2^{2}\cdot 61$	$2$	$6.24879976953$	$15.620499351813308$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$3.30267551049$
3.1.671.1	$x^{3} - x - 5$	$3$	[1,1]	$-\,11\cdot 61$	$2$	$8.7546913618$	$25.903667693977237$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.34508410892$
3.1.1464.1	$x^{3} - x^{2} - 7 x + 19$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 61$	$3$	$11.3548227417$	$38.262252939417984$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$12.8372938839$
3.1.1647.1	$x^{3} - 9 x - 13$	$3$	[1,1]	$-\,3^{3}\cdot 61$	$2$	$11.8094915493$	$28.1388523948431$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$4.04672973703$
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	$2$	$2$	$12.8730037743$
3.1.2440.1	$x^{3} + 13 x - 6$	$3$	[1,1]	$-\,2^{3}\cdot 5\cdot 61$	$3$	$13.4626309945$	$49.39635614091387$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$16.8515410889$
3.1.2867.1	$x^{3} - x^{2} + 2 x + 20$	$3$	[1,1]	$-\,47\cdot 61$	$2$	$14.2061336988$	$53.54437412091022$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$8.91811157377$
3.1.3599.1	$x^{3} - x^{2} - 6 x - 11$	$3$	[1,1]	$-\,59\cdot 61$	$2$	$15.3247694249$	$59.99166608788257$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$3.89349056616$
3.1.3660.1	$x^{3} - x^{2} - 11 x - 15$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 5\cdot 61$	$4$	$15.4108654417$	$48.01724191691582$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$16.0129654422$
3.1.4087.1	$x^{3} - x^{2} + 2 x + 24$	$3$	[1,1]	$-\,61\cdot 67$	$2$	$15.9882726564$	$63.92964883369844$		$S_3$ (as 3T2)	trivial	$2$	$1$	$32.448369121$
3.1.4575.1	$x^{3} - 15 x - 45$	$3$	[1,1]	$-\,3\cdot 5^{2}\cdot 61$	$3$	$16.6008515555$	$39.55537878985837$		$S_3$ (as 3T2)	trivial	$2$	$1$	$8.99998754068$
3.1.4819.1	$x^{3} + 4 x - 13$	$3$	[1,1]	$-\,61\cdot 79$	$2$	$16.8908812541$	$69.41901756723442$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$12.8994197988$
3.1.5063.1	$x^{3} + 5 x - 13$	$3$	[1,1]	$-\,61\cdot 83$	$2$	$17.1712789107$	$71.15476090888086$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$4.7885323619$
3.1.5124.1	$x^{3} - x^{2} - 7 x + 31$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 7\cdot 61$	$4$	$17.2399647575$	$71.58212067269312$		$S_3$ (as 3T2)	trivial	$2$	$1$	$28.3325579948$
3.1.5307.1	$x^{3} - 12 x - 45$	$3$	[1,1]	$-\,3\cdot 29\cdot 61$	$3$	$17.442806481$	$72.84915922644544$		$S_3$ (as 3T2)	trivial	$2$	$1$	$20.8461538378$
3.3.5368.1	$x^{3} - x^{2} - 19 x + 23$	$3$	[3,0]	$2^{3}\cdot 11\cdot 61$	$3$	$17.5093827236$	$73.2666363360568$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$24.4623695699$
3.1.5612.1	$x^{3} - x^{2} + 7 x - 45$	$3$	[1,1]	$-\,2^{2}\cdot 23\cdot 61$	$3$	$17.7707553283$	$59.45871228007729$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$22.1826499538$
3.1.6283.1	$x^{3} - x^{2} + 7 x - 16$	$3$	[1,1]	$-\,61\cdot 103$	$2$	$18.4525200789$	$79.26537705707328$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$8.49514925975$
3.3.6588.1	$x^{3} - 15 x - 16$	$3$	[3,0]	$2^{2}\cdot 3^{3}\cdot 61$	$3$	$18.7463993086$	$56.2777047896862$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$33.6013867385$
3.1.7259.1	$x^{3} - x^{2} - x - 16$	$3$	[1,1]	$-\,7\cdot 17\cdot 61$	$3$	$19.3623886018$	$85.19976525789258$	✓	$S_3$ (as 3T2)	$[4]$	$2$	$1$	$3.01527328093$
3.1.7320.1	$x^{3} - x^{2} + 14 x - 50$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 5\cdot 61$	$4$	$19.4164737671$	$85.55699854482975$		$S_3$ (as 3T2)	$[7]$	$2$	$1$	$5.45094642024$
3.1.7564.1	$x^{3} - x^{2} + 25 x + 9$	$3$	[1,1]	$-\,2^{2}\cdot 31\cdot 61$	$3$	$19.6298586967$	$69.02913450639817$		$S_3$ (as 3T2)	trivial	$2$	$1$	$13.7144751218$
3.3.8113.1	$x^{3} - 13 x - 5$	$3$	[3,0]	$7\cdot 19\cdot 61$	$3$	$20.0937267454$	$90.07219326740079$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$12.5531294899$
3.1.8235.1	$x^{3} + 12 x - 7$	$3$	[1,1]	$-\,3^{3}\cdot 5\cdot 61$	$3$	$20.1939464918$	$62.92038676370192$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$25.7545433451$
3.1.9028.1	$x^{3} - x^{2} - 23 x - 49$	$3$	[1,1]	$-\,2^{2}\cdot 37\cdot 61$	$3$	$20.8223871386$	$95.01578816175763$		$S_3$ (as 3T2)	$[2]$	$2$	$1$	$18.033648605$
3.1.9211.1	$x^{3} - x^{2} + 14 x - 36$	$3$	[1,1]	$-\,61\cdot 151$	$2$	$20.9621388859$	$95.97395480024775$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$17.534664529$
3.1.9272.1	$x^{3} + 5 x - 74$	$3$	[1,1]	$-\,2^{3}\cdot 19\cdot 61$	$3$	$21.0083111472$	$96.29122493768578$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$20.7444266484$
3.1.9455.1	$x^{3} - x^{2} - 6 x + 40$	$3$	[1,1]	$-\,5\cdot 31\cdot 61$	$3$	$21.1456242683$	$97.23682430026189$		$S_3$ (as 3T2)	trivial	$2$	$1$	$45.7243474198$
3.1.9516.1	$x^{3} + 12 x - 54$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 13\cdot 61$	$4$	$21.1910012835$	$77.4254761393161$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$18.3335332159$
3.3.10004.1	$x^{3} - 14 x - 6$	$3$	[3,0]	$2^{2}\cdot 41\cdot 61$	$3$	$21.547219097$	$79.385925021846$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$26.7466525425$
3.1.10187.1	$x^{3} - x^{2} + 7 x + 16$	$3$	[1,1]	$-\,61\cdot 167$	$2$	$21.6778114919$	$100.93066927351666$	✓	$S_3$ (as 3T2)	$[4]$	$2$	$1$	$4.20223648643$
3.1.10248.1	$x^{3} + 15 x - 54$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 7\cdot 61$	$4$	$21.7209944974$	$101.23240587875011$		$S_3$ (as 3T2)	trivial	$2$	$1$	$45.6393407745$
3.3.10309.1	$x^{3} - x^{2} - 17 x - 14$	$3$	[3,0]	$13^{2}\cdot 61$	$2$	$21.7640064801$	$43.181111696696284$	✓	$S_3$ (as 3T2)	$[3]$	$2$	$2$	$6.52571965153$
3.1.10675.1	$x^{3} - 5 x - 40$	$3$	[1,1]	$-\,5^{2}\cdot 7\cdot 61$	$3$	$22.0185793464$	$60.42183914905891$		$S_3$ (as 3T2)	trivial	$2$	$1$	$26.0866933156$
3.1.10919.1	$x^{3} - x^{2} - 4 x - 19$	$3$	[1,1]	$-\,61\cdot 179$	$2$	$22.1850777624$	$104.4940189675945$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$6.14619168058$
3.1.11224.1	$x^{3} - x^{2} + 5 x + 39$	$3$	[1,1]	$-\,2^{3}\cdot 23\cdot 61$	$3$	$22.3897487106$	$105.94338110519222$		$S_3$ (as 3T2)	trivial	$2$	$1$	$29.2442537073$
3.1.11407.1	$x^{3} - 5 x - 21$	$3$	[1,1]	$-\,11\cdot 17\cdot 61$	$3$	$22.5107767305$	$106.803557993168$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$6.79343973989$
3.1.11651.1	$x^{3} + 14 x - 5$	$3$	[1,1]	$-\,61\cdot 191$	$2$	$22.6701503681$	$107.93979803575695$	✓	$S_3$ (as 3T2)	$[3]$	$2$	$1$	$6.53673780621$
3.1.11651.2	$x^{3} - x^{2} + 5 x - 22$	$3$	[1,1]	$-\,61\cdot 191$	$2$	$22.6701503681$	$107.93979803575695$	✓	$S_3$ (as 3T2)	$[3]$	$2$	$1$	$4.31979360233$
3.1.11651.3	$x^{3} - 4 x - 21$	$3$	[1,1]	$-\,61\cdot 191$	$2$	$22.6701503681$	$107.93979803575695$	✓	$S_3$ (as 3T2)	$[9]$	$2$	$1$	$4.70081153082$
3.1.11651.4	$x^{3} - x^{2} - 15 x - 26$	$3$	[1,1]	$-\,61\cdot 191$	$2$	$22.6701503681$	$107.93979803575695$	✓	$S_3$ (as 3T2)	$[3]$	$2$	$1$	$3.91758436668$
3.1.11895.1	$x^{3} - 3 x - 63$	$3$	[1,1]	$-\,3\cdot 5\cdot 13\cdot 61$	$4$	$22.8273141409$	$109.0642012761291$		$S_3$ (as 3T2)	trivial	$2$	$1$	$19.5445984493$
3.1.12200.1	$x^{3} - x^{2} - 23 x - 53$	$3$	[1,1]	$-\,2^{3}\cdot 5^{2}\cdot 61$	$3$	$23.0207751796$	$64.59366307844091$		$S_3$ (as 3T2)	trivial	$2$	$1$	$23.0750360897$
3.1.12871.1	$x^{3} - x^{2} - 4 x + 67$	$3$	[1,1]	$-\,61\cdot 211$	$2$	$23.4353132775$	$113.45042970390196$		$S_3$ (as 3T2)	trivial	$2$	$1$	$11.5588065848$
3.1.13359.1	$x^{3} - x^{2} - 6 x - 21$	$3$	[1,1]	$-\,3\cdot 61\cdot 73$	$3$	$23.7278278762$	$115.58114033007288$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$8.26230880019$
3.1.13420.1	$x^{3} - x^{2} - x + 45$	$3$	[1,1]	$-\,2^{2}\cdot 5\cdot 11\cdot 61$	$4$	$23.7638884603$	$91.94601810189825$		$S_3$ (as 3T2)	trivial	$2$	$1$	$25.6537476274$
3.3.14457.1	$x^{3} - x^{2} - 32 x - 3$	$3$	[3,0]	$3\cdot 61\cdot 79$	$3$	$24.3608662309$	$120.23726543796644$		$S_3$ (as 3T2)	trivial	$2$	$2$	$29.1247947375$
3.1.14823.1	$x^{3} - 9 x - 48$	$3$	[1,1]	$-\,3^{5}\cdot 61$	$2$	$24.5647323302$	$58.531171665758464$		$S_3$ (as 3T2)	trivial	$2$	$1$	$68.4534913133$
3.1.14823.2	$x^{3} + 9 x - 21$	$3$	[1,1]	$-\,3^{5}\cdot 61$	$2$	$24.5647323302$	$58.531171665758464$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$11.7469140114$

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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.