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 153519 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
3.1.876.1	$x^{3} - x^{2} - x - 11$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 73$	$3$	$9.56829820512$	$23.491390334630143$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$5.94796000997$
3.3.1825.1	$x^{3} - x^{2} - 8 x + 7$	$3$	[3,0]	$5^{2}\cdot 73$	$2$	$12.2204619442$	$24.982818506665623$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$4.48807298189$
3.1.1971.1	$x^{3} - 9 x - 20$	$3$	[1,1]	$-\,3^{3}\cdot 73$	$2$	$12.5380175891$	$30.782429528741996$		$S_3$ (as 3T2)	trivial	$2$	$1$	$11.1179700318$
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	$2$	$1$	$9.76411086644$
3.1.2920.1	$x^{3} - x^{2} - 6 x - 10$	$3$	[1,1]	$-\,2^{3}\cdot 5\cdot 73$	$3$	$14.2931389976$	$54.037024344425184$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$18.9538117722$
3.3.2920.1	$x^{3} - x^{2} - 15 x - 5$	$3$	[3,0]	$2^{3}\cdot 5\cdot 73$	$3$	$14.2931389976$	$54.037024344425184$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$17.9415860841$
3.3.2993.1	$x^{3} - x^{2} - 12 x + 17$	$3$	[3,0]	$41\cdot 73$	$2$	$14.4112694704$	$54.70831746635972$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$7.51407984191$
3.1.3212.1	$x^{3} + 8 x - 20$	$3$	[1,1]	$-\,2^{2}\cdot 11\cdot 73$	$3$	$14.7545231747$	$44.982587810520265$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$6.02364102111$
3.1.3431.1	$x^{3} - x^{2} + 14 x - 16$	$3$	[1,1]	$-\,47\cdot 73$	$2$	$15.0825082896$	$58.57473858242988$		$S_3$ (as 3T2)	$[2]$	$2$	$1$	$10.9231828967$
3.1.3723.1	$x^{3} - x^{2} + x - 12$	$3$	[1,1]	$-\,3\cdot 17\cdot 73$	$3$	$15.4987858917$	$61.0163912403872$	✓	$S_3$ (as 3T2)	$[4]$	$2$	$1$	$4.08125605174$
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	$2$	$1$	$17.9412342226$
3.1.4307.1	$x^{3} - 4 x - 13$	$3$	[1,1]	$-\,59\cdot 73$	$2$	$16.2701525122$	$65.62773803811922$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$10.2743270005$
3.1.5548.1	$x^{3} + 4 x - 14$	$3$	[1,1]	$-\,2^{2}\cdot 19\cdot 73$	$3$	$17.7029435346$	$59.11870256025904$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$6.78336752394$
3.3.5621.1	$x^{3} - x^{2} - 21 x - 28$	$3$	[3,0]	$7\cdot 11\cdot 73$	$3$	$17.7802499456$	$74.97332859090625$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$22.7920161354$
3.1.5767.1	$x^{3} - x^{2} + 12 x - 45$	$3$	[1,1]	$-\,73\cdot 79$	$2$	$17.9328775821$	$75.94076639065476$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$7.75919344984$
3.1.6132.1	$x^{3} + 9 x - 44$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 7\cdot 73$	$4$	$18.3034960026$	$78.30708780180757$		$S_3$ (as 3T2)	trivial	$2$	$1$	$34.5756663055$
3.1.6643.1	$x^{3} - x^{2} + 13 x - 48$	$3$	[1,1]	$-\,7\cdot 13\cdot 73$	$3$	$18.7984229853$	$81.50460109711598$		$S_3$ (as 3T2)	trivial	$2$	$1$	$19.7618971606$
3.1.7300.1	$x^{3} - x^{2} + 12 x + 2$	$3$	[1,1]	$-\,2^{2}\cdot 5^{2}\cdot 73$	$3$	$19.3987741458$	$49.965637013331246$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$32.5162033253$
3.3.7665.1	$x^{3} - x^{2} - 26 x + 21$	$3$	[3,0]	$3\cdot 5\cdot 7\cdot 73$	$4$	$19.7168433685$	$87.54998572244315$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$21.4003853711$
3.1.7811.1	$x^{3} - x^{2} + 3 x + 16$	$3$	[1,1]	$-\,73\cdot 107$	$2$	$19.8412431475$	$88.37986195961159$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$9.47968037272$
3.1.7884.1	$x^{3} - 18 x - 34$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}\cdot 73$	$3$	$19.9028623106$	$48.86406101606202$	✓	$S_3$ (as 3T2)	$[3]$	$2$	$1$	$4.03840765945$
3.1.7884.2	$x^{3} + 12 x - 6$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}\cdot 73$	$3$	$19.9028623106$	$48.86406101606202$	✓	$S_3$ (as 3T2)	$[3]$	$2$	$1$	$3.93067780292$
3.1.8103.1	$x^{3} - 9 x - 53$	$3$	[1,1]	$-\,3\cdot 37\cdot 73$	$3$	$20.0854675777$	$90.0166651237425$		$S_3$ (as 3T2)	trivial	$2$	$1$	$11.4759709888$
3.1.8395.1	$x^{3} - x^{2} - 5 x + 20$	$3$	[1,1]	$-\,5\cdot 23\cdot 73$	$3$	$20.3238930169$	$91.62423260251624$	✓	$S_3$ (as 3T2)	$[7]$	$2$	$1$	$3.31924620059$
3.1.8468.1	$x^{3} - x^{2} + 12 x + 28$	$3$	[1,1]	$-\,2^{2}\cdot 29\cdot 73$	$3$	$20.3826329187$	$92.02173656261873$		$S_3$ (as 3T2)	trivial	$2$	$1$	$25.5123361824$
3.3.8468.1	$x^{3} - x^{2} - 27 x - 43$	$3$	[3,0]	$2^{2}\cdot 29\cdot 73$	$3$	$20.3826329187$	$73.03770071172075$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$2$	$11.8995903551$
3.1.8687.1	$x^{3} - x^{2} - 22 x - 37$	$3$	[1,1]	$-\,7\cdot 17\cdot 73$	$3$	$20.5568519054$	$93.20407716403827$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$8.74257411025$
3.1.8760.1	$x^{3} + 3 x - 54$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 5\cdot 73$	$4$	$20.6142735777$	$93.59487165438073$		$S_3$ (as 3T2)	trivial	$2$	$1$	$42.8982977897$
3.1.8979.1	$x^{3} - x^{2} + 25 x - 36$	$3$	[1,1]	$-\,3\cdot 41\cdot 73$	$3$	$20.7846472012$	$94.75758544834287$		$S_3$ (as 3T2)	trivial	$2$	$1$	$28.4953200144$
3.3.9052.1	$x^{3} - x^{2} - 23 x + 31$	$3$	[3,0]	$2^{2}\cdot 31\cdot 73$	$3$	$20.8408221966$	$95.14199913813037$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$30.7894192003$
3.1.9271.1	$x^{3} - x^{2} - 12 x + 29$	$3$	[1,1]	$-\,73\cdot 127$	$2$	$21.00755586$	$96.28603221651622$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$7.10991083362$
3.1.9563.1	$x^{3} - 16 x - 31$	$3$	[1,1]	$-\,73\cdot 131$	$2$	$21.2258318082$	$97.79059259458448$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$14.8770936504$
3.1.9855.1	$x^{3} + 3 x - 19$	$3$	[1,1]	$-\,3^{3}\cdot 5\cdot 73$	$3$	$21.4397084964$	$68.83160493886392$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$12.4434149295$
3.1.9928.1	$x^{3} - x^{2} - 8 x + 80$	$3$	[1,1]	$-\,2^{3}\cdot 17\cdot 73$	$3$	$21.4925158735$	$99.6393496566492$		$S_3$ (as 3T2)	trivial	$2$	$1$	$32.1865041609$
3.1.10804.1	$x^{3} + x - 20$	$3$	[1,1]	$-\,2^{2}\cdot 37\cdot 73$	$3$	$22.1069175672$	$103.94229168149026$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$44.0441751666$
3.1.11023.1	$x^{3} - x^{2} - 6 x + 63$	$3$	[1,1]	$-\,73\cdot 151$	$2$	$22.2552905822$	$104.99047575851822$		$S_3$ (as 3T2)	trivial	$2$	$1$	$14.5307591928$
3.1.11607.1	$x^{3} - x^{2} + 30 x - 9$	$3$	[1,1]	$-\,3\cdot 53\cdot 73$	$3$	$22.641576428$	$107.73578792583271$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$8.34829656904$
3.1.11899.1	$x^{3} - x^{2} + 3 x + 20$	$3$	[1,1]	$-\,73\cdot 163$	$2$	$22.8298726115$	$109.0825375575761$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$29.2384202354$
3.1.11972.1	$x^{3} - x^{2} + 8 x - 22$	$3$	[1,1]	$-\,2^{2}\cdot 41\cdot 73$	$3$	$22.8764643175$	$109.41663493271945$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$23.5001594976$
3.1.12556.1	$x^{3} - 8 x - 44$	$3$	[1,1]	$-\,2^{2}\cdot 43\cdot 73$	$3$	$23.2425497384$	$88.93696841718867$		$S_3$ (as 3T2)	trivial	$2$	$1$	$14.6093493212$
3.1.13067.1	$x^{3} - x - 44$	$3$	[1,1]	$-\,73\cdot 179$	$2$	$23.5536724033$	$114.3109793501919$		$S_3$ (as 3T2)	trivial	$2$	$1$	$19.1576187174$
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)	$[2]$	$2$	$1$	$8.26230880019$
3.3.14089.1	$x^{3} - x^{2} - 22 x + 41$	$3$	[3,0]	$73\cdot 193$	$2$	$24.1523868605$	$118.69709347747316$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$7.0215106937$
3.1.14235.1	$x^{3} + 12 x - 67$	$3$	[1,1]	$-\,3\cdot 5\cdot 13\cdot 73$	$4$	$24.2355282627$	$119.31051923447488$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$42.6861700546$
3.1.14308.1	$x^{3} - x^{2} - 23 x + 71$	$3$	[1,1]	$-\,2^{2}\cdot 7^{2}\cdot 73$	$3$	$24.2768858997$	$62.53024338339619$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$13.3732823976$
3.1.14819.1	$x^{3} - x^{2} + 13 x - 20$	$3$	[1,1]	$-\,7\cdot 29\cdot 73$	$3$	$24.5625225263$	$121.73331507849443$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$28.3057897999$
3.1.14892.1	$x^{3} + 24 x - 54$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 17\cdot 73$	$4$	$24.6027890287$	$96.85748364229387$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$9.33635774803$
3.1.14892.2	$x^{3} - x^{2} - 23 x - 75$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 17\cdot 73$	$4$	$24.6027890287$	$96.85748364229387$		$S_3$ (as 3T2)	$[9]$	$2$	$1$	$4.07610286751$
3.1.14892.3	$x^{3} - x^{2} + 17 x - 71$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 17\cdot 73$	$4$	$24.6027890287$	$96.85748364229387$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$9.5593752021$
3.1.15695.1	$x^{3} - x^{2} + 30 x + 27$	$3$	[1,1]	$-\,5\cdot 43\cdot 73$	$3$	$25.0372777206$	$125.27968710050325$		$S_3$ (as 3T2)	trivial	$2$	$1$	$31.1451705543$

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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.