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 56672 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
3.1.1351.1	$x^{3} - x^{2} - 7$	$3$	[1,1]	$-\,7\cdot 193$	$2$	$11.0548227346$	$36.75595189897821$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$3.5340812844$
3.1.2316.1	$x^{3} - x^{2} + x + 9$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 193$	$3$	$13.2305993516$	$38.1967089652795$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$12.8139641808$
3.1.2895.1	$x^{3} + 3 x - 31$	$3$	[1,1]	$-\,3\cdot 5\cdot 193$	$3$	$14.2522311065$	$53.80520420925842$		$S_3$ (as 3T2)	trivial	$2$	$1$	$8.23722212582$
3.3.3281.1	$x^{3} - x^{2} - 14 x - 13$	$3$	[3,0]	$17\cdot 193$	$2$	$14.8594274805$	$57.28001396647874$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$7.45000920901$
3.1.4632.1	$x^{3} - x^{2} + 17 x - 5$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 193$	$3$	$16.6695106258$	$68.05879810869422$		$S_3$ (as 3T2)	trivial	$2$	$1$	$26.4997634548$
3.3.4825.1	$x^{3} - x^{2} - 18 x - 8$	$3$	[3,0]	$5^{2}\cdot 193$	$2$	$16.8978884656$	$40.621752652279945$		$S_3$ (as 3T2)	trivial	$2$	$2$	$36.3964999556$
3.1.5211.1	$x^{3} - 6 x - 15$	$3$	[1,1]	$-\,3^{3}\cdot 193$	$2$	$17.3369896955$	$50.05184815392902$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$19.937704382$
3.1.8492.1	$x^{3} - 4 x - 18$	$3$	[1,1]	$-\,2^{2}\cdot 11\cdot 193$	$3$	$20.4018709036$	$73.14112918087628$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$15.2345932099$
3.1.9071.1	$x^{3} - x^{2} + 8 x - 19$	$3$	[1,1]	$-\,47\cdot 193$	$2$	$20.8553935214$	$95.24179754708538$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$2.75272362745$
3.1.9843.1	$x^{3} - x^{2} + 11 x + 10$	$3$	[1,1]	$-\,3\cdot 17\cdot 193$	$3$	$21.4310028987$	$99.21189444819608$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$15.2145212229$
3.1.10615.1	$x^{3} - x^{2} - 10 x + 27$	$3$	[1,1]	$-\,5\cdot 11\cdot 193$	$3$	$21.9772492082$	$103.02912209661888$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$16.6374486927$
3.1.10808.1	$x^{3} + 2 x - 40$	$3$	[1,1]	$-\,2^{3}\cdot 7\cdot 193$	$3$	$22.1096454692$	$103.96153134693621$		$S_3$ (as 3T2)	trivial	$2$	$1$	$24.8600363572$
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	$2$	$2$	$7.0215106937$
3.1.14668.1	$x^{3} - x^{2} + 29 x - 47$	$3$	[1,1]	$-\,2^{2}\cdot 19\cdot 193$	$3$	$24.478809944$	$96.12627622002738$		$S_3$ (as 3T2)	trivial	$2$	$1$	$23.1721229145$
3.3.14668.1	$x^{3} - x^{2} - 28 x + 44$	$3$	[3,0]	$2^{2}\cdot 19\cdot 193$	$3$	$24.478809944$	$121.11151885762146$		$S_3$ (as 3T2)	trivial	$2$	$2$	$44.4546201216$
3.1.16019.1	$x^{3} - x^{2} - 21 x - 38$	$3$	[1,1]	$-\,83\cdot 193$	$2$	$25.2083914273$	$126.56618821786489$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$12.6839080299$
3.1.16984.1	$x^{3} - x^{2} - 32 x - 76$	$3$	[1,1]	$-\,2^{3}\cdot 11\cdot 193$	$3$	$25.7047466087$	$130.3226764611593$		$S_3$ (as 3T2)	trivial	$2$	$1$	$43.5339028812$
3.1.19300.1	$x^{3} - x^{2} - 8 x - 78$	$3$	[1,1]	$-\,2^{2}\cdot 5^{2}\cdot 193$	$3$	$26.8237259263$	$81.24350530455989$		$S_3$ (as 3T2)	trivial	$2$	$1$	$45.9660857236$
3.1.20651.1	$x^{3} - x^{2} + 20 x - 112$	$3$	[1,1]	$-\,107\cdot 193$	$2$	$27.4355515573$	$143.7045580348793$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$20.7682881823$
3.1.20844.1	$x^{3} + 12 x - 138$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}\cdot 193$	$3$	$27.5207556805$	$79.45235641249951$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$10.5096872292$
3.1.20844.2	$x^{3} + 24 x - 70$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}\cdot 193$	$3$	$27.5207556805$	$79.45235641249951$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$13.6391945297$
3.3.21809.1	$x^{3} - x^{2} - 40 x - 81$	$3$	[3,0]	$113\cdot 193$	$2$	$27.9390681805$	$147.6787053031005$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$16.1428392525$
3.1.22195.1	$x^{3} - x^{2} + 5 x - 30$	$3$	[1,1]	$-\,5\cdot 23\cdot 193$	$3$	$28.1029374301$	$148.9798644112687$	✓	$S_3$ (as 3T2)	$[4]$	$2$	$1$	$10.2616866976$
3.1.22967.1	$x^{3} + 23 x - 40$	$3$	[1,1]	$-\,7\cdot 17\cdot 193$	$3$	$28.4250621855$	$151.5486720496092$		$S_3$ (as 3T2)	trivial	$2$	$1$	$69.0731242478$
3.3.23353.1	$x^{3} - x^{2} - 18 x + 13$	$3$	[3,0]	$11^{2}\cdot 193$	$2$	$28.5834223455$	$68.71324277225357$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$15.233994078$
3.1.25283.1	$x^{3} + 2 x - 153$	$3$	[1,1]	$-\,131\cdot 193$	$2$	$29.3500965938$	$159.00628918379297$		$S_3$ (as 3T2)	trivial	$2$	$1$	$64.3555243938$
3.1.25476.1	$x^{3} - x^{2} + 36 x - 54$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 11\cdot 193$	$4$	$29.4245895442$	$159.6120296218302$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$29.3549627517$
3.1.26248.1	$x^{3} - x^{2} + 17 x + 51$	$3$	[1,1]	$-\,2^{3}\cdot 17\cdot 193$	$3$	$29.7188549609$	$162.01234520862909$		$S_3$ (as 3T2)	trivial	$2$	$1$	$49.5281287056$
3.1.27020.1	$x^{3} + 32 x - 142$	$3$	[1,1]	$-\,2^{2}\cdot 5\cdot 7\cdot 193$	$4$	$30.0074055792$	$130.46659872963554$		$S_3$ (as 3T2)	trivial	$2$	$1$	$28.8409803226$
3.3.27213.1	$x^{3} - x^{2} - 33 x + 78$	$3$	[3,0]	$3\cdot 47\cdot 193$	$3$	$30.0786823448$	$164.96363235574077$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$69.8448138597$
3.1.28371.1	$x^{3} - x^{2} - 37 x + 106$	$3$	[1,1]	$-\,3\cdot 7^{2}\cdot 193$	$3$	$30.4994177081$	$88.05174662535941$	✓	$S_3$ (as 3T2)	$[6]$	$2$	$1$	$6.93538558687$
3.1.28564.1	$x^{3} - x^{2} + 21 x - 61$	$3$	[1,1]	$-\,2^{2}\cdot 37\cdot 193$	$3$	$30.5684211508$	$169.00887550658396$		$S_3$ (as 3T2)	trivial	$2$	$1$	$75.4303154331$
3.1.29915.1	$x^{3} - x^{2} + 29 x + 146$	$3$	[1,1]	$-\,5\cdot 31\cdot 193$	$3$	$31.0429512153$	$172.9595328393321$		$S_3$ (as 3T2)	trivial	$2$	$1$	$31.7950192072$
3.1.30687.1	$x^{3} - x^{2} - 2 x - 33$	$3$	[1,1]	$-\,3\cdot 53\cdot 193$	$3$	$31.3077226468$	$175.17705329180532$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$35.7153374263$
3.1.31652.1	$x^{3} - x^{2} + 48 x - 68$	$3$	[1,1]	$-\,2^{2}\cdot 41\cdot 193$	$3$	$31.6325147353$	$177.9100896520487$		$S_3$ (as 3T2)	trivial	$2$	$1$	$36.784138016$
3.1.33196.1	$x^{3} - 32 x - 78$	$3$	[1,1]	$-\,2^{2}\cdot 43\cdot 193$	$3$	$32.1387206905$	$144.61040621668658$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$24.6535713858$
3.1.35319.1	$x^{3} - 9 x - 109$	$3$	[1,1]	$-\,3\cdot 61\cdot 193$	$3$	$32.8097408111$	$187.9334988765973$		$S_3$ (as 3T2)	trivial	$2$	$1$	$20.2676202882$
3.3.36477.1	$x^{3} - 48 x - 65$	$3$	[3,0]	$3^{3}\cdot 7\cdot 193$	$3$	$33.1644682038$	$132.42474287446353$		$S_3$ (as 3T2)	trivial	$2$	$2$	$71.6118594825$
3.1.38407.1	$x^{3} - x^{2} - 2 x - 37$	$3$	[1,1]	$-\,193\cdot 199$	$2$	$33.7393564975$	$195.97703947146462$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$13.7025736369$
3.1.41495.1	$x^{3} - x^{2} + 64 x - 35$	$3$	[1,1]	$-\,5\cdot 43\cdot 193$	$3$	$34.6203873744$	$203.70321548763044$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$20.0750713048$
3.1.42267.1	$x^{3} - x^{2} - 15 x + 126$	$3$	[1,1]	$-\,3\cdot 73\cdot 193$	$3$	$34.8337695714$	$205.5893966137359$		$S_3$ (as 3T2)	$[8]$	$2$	$1$	$7.91718526966$
3.1.44004.1	$x^{3} + 9 x - 242$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 19\cdot 193$	$4$	$35.3045531234$	$209.77130404323657$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$26.0665573203$
3.1.44004.2	$x^{3} - 18 x - 244$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 19\cdot 193$	$4$	$35.3045531234$	$209.77130404323657$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$28.6821975302$
3.1.44004.3	$x^{3} - x^{2} - 52 x + 184$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 19\cdot 193$	$4$	$35.3045531234$	$209.77130404323657$		$S_3$ (as 3T2)	$[21]$	$2$	$1$	$4.39307647346$
3.1.44004.4	$x^{3} - x^{2} + x - 81$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 19\cdot 193$	$4$	$35.3045531234$	$209.77130404323657$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$42.1205586456$
3.1.45355.1	$x^{3} - 2 x - 41$	$3$	[1,1]	$-\,5\cdot 47\cdot 193$	$3$	$35.66222128$	$212.96713361455565$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$54.9173230974$
3.1.45548.1	$x^{3} - 28 x - 100$	$3$	[1,1]	$-\,2^{2}\cdot 59\cdot 193$	$3$	$35.7127344088$	$169.39138700154103$		$S_3$ (as 3T2)	trivial	$2$	$1$	$24.8106049919$
3.1.46899.1	$x^{3} - 18 x - 51$	$3$	[1,1]	$-\,3^{5}\cdot 193$	$2$	$36.0623918059$	$104.11203965883807$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$32.9909731168$
3.1.46899.2	$x^{3} - 36 x - 93$	$3$	[1,1]	$-\,3^{5}\cdot 193$	$2$	$36.0623918059$	$104.11203965883807$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$29.9331897748$
3.1.46899.3	$x^{3} - 9 x - 84$	$3$	[1,1]	$-\,3^{5}\cdot 193$	$2$	$36.0623918059$	$104.11203965883807$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$45.0290940806$

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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.