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 111226 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
3.1.356.1	$x^{3} - x^{2} + x + 7$	$3$	[1,1]	$-\,2^{2}\cdot 89$	$2$	$7.0873410615$	$18.867962264113206$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$3.75521003542$
3.1.1068.1	$x^{3} - x^{2} - x + 13$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 89$	$3$	$10.2217146006$	$25.93834691424768$		$S_3$ (as 3T2)	trivial	$2$	$1$	$7.16938023975$
3.1.1691.1	$x^{3} - x^{2} + x - 24$	$3$	[1,1]	$-\,19\cdot 89$	$2$	$11.9137331742$	$41.12177038990418$		$S_3$ (as 3T2)	trivial	$2$	$1$	$12.2275307861$
3.1.2047.1	$x^{3} + 7 x - 5$	$3$	[1,1]	$-\,23\cdot 89$	$2$	$12.6971411508$	$45.24378410345447$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$4.31855543785$
3.1.2403.1	$x^{3} - 6 x - 11$	$3$	[1,1]	$-\,3^{3}\cdot 89$	$2$	$13.3942352868$	$33.98884972775975$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$10.2990036664$
3.1.2759.1	$x^{3} - x^{2} + 4 x - 11$	$3$	[1,1]	$-\,31\cdot 89$	$2$	$14.0254638611$	$52.52618394667559$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.44889225965$
3.1.3115.1	$x^{3} - x^{2} - 5 x - 10$	$3$	[1,1]	$-\,5\cdot 7\cdot 89$	$3$	$14.6044772657$	$55.812185049503306$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$14.3337029794$
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	$2$	$1$	$33.3613959484$
3.1.3560.1	$x^{3} - x^{2} - 11 x + 31$	$3$	[1,1]	$-\,2^{3}\cdot 5\cdot 89$	$3$	$15.269213443$	$59.665735560705194$		$S_3$ (as 3T2)	trivial	$2$	$1$	$15.4341727951$
3.1.3916.1	$x^{3} - x^{2} + 3 x + 11$	$3$	[1,1]	$-\,2^{2}\cdot 11\cdot 89$	$3$	$15.7621054159$	$49.66815293217739$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$12.2557528981$
3.3.4628.1	$x^{3} - x^{2} - 13 x - 9$	$3$	[3,0]	$2^{2}\cdot 13\cdot 89$	$3$	$16.6647108816$	$53.99497485374668$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$16.7152101414$
3.3.5073.1	$x^{3} - x^{2} - 18 x + 33$	$3$	[3,0]	$3\cdot 19\cdot 89$	$3$	$17.1825765512$	$71.22499561249548$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$14.2864682842$
3.1.5963.1	$x^{3} - x^{2} + 11 x - 10$	$3$	[1,1]	$-\,67\cdot 89$	$2$	$18.1337769619$	$77.22046360907191$	✓	$S_3$ (as 3T2)	$[4]$	$2$	$1$	$2.4713308177$
3.1.6675.1	$x^{3} - x^{2} + 7 x + 12$	$3$	[1,1]	$-\,3\cdot 5^{2}\cdot 89$	$3$	$18.8285592793$	$47.77884352737507$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$24.0659240733$
3.1.7031.1	$x^{3} - x^{2} + 18 x - 45$	$3$	[1,1]	$-\,79\cdot 89$	$2$	$19.1575087525$	$83.85105843100611$		$S_3$ (as 3T2)	trivial	$2$	$1$	$19.7096002661$
3.3.8277.1	$x^{3} - x^{2} - 22 x + 28$	$3$	[3,0]	$3\cdot 31\cdot 89$	$3$	$20.2282192271$	$90.97801932335085$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$35.3501486324$
3.1.9167.1	$x^{3} - x^{2} + 2 x - 19$	$3$	[1,1]	$-\,89\cdot 103$	$2$	$20.9287076026$	$95.74445153636842$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$7.24134216561$
3.1.9523.1	$x^{3} - x^{2} - 3 x + 20$	$3$	[1,1]	$-\,89\cdot 107$	$2$	$21.196196066$	$97.58585963140357$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$26.257594399$
3.1.9612.1	$x^{3} + 12 x - 10$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}\cdot 89$	$3$	$21.2620231845$	$53.95393581303488$	✓	$S_3$ (as 3T2)	$[3]$	$2$	$1$	$5.80843997917$
3.1.9612.2	$x^{3} + 24 x - 34$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}\cdot 89$	$3$	$21.2620231845$	$53.95393581303488$	✓	$S_3$ (as 3T2)	$[3]$	$2$	$1$	$5.60910937229$
3.3.9612.1	$x^{3} - 15 x - 12$	$3$	[3,0]	$2^{2}\cdot 3^{3}\cdot 89$	$3$	$21.2620231845$	$67.9776994555195$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$41.0496580113$
3.1.10324.1	$x^{3} - 14 x - 44$	$3$	[1,1]	$-\,2^{2}\cdot 29\cdot 89$	$3$	$21.7745571926$	$101.6070863670443$		$S_3$ (as 3T2)	trivial	$2$	$1$	$29.8651021081$
3.3.10324.1	$x^{3} - x^{2} - 29 x + 27$	$3$	[3,0]	$2^{2}\cdot 29\cdot 89$	$3$	$21.7745571926$	$80.6455978932349$		$S_3$ (as 3T2)	trivial	$2$	$2$	$41.2769285833$
3.1.11748.1	$x^{3} - x^{2} - 8 x + 66$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 11\cdot 89$	$4$	$22.7328897632$	$108.38819123871383$		$S_3$ (as 3T2)	trivial	$2$	$1$	$48.1693070998$
3.1.12015.1	$x^{3} + 3 x - 21$	$3$	[1,1]	$-\,3^{3}\cdot 5\cdot 89$	$3$	$22.9038201645$	$76.00137846829602$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$16.4870473858$
3.1.12104.1	$x^{3} - x^{2} + 16 x + 76$	$3$	[1,1]	$-\,2^{3}\cdot 17\cdot 89$	$3$	$22.9602337425$	$110.01818031580054$		$S_3$ (as 3T2)	trivial	$2$	$1$	$20.3973495286$
3.1.12727.1	$x^{3} - x^{2} + 6 x + 63$	$3$	[1,1]	$-\,11\cdot 13\cdot 89$	$3$	$23.3475876621$	$112.81400622263178$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$14.3151822065$
3.3.13172.1	$x^{3} - 26 x - 46$	$3$	[3,0]	$2^{2}\cdot 37\cdot 89$	$3$	$23.6165927552$	$91.09248066996328$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$29.4583124183$
3.1.13439.1	$x^{3} - 13 x - 113$	$3$	[1,1]	$-\,89\cdot 151$	$2$	$23.7750981216$	$115.92670097954138$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$7.67229137205$
3.1.14507.1	$x^{3} - x^{2} - 7 x - 22$	$3$	[1,1]	$-\,89\cdot 163$	$2$	$24.3889181962$	$120.44500819876264$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$14.8378219074$
3.1.14863.1	$x^{3} + 13 x - 15$	$3$	[1,1]	$-\,89\cdot 167$	$2$	$24.5868085355$	$121.91390404707742$	✓	$S_3$ (as 3T2)	$[4]$	$2$	$1$	$2.78433972654$
3.1.14952.1	$x^{3} - x^{2} + 2 x + 70$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 7\cdot 89$	$4$	$24.6357863712$	$122.27837094106218$		$S_3$ (as 3T2)	trivial	$2$	$1$	$58.3841721388$
3.1.15308.1	$x^{3} - x^{2} - 3 x + 25$	$3$	[1,1]	$-\,2^{2}\cdot 43\cdot 89$	$3$	$24.8297769282$	$98.20099651172262$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$8.74877965544$
3.1.15575.1	$x^{3} - x^{2} - 8 x - 23$	$3$	[1,1]	$-\,5^{2}\cdot 7\cdot 89$	$3$	$24.9733048382$	$72.98338902721571$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$9.08114019341$
3.1.15931.1	$x^{3} - x^{2} + 15 x - 16$	$3$	[1,1]	$-\,89\cdot 179$	$2$	$25.1621460724$	$126.21806526801146$	✓	$S_3$ (as 3T2)	$[8]$	$2$	$1$	$2.78054781595$
3.1.16287.1	$x^{3} - 15 x - 77$	$3$	[1,1]	$-\,3\cdot 61\cdot 89$	$3$	$25.3481945736$	$127.62053126358627$		$S_3$ (as 3T2)	$[2]$	$2$	$1$	$12.3131047684$
3.1.16376.1	$x^{3} - x^{2} + 6 x - 26$	$3$	[1,1]	$-\,2^{3}\cdot 23\cdot 89$	$3$	$25.3942823016$	$127.96874618437113$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$39.2487360672$
3.3.16376.1	$x^{3} - x^{2} - 40 x - 72$	$3$	[3,0]	$2^{3}\cdot 23\cdot 89$	$3$	$25.3942823016$	$127.96874618437113$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$53.787870419$
3.1.16999.1	$x^{3} - x^{2} + 16 x - 13$	$3$	[1,1]	$-\,89\cdot 191$	$2$	$25.7123117238$	$130.3802132227126$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$10.3370539269$
3.1.17800.1	$x^{3} - x^{2} - 3 x - 153$	$3$	[1,1]	$-\,2^{3}\cdot 5^{2}\cdot 89$	$3$	$26.1099877122$	$78.02252476156511$		$S_3$ (as 3T2)	trivial	$2$	$1$	$45.1991702637$
3.3.17800.1	$x^{3} - x^{2} - 18 x + 22$	$3$	[3,0]	$2^{3}\cdot 5^{2}\cdot 89$	$3$	$26.1099877122$	$78.02252476156511$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$45.0496156495$
3.3.17889.1	$x^{3} - x^{2} - 30 x + 69$	$3$	[3,0]	$3\cdot 67\cdot 89$	$3$	$26.1534320314$	$133.7497663549361$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$22.4985823721$
3.1.18156.1	$x^{3} - 24 x - 90$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 17\cdot 89$	$4$	$26.2829065774$	$106.9465440813571$		$S_3$ (as 3T2)	trivial	$2$	$1$	$40.6290819972$
3.1.18779.1	$x^{3} + 14 x - 17$	$3$	[1,1]	$-\,89\cdot 211$	$2$	$26.5801536626$	$137.03649149040558$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$11.5539522649$
3.1.18868.1	$x^{3} - x^{2} + 4 x + 78$	$3$	[1,1]	$-\,2^{2}\cdot 53\cdot 89$	$3$	$26.6220782646$	$137.3608386695422$		$S_3$ (as 3T2)	trivial	$2$	$1$	$43.3264566525$
3.1.19224.1	$x^{3} + 27 x - 122$	$3$	[1,1]	$-\,2^{3}\cdot 3^{3}\cdot 89$	$3$	$26.7884705735$	$96.13498450891784$		$S_3$ (as 3T2)	trivial	$2$	$1$	$58.0561057242$
3.1.19847.1	$x^{3} - x^{2} - 18 x - 56$	$3$	[1,1]	$-\,89\cdot 223$	$2$	$27.0747812577$	$140.87938103214395$		$S_3$ (as 3T2)	trivial	$2$	$1$	$60.2093602895$
3.1.20292.1	$x^{3} - x^{2} - 16 x - 134$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 19\cdot 89$	$4$	$27.2756400929$	$142.44999122499095$		$S_3$ (as 3T2)	trivial	$2$	$1$	$56.4525671366$
3.3.20381.1	$x^{3} - 26 x - 43$	$3$	[3,0]	$89\cdot 229$	$2$	$27.3154586022$	$142.76203977248295$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$36.5161764746$
3.1.20915.1	$x^{3} + 23 x - 36$	$3$	[1,1]	$-\,5\cdot 47\cdot 89$	$3$	$27.5519678481$	$144.62019222777985$		$S_3$ (as 3T2)	trivial	$2$	$1$	$63.9456953871$

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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.