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	Galois	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 146633 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
3.1.268.1	$x^{3} - x^{2} - 3 x + 5$	$3$	[1,1]	$-\,2^{2}\cdot 67$	$2$	$6.44730572727$	$12.993437600801144$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.52123333635$
3.1.335.1	$x^{3} - x^{2} + 4 x + 1$	$3$	[1,1]	$-\,5\cdot 67$	$2$	$6.94514955803$	$18.303005217723125$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$1.45573302761$
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	$2$	$2$	$3.85282661562$
3.1.804.1	$x^{3} - x^{2} + 4 x - 6$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 67$	$3$	$9.29862391479$	$28.35489375751565$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$8.57148683343$
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	$2$	$1$	$5.37065976135$
3.1.1675.1	$x^{3} - x^{2} + 7 x + 2$	$3$	[1,1]	$-\,5^{2}\cdot 67$	$2$	$11.8760386903$	$23.934116698484896$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$8.65921215673$
3.1.2479.1	$x^{3} - 11 x - 17$	$3$	[1,1]	$-\,37\cdot 67$	$2$	$13.5339793318$	$49.78955713801841$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$4.26288098313$
3.1.2680.1	$x^{3} - x^{2} + 20$	$3$	[1,1]	$-\,2^{3}\cdot 5\cdot 67$	$3$	$13.8902991161$	$51.76871642217914$		$S_3$ (as 3T2)	trivial	$2$	$1$	$16.9098926558$
3.1.2747.1	$x^{3} - x^{2} + x + 30$	$3$	[1,1]	$-\,41\cdot 67$	$2$	$14.0051001826$	$52.41183072551463$		$S_3$ (as 3T2)	trivial	$2$	$1$	$17.1384895724$
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	$2$	$1$	$32.448369121$
3.1.4556.1	$x^{3} - x^{2} + 3 x + 25$	$3$	[1,1]	$-\,2^{2}\cdot 17\cdot 67$	$3$	$16.5778385259$	$53.57331566797524$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$13.3470593242$
3.1.4623.1	$x^{3} - x^{2} - 18 x + 39$	$3$	[1,1]	$-\,3\cdot 23\cdot 67$	$3$	$16.6587073106$	$67.9926466612383$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$11.5813969193$
3.1.5427.1	$x^{3} + 6 x - 13$	$3$	[1,1]	$-\,3^{4}\cdot 67$	$2$	$17.573298008$	$35.41596455414278$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$15.6284604167$
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)	$[4]$	$2$	$1$	$2.4713308177$
3.1.6164.1	$x^{3} - x^{2} + 13 x - 29$	$3$	[1,1]	$-\,2^{2}\cdot 23\cdot 67$	$3$	$18.3352798668$	$78.51114570556209$		$S_3$ (as 3T2)	$[5]$	$2$	$1$	$3.03309980274$
3.1.6700.1	$x^{3} - x^{2} - 3 x + 17$	$3$	[1,1]	$-\,2^{2}\cdot 5^{2}\cdot 67$	$3$	$18.8520363102$	$37.99304202510457$	✓	$S_3$ (as 3T2)	$[3]$	$2$	$1$	$5.20909790287$
3.1.6700.2	$x^{3} + 10 x - 10$	$3$	[1,1]	$-\,2^{2}\cdot 5^{2}\cdot 67$	$3$	$18.8520363102$	$37.99304202510457$	✓	$S_3$ (as 3T2)	$[6]$	$2$	$1$	$2.54719483066$
3.3.6901.1	$x^{3} - x^{2} - 25 x - 2$	$3$	[3,0]	$67\cdot 103$	$2$	$19.0387022749$	$83.07225770376053$		$S_3$ (as 3T2)	trivial	$2$	$2$	$37.0238862725$
3.1.7303.1	$x^{3} - 5 x - 17$	$3$	[1,1]	$-\,67\cdot 109$	$2$	$19.4014311482$	$85.45759182190895$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$9.88200894103$
3.1.7571.1	$x^{3} - x^{2} - 7 x - 16$	$3$	[1,1]	$-\,67\cdot 113$	$2$	$19.6359122241$	$87.01149349367589$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$5.4894496635$
3.1.8107.1	$x^{3} - x^{2} + 4 x + 68$	$3$	[1,1]	$-\,11^{2}\cdot 67$	$2$	$20.0887720598$	$40.48547056351927$		$S_3$ (as 3T2)	trivial	$2$	$1$	$21.3600423831$
3.3.8308.1	$x^{3} - x^{2} - 13 x + 11$	$3$	[3,0]	$2^{2}\cdot 31\cdot 67$	$3$	$20.2534414717$	$72.34439882439624$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$17.7154841799$
3.1.8844.1	$x^{3} + 6 x - 54$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 11\cdot 67$	$4$	$20.6799544562$	$74.64161629168495$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$18.4569960385$
3.3.9045.1	$x^{3} - 18 x - 23$	$3$	[3,0]	$3^{3}\cdot 5\cdot 67$	$3$	$20.8354486741$	$65.94226607022871$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$30.4820593599$
3.1.9112.1	$x^{3} - x^{2} + x - 37$	$3$	[1,1]	$-\,2^{3}\cdot 17\cdot 67$	$3$	$20.8867677205$	$95.4567965102538$		$S_3$ (as 3T2)	trivial	$2$	$1$	$29.8492112373$
3.1.9715.1	$x^{3} - 2 x - 19$	$3$	[1,1]	$-\,5\cdot 29\cdot 67$	$3$	$21.337699844$	$98.56469956328178$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$22.5645755068$
3.1.10452.1	$x^{3} - 27 x - 80$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 13\cdot 67$	$4$	$21.8641769589$	$102.23502335305646$		$S_3$ (as 3T2)	trivial	$2$	$1$	$44.2404218528$
3.1.11055.1	$x^{3} - x^{2} + 14 x + 1$	$3$	[1,1]	$-\,3\cdot 5\cdot 11\cdot 67$	$4$	$22.2768056343$	$105.14276009312292$	✓	$S_3$ (as 3T2)	$[7]$	$2$	$1$	$2.64447757701$
3.1.11591.1	$x^{3} - x^{2} + 14 x + 32$	$3$	[1,1]	$-\,67\cdot 173$	$2$	$22.6311680018$	$107.66150658429409$		$S_3$ (as 3T2)	trivial	$2$	$1$	$34.6671971543$
3.1.12395.1	$x^{3} - 7 x - 86$	$3$	[1,1]	$-\,5\cdot 37\cdot 67$	$3$	$23.1427791202$	$111.33283433021904$		$S_3$ (as 3T2)	trivial	$2$	$1$	$18.4183086728$
3.1.12596.1	$x^{3} - x^{2} + 4 x - 66$	$3$	[1,1]	$-\,2^{2}\cdot 47\cdot 67$	$3$	$23.2672050552$	$112.2319027727856$		$S_3$ (as 3T2)	trivial	$2$	$1$	$54.0978537414$
3.1.12663.1	$x^{3} + 9 x - 86$	$3$	[1,1]	$-\,3^{3}\cdot 7\cdot 67$	$3$	$23.308386035$	$78.02394142997116$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$22.1851845219$
3.1.12663.2	$x^{3} - 3 x - 65$	$3$	[1,1]	$-\,3^{3}\cdot 7\cdot 67$	$3$	$23.308386035$	$78.02394142997116$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$4.23157951777$
3.1.12663.3	$x^{3} + 15 x - 61$	$3$	[1,1]	$-\,3^{3}\cdot 7\cdot 67$	$3$	$23.308386035$	$78.02394142997116$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$4.4643204491$
3.3.13333.1	$x^{3} - 34 x - 73$	$3$	[3,0]	$67\cdot 199$	$2$	$23.7124244231$	$115.46861045323097$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$25.7079123679$
3.1.13668.1	$x^{3} - x^{2} + 17 x + 31$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 17\cdot 67$	$4$	$23.9093804906$	$116.91022196540386$		$S_3$ (as 3T2)	$[11]$	$2$	$1$	$4.67186748369$
3.1.14003.1	$x^{3} - x^{2} + 11 x - 22$	$3$	[1,1]	$-\,11\cdot 19\cdot 67$	$3$	$24.103144049$	$118.33427229674419$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$12.5446854446$
3.1.14472.1	$x^{3} + 15 x - 6$	$3$	[1,1]	$-\,2^{3}\cdot 3^{3}\cdot 67$	$3$	$24.3692886027$	$83.41110194190541$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$20.8110733447$
3.1.14539.1	$x^{3} + 4 x - 23$	$3$	[1,1]	$-\,7\cdot 31\cdot 67$	$3$	$24.406837643$	$120.57777573002414$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$24.8336033831$
3.1.15276.1	$x^{3} - x^{2} - 27 x + 69$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 19\cdot 67$	$4$	$24.8124633851$	$98.09830263107024$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$23.5177394991$
3.1.15544.1	$x^{3} - 11 x - 50$	$3$	[1,1]	$-\,2^{3}\cdot 29\cdot 67$	$3$	$24.9567251347$	$124.6755790040696$		$S_3$ (as 3T2)	$[6]$	$2$	$1$	$9.20180604487$
3.1.15544.2	$x^{3} - x^{2} - 7 x + 51$	$3$	[1,1]	$-\,2^{3}\cdot 29\cdot 67$	$3$	$24.9567251347$	$124.6755790040696$		$S_3$ (as 3T2)	$[9]$	$2$	$1$	$3.88637006452$
3.1.15544.3	$x^{3} - 2 x - 48$	$3$	[1,1]	$-\,2^{3}\cdot 29\cdot 67$	$3$	$24.9567251347$	$124.6755790040696$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$13.4193713248$
3.1.15544.4	$x^{3} - x^{2} + 13 x - 49$	$3$	[1,1]	$-\,2^{3}\cdot 29\cdot 67$	$3$	$24.9567251347$	$124.6755790040696$		$S_3$ (as 3T2)	$[6]$	$2$	$1$	$6.18014094868$
3.1.15812.1	$x^{3} + 14 x - 44$	$3$	[1,1]	$-\,2^{2}\cdot 59\cdot 67$	$3$	$25.0993380883$	$125.7457752769452$		$S_3$ (as 3T2)	trivial	$2$	$1$	$23.5975646509$
3.3.16013.1	$x^{3} - x^{2} - 21 x - 22$	$3$	[3,0]	$67\cdot 239$	$2$	$25.2052437228$	$126.54248298496438$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$38.2626783691$
3.1.16616.1	$x^{3} - x^{2} - 27 x - 65$	$3$	[1,1]	$-\,2^{3}\cdot 31\cdot 67$	$3$	$25.517737243$	$128.90306435457614$		$S_3$ (as 3T2)	trivial	$2$	$1$	$36.5916170011$
3.1.17420.1	$x^{3} - x^{2} - x - 25$	$3$	[1,1]	$-\,2^{2}\cdot 5\cdot 13\cdot 67$	$4$	$25.9228482111$	$104.75644297409275$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$19.0271066418$
3.3.17420.1	$x^{3} - x^{2} - 16 x + 6$	$3$	[3,0]	$2^{2}\cdot 5\cdot 13\cdot 67$	$4$	$25.9228482111$	$131.98484761517133$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$69.1658050392$
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	$2$	$2$	$22.4985823721$

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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.