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 233741 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
3.1.451.1	$x^{3} - x^{2} - 5 x + 8$	$3$	[1,1]	$-\,11\cdot 41$	$2$	$7.66876649057$	$21.236760581595302$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$3.57573413339$
3.1.492.1	$x^{3} - x^{2} + 3 x + 3$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 41$	$3$	$7.89444677297$	$17.60512931716603$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$4.42072409068$
3.3.697.1	$x^{3} - 7 x - 5$	$3$	[3,0]	$17\cdot 41$	$2$	$8.86633751079$	$26.40075756488817$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$2.71184965988$
3.1.984.1	$x^{3} - x^{2} - 12$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 41$	$3$	$9.94637966654$	$31.368774282716245$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$9.97919841126$
3.1.1107.1	$x^{3} + 6 x - 3$	$3$	[1,1]	$-\,3^{3}\cdot 41$	$2$	$10.3446517211$	$23.06924557594881$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$3.29174765018$
3.1.1599.1	$x^{3} + 3 x - 23$	$3$	[1,1]	$-\,3\cdot 13\cdot 41$	$3$	$11.6936337636$	$39.98749804626441$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$4.46392196445$
3.1.1763.1	$x^{3} + 11 x - 8$	$3$	[1,1]	$-\,41\cdot 43$	$2$	$12.0804775917$	$41.98809355043403$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$6.01867017906$
3.1.1927.1	$x^{3} - x^{2} + 2 x - 9$	$3$	[1,1]	$-\,41\cdot 47$	$2$	$12.4440163071$	$43.89760813529594$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$2.35003381942$
3.1.2091.1	$x^{3} - x^{2} - x - 26$	$3$	[1,1]	$-\,3\cdot 17\cdot 41$	$3$	$12.7874714651$	$45.727453460694704$		$S_3$ (as 3T2)	trivial	$2$	$1$	$10.5941817144$
3.1.2132.1	$x^{3} - x^{2} - 3 x + 19$	$3$	[1,1]	$-\,2^{2}\cdot 13\cdot 41$	$3$	$12.8705093209$	$46.17358552246078$		$S_3$ (as 3T2)	trivial	$2$	$1$	$10.7300504916$
3.3.2296.1	$x^{3} - x^{2} - 14 x - 14$	$3$	[3,0]	$2^{3}\cdot 7\cdot 41$	$3$	$13.1924045682$	$47.916594202843754$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$14.2655509459$
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	$2$	$1$	$17.1384895724$
3.1.2911.1	$x^{3} - x^{2} + 8 x - 9$	$3$	[1,1]	$-\,41\cdot 71$	$2$	$14.2784391528$	$53.95368384086484$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$2.22229249028$
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	$2$	$2$	$7.51407984191$
3.1.3075.1	$x^{3} - x^{2} - 3 x + 12$	$3$	[1,1]	$-\,3\cdot 5^{2}\cdot 41$	$3$	$14.541695288$	$32.42892547103848$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$15.1066430535$
3.1.3116.1	$x^{3} - x^{2} + x - 11$	$3$	[1,1]	$-\,2^{2}\cdot 19\cdot 41$	$3$	$14.6060399104$	$44.305270518711566$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$9.32643149743$
3.1.3444.1	$x^{3} - x^{2} - 12 x - 24$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 7\cdot 41$	$4$	$15.1015334027$	$58.68560300448484$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$22.2657969372$
3.1.4428.1	$x^{3} + 6 x - 38$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}\cdot 41$	$3$	$16.4211110244$	$36.620144695373874$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$3.65760853101$
3.1.4428.2	$x^{3} + 12 x - 20$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}\cdot 41$	$3$	$16.4211110244$	$36.620144695373874$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$4.14689670911$
3.1.4715.1	$x^{3} - 13 x - 32$	$3$	[1,1]	$-\,5\cdot 23\cdot 41$	$3$	$16.7684877919$	$68.66585760041157$		$S_3$ (as 3T2)	trivial	$2$	$1$	$13.1483780435$
3.1.4920.1	$x^{3} - x^{2} - 10 x + 22$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 5\cdot 41$	$4$	$17.0080699863$	$70.14271166700073$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$30.5407656957$
3.1.5371.1	$x^{3} - x^{2} - 15 x - 22$	$3$	[1,1]	$-\,41\cdot 131$	$2$	$17.5126439237$	$73.287106642301$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$7.37403019552$
3.1.5535.1	$x^{3} - 3 x - 43$	$3$	[1,1]	$-\,3^{3}\cdot 5\cdot 41$	$3$	$17.6891056199$	$51.58440129745782$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$6.85433781231$
3.1.5576.1	$x^{3} - x^{2} + x - 29$	$3$	[1,1]	$-\,2^{3}\cdot 17\cdot 41$	$3$	$17.7326750216$	$74.67261881037788$		$S_3$ (as 3T2)	trivial	$2$	$1$	$13.6693037397$
3.1.5699.1	$x^{3} - x^{2} + 11 x + 2$	$3$	[1,1]	$-\,41\cdot 139$	$2$	$17.8621151868$	$75.49172140042907$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$9.21811203901$
3.1.5740.1	$x^{3} - x^{2} - x + 15$	$3$	[1,1]	$-\,2^{2}\cdot 5\cdot 7\cdot 41$	$4$	$17.9048476826$	$60.13296444711741$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$17.7273422451$
3.1.6068.1	$x^{3} - x^{2} - 3 x - 29$	$3$	[1,1]	$-\,2^{2}\cdot 37\cdot 41$	$3$	$18.2395949946$	$77.89736837660179$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$20.1752552719$
3.1.6232.1	$x^{3} - x^{2} + 17 x - 21$	$3$	[1,1]	$-\,2^{3}\cdot 19\cdot 41$	$3$	$18.4024571388$	$78.9430174239622$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$25.8810435226$
3.3.6396.1	$x^{3} - x^{2} - 20 x + 24$	$3$	[3,0]	$2^{2}\cdot 3\cdot 13\cdot 41$	$4$	$18.5624865376$	$79.97499609252883$		$S_3$ (as 3T2)	trivial	$2$	$2$	$40.0307498197$
3.3.6601.1	$x^{3} - 13 x - 9$	$3$	[3,0]	$7\cdot 23\cdot 41$	$3$	$18.7587218682$	$81.24653838779841$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$20.2402177624$
3.1.6888.1	$x^{3} - x^{2} - 8 x + 36$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 7\cdot 41$	$4$	$19.0267398198$	$82.99397568498571$		$S_3$ (as 3T2)	trivial	$2$	$1$	$36.710086358$
3.1.7052.1	$x^{3} - x^{2} + 7 x - 17$	$3$	[1,1]	$-\,2^{2}\cdot 41\cdot 43$	$3$	$19.1765628374$	$66.65194387209816$	✓	$S_3$ (as 3T2)	$[3]$	$2$	$1$	$2.68537313887$
3.1.7052.2	$x^{3} - x^{2} + 3 x - 17$	$3$	[1,1]	$-\,2^{2}\cdot 41\cdot 43$	$3$	$19.1765628374$	$66.65194387209816$	✓	$S_3$ (as 3T2)	$[3]$	$2$	$1$	$4.20757937446$
3.1.7052.3	$x^{3} - x^{2} + 11 x + 43$	$3$	[1,1]	$-\,2^{2}\cdot 41\cdot 43$	$3$	$19.1765628374$	$66.65194387209816$	✓	$S_3$ (as 3T2)	$[3]$	$2$	$1$	$8.5827972619$
3.1.7175.1	$x^{3} - x^{2} + 2 x + 32$	$3$	[1,1]	$-\,5^{2}\cdot 7\cdot 41$	$3$	$19.2874124836$	$49.53600189237178$		$S_3$ (as 3T2)	trivial	$2$	$1$	$38.1878377973$
3.1.7339.1	$x^{3} - x^{2} + 11 x + 6$	$3$	[1,1]	$-\,41\cdot 179$	$2$	$19.4332585701$	$85.66796367370944$	✓	$S_3$ (as 3T2)	$[5]$	$2$	$1$	$3.93374993166$
3.1.7831.1	$x^{3} + 19 x - 12$	$3$	[1,1]	$-\,41\cdot 191$	$2$	$19.8581631601$	$88.4929375713113$		$S_3$ (as 3T2)	$[7]$	$2$	$1$	$6.0476012547$
3.1.8036.1	$x^{3} + 14 x - 28$	$3$	[1,1]	$-\,2^{2}\cdot 7^{2}\cdot 41$	$3$	$20.0299551122$	$46.86197816804901$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$5.30367610236$
3.1.8200.1	$x^{3} - x^{2} + 2 x - 18$	$3$	[1,1]	$-\,2^{3}\cdot 5^{2}\cdot 41$	$3$	$20.1652967522$	$52.956213540525944$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$27.7623638038$
3.1.8323.1	$x^{3} - x^{2} + 7 x + 14$	$3$	[1,1]	$-\,7\cdot 29\cdot 41$	$3$	$20.2656232631$	$91.23047736365298$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$21.6302707516$
3.3.8692.1	$x^{3} - x^{2} - 13 x - 1$	$3$	[3,0]	$2^{2}\cdot 41\cdot 53$	$3$	$20.5607951365$	$73.99741132528298$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$16.9558131292$
3.1.8815.1	$x^{3} + 13 x - 1$	$3$	[1,1]	$-\,5\cdot 41\cdot 43$	$3$	$20.6573261063$	$93.888231424391$	✓	$S_3$ (as 3T2)	$[5]$	$2$	$1$	$2.56540400654$
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	$2$	$1$	$28.4953200144$
3.1.9143.1	$x^{3} - x^{2} - 10 x - 19$	$3$	[1,1]	$-\,41\cdot 223$	$2$	$20.9104272503$	$95.61903576171431$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$7.45296754252$
3.1.9471.1	$x^{3} - x^{2} - 12 x - 21$	$3$	[1,1]	$-\,3\cdot 7\cdot 11\cdot 41$	$4$	$21.1575452739$	$97.31906288081488$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$13.6166546597$
3.1.9512.1	$x^{3} - x^{2} + 21 x - 17$	$3$	[1,1]	$-\,2^{3}\cdot 29\cdot 41$	$3$	$21.1880316925$	$97.52948272189288$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$18.7585968591$
3.1.9676.1	$x^{3} - x^{2} + 11 x - 39$	$3$	[1,1]	$-\,2^{2}\cdot 41\cdot 59$	$3$	$21.3091087875$	$78.0736705899716$		$S_3$ (as 3T2)	trivial	$2$	$1$	$15.8255434722$
3.3.9676.1	$x^{3} - 22 x - 12$	$3$	[3,0]	$2^{2}\cdot 41\cdot 59$	$3$	$21.3091087875$	$98.3666610188635$		$S_3$ (as 3T2)	trivial	$2$	$2$	$61.8715966852$
3.1.9799.1	$x^{3} - x^{2} + 26 x + 19$	$3$	[1,1]	$-\,41\cdot 239$	$2$	$21.3990217039$	$98.98989847454133$		$S_3$ (as 3T2)	trivial	$2$	$1$	$13.6687063253$
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	$2$	$2$	$26.7466525425$

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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.