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 162491 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
3.3.564.1	$x^{3} - x^{2} - 5 x + 3$	$3$	[3,0]	$2^{2}\cdot 3\cdot 47$	$3$	$8.26214922567$	$18.849343120394256$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$5.40274790998$
3.1.940.1	$x^{3} - 2 x - 6$	$3$	[1,1]	$-\,2^{2}\cdot 5\cdot 47$	$3$	$9.79586108716$	$24.334397330745432$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$5.83785794545$
3.3.940.1	$x^{3} - 7 x - 4$	$3$	[3,0]	$2^{2}\cdot 5\cdot 47$	$3$	$9.79586108716$	$30.659419433511783$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$8.90818681934$
3.1.1175.1	$x^{3} + 5 x - 5$	$3$	[1,1]	$-\,5^{2}\cdot 47$	$2$	$10.5522714725$	$20.046055658633293$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$2.03126124223$
3.1.1316.1	$x^{3} - x^{2} + 9 x + 7$	$3$	[1,1]	$-\,2^{2}\cdot 7\cdot 47$	$3$	$10.9585214764$	$36.27671429443411$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$7.74628402629$
3.1.1363.1	$x^{3} - x^{2} - 9 x - 10$	$3$	[1,1]	$-\,29\cdot 47$	$2$	$11.087457087$	$36.91882988394946$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$7.21522159627$
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)	$[2]$	$2$	$1$	$2.35003381942$
3.3.2021.1	$x^{3} - 8 x - 1$	$3$	[3,0]	$43\cdot 47$	$2$	$12.6431542894$	$44.955533585978046$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$11.5183824927$
3.1.2068.1	$x^{3} - x^{2} - 8 x - 10$	$3$	[1,1]	$-\,2^{2}\cdot 11\cdot 47$	$3$	$12.7404131445$	$45.47526800360829$	✓	$S_3$ (as 3T2)	$[4]$	$2$	$1$	$3.3613387902$
3.1.2303.1	$x^{3} - x^{2} - 2 x - 27$	$3$	[1,1]	$-\,7^{2}\cdot 47$	$2$	$13.2057978815$	$25.086936025192795$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$2.98312268036$
3.1.2444.1	$x^{3} - 4 x - 10$	$3$	[1,1]	$-\,2^{2}\cdot 13\cdot 47$	$3$	$13.4699836071$	$39.23803668599558$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$8.27970067148$
3.1.2491.1	$x^{3} - x^{2} - 2 x + 20$	$3$	[1,1]	$-\,47\cdot 53$	$2$	$13.5557819943$	$49.90991885387112$		$S_3$ (as 3T2)	trivial	$2$	$1$	$12.7194529978$
3.1.2820.1	$x^{3} - x^{2} + 9 x + 15$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 5\cdot 47$	$4$	$14.1280764437$	$53.103672189407014$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$21.5800609877$
3.1.2867.1	$x^{3} - x^{2} + 2 x + 20$	$3$	[1,1]	$-\,47\cdot 61$	$2$	$14.2061336988$	$53.54437412091022$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$8.91811157377$
3.1.3055.1	$x^{3} - x^{2} + 10 x - 33$	$3$	[1,1]	$-\,5\cdot 13\cdot 47$	$3$	$14.5100999787$	$55.27205442174191$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$7.17246698248$
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)	$[2]$	$2$	$1$	$10.9231828967$
3.1.3619.1	$x^{3} - x^{2} - x + 12$	$3$	[1,1]	$-\,7\cdot 11\cdot 47$	$3$	$15.3531040913$	$60.15812497078013$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$13.0658946129$
3.1.3807.1	$x^{3} + 15 x - 8$	$3$	[1,1]	$-\,3^{4}\cdot 47$	$2$	$15.6144835902$	$29.66269470481324$		$S_3$ (as 3T2)	$[3]$	$2$	$1$	$10.2390662993$
3.1.3948.1	$x^{3} - x^{2} + 7 x + 21$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 7\cdot 47$	$4$	$15.8049228905$	$49.87067427348942$		$S_3$ (as 3T2)	trivial	$2$	$1$	$16.083578634$
3.1.4559.1	$x^{3} - x - 13$	$3$	[1,1]	$-\,47\cdot 97$	$2$	$16.5814764102$	$67.52036729757918$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$4.25969373476$
3.3.4841.1	$x^{3} - x^{2} - 16 x + 27$	$3$	[3,0]	$47\cdot 103$	$2$	$16.9165460072$	$69.57729514719583$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$9.10662727248$
3.1.4888.1	$x^{3} + 10 x - 24$	$3$	[1,1]	$-\,2^{3}\cdot 13\cdot 47$	$3$	$16.9711158884$	$69.91423317179414$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$23.4065691367$
3.1.4935.1	$x^{3} - x^{2} + 14 x + 31$	$3$	[1,1]	$-\,3\cdot 5\cdot 7\cdot 47$	$4$	$17.0253370743$	$70.24955515873393$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$12.0974568534$
3.1.5076.1	$x^{3} - 6 x - 28$	$3$	[1,1]	$-\,2^{2}\cdot 3^{3}\cdot 47$	$3$	$17.1859629479$	$49.399253769263915$		$S_3$ (as 3T2)	trivial	$2$	$1$	$19.9460559699$
3.1.5687.1	$x^{3} - x^{2} + 4 x + 13$	$3$	[1,1]	$-\,11^{2}\cdot 47$	$2$	$17.8495693598$	$33.90866713429379$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$7.34146411119$
3.1.5828.1	$x^{3} - x^{2} + 4 x + 28$	$3$	[1,1]	$-\,2^{2}\cdot 31\cdot 47$	$3$	$17.9958838325$	$76.34133873596926$		$S_3$ (as 3T2)	trivial	$2$	$1$	$17.1531412128$
3.1.6063.1	$x^{3} - 3 x - 45$	$3$	[1,1]	$-\,3\cdot 43\cdot 47$	$3$	$18.2345838413$	$77.86526825228306$		$S_3$ (as 3T2)	trivial	$2$	$1$	$10.2971839501$
3.1.6251.1	$x^{3} - x^{2} + 14 x - 28$	$3$	[1,1]	$-\,7\cdot 19\cdot 47$	$3$	$18.4211398492$	$79.06326580656784$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$11.6039902301$
3.1.6956.1	$x^{3} - x^{2} - 9 x + 37$	$3$	[1,1]	$-\,2^{2}\cdot 37\cdot 47$	$3$	$19.0891470922$	$66.1967175283969$		$S_3$ (as 3T2)	trivial	$2$	$1$	$9.96641539563$
3.1.7332.1	$x^{3} - x^{2} + 8 x - 68$	$3$	[1,1]	$-\,2^{2}\cdot 3\cdot 13\cdot 47$	$4$	$19.4270780694$	$85.62709851443059$		$S_3$ (as 3T2)	trivial	$2$	$1$	$34.1659803762$
3.3.7473.1	$x^{3} - x^{2} - 12 x + 3$	$3$	[3,0]	$3\cdot 47\cdot 53$	$3$	$19.5508207564$	$86.44651525654461$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$15.7015714216$
3.1.8131.1	$x^{3} - x^{2} + 13 x - 4$	$3$	[1,1]	$-\,47\cdot 173$	$2$	$20.1085761609$	$90.17205775626948$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$8.79482373426$
3.1.8507.1	$x^{3} - x^{2} + 13 x - 8$	$3$	[1,1]	$-\,47\cdot 181$	$2$	$20.4138762441$	$92.233399590387$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$14.3791456869$
3.1.8695.1	$x^{3} - x^{2} + 4 x - 19$	$3$	[1,1]	$-\,5\cdot 37\cdot 47$	$3$	$20.5631603493$	$93.24698386543128$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$12.2175218312$
3.3.8789.1	$x^{3} - 14 x - 9$	$3$	[3,0]	$11\cdot 17\cdot 47$	$3$	$20.636996388$	$93.74966666607408$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$2$	$12.6784590969$
3.1.8883.1	$x^{3} - 6 x - 19$	$3$	[1,1]	$-\,3^{3}\cdot 7\cdot 47$	$3$	$20.7103078257$	$65.3490702128212$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$21.5909991945$
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)	$[2]$	$2$	$1$	$2.75272362745$
3.1.9259.1	$x^{3} - x^{2} - 12 x + 80$	$3$	[1,1]	$-\,47\cdot 197$	$2$	$20.9984881754$	$96.22369770487933$		$S_3$ (as 3T2)	trivial	$2$	$1$	$25.9557585894$
3.1.9823.1	$x^{3} - x^{2} - 19$	$3$	[1,1]	$-\,11\cdot 19\cdot 47$	$3$	$21.4164778321$	$99.11104882907858$	✓	$S_3$ (as 3T2)	$[4]$	$2$	$1$	$3.06228902273$
3.1.10011.1	$x^{3} - x^{2} + 9 x + 54$	$3$	[1,1]	$-\,3\cdot 47\cdot 71$	$3$	$21.5522435994$	$100.05498488331304$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$25.1815068947$
3.1.10152.1	$x^{3} - 9 x - 22$	$3$	[1,1]	$-\,2^{3}\cdot 3^{3}\cdot 47$	$3$	$21.6529564808$	$69.86109465160327$	✓	$S_3$ (as 3T2)	$[2]$	$2$	$1$	$17.3205498928$
3.1.10763.1	$x^{3} - x^{2} - 18 x - 44$	$3$	[1,1]	$-\,47\cdot 229$	$2$	$22.0789176726$	$103.74487939170781$		$S_3$ (as 3T2)	trivial	$2$	$1$	$18.7267086625$
3.3.10904.1	$x^{3} - 17 x - 18$	$3$	[3,0]	$2^{3}\cdot 29\cdot 47$	$3$	$22.174914174$	$104.4222198576529$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$45.377097138$
3.1.10951.1	$x^{3} - x^{2} + 8 x - 21$	$3$	[1,1]	$-\,47\cdot 233$	$2$	$22.2067290155$	$104.64702575802143$	✓	$S_3$ (as 3T2)	$[4]$	$2$	$1$	$2.79179763938$
3.3.11092.1	$x^{3} - 16 x - 14$	$3$	[3,0]	$2^{2}\cdot 47\cdot 59$	$3$	$22.3016307199$	$83.59139993275006$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$21.5458426211$
3.1.11139.1	$x^{3} - 12 x - 63$	$3$	[1,1]	$-\,3\cdot 47\cdot 79$	$3$	$22.333085812$	$105.54146104730596$		$S_3$ (as 3T2)	trivial	$2$	$1$	$26.3958891085$
3.3.11421.1	$x^{3} - 18 x - 21$	$3$	[3,0]	$3^{5}\cdot 47$	$2$	$22.5199822485$	$51.377294318140834$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$33.3588143865$
3.1.11656.1	$x^{3} - x^{2} + 10 x + 14$	$3$	[1,1]	$-\,2^{3}\cdot 31\cdot 47$	$3$	$22.673392852$	$107.96295661012624$	✓	$S_3$ (as 3T2)	trivial	$2$	$1$	$30.6375521696$
3.3.11656.1	$x^{3} - 34 x - 64$	$3$	[3,0]	$2^{3}\cdot 31\cdot 47$	$3$	$22.673392852$	$107.96295661012624$	✓	$S_3$ (as 3T2)	trivial	$2$	$2$	$29.0842105175$
3.1.12408.1	$x^{3} - x^{2} - 32 x - 72$	$3$	[1,1]	$-\,2^{3}\cdot 3\cdot 11\cdot 47$	$4$	$23.1508670861$	$111.39120252515457$		$S_3$ (as 3T2)	trivial	$2$	$1$	$50.5848462995$

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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.