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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Label	Polynomial	Degree	Signature	Discriminant	Ram. prime count	Root discriminant	Galois root discriminant	CM field	Galois	Monogenic	Galois group	Class group	Unit group torsion	Unit group rank	Regulator
4.2.5424.1	$x^{4} - 2 x^{3} + 2 x^{2} - 3$	$4$	[2,1]	$-\,2^{4}\cdot 3\cdot 113$	$3$	$8.58183025689$	$41.33343614626599$			✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$4.99023220758$
4.2.7571.1	$x^{4} - x^{3} + x^{2} - 5 x + 3$	$4$	[2,1]	$-\,67\cdot 113$	$2$	$9.32799514867$	$87.01149349367589$			✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$3.52479236354$
4.2.10735.1	$x^{4} - 2 x^{3} + x^{2} - 3 x - 2$	$4$	[2,1]	$-\,5\cdot 19\cdot 113$	$3$	$10.1788921348$	$103.60984509205677$			✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$7.07169641669$
4.0.12317.1	$x^{4} - x^{3} + 3 x^{2} - 3 x + 5$	$4$	[0,2]	$109\cdot 113$	$2$	$10.5347985514$	$110.98198051936178$			✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$2.95906292048$
4.0.13673.1	$x^{4} - x^{3} + 7 x^{2} - 2 x + 4$	$4$	[0,2]	$11^{2}\cdot 113$	$2$	$10.8134917552$	$35.25620512760839$				$D_{4}$ (as 4T3)	trivial	$2$	$1$	$7.19209429021$
4.0.16272.1	$x^{4} - 2 x^{3} - 9 x^{2} + 10 x + 61$	$4$	[0,2]	$2^{4}\cdot 3^{2}\cdot 113$	$3$	$11.2943237847$	$36.823905279043935$				$D_{4}$ (as 4T3)	$[2]$	$4$	$1$	$3.4148483592$
4.0.17176.1	$x^{4} + 3 x^{2} - 8 x + 16$	$4$	[0,2]	$2^{3}\cdot 19\cdot 113$	$3$	$11.4480233841$	$131.057239403247$			✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$9.18027151691$
4.0.19097.1	$x^{4} - x^{3} + 7 x^{2} + 8 x + 12$	$4$	[0,2]	$13^{2}\cdot 113$	$2$	$11.7555049047$	$38.3275357934736$	✓			$D_{4}$ (as 4T3)	$[3]$	$2$	$1$	$2.38952643457$
4.2.21696.1	$x^{4} - 2 x^{3} - 4 x^{2} + 4 x - 2$	$4$	[2,1]	$-\,2^{6}\cdot 3\cdot 113$	$3$	$12.1365407393$	$52.076866265166146$			✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$11.5409864523$
4.0.21809.1	$x^{4} - x^{3} - 2 x^{2} + 7 x + 8$	$4$	[0,2]	$113\cdot 193$	$2$	$12.1523127553$	$147.6787053031005$			✓	$S_4$ (as 4T5)	$[3]$	$2$	$1$	$2.2541185776$
4.0.23617.1	$x^{4} - 2 x^{3} + x^{2} - 3 x + 11$	$4$	[0,2]	$11\cdot 19\cdot 113$	$3$	$12.3967025972$	$153.6782352839855$				$S_4$ (as 4T5)	trivial	$2$	$1$	$6.36306376294$
4.0.23956.1	$x^{4} + 8 x^{2} - 2 x + 12$	$4$	[0,2]	$2^{2}\cdot 53\cdot 113$	$3$	$12.4409509006$	$122.846792125968$			✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$13.2259888816$
4.2.26216.1	$x^{4} + 3 x^{2} - 10 x + 4$	$4$	[2,1]	$-\,2^{3}\cdot 29\cdot 113$	$3$	$12.7245258137$	$161.91355718407277$				$S_4$ (as 4T5)	trivial	$2$	$2$	$12.3996759189$
4.0.28024.1	$x^{4} - x^{3} - x^{2} + 8$	$4$	[0,2]	$2^{3}\cdot 31\cdot 113$	$3$	$12.9384583186$	$167.40370366273262$				$S_4$ (as 4T5)	trivial	$2$	$1$	$10.2704138711$
4.0.28137.1	$x^{4} - x^{2} - x + 5$	$4$	[0,2]	$3\cdot 83\cdot 113$	$3$	$12.9514814436$	$167.74087158471545$			✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$10.4186150052$
4.0.30397.1	$x^{4} - x^{3} + 5 x^{2} + 2 x + 8$	$4$	[0,2]	$113\cdot 269$	$2$	$13.2040658299$	$174.34735443934903$				$S_4$ (as 4T5)	$[3]$	$2$	$1$	$6.96901021465$
4.0.30736.1	$x^{4} - 19 x^{2} - 10 x + 125$	$4$	[0,2]	$2^{4}\cdot 17\cdot 113$	$3$	$13.2407271706$	$87.65842800324451$				$D_{4}$ (as 4T3)	$[2]$	$4$	$1$	$7.95400320998$
4.2.36160.1	$x^{4} - 2 x^{3} - 3 x^{2} - 12 x - 4$	$4$	[2,1]	$-\,2^{6}\cdot 5\cdot 113$	$3$	$13.7897726299$	$95.0789145920377$				$S_4$ (as 4T5)	trivial	$2$	$2$	$23.6646576021$
4.0.36160.2	$x^{4} - 2 x^{3} - 4 x^{2} - 18 x + 153$	$4$	[0,2]	$2^{6}\cdot 5\cdot 113$	$3$	$13.7897726299$	$67.23094525588644$				$D_{4}$ (as 4T3)	$[4]$	$4$	$1$	$5.46317227893$
4.0.44296.1	$x^{4} - x^{3} - 7 x^{2} - 13 x + 64$	$4$	[0,2]	$2^{3}\cdot 7^{2}\cdot 113$	$3$	$14.5074515094$	$79.54872720540537$				$D_{4}$ (as 4T3)	$[6]$	$2$	$1$	$4.12794819647$
4.0.44296.2	$x^{4} - x^{3} + 7 x^{2} + 15 x + 50$	$4$	[0,2]	$2^{3}\cdot 7^{2}\cdot 113$	$3$	$14.5074515094$	$79.54872720540537$				$D_{4}$ (as 4T3)	$[2]$	$2$	$1$	$12.4462841895$
4.0.45200.1	$x^{4} - 17 x^{2} - 20 x + 181$	$4$	[0,2]	$2^{4}\cdot 5^{2}\cdot 113$	$3$	$14.5809093082$	$47.53945729601885$			✓	$D_{4}$ (as 4T3)	$[2]$	$4$	$1$	$6.17311128469$
4.0.45652.1	$x^{4} - x^{3} - x^{2} + 5 x + 6$	$4$	[0,2]	$2^{2}\cdot 101\cdot 113$	$3$	$14.6172256774$	$169.58466288229127$			✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$22.8153261458$
4.2.48816.1	$x^{4} - 2 x^{3} + 3 x^{2} + 10 x - 11$	$4$	[2,1]	$-\,2^{4}\cdot 3^{3}\cdot 113$	$3$	$14.8641660269$	$48.462984793173824$				$S_4$ (as 4T5)	$[2]$	$2$	$2$	$6.25522385218$
4.0.49833.1	$x^{4} - 2 x^{3} + 3 x^{2} - 2 x + 64$	$4$	[0,2]	$3^{2}\cdot 7^{2}\cdot 113$	$3$	$14.9409859804$	$48.7134478352744$				$D_{4}$ (as 4T3)	$[2]$	$2$	$1$	$23.8188289599$
4.2.50624.1	$x^{4} - 2 x^{3} + 3 x^{2} - 2 x - 49$	$4$	[2,1]	$-\,2^{6}\cdot 7\cdot 113$	$3$	$14.9999259254$	$79.54872720540537$				$D_{4}$ (as 4T3)	$[2]$	$2$	$2$	$5.17248008181$
4.2.50624.2	$x^{4} - 2 x^{3} + 11 x^{2} - 10 x - 47$	$4$	[2,1]	$-\,2^{6}\cdot 7\cdot 113$	$3$	$14.9999259254$	$79.54872720540537$				$D_{4}$ (as 4T3)	$[2]$	$2$	$2$	$4.70535094006$
4.0.51528.1	$x^{4} - 2 x^{3} + 5 x^{2} + 24$	$4$	[0,2]	$2^{3}\cdot 3\cdot 19\cdot 113$	$4$	$15.0664460755$	$226.99779734614165$				$S_4$ (as 4T5)	trivial	$2$	$1$	$36.3432903148$
4.0.51980.1	$x^{4} - 2 x^{3} + x^{2} - 8 x + 28$	$4$	[0,2]	$2^{2}\cdot 5\cdot 23\cdot 113$	$4$	$15.0993783945$	$227.99122790142607$				$S_4$ (as 4T5)	trivial	$2$	$1$	$16.0592969669$
4.2.56500.1	$x^{4} - 5 x^{2} - 10 x + 10$	$4$	[2,1]	$-\,2^{2}\cdot 5^{3}\cdot 113$	$3$	$15.4174344974$	$71.08806952752005$				$S_4$ (as 4T5)	$[2]$	$2$	$2$	$8.5256922445$
4.2.57291.2	$x^{4} - x^{3} + 6 x^{2} - 24 x - 9$	$4$	[2,1]	$-\,3\cdot 13^{2}\cdot 113$	$3$	$15.4711145143$	$66.385239323211$				$D_{4}$ (as 4T3)	trivial	$2$	$2$	$15.1260200742$
4.0.59212.1	$x^{4} - x^{3} + 9 x^{2} - 4 x + 8$	$4$	[0,2]	$2^{2}\cdot 113\cdot 131$	$3$	$15.5992038198$	$243.33515981049678$				$S_4$ (as 4T5)	trivial	$2$	$1$	$11.4871143986$
4.2.61924.1	$x^{4} - x^{3} + 6 x^{2} + 7 x + 1$	$4$	[2,1]	$-\,2^{2}\cdot 113\cdot 137$	$3$	$15.7748322809$	$248.8453334905037$			✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$19.2486144337$
4.2.62828.1	$x^{4} - x^{3} + 2 x^{2} + 2 x - 6$	$4$	[2,1]	$-\,2^{2}\cdot 113\cdot 139$	$3$	$15.8320921416$	$198.94511771188945$			✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$31.4071239589$
4.2.63280.1	$x^{4} - x^{2} - 12 x + 9$	$4$	[2,1]	$-\,2^{4}\cdot 5\cdot 7\cdot 113$	$4$	$15.8604906272$	$125.777581468241$				$S_4$ (as 4T5)	trivial	$2$	$2$	$19.5779772433$
4.0.64636.1	$x^{4} - x^{3} - 7 x^{2} + 22$	$4$	[0,2]	$2^{2}\cdot 11\cdot 13\cdot 113$	$4$	$15.9447831823$	$254.23611073173691$			✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$16.9658008611$
4.0.65088.1	$x^{4} - 2 x^{3} + 15 x^{2} - 14 x + 67$	$4$	[0,2]	$2^{6}\cdot 3^{2}\cdot 113$	$3$	$15.9725858741$	$52.076866265166146$				$D_{4}$ (as 4T3)	$[2]$	$2$	$1$	$5.85476969807$
4.0.65088.2	$x^{4} - 2 x^{3} + 17 x^{2} - 10 x + 73$	$4$	[0,2]	$2^{6}\cdot 3^{2}\cdot 113$	$3$	$15.9725858741$	$52.076866265166146$	✓			$D_{4}$ (as 4T3)	$[2]$	$2$	$1$	$1.76274717404$
4.2.66444.1	$x^{4} - 2 x^{3} + 2 x^{2} + 6 x - 12$	$4$	[2,1]	$-\,2^{2}\cdot 3\cdot 7^{2}\cdot 113$	$4$	$16.0551342864$	$77.32777833871259$				$S_4$ (as 4T5)	trivial	$2$	$2$	$34.8798325677$
4.4.67348.1	$x^{4} - 2 x^{3} - 10 x^{2} + 10 x + 24$	$4$	[4,0]	$2^{2}\cdot 113\cdot 149$	$3$	$16.1094671586$	$205.9771381346968$			✓	$S_4$ (as 4T5)	trivial	$2$	$3$	$43.3183818724$
4.0.68365.1	$x^{4} - x^{3} + 11 x^{2} + 7 x + 27$	$4$	[0,2]	$5\cdot 11^{2}\cdot 113$	$3$	$16.1699417169$	$78.835271294009$				$D_{4}$ (as 4T3)	$[6]$	$2$	$1$	$3.23259762675$
4.0.69269.1	$x^{4} + 4 x^{2} - 5 x + 6$	$4$	[0,2]	$113\cdot 613$	$2$	$16.2231330329$	$263.1900454044567$			✓	$S_4$ (as 4T5)	trivial	$2$	$1$	$38.3852248608$
4.2.70964.1	$x^{4} - 2 x^{3} + 5 x^{2} + 6 x - 4$	$4$	[2,1]	$-\,2^{2}\cdot 113\cdot 157$	$3$	$16.3214794222$	$266.39069052802876$				$S_4$ (as 4T5)	trivial	$2$	$2$	$21.5484167839$
4.2.71755.1	$x^{4} - x^{3} - 3 x^{2} - 5 x - 17$	$4$	[2,1]	$-\,5\cdot 113\cdot 127$	$3$	$16.3667723673$	$267.8712377243962$				$S_4$ (as 4T5)	trivial	$2$	$2$	$11.1159254992$
4.0.74128.1	$x^{4} - 33 x^{2} - 6 x + 298$	$4$	[0,2]	$2^{4}\cdot 41\cdot 113$	$3$	$16.5004417275$	$136.13228860193308$				$D_{4}$ (as 4T3)	$[2]$	$4$	$1$	$24.9620003092$
4.0.74128.2	$x^{4} - x^{2} - 34 x + 290$	$4$	[0,2]	$2^{4}\cdot 41\cdot 113$	$3$	$16.5004417275$	$136.13228860193308$				$D_{4}$ (as 4T3)	$[2]$	$4$	$1$	$23.425691886$
4.2.80795.1	$x^{4} - x^{3} + x^{2} - 13 x - 13$	$4$	[2,1]	$-\,5\cdot 11\cdot 13\cdot 113$	$4$	$16.8595555388$	$284.2446129656638$				$S_4$ (as 4T5)	trivial	$2$	$2$	$13.5515196861$
4.2.80908.1	$x^{4} - x^{3} + x^{2} + 7 x - 2$	$4$	[2,1]	$-\,2^{2}\cdot 113\cdot 179$	$3$	$16.865447399$	$225.76280949803757$			✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$45.4814315943$
4.2.81360.1	$x^{4} - 2 x^{3} - x^{2} - 4 x + 7$	$4$	[2,1]	$-\,2^{4}\cdot 3^{2}\cdot 5\cdot 113$	$4$	$16.8889533061$	$116.4474130240771$			✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$12.1375937267$
4.2.84072.1	$x^{4} - x^{3} - 6 x^{2} - 21 x - 9$	$4$	[2,1]	$-\,2^{3}\cdot 3\cdot 31\cdot 113$	$4$	$17.0279687608$	$289.9517201190571$			✓	$S_4$ (as 4T5)	trivial	$2$	$2$	$41.1718352628$

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