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.

Results (1-50 of )

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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	Narrow class group	Unit group torsion	Unit group rank	Regulator
4.2.4775.1	$x^{4} - x^{3} + 2 x^{2} - 9 x - 9$	$4$	[2,1]	$-\,5^{2}\cdot 191$	$2$	$8.3127236695$	$30.903074280724887$				$D_{4}$ (as 4T3)	trivial	trivial	$2$	$2$	$2.72731892757$
4.2.5348.1	$x^{4} - x^{3} + 4 x^{2} - x - 1$	$4$	[2,1]	$-\,2^{2}\cdot 7\cdot 191$	$3$	$8.55160927517$	$73.13002119512889$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$5.12711239714$
4.0.6685.1	$x^{4} - 2 x^{3} + 4 x^{2} - x + 3$	$4$	[0,2]	$5\cdot 7\cdot 191$	$3$	$9.04222590373$	$81.7618492941543$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$2.49243428852$
4.2.20055.1	$x^{4} - x^{3} + x^{2} + 6 x - 9$	$4$	[2,1]	$-\,3\cdot 5\cdot 7\cdot 191$	$4$	$11.9002385311$	$141.6156770982648$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$11.4047943035$
4.0.20628.1	$x^{4} - x^{3} + 3 x^{2} + 5 x + 16$	$4$	[0,2]	$2^{2}\cdot 3^{3}\cdot 191$	$3$	$11.9843443894$	$50.00855300493292$				$S_4$ (as 4T5)	$[4]$	$[4]$	$2$	$1$	$3.30120913265$
4.0.20628.2	$x^{4} - x^{3} + 8 x + 16$	$4$	[0,2]	$2^{2}\cdot 3^{3}\cdot 191$	$3$	$11.9843443894$	$50.00855300493292$				$S_4$ (as 4T5)	$[4]$	$[4]$	$2$	$1$	$3.64246303658$
4.2.20819.1	$x^{4} - 2 x^{3} - 4 x^{2} + 3 x + 7$	$4$	[2,1]	$-\,109\cdot 191$	$2$	$12.0119901195$	$144.28790663115188$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$7.04070067301$
4.2.22920.1	$x^{4} - x^{3} - 4 x^{2} - 3 x - 3$	$4$	[2,1]	$-\,2^{3}\cdot 3\cdot 5\cdot 191$	$4$	$12.3042076926$	$151.3935269422045$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$13.3760346896$
4.2.23875.1	$x^{4} - 2 x^{3} + 4 x^{2} - 3 x - 59$	$4$	[2,1]	$-\,5^{3}\cdot 191$	$2$	$12.4304212078$	$46.210874461670805$				$D_{4}$ (as 4T3)	trivial	$[2]$	$2$	$2$	$4.20060067157$
4.2.27504.2	$x^{4} - x^{2} - 12 x - 11$	$4$	[2,1]	$-\,2^{4}\cdot 3^{2}\cdot 191$	$3$	$12.8780161334$	$47.8748368143433$				$D_{4}$ (as 4T3)	trivial	$[2]$	$2$	$2$	$5.15093527712$
4.2.29032.1	$x^{4} - x^{3} + 2 x^{2} - 8 x - 16$	$4$	[2,1]	$-\,2^{3}\cdot 19\cdot 191$	$3$	$13.0532675216$	$170.3877929899909$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$25.4557318757$
4.0.31897.1	$x^{4} - x^{3} + 5 x^{2} - 4 x + 9$	$4$	[0,2]	$167\cdot 191$	$2$	$13.3640305451$	$178.59731240978965$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$10.6575414683$
4.2.32279.1	$x^{4} - x^{3} + 3 x^{2} - 16 x - 4$	$4$	[2,1]	$-\,13^{2}\cdot 191$	$2$	$13.4038641628$	$49.82971001320397$				$D_{4}$ (as 4T3)	trivial	trivial	$2$	$2$	$7.35197021222$
4.2.33043.1	$x^{4} - x^{3} + 6 x - 1$	$4$	[2,1]	$-\,173\cdot 191$	$2$	$13.4824825726$	$181.77733632111568$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$5.60479203921$
4.2.35335.1	$x^{4} - x^{3} + 3 x^{2} + 6 x - 4$	$4$	[2,1]	$-\,5\cdot 37\cdot 191$	$3$	$13.7104362551$	$187.97606230581596$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$11.5163300791$
4.0.37436.2	$x^{4} - x^{3} + 7 x^{2} - 20 x + 50$	$4$	[0,2]	$2^{2}\cdot 7^{2}\cdot 191$	$3$	$13.9098472118$	$73.13002119512889$				$D_{4}$ (as 4T3)	trivial	trivial	$2$	$1$	$5.7236496646$
4.2.39728.1	$x^{4} - 4 x^{2} - 10 x + 2$	$4$	[2,1]	$-\,2^{4}\cdot 13\cdot 191$	$3$	$14.1180324427$	$111.863916736166$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$12.6218009874$
4.2.40492.1	$x^{4} - 8 x^{2} - 6 x + 8$	$4$	[2,1]	$-\,2^{2}\cdot 53\cdot 191$	$3$	$14.1854235338$	$159.7133731913881$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$24.2212565596$
4.2.45076.1	$x^{4} - x^{3} - 8 x^{2} + 11 x + 3$	$4$	[2,1]	$-\,2^{2}\cdot 59\cdot 191$	$3$	$14.5708988228$	$212.31109250343$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$23.6129237616$
4.0.46413.1	$x^{4} - x^{3} + 9 x^{2} - 6 x + 24$	$4$	[0,2]	$3^{5}\cdot 191$	$2$	$14.6777643279$	$103.57119423603586$				$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$31.267483066$
4.2.48896.1	$x^{4} + 2 x^{2} - 20 x - 46$	$4$	[2,1]	$-\,2^{8}\cdot 191$	$2$	$14.8702521625$	$55.281099844341014$				$D_{4}$ (as 4T3)	trivial	$[2]$	$2$	$2$	$12.6313549529$
4.2.49851.1	$x^{4} - 2 x^{3} + 4 x^{2} + 9 x - 3$	$4$	[2,1]	$-\,3^{2}\cdot 29\cdot 191$	$3$	$14.9423349928$	$128.90694317995442$				$S_4$ (as 4T5)	trivial	$[2]$	$2$	$2$	$12.0845804652$
4.2.51188.1	$x^{4} - x^{3} - 2 x^{2} + 10 x - 12$	$4$	[2,1]	$-\,2^{2}\cdot 67\cdot 191$	$3$	$15.0415309035$	$226.2476519215172$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$44.7243586662$
4.2.52907.1	$x^{4} - 4 x^{2} - 5 x + 3$	$4$	[2,1]	$-\,191\cdot 277$	$2$	$15.1662525657$	$230.01521688792678$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$12.7923117454$
4.2.56536.2	$x^{4} - x^{3} + x^{2} + 8 x - 8$	$4$	[2,1]	$-\,2^{3}\cdot 37\cdot 191$	$3$	$15.4198897853$	$237.77300099044047$				$S_4$ (as 4T5)	trivial	$[2]$	$2$	$2$	$15.1427813656$
4.2.56536.3	$x^{4} - 5 x^{2} - 14 x - 8$	$4$	[2,1]	$-\,2^{3}\cdot 37\cdot 191$	$3$	$15.4198897853$	$237.77300099044047$				$S_4$ (as 4T5)	trivial	$[2]$	$2$	$2$	$44.1402437115$
4.2.56727.1	$x^{4} - x^{3} - 3 x^{2} - 4 x - 5$	$4$	[2,1]	$-\,3^{3}\cdot 11\cdot 191$	$3$	$15.4328968739$	$104.4850048576667$			✓	$S_4$ (as 4T5)	trivial	$[2]$	$2$	$2$	$21.2744406288$
4.2.56727.2	$x^{4} - x^{3} + 7 x + 2$	$4$	[2,1]	$-\,3^{3}\cdot 11\cdot 191$	$3$	$15.4328968739$	$165.14084026957954$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$23.4309349075$
4.2.59019.1	$x^{4} - 8 x^{2} - 3 x + 12$	$4$	[2,1]	$-\,3\cdot 103\cdot 191$	$3$	$15.5864769516$	$242.93826376262757$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$37.617580967$
4.2.64940.1	$x^{4} - x^{3} + 8 x^{2} + 8 x - 6$	$4$	[2,1]	$-\,2^{2}\cdot 5\cdot 17\cdot 191$	$4$	$15.9634983269$	$202.261307446794$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$40.5559395858$
4.0.64940.1	$x^{4} - x^{3} - 4 x^{2} + 11 x + 31$	$4$	[0,2]	$2^{2}\cdot 5\cdot 17\cdot 191$	$4$	$15.9634983269$	$254.83327883147444$				$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$24.0164360306$
4.0.69333.2	$x^{4} - x^{3} - 6 x^{2} - 12 x + 45$	$4$	[0,2]	$3\cdot 11^{2}\cdot 191$	$3$	$16.2268790123$	$79.39143530633515$				$D_{4}$ (as 4T3)	trivial	trivial	$2$	$1$	$15.1515174801$
4.0.71816.1	$x^{4} - 2 x^{3} + 10 x^{2} + 10 x + 13$	$4$	[0,2]	$2^{3}\cdot 47\cdot 191$	$3$	$16.3702496686$	$267.98507421123287$				$S_4$ (as 4T5)	$[7]$	$[7]$	$2$	$1$	$5.24534984609$
4.0.74872.1	$x^{4} - x^{3} + 12 x^{2} - 6 x + 28$	$4$	[0,2]	$2^{3}\cdot 7^{2}\cdot 191$	$3$	$16.5416892729$	$143.04094494561156$				$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$44.8504799609$
4.0.74872.3	$x^{4} - x^{3} + 5 x^{2} + 23 x + 88$	$4$	[0,2]	$2^{3}\cdot 7^{2}\cdot 191$	$3$	$16.5416892729$	$103.42146779078317$				$D_{4}$ (as 4T3)	trivial	trivial	$2$	$1$	$32.6578438342$
4.4.76400.1	$x^{4} - 28 x^{2} + 191$	$4$	[4,0]	$2^{4}\cdot 5^{2}\cdot 191$	$3$	$16.625447339$	$87.40709353364863$				$D_{4}$ (as 4T3)	trivial	$[2]$	$2$	$3$	$16.0836857039$
4.0.77928.1	$x^{4} - x^{3} + 5 x^{2} - 6 x + 18$	$4$	[0,2]	$2^{3}\cdot 3\cdot 17\cdot 191$	$4$	$16.7079582965$	$279.15587043800457$				$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$19.2393174363$
4.0.82321.1	$x^{4} - x^{3} + 11 x^{2} + 9 x + 20$	$4$	[0,2]	$191\cdot 431$	$2$	$16.9386057313$	$286.9163641202781$				$S_4$ (as 4T5)	$[7]$	$[7]$	$2$	$1$	$3.55508246392$
4.0.85377.1	$x^{4} - x^{3} + 2 x^{2} + 3 x + 6$	$4$	[0,2]	$3\cdot 149\cdot 191$	$3$	$17.0936663441$	$292.1934290842284$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$16.8981950952$
4.2.85568.2	$x^{4} - 2 x^{3} - 9 x^{2} + 6 x + 19$	$4$	[2,1]	$-\,2^{6}\cdot 7\cdot 191$	$3$	$17.1032185503$	$146.26004239025778$			✓	$S_4$ (as 4T5)	trivial	$[2]$	$2$	$2$	$18.8373797713$
4.0.85568.1	$x^{4} - 2 x^{3} + 23 x^{2} - 22 x + 89$	$4$	[0,2]	$2^{6}\cdot 7\cdot 191$	$3$	$17.1032185503$	$103.42146779078317$	✓			$D_{4}$ (as 4T3)	$[2]$	$[2]$	$2$	$1$	$1.76274717404$
4.4.85568.1	$x^{4} - 2 x^{3} - 17 x^{2} + 18 x + 79$	$4$	[4,0]	$2^{6}\cdot 7\cdot 191$	$3$	$17.1032185503$	$103.42146779078317$			✓	$D_{4}$ (as 4T3)	trivial	$[2]$	$2$	$3$	$16.0845347739$
4.2.89388.1	$x^{4} - x^{3} - 4 x^{2} - 8 x + 4$	$4$	[2,1]	$-\,2^{2}\cdot 3^{2}\cdot 13\cdot 191$	$4$	$17.2909878283$	$137.00475831639505$				$S_4$ (as 4T5)	trivial	$[2]$	$2$	$2$	$37.6870196495$
4.0.90725.1	$x^{4} - x^{3} + 30 x^{2} - 18 x + 239$	$4$	[0,2]	$5^{2}\cdot 19\cdot 191$	$3$	$17.3552847754$	$134.70337783441067$	✓			$D_{4}$ (as 4T3)	$[4]$	$[4]$	$2$	$1$	$0.962423650119$
4.4.90725.1	$x^{4} - x^{3} - 33 x^{2} + 31 x + 211$	$4$	[4,0]	$5^{2}\cdot 19\cdot 191$	$3$	$17.3552847754$	$134.70337783441067$				$D_{4}$ (as 4T3)	trivial	$[2]$	$2$	$3$	$10.1360083557$
4.0.93972.1	$x^{4} - x^{3} - 4 x^{2} + 6 x + 18$	$4$	[0,2]	$2^{2}\cdot 3\cdot 41\cdot 191$	$4$	$17.508527293$	$243.3077278886976$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$38.9501895183$
4.2.95500.1	$x^{4} - 2 x^{3} - 6 x^{2} + 2 x + 16$	$4$	[2,1]	$-\,2^{2}\cdot 5^{3}\cdot 191$	$3$	$17.5792702581$	$73.35519073282664$				$S_4$ (as 4T5)	$[2]$	$[4]$	$2$	$2$	$17.7758558992$
4.2.95691.1	$x^{4} - 2 x^{3} + 4 x^{2} + 3 x - 15$	$4$	[2,1]	$-\,3\cdot 167\cdot 191$	$3$	$17.5880533087$	$309.33961918900724$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$28.4066605077$
4.0.96837.1	$x^{4} - 2 x^{3} + 14 x^{2} - 13 x + 39$	$4$	[0,2]	$3\cdot 13^{2}\cdot 191$	$3$	$17.6404772978$	$86.3075894692929$	✓		✓	$D_{4}$ (as 4T3)	$[6]$	$[6]$	$2$	$1$	$2.38952643457$
4.2.99511.1	$x^{4} - x^{3} - 11 x^{2} + 2 x + 25$	$4$	[2,1]	$-\,191\cdot 521$	$2$	$17.7610146558$	$315.45364160205855$			✓	$S_4$ (as 4T5)	trivial	$[2]$	$2$	$2$	$24.9269841987$

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