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
4.0.2133.1	$x^{4} - 2 x^{3} - 6 x^{2} + 7 x + 13$	$4$	[0,2]	$3^{3}\cdot 79$	$2$	$6.79591148817$	$20.260701897856944$			✓	$D_{4}$ (as 4T3)	trivial	trivial	$6$	$1$	$3.86222722813$
4.0.4345.1	$x^{4} - x^{3} + x^{2} - 2 x + 3$	$4$	[0,2]	$5\cdot 11\cdot 79$	$3$	$8.11890472852$	$65.91661399070799$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$3.62072447578$
4.0.4977.1	$x^{4} - x^{3} - 10 x^{2} + x + 37$	$4$	[0,2]	$3^{2}\cdot 7\cdot 79$	$3$	$8.39927711758$	$40.7308237088326$				$D_{4}$ (as 4T3)	trivial	trivial	$6$	$1$	$7.03031087594$
4.0.4977.2	$x^{4} - x^{3} - 11 x^{2} + 6 x + 36$	$4$	[0,2]	$3^{2}\cdot 7\cdot 79$	$3$	$8.39927711758$	$40.7308237088326$				$D_{4}$ (as 4T3)	trivial	trivial	$6$	$1$	$10.2452243803$
4.0.5056.1	$x^{4} - 2 x^{3} + x^{2} - 4 x + 6$	$4$	[0,2]	$2^{6}\cdot 79$	$2$	$8.43241100389$	$35.552777669262355$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$5.93141729959$
4.0.5056.2	$x^{4} - 2 x^{3} + 5 x^{2} - 2 x + 3$	$4$	[0,2]	$2^{6}\cdot 79$	$2$	$8.43241100389$	$35.552777669262355$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$2.50100435014$
4.0.5925.1	$x^{4} - x^{3} - 6 x + 9$	$4$	[0,2]	$3\cdot 5^{2}\cdot 79$	$3$	$8.77348400533$	$45.0146809031441$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$4.15032310632$
4.0.8532.1	$x^{4} - x^{3} + 3 x^{2} - 5 x + 8$	$4$	[0,2]	$2^{2}\cdot 3^{3}\cdot 79$	$3$	$9.61087019526$	$50.83252387012967$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$9.23503785722$
4.2.9875.1	$x^{4} - x^{3} - 4 x^{2} + 14 x - 29$	$4$	[2,1]	$-\,5^{3}\cdot 79$	$2$	$9.96860243824$	$29.719469226626792$				$D_{4}$ (as 4T3)	trivial	$[2]$	$2$	$2$	$3.00734262925$
4.0.11376.1	$x^{4} - 17 x^{2} + 79$	$4$	[0,2]	$2^{4}\cdot 3^{2}\cdot 79$	$3$	$10.3275521305$	$43.54308211415448$				$D_{4}$ (as 4T3)	trivial	trivial	$6$	$1$	$16.3413616696$
4.2.13351.1	$x^{4} - x^{3} - 2 x^{2} - 7 x - 3$	$4$	[2,1]	$-\,13^{2}\cdot 79$	$2$	$10.7492570639$	$32.046840717924134$				$D_{4}$ (as 4T3)	trivial	trivial	$2$	$2$	$7.72158948671$
4.0.13509.2	$x^{4} - x^{3} + 19 x^{2} - 15 x + 93$	$4$	[0,2]	$3^{2}\cdot 19\cdot 79$	$3$	$10.7809194305$	$67.10439627923047$				$D_{4}$ (as 4T3)	trivial	trivial	$6$	$1$	$13.2352554725$
4.0.15089.1	$x^{4} - x^{3} + 3 x^{2} + 2 x + 3$	$4$	[0,2]	$79\cdot 191$	$2$	$11.0831985796$	$122.83729075488436$			✓	$S_4$ (as 4T5)	$[3]$	$[3]$	$2$	$1$	$3.22243874472$
4.0.15484.1	$x^{4} - x^{3} - 7 x^{2} + 8 x + 22$	$4$	[0,2]	$2^{2}\cdot 7^{2}\cdot 79$	$3$	$11.1550312345$	$47.031904065219386$				$D_{4}$ (as 4T3)	trivial	trivial	$2$	$1$	$5.27691511571$
4.2.16827.1	$x^{4} - x^{3} - 2 x^{2} + 6 x - 1$	$4$	[2,1]	$-\,3\cdot 71\cdot 79$	$3$	$11.3894217112$	$129.71892691508052$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$5.00354800007$
4.2.17064.1	$x^{4} - 3 x^{2} - 4 x - 6$	$4$	[2,1]	$-\,2^{3}\cdot 3^{3}\cdot 79$	$3$	$11.4293152176$	$90.57326071147344$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$13.4428710485$
4.2.19355.1	$x^{4} - x^{3} - 3 x^{2} + 5 x + 5$	$4$	[2,1]	$-\,5\cdot 7^{2}\cdot 79$	$3$	$11.7950094701$	$72.72726256614953$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$5.57160152882$
4.2.20224.1	$x^{4} - 8 x^{2} - 16 x - 23$	$4$	[2,1]	$-\,2^{8}\cdot 79$	$2$	$11.9252300052$	$35.552777669262355$				$D_{4}$ (as 4T3)	trivial	$[2]$	$2$	$2$	$7.34508917134$
4.2.20303.1	$x^{4} - 2 x^{3} + 5 x^{2} - 3 x - 2$	$4$	[2,1]	$-\,79\cdot 257$	$2$	$11.9368587172$	$142.4885960349108$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$10.2199821345$
4.2.20619.1	$x^{4} - 4 x^{2} - 3 x + 4$	$4$	[2,1]	$-\,3^{2}\cdot 29\cdot 79$	$3$	$11.9830369826$	$82.90355842785037$			✓	$S_4$ (as 4T5)	trivial	$[2]$	$2$	$2$	$15.9622103713$
4.4.21725.1	$x^{4} - x^{3} - 15 x^{2} + 7 x + 49$	$4$	[4,0]	$5^{2}\cdot 11\cdot 79$	$3$	$12.1405942906$	$65.91661399070799$				$D_{4}$ (as 4T3)	trivial	$[2]$	$2$	$3$	$5.53681998518$
4.0.22041.2	$x^{4} - x^{3} + 23 x^{2} - 2 x + 148$	$4$	[0,2]	$3^{2}\cdot 31\cdot 79$	$3$	$12.184503124$	$85.71464285639881$				$D_{4}$ (as 4T3)	trivial	trivial	$6$	$1$	$13.9601967049$
4.0.24648.1	$x^{4} - x^{3} + 2 x^{2} + 8 x + 16$	$4$	[0,2]	$2^{3}\cdot 3\cdot 13\cdot 79$	$4$	$12.5298370003$	$156.9968152543229$				$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$23.510135379$
4.2.24727.1	$x^{4} - 5 x^{2} - 14 x - 5$	$4$	[2,1]	$-\,79\cdot 313$	$2$	$12.5398648891$	$157.24821143656928$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$7.99597844581$
4.2.26623.1	$x^{4} + 5 x^{2} - 3 x - 1$	$4$	[2,1]	$-\,79\cdot 337$	$2$	$12.7736275228$	$163.16556009158305$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$7.37179479958$
4.0.28361.1	$x^{4} - x^{3} - x^{2} + 4 x + 5$	$4$	[0,2]	$79\cdot 359$	$2$	$12.9771816855$	$168.40724449975423$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$1$	$5.5921954963$
4.4.28677.1	$x^{4} - x^{3} - 8 x^{2} + 12 x - 1$	$4$	[4,0]	$3\cdot 11^{2}\cdot 79$	$3$	$13.0131797614$	$76.14404833021558$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$3$	$9.26369629318$
4.2.29467.1	$x^{4} - x^{3} - 5 x^{2} - 2 x + 8$	$4$	[2,1]	$-\,79\cdot 373$	$2$	$13.1018909619$	$171.65954677791737$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$19.8152485642$
4.0.30968.1	$x^{4} - x^{3} - 9 x^{2} - 5 x + 46$	$4$	[0,2]	$2^{3}\cdot 7^{2}\cdot 79$	$3$	$13.2656425121$	$66.51315659326356$				$D_{4}$ (as 4T3)	trivial	trivial	$2$	$1$	$19.7509802224$
4.2.31600.1	$x^{4} + 2 x^{2} - 79$	$4$	[2,1]	$-\,2^{4}\cdot 5^{2}\cdot 79$	$3$	$13.3328124695$	$39.74921382870358$				$D_{4}$ (as 4T3)	trivial	$[2]$	$2$	$2$	$8.21960906412$
4.0.31600.1	$x^{4} + 19 x^{2} + 79$	$4$	[0,2]	$2^{4}\cdot 5^{2}\cdot 79$	$3$	$13.3328124695$	$56.213877290220786$	✓			$D_{4}$ (as 4T3)	$[4]$	$[4]$	$2$	$1$	$0.962423650119$
4.2.32864.1	$x^{4} - 3 x^{2} - 6 x + 10$	$4$	[2,1]	$-\,2^{5}\cdot 13\cdot 79$	$3$	$13.4641860912$	$128.18736287169654$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$19.9209356484$
4.2.33259.1	$x^{4} - 5 x - 4$	$4$	[2,1]	$-\,79\cdot 421$	$2$	$13.5044622996$	$182.37050200073475$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$18.6218677646$
4.2.34128.1	$x^{4} + 3 x^{2} - 8 x + 3$	$4$	[2,1]	$-\,2^{4}\cdot 3^{3}\cdot 79$	$3$	$13.5918229763$	$90.57326071147344$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$13.9638124583$
4.2.34128.2	$x^{4} - 3 x^{2} - 12 x + 15$	$4$	[2,1]	$-\,2^{4}\cdot 3^{3}\cdot 79$	$3$	$13.5918229763$	$57.30591881429519$				$S_4$ (as 4T5)	trivial	$[2]$	$2$	$2$	$11.5611438529$
4.0.35392.1	$x^{4} - 2 x^{3} + 13 x^{2} - 6 x + 41$	$4$	[0,2]	$2^{6}\cdot 7\cdot 79$	$3$	$13.7159620975$	$66.51315659326356$	✓			$D_{4}$ (as 4T3)	$[2]$	$[2]$	$2$	$1$	$1.76274717404$
4.4.35392.1	$x^{4} - 2 x^{3} - 13 x^{2} + 14 x + 31$	$4$	[4,0]	$2^{6}\cdot 7\cdot 79$	$3$	$13.7159620975$	$66.51315659326356$				$D_{4}$ (as 4T3)	trivial	$[2]$	$2$	$3$	$11.5035438122$
4.4.37525.1	$x^{4} - 2 x^{3} - 18 x^{2} + 19 x + 89$	$4$	[4,0]	$5^{2}\cdot 19\cdot 79$	$3$	$13.918107137$	$86.63140308225418$			✓	$D_{4}$ (as 4T3)	trivial	$[2]$	$2$	$3$	$11.3898255303$
4.0.42581.2	$x^{4} - 2 x^{3} + 8 x^{2} - 7 x + 56$	$4$	[0,2]	$7^{2}\cdot 11\cdot 79$	$3$	$14.3649448875$	$77.99358948016176$				$D_{4}$ (as 4T3)	trivial	trivial	$2$	$1$	$17.00333264$
4.2.44635.2	$x^{4} - 8 x^{2} - 15 x - 10$	$4$	[2,1]	$-\,5\cdot 79\cdot 113$	$3$	$14.5351288082$	$211.2699694703438$			✓	$S_4$ (as 4T5)	trivial	$[2]$	$2$	$2$	$26.0150793135$
4.2.44635.3	$x^{4} - x^{3} - 8 x^{2} + 20 x - 15$	$4$	[2,1]	$-\,5\cdot 79\cdot 113$	$3$	$14.5351288082$	$211.2699694703438$			✓	$S_4$ (as 4T5)	trivial	$[2]$	$2$	$2$	$13.8372557958$
4.2.45267.1	$x^{4} - x^{3} + 2 x^{2} - 6 x - 9$	$4$	[2,1]	$-\,3\cdot 79\cdot 191$	$3$	$14.586309631$	$212.76042865157044$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$11.256437625$
4.0.45504.1	$x^{4} - 34 x^{2} + 316$	$4$	[0,2]	$2^{6}\cdot 3^{2}\cdot 79$	$3$	$14.605364289$	$61.57921727336261$				$D_{4}$ (as 4T3)	trivial	trivial	$6$	$1$	$23.0103087205$
4.0.47637.2	$x^{4} - x^{3} - 31 x^{2} + 31 x + 331$	$4$	[0,2]	$3^{2}\cdot 67\cdot 79$	$3$	$14.7735919949$	$126.01190419956362$				$D_{4}$ (as 4T3)	trivial	trivial	$6$	$1$	$7.05963755541$
4.2.49059.1	$x^{4} - x^{3} - 6 x^{2} + 6 x - 3$	$4$	[2,1]	$-\,3^{3}\cdot 23\cdot 79$	$3$	$14.8826295907$	$153.57443544821206$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$14.5418015766$
4.2.49296.1	$x^{4} - 2 x^{3} - x^{2} - 12 x - 15$	$4$	[2,1]	$-\,2^{4}\cdot 3\cdot 13\cdot 79$	$4$	$14.9005713106$	$156.9968152543229$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$14.2669355495$
4.2.51508.1	$x^{4} - 5 x^{2} - 4 x - 20$	$4$	[2,1]	$-\,2^{2}\cdot 79\cdot 163$	$3$	$15.0649838958$	$226.9537397797181$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$19.1533305113$
4.0.53325.1	$x^{4} - 2 x^{3} - 36 x^{2} + 37 x + 361$	$4$	[0,2]	$3^{3}\cdot 5^{2}\cdot 79$	$3$	$15.1961200566$	$45.30430671546713$				$D_{4}$ (as 4T3)	$[2]$	$[2]$	$6$	$1$	$8.23492663224$
4.2.54115.1	$x^{4} - x^{3} - 2 x^{2} - 5$	$4$	[2,1]	$-\,5\cdot 79\cdot 137$	$3$	$15.2520919803$	$232.62630977600105$			✓	$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$10.6678410697$
4.2.54668.1	$x^{4} - 2 x^{3} - 2 x^{2} - 6 x - 16$	$4$	[2,1]	$-\,2^{2}\cdot 79\cdot 173$	$3$	$15.2909087435$	$185.57662023494893$				$S_4$ (as 4T5)	trivial	trivial	$2$	$2$	$28.9053075584$

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