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	Discriminant	Root discriminant	Galois root discriminant

Ramified prime count	Ramified	Unramified primes	Maximum ramified prime	Galois group

Class group structure	Class number	Regulator	Intermediate field	Field is

Narrow class group	Narrow class number	Relative class number	CM field	$p$-adic completions

Monogenic	Index	Ines. $p$	Minimal sibling	Unit signature rank

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Results (1-50 of at least 1000)

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Label	Polynomial	Degree	Signature	Discriminant	Ram. prime count	Root discriminant	Galois root discriminant	Monogenic	Galois group	Class group	Narrow class group	Unit group torsion	Unit group rank	Regulator	Unit signature rank	Max $p$
7.1.383687.1	$x^{7} - x^{6} + 2 x^{5} - 2 x^{4} + x^{3} - x + 1$	$7$	(1, 3)	$-\,31\cdot 12377$	$2$	$6.27640575735$	$619.4247331193678$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$0.652352405095$	$1$	$12377$
7.1.858607.1	$x^{7} - 3 x^{6} + 4 x^{5} - 4 x^{4} + x^{3} + 2 x^{2} - x + 1$	$7$	(1, 3)	$-\,31\cdot 27697$	$2$	$7.04181997097$	$926.6104899039294$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$1.31752977322$	$1$	$27697$
7.1.868403.1	$x^{7} - x^{6} + x^{5} - x^{4} + x^{3} + x - 1$	$7$	(1, 3)	$-\,31\cdot 109\cdot 257$	$3$	$7.05324158352$	$931.8814302259703$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$1.17971262945$	$1$	$257$
7.1.896551.1	$x^{7} - 2 x^{6} + 2 x^{4} - 3 x^{3} + 4 x^{2} - 2 x + 1$	$7$	(1, 3)	$-\,31\cdot 28921$	$2$	$7.08545684167$	$946.8637705604751$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$1.31363635585$	$1$	$28921$
7.1.1046963.1	$x^{7} - x^{6} + 2 x^{5} - x^{4} + x^{3} - 2 x^{2} - x - 1$	$7$	(1, 3)	$-\,31\cdot 33773$	$2$	$7.24419597356$	$1023.2120992247893$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$1.83521662019$	$1$	$33773$
7.1.1074615.1	$x^{7} - x^{6} + 2 x^{5} - 2 x^{4} + x^{3} - 2 x^{2} + x + 1$	$7$	(1, 3)	$-\,3\cdot 5\cdot 31\cdot 2311$	$4$	$7.27122455292$	$1036.6363875535144$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$1.53074412702$	$1$	$2311$
7.1.1440911.1	$x^{7} - 2 x^{6} + x^{5} - x^{4} + 3 x^{3} - 5 x^{2} + 3 x - 1$	$7$	(1, 3)	$-\,31\cdot 53\cdot 877$	$3$	$7.58237584987$	$1200.3795233175215$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$1.62111069645$	$1$	$877$
7.1.1533911.1	$x^{7} + x^{5} - x^{4} - x^{3} - 3 x^{2} - x - 1$	$7$	(1, 3)	$-\,31\cdot 49481$	$2$	$7.65042809079$	$1238.5116067280112$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$1.67035022761$	$1$	$49481$
7.1.1782035.1	$x^{7} - x^{6} + 3 x^{5} - 3 x^{4} + 5 x^{3} - 4 x^{2} + 2 x - 1$	$7$	(1, 3)	$-\,5\cdot 31\cdot 11497$	$3$	$7.81606266725$	$1334.9288370546199$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$2.67631469189$	$1$	$11497$
7.1.1931672.1	$x^{7} - x^{6} - x^{3} + 2 x + 1$	$7$	(1, 3)	$-\,2^{3}\cdot 31\cdot 7789$	$3$	$7.90661302447$	$1389.8460346383695$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$2.64568847317$	$1$	$7789$
7.1.2173379.1	$x^{7} - 2 x^{5} - x^{2} + 2 x + 1$	$7$	(1, 3)	$-\,13\cdot 31\cdot 5393$	$3$	$8.04090750872$	$1474.238447470422$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$2.21840085161$	$1$	$5393$
7.1.2214671.1	$x^{7} - x^{6} - 2 x^{5} + 5 x^{4} - 2 x^{3} - 2 x^{2} + 2 x + 1$	$7$	(1, 3)	$-\,31\cdot 199\cdot 359$	$3$	$8.06255602633$	$1488.1770727974545$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$3.20420171394$	$1$	$359$
7.1.2289939.1	$x^{7} - x^{6} + x^{2} + x - 1$	$7$	(1, 3)	$-\,3\cdot 31\cdot 24623$	$3$	$8.10114253451$	$1513.2544399406202$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$2.31145304323$	$1$	$24623$
7.3.2452069.1	$x^{7} - x^{6} - x^{5} + 3 x^{4} - 5 x^{3} + 4 x^{2} - 4 x + 1$	$7$	(3, 2)	$31\cdot 83\cdot 953$	$3$	$8.18069844012$	$1565.908362580646$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$3.76884529931$	$3$	$953$
7.3.2516177.1	$x^{7} - 3 x^{6} + 3 x^{5} - 2 x^{4} - x^{3} + 5 x^{2} - 3 x + 1$	$7$	(3, 2)	$23\cdot 31\cdot 3529$	$3$	$8.21091579428$	$1586.2461977889814$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$2.65350604519$	$3$	$3529$
7.1.2601179.1	$x^{7} - 2 x^{6} + x^{5} - 2 x^{4} + 3 x^{3} - x^{2} + 1$	$7$	(1, 3)	$-\,7\cdot 31\cdot 11987$	$3$	$8.24997991637$	$1612.817100603785$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$2.57331650246$	$1$	$11987$
7.3.2777693.1	$x^{7} - 2 x^{6} - x^{5} + 5 x^{4} - 5 x^{2} + 1$	$7$	(3, 2)	$31\cdot 89603$	$2$	$8.32772395022$	$1666.641233139274$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$2.75986306895$	$3$	$89603$
7.1.2821527.1	$x^{7} - 2 x^{6} + 2 x^{5} - 2 x^{4} + 3 x^{3} - 2 x^{2} + 4 x - 1$	$7$	(1, 3)	$-\,3^{3}\cdot 31\cdot 3371$	$3$	$8.34637213323$	$1164.6667768594803$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$2.88261412882$	$1$	$3371$
7.1.3007403.1	$x^{7} + x^{5} - x^{4} - 2 x^{2} + x - 1$	$7$	(1, 3)	$-\,7\cdot 31\cdot 13859$	$3$	$8.42278949775$	$1734.186552825272$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$2.46418059666$	$1$	$13859$
7.3.3080873.1	$x^{7} - 3 x^{6} + 3 x^{5} - 4 x^{4} + 3 x^{3} + x^{2} - 3 x + 1$	$7$	(3, 2)	$23\cdot 29\cdot 31\cdot 149$	$4$	$8.45188150146$	$1755.2415788147225$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$2.82560973852$	$3$	$149$
7.1.3168448.1	$x^{7} - x^{6} - 2 x^{5} - 2 x^{3} - 2 x^{2} - 2$	$7$	(1, 3)	$-\,2^{6}\cdot 31\cdot 1597$	$3$	$8.48579174229$	$499.4993950632793$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$5.58987012373$	$1$	$1597$
7.1.3170432.1	$x^{7} - x^{6} + x^{5} - 3 x^{4} - x^{3} + 3 x^{2} + 3 x + 1$	$7$	(1, 3)	$-\,2^{7}\cdot 17\cdot 31\cdot 47$	$4$	$8.48655062192$	$587.3692794456543$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$5.57909022246$	$1$	$47$
7.1.3233579.1	$x^{7} - x^{6} - 3 x^{5} + 3 x^{4} + x^{3} + 2 x^{2} - 3 x + 1$	$7$	(1, 3)	$-\,31\cdot 104309$	$2$	$8.51049424258$	$1798.215504326442$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$3.88411006402$	$1$	$104309$
7.3.3246289.1	$x^{7} - x^{6} - 2 x^{5} - 2 x^{4} + 2 x^{3} + 4 x^{2} - 1$	$7$	(3, 2)	$23\cdot 29\cdot 31\cdot 157$	$4$	$8.51526501154$	$1801.7460975398283$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$3.04229175688$	$3$	$157$
7.1.3473488.1	$x^{7} - x^{6} + 5 x^{5} - 2 x^{4} + 7 x^{3} + 4 x + 1$	$7$	(1, 3)	$-\,2^{4}\cdot 31\cdot 47\cdot 149$	$4$	$8.59795394907$	$1317.8558343005504$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$3.64250423577$	$1$	$149$
7.3.3533721.1	$x^{7} - 2 x^{6} + 3 x^{5} - 3 x^{4} - 3 x^{3} + 2 x^{2} - 2 x + 1$	$7$	(3, 2)	$3\cdot 31\cdot 37997$	$3$	$8.61909665736$	$1879.8194062196508$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$3.77055696847$	$3$	$37997$
7.3.3544757.1	$x^{7} - x^{6} - 2 x^{5} + 4 x^{4} - x^{3} - 3 x^{2} + 2 x + 1$	$7$	(3, 2)	$29\cdot 31\cdot 3943$	$3$	$8.62293693476$	$1882.752506305595$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$3.56838372775$	$3$	$3943$
7.1.3589056.1	$x^{7} - x^{6} + 3 x^{5} - x^{4} + 3 x^{3} - 3 x^{2} + x + 1$	$7$	(1, 3)	$-\,2^{6}\cdot 3^{3}\cdot 31\cdot 67$	$4$	$8.63824958192$	$297.4304462349062$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$4.92184500866$	$1$	$67$
7.1.3641911.1	$x^{7} - x^{6} + 2 x^{5} - x^{4} + 2 x^{3} - 2 x^{2} - x + 1$	$7$	(1, 3)	$-\,7\cdot 13\cdot 31\cdot 1291$	$4$	$8.65630920665$	$1908.3791551995112$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$2.73582331411$	$1$	$1291$
7.1.3662619.1	$x^{7} - x^{6} + x^{5} + x^{3} + 2 x^{2} - 3 x + 3$	$7$	(1, 3)	$-\,3\cdot 31\cdot 39383$	$3$	$8.66332356029$	$1913.7970111796078$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$3.68587370958$	$1$	$39383$
7.1.3713087.1	$x^{7} - x^{6} - x^{5} + x^{4} + 3 x^{3} - x^{2} - 2 x - 1$	$7$	(1, 3)	$-\,7\cdot 31\cdot 71\cdot 241$	$4$	$8.680277099$	$1926.9372070724048$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$3.17012084293$	$1$	$241$
7.3.3750101.1	$x^{7} - 3 x^{6} + 4 x^{5} - 2 x^{4} - 2 x^{3} + 3 x^{2} - 2 x - 1$	$7$	(3, 2)	$31\cdot 137\cdot 883$	$3$	$8.69258597747$	$1936.5177510159829$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$4.99848756403$	$3$	$883$
7.1.4073648.1	$x^{7} - x^{6} - 2 x^{5} + x^{4} + 4 x^{3} + x^{2} - 3 x - 2$	$7$	(1, 3)	$-\,2^{4}\cdot 31\cdot 43\cdot 191$	$4$	$8.79596225471$	$800.9740059032932$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$4.66181667846$	$1$	$191$
7.1.4096960.1	$x^{7} - 2 x^{6} + 4 x^{4} - 4 x^{3} + 4 x - 2$	$7$	(1, 3)	$-\,2^{6}\cdot 5\cdot 7\cdot 31\cdot 59$	$5$	$8.8031355495$	$458.317653226794$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$4.62531548594$	$1$	$59$
7.1.4116211.1	$x^{7} - x^{6} + 2 x^{5} - 2 x^{4} + 3 x - 2$	$7$	(1, 3)	$-\,11\cdot 31\cdot 12071$	$3$	$8.80903291438$	$2028.8447451690333$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$4.22103272696$	$1$	$12071$
7.3.4290369.1	$x^{7} - 2 x^{6} - x^{5} + 4 x^{4} - x^{3} - x - 1$	$7$	(3, 2)	$3\cdot 31\cdot 46133$	$3$	$8.86133663039$	$2071.3205932448022$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$4.31901151989$	$3$	$46133$
7.1.4297871.1	$x^{7} - 2 x^{6} + 3 x^{5} - 4 x^{4} + 5 x^{3} - x^{2} + x + 1$	$7$	(1, 3)	$-\,31\cdot 138641$	$2$	$8.86354849421$	$2073.1307242911626$		$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$3.67377693332$	$1$	$138641$
7.1.4440719.1	$x^{7} - x^{6} + 2 x^{5} - 2 x^{4} + 4 x^{3} - 4 x^{2} + 2 x - 1$	$7$	(1, 3)	$-\,31\cdot 143249$	$2$	$8.90504635833$	$2107.301354813782$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$2.77916272192$	$1$	$143249$
7.3.4673653.1	$x^{7} - x^{6} + 2 x^{5} + x^{4} + 2 x^{3} + 5 x^{2} - 3 x - 2$	$7$	(3, 2)	$31\cdot 107\cdot 1409$	$3$	$8.97032269303$	$2161.863316678462$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$6.07399828167$	$3$	$1409$
7.3.4713581.1	$x^{7} + 3 x^{5} - x^{4} - x^{2} - 3 x + 2$	$7$	(3, 2)	$31\cdot 383\cdot 397$	$3$	$8.98123072682$	$2171.078303516481$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$6.10488521876$	$3$	$397$
7.3.4729577.1	$x^{7} - 2 x^{6} + x^{5} - 2 x^{4} + 4 x^{3} + x^{2} - 3 x - 1$	$7$	(3, 2)	$31\cdot 152567$	$2$	$8.98557850584$	$2174.7590671152516$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$3.94560170521$	$3$	$152567$
7.1.4763243.1	$x^{7} - x^{6} + 4 x^{5} + 3 x^{3} + 3 x^{2} + 1$	$7$	(1, 3)	$-\,19\cdot 31\cdot 8087$	$3$	$8.99468803944$	$2182.4855096884376$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$3.54881172092$	$1$	$8087$
7.1.4949708.1	$x^{7} - 3 x^{6} + 8 x^{5} - 9 x^{4} + 9 x^{3} - 3 x^{2} + 1$	$7$	(1, 3)	$-\,2^{2}\cdot 31\cdot 179\cdot 223$	$4$	$9.04416571602$	$1765.8201069238414$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$7.59882319291$	$1$	$223$
7.1.5105359.1	$x^{7} - x^{6} + x^{5} + 2 x^{4} - x^{2} + 2 x + 1$	$7$	(1, 3)	$-\,7^{2}\cdot 31\cdot 3361$	$3$	$9.08425819771$	$854.012294993462$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$3.81104298037$	$1$	$3361$
7.1.5135336.1	$x^{7} - 2 x^{6} + 4 x^{5} - 2 x^{4} - x^{3} + 3 x^{2} - 2 x - 2$	$7$	(1, 3)	$-\,2^{3}\cdot 31\cdot 20707$	$3$	$9.09185906285$	$2266.127975203519$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$5.92112506861$	$1$	$20707$
7.3.5218385.1	$x^{7} - x^{6} + x^{5} - 3 x^{4} - 2 x^{3} + x^{2} - 3 x + 1$	$7$	(3, 2)	$5\cdot 31\cdot 131\cdot 257$	$4$	$9.11271980093$	$2284.3784712695924$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$4.60055698073$	$3$	$257$
7.3.5225329.1	$x^{7} - x^{6} - 2 x^{5} + 4 x^{4} - 2 x^{3} - 2 x + 1$	$7$	(3, 2)	$31\cdot 168559$	$2$	$9.11445111569$	$2285.8978542358363$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$4.25072274026$	$3$	$168559$
7.3.5251493.1	$x^{7} - 3 x^{6} + 6 x^{5} - 6 x^{4} + 3 x^{3} + 3 x^{2} - 4 x + 1$	$7$	(3, 2)	$13\cdot 31\cdot 83\cdot 157$	$4$	$9.12095679875$	$2291.6136236285556$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$4.46285831416$	$3$	$157$
7.3.5431169.1	$x^{7} - x^{6} + 2 x^{5} - x^{4} - 2 x^{3} + 2 x^{2} - 3 x + 1$	$7$	(3, 2)	$19\cdot 31\cdot 9221$	$3$	$9.16489759499$	$2330.4868590060746$	✓	$S_7$ (as 7T7)	trivial	trivial	$2$	$4$	$4.8900987886$	$3$	$9221$
7.1.5446576.1	$x^{7} - 2 x^{6} + x^{5} - 3 x^{4} + 5 x^{3} - 3 x^{2} + 4 x - 1$	$7$	(1, 3)	$-\,2^{4}\cdot 31\cdot 79\cdot 139$	$4$	$9.16860719385$	$926.1651953250712$	?	$S_7$ (as 7T7)	trivial	trivial	$2$	$3$	$6.09795327333$	$1$	$139$

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