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	Ramified prime count	Ramified	Unramified primes

Galois group	Field is	Intermediate field	Class group structure	Class number

Discriminant	Root discriminant	Galois root discriminant	CM field	Relative class number

Regulator	$p$-adic completions	Monogenic	Index	Ines. $p$

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Results (1-50 of 1192 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
10.0.1236938287.1	$x^{10} - x^{9} + 2 x^{8} + x^{7} + x^{6} + 3 x^{5} + 2 x^{4} + 2 x^{3} + 2 x^{2} + 2 x + 1$	$10$	[0,5]	$-\,13\cdot 89\cdot 131\cdot 8161$	$4$	$8.11399626109$	$35170.13345155233$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.7269076221085293$
10.2.1377579113.1	$x^{10} - x^{9} + x^{8} + x^{7} - 2 x^{6} - x^{4} - 3 x^{3} + x^{2} + 3 x + 1$	$10$	[2,4]	$89\cdot 251\cdot 61667$	$3$	$8.20184683724$	$37115.75289550247$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.316987364606681$
10.0.1460182327.1	$x^{10} - x^{9} + 2 x^{8} - 2 x^{7} + 2 x^{6} - 2 x^{5} + 2 x^{4} - 2 x^{3} + x^{2} + 1$	$10$	[0,5]	$-\,59\cdot 89\cdot 487\cdot 571$	$4$	$8.24974849673$	$38212.332132441225$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.03230340984159$
10.2.1467445261.1	$x^{10} - x^{9} + 2 x^{8} - 2 x^{7} + 2 x^{6} - x^{5} + x^{4} - x^{2} - x - 1$	$10$	[2,4]	$89\cdot 101\cdot 163249$	$3$	$8.25384275852$	$38307.24815227531$	?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.4486828467857595$
10.2.1494784993.1	$x^{10} - 2 x^{9} + 3 x^{7} - 4 x^{6} + x^{5} + 3 x^{4} - x^{3} - x - 1$	$10$	[2,4]	$13\cdot 17\cdot 89\cdot 75997$	$4$	$8.2690929347$	$38662.44939214276$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.4508662630659335$
10.2.1791395293.1	$x^{10} - x^{9} - x^{8} + 3 x^{7} + x^{6} - 3 x^{5} + x^{4} + 2 x^{3} - x - 1$	$10$	[2,4]	$37\cdot 89\cdot 544001$	$3$	$8.42013671152$	$42324.87794430127$	?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.831850455461943$
10.0.1913608936.1	$x^{10} + x^{8} + x^{6} - x^{5} - x^{4} + 3 x^{2} + 3 x + 1$	$10$	[0,5]	$-\,2^{3}\cdot 89\cdot 2687653$	$3$	$8.47589014106$	$43744.81610431115$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.8483144252652295$
10.2.1982289613.1	$x^{10} - x^{8} - x^{7} + 2 x^{5} + 2 x^{4} + 2 x^{3} - x^{2} - 2 x - 1$	$10$	[2,4]	$89\cdot 22272917$	$2$	$8.50583023508$	$44522.91110203824$	?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.0173566592714693$
10.0.2004910832.1	$x^{10} - x^{9} + 2 x^{8} - x^{7} + x^{6} + 2 x^{5} - 3 x^{4} + 4 x^{3} - 2 x + 2$	$10$	[0,5]	$-\,2^{4}\cdot 89\cdot 787\cdot 1789$	$4$	$8.51548731271$	$25129.80979036174$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.9690436919094463$
10.2.2041086217.1	$x^{10} - 3 x^{7} + x^{6} + x^{5} + 5 x^{4} - 2 x^{3} - x^{2} - 2 x + 1$	$10$	[2,4]	$23\cdot 89\cdot 997111$	$3$	$8.53072878944$	$45178.38218661664$	?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$2.970660401018618$
10.0.2260206175.1	$x^{10} - x^{9} + 3 x^{8} - 3 x^{7} + 3 x^{6} - 4 x^{5} + 3 x^{4} - 2 x^{3} + 3 x^{2} + x + 1$	$10$	[0,5]	$-\,5^{2}\cdot 89\cdot 1015823$	$3$	$8.61816502078$	$27802.511374982445$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.733540004004307$
10.2.2405914661.1	$x^{10} - 3 x^{9} + 2 x^{8} + 4 x^{7} - 8 x^{6} + 3 x^{5} + 6 x^{4} - 6 x^{3} - 2 x^{2} + 3 x + 1$	$10$	[2,4]	$89\cdot 227\cdot 119087$	$3$	$8.67217477521$	$49050.12396518484$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.3474475908407824$
10.0.2547692907.1	$x^{10} - 3 x^{9} + 7 x^{8} - 11 x^{7} + 15 x^{6} - 15 x^{5} + 15 x^{4} - 11 x^{3} + 7 x^{2} - 3 x + 1$	$10$	[0,5]	$-\,3\cdot 89\cdot 3089^{2}$	$3$	$8.72197239494$	$908.1646326520319$	✓	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$4$	$3.24686884473$
10.2.2639387293.1	$x^{10} - 4 x^{9} + 8 x^{8} - 9 x^{7} + 5 x^{6} + 4 x^{5} - 11 x^{4} + 10 x^{3} - 3 x^{2} - x + 1$	$10$	[2,4]	$89\cdot 1429\cdot 20753$	$3$	$8.75286665228$	$51374.96757176592$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.505779117636424$
10.0.2725400987.1	$x^{10} - 3 x^{9} + 5 x^{8} - 4 x^{7} + 2 x^{6} + x^{4} - x^{3} + x^{2} + 1$	$10$	[0,5]	$-\,89\cdot 30622483$	$2$	$8.78098106963$	$52205.3731621564$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$3.4595698951756733$
10.2.2854517381.1	$x^{10} - x^{9} + 3 x^{7} - 7 x^{5} + 10 x^{4} - 5 x^{3} - x^{2} + 2 x - 1$	$10$	[2,4]	$89\cdot 32073229$	$2$	$8.82171999902$	$53427.68365744485$	?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.7468652847862187$
10.2.2862439093.1	$x^{10} - x^{9} - x^{7} - x^{6} + 2 x^{5} + x^{4} - x - 1$	$10$	[2,4]	$43\cdot 89\cdot 139\cdot 5381$	$4$	$8.82416510611$	$53501.76719511235$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.6641063144013115$
10.2.2969199221.1	$x^{10} - 3 x^{8} - x^{7} + 5 x^{6} + 2 x^{5} - 4 x^{4} - 2 x^{3} + 3 x^{2} + x - 1$	$10$	[2,4]	$89\cdot 151\cdot 220939$	$3$	$8.85653684643$	$54490.358972941256$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$4.032791829788733$
10.0.2976232179.1	$x^{10} - x^{9} + x^{8} + 2 x^{7} - x^{6} - x^{5} + 3 x^{4} - x^{3} - 2 x^{2} + x + 1$	$10$	[0,5]	$-\,3\cdot 89\cdot 11146937$	$3$	$8.85863240671$	$54554.85477022187$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$4.5620433066747195$
10.0.3024769219.1	$x^{10} - 2 x^{9} + 2 x^{8} + x^{7} - 3 x^{6} + x^{5} + 2 x^{4} - 2 x^{3} - x^{2} + x + 1$	$10$	[0,5]	$-\,41\cdot 89\cdot 359\cdot 2309$	$4$	$8.8729743178$	$54997.901950892636$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$4.038735187533842$
10.0.3039896816.1	$x^{10} - x^{9} - x^{8} + 2 x^{7} - x^{5} + 2 x^{3} + x^{2} - 2 x + 1$	$10$	[0,5]	$-\,2^{4}\cdot 11\cdot 89\cdot 194069$	$4$	$8.87740194974$	$30943.618033395844$	?	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$4.956867406769164$
10.0.3042995707.1	$x^{10} - 2 x^{9} + x^{8} + x^{7} - 2 x^{6} + x^{5} + 1$	$10$	[0,5]	$-\,79\cdot 89\cdot 432797$	$3$	$8.87830650312$	$55163.35474751332$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$4$	$2.735477482470533$
10.2.3140714681.1	$x^{10} - x^{9} + 2 x^{8} + 2 x^{7} - 3 x^{6} + 6 x^{5} - 2 x^{4} + x^{3} + x^{2} + 1$	$10$	[2,4]	$13\cdot 89\cdot 409\cdot 6637$	$4$	$8.90641337926$	$56042.07955634766$	?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$3.9465849729207885$
10.4.3412809931.1	$x^{10} - x^{9} - 2 x^{8} + 5 x^{7} + x^{6} - 8 x^{5} + x^{4} + 7 x^{3} - 3 x^{2} - 3 x + 1$	$10$	[4,3]	$-\,89\cdot 103\cdot 372293$	$3$	$8.98072111618$	$58419.25993197791$	?	$S_{10}$ (as 10T45)	trivial	$2$	$6$	$5.099985297480563$
10.2.3435256621.1	$x^{10} - 2 x^{9} + 2 x^{8} - x^{7} - 3 x^{6} + 4 x^{5} - 2 x^{4} - x^{3} + 2 x^{2} + 2 x - 1$	$10$	[2,4]	$89\cdot 38598389$	$2$	$8.98661049459$	$58611.06227496649$	?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$5.047283288089158$
10.2.3726609157.1	$x^{10} - 2 x^{9} + 3 x^{8} - x^{7} - 4 x^{6} + 8 x^{5} - 9 x^{4} + 6 x^{3} - 3 x^{2} + 3 x - 1$	$10$	[2,4]	$89\cdot 2297\cdot 18229$	$3$	$9.06006648561$	$61045.95938307465$	?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$5.28391871382183$
10.2.3779323501.1	$x^{10} - x^{9} + 3 x^{8} - x^{7} + 2 x^{6} + 4 x^{5} + 6 x^{3} + 2 x + 1$	$10$	[2,4]	$89\cdot 42464309$	$2$	$9.07280144927$	$61476.20272105296$	?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$5.331681681422647$
10.2.3819701377.1	$x^{10} - 2 x^{9} + x^{8} + x^{6} - 5 x^{5} + 3 x^{4} + x^{3} + x^{2} - 3 x + 1$	$10$	[2,4]	$89\cdot 3467\cdot 12379$	$3$	$9.08244844044$	$61803.732710897$	?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$4.662683071350745$
10.2.3880593397.1	$x^{10} - 3 x^{9} + x^{8} + 6 x^{7} - 8 x^{6} + 2 x^{5} + x^{4} + 1$	$10$	[2,4]	$89\cdot 3727\cdot 11699$	$3$	$9.09682445494$	$62294.409034840355$	?	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$4.598612187387434$
10.2.3887902433.1	$x^{10} + x^{8} - 2 x^{7} - 2 x^{6} + 4 x^{3} + x - 1$	$10$	[2,4]	$89\cdot 43684297$	$2$	$9.0985363769$	$62353.046701825246$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$5$	$6.964863020735921$
10.4.6121452307.1	$x^{10} - x^{9} - 5 x^{8} + 6 x^{7} + 6 x^{6} - 10 x^{5} + x^{4} + 4 x^{3} - x^{2} + x - 1$	$10$	[4,3]	$-\,89\cdot 4049\cdot 16987$	$3$	$9.52106317904$	$78239.71055033371$	?	$S_{10}$ (as 10T45)	trivial	$2$	$6$	$7.461116842582305$
10.4.6734591819.1	$x^{10} - x^{9} - x^{8} + 4 x^{7} - 5 x^{6} + x^{5} - x^{4} + x^{3} - 2 x^{2} + x + 1$	$10$	[4,3]	$-\,19^{2}\cdot 89\cdot 263\cdot 797$	$4$	$9.6123843534$	$18826.90099299404$	?	$S_{10}$ (as 10T45)	trivial	$2$	$6$	$7.879352960587293$
10.4.7306488019.1	$x^{10} - x^{9} - 2 x^{8} + 3 x^{7} - 2 x^{6} + x^{5} - x^{4} - 2 x^{3} + 5 x^{2} - 4 x + 1$	$10$	[4,3]	$-\,19^{3}\cdot 89\cdot 11969$	$3$	$9.69105074471$	$4498.841962105359$	?	$S_{10}$ (as 10T45)	trivial	$2$	$6$	$8.84630035400866$
10.4.10339269107.1	$x^{10} - 2 x^{9} - x^{8} + 8 x^{7} - 6 x^{6} - 10 x^{5} + 10 x^{4} + 5 x^{3} - 5 x^{2} - 2 x + 1$	$10$	[4,3]	$-\,89\cdot 227\cdot 307\cdot 1667$	$4$	$10.0334198077$	$101682.19660786248$	?	$S_{10}$ (as 10T45)	trivial	$2$	$6$	$10.133964984855343$
10.2.11100455936.1	$x^{10} - 2 x^{9} + 6 x^{6} - 8 x^{5} - 2 x^{3} + 12 x^{2} - 10 x + 2$	$10$	[2,4]	$2^{10}\cdot 89\cdot 349^{2}$	$3$	$10.1049479707$	$397.94033545326164$	?	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$5$	$13.3352624222$
10.4.11404512599.1	$x^{10} - 2 x^{9} - 2 x^{8} + 7 x^{7} - 5 x^{6} + 5 x^{4} - 5 x^{3} + x^{2} - 1$	$10$	[4,3]	$-\,53\cdot 89\cdot 2417747$	$3$	$10.1322914356$	$106791.91261045942$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$6$	$12.165531580178735$
10.4.11713776563.1	$x^{10} - 3 x^{9} + 2 x^{8} + x^{6} - 6 x^{5} + 5 x^{4} + x^{3} - 4 x^{2} + x + 1$	$10$	[4,3]	$-\,89\cdot 131615467$	$2$	$10.1594382169$	$108230.20171375456$	?	$S_{10}$ (as 10T45)	trivial	$2$	$6$	$11.00419334580777$
10.4.13415366851.1	$x^{10} - 4 x^{9} + 6 x^{8} - 11 x^{6} + 9 x^{5} + 4 x^{4} - 9 x^{3} + x^{2} + 3 x - 1$	$10$	[4,3]	$-\,89\cdot 150734459$	$2$	$10.2981747076$	$115824.72469641358$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$6$	$12.053836915812875$
10.0.14270992648.1	$x^{10} + 2 x^{8} - 5 x^{7} + 5 x^{6} - 7 x^{5} + 11 x^{4} - 8 x^{3} + 6 x^{2} - 4 x + 1$	$10$	[0,5]	$-\,2^{3}\cdot 11^{4}\cdot 37^{2}\cdot 89$	$4$	$10.3620436697$	$802.7912845600272$	?	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$4$	$10.8801145261$
10.2.15046275208.1	$x^{10} - 3 x^{9} + x^{8} + 6 x^{7} - 3 x^{6} - 8 x^{5} + 3 x^{4} + 6 x^{3} - x^{2} - 3 x - 1$	$10$	[2,4]	$2^{3}\cdot 89\cdot 4597^{2}$	$3$	$10.4170056596$	$1809.1611315745206$	✓	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$5$	$16.0833130426$
10.4.15193488239.1	$x^{10} - x^{8} - 4 x^{7} - 6 x^{6} - 5 x^{5} + x^{4} + 11 x^{3} + 12 x^{2} + 7 x + 1$	$10$	[4,3]	$-\,89\cdot 170713351$	$2$	$10.4271530795$	$123261.86855228181$	?	$S_{10}$ (as 10T45)	trivial	$2$	$6$	$13.173297862933424$
10.4.15382321408.1	$x^{10} - x^{9} - x^{8} - 3 x^{6} - x^{5} + 7 x^{4} - 3 x^{2} + 3 x - 1$	$10$	[4,3]	$-\,2^{8}\cdot 59\cdot 89\cdot 11443$	$4$	$10.4400406137$	$28432.968243602874$	?	$S_{10}$ (as 10T45)	trivial	$2$	$6$	$14.634461215941819$
10.0.17401278843.1	$x^{10} - 3 x^{9} + 3 x^{8} + x^{7} - 2 x^{6} - 6 x^{5} + 16 x^{4} - 14 x^{3} + 6 x^{2} + 1$	$10$	[0,5]	$-\,3^{7}\cdot 13^{2}\cdot 23^{2}\cdot 89$	$4$	$10.569589404$	$587.7221536058123$	?	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$4$	$9.80527029428$
10.2.17925744896.1	$x^{10} - 3 x^{9} + 5 x^{8} - 3 x^{7} - 4 x^{6} + 10 x^{5} + 4 x^{4} - 3 x^{3} - 5 x^{2} - 3 x - 1$	$10$	[2,4]	$2^{8}\cdot 89\cdot 887^{2}$	$3$	$10.6010216429$	$1123.8718788189337$	?	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$5$	$18.8028929429$
10.6.19299535813.1	$x^{10} - x^{9} - 5 x^{8} + 4 x^{7} + 9 x^{6} - 8 x^{5} - 5 x^{4} + 9 x^{3} - 4 x - 1$	$10$	[6,2]	$61^{2}\cdot 89\cdot 101\cdot 577$	$4$	$10.679592615$	$35290.961171069495$	✓	$S_{10}$ (as 10T45)	trivial	$2$	$7$	$18.781741633198425$
10.0.21200543684.1	$x^{10} - x^{9} + 2 x^{8} + 2 x^{6} - 3 x^{5} + 3 x^{4} - 2 x^{3} + 5 x^{2} + 2 x + 2$	$10$	[0,5]	$-\,2^{2}\cdot 89\cdot 7717^{2}$	$3$	$10.7803956429$	$1657.4836349116692$	?	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$4$	$14.1931146954$
10.2.21951585225.1	$x^{10} - x^{9} - 2 x^{8} + x^{7} + x^{6} + 9 x^{5} - x^{4} + x^{3} + 2 x^{2} - x - 1$	$10$	[2,4]	$3^{4}\cdot 5^{2}\cdot 89\cdot 349^{2}$	$4$	$10.8179903278$	$682.5796656801315$	?	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$5$	$12.3796743856$
10.2.22553895929.1	$x^{10} - 5 x^{9} + 10 x^{8} - 10 x^{7} + 3 x^{6} + 5 x^{5} - 3 x^{4} - 4 x^{3} + 3 x^{2} - 1$	$10$	[2,4]	$89\cdot 15919^{2}$	$2$	$10.8473126323$	$1190.2903007249954$	?	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$5$	$19.4217627898$
10.4.25428567627.1	$x^{10} - 3 x^{8} - x^{7} + x^{6} - x^{5} + 3 x^{4} - 2 x^{3} - 5 x^{2} + 3 x + 1$	$10$	[4,3]	$-\,3^{3}\cdot 89\cdot 3253^{2}$	$3$	$10.9782265744$	$1226.529437463592$	?	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$6$	$21.5918248637$
10.2.29232506125.1	$x^{10} - 3 x^{9} + 4 x^{7} - 7 x^{5} + 4 x^{3} - 3 x - 1$	$10$	[2,4]	$5^{3}\cdot 89\cdot 1621^{2}$	$3$	$11.1323436656$	$849.320316488426$	✓	$C_2 \wr S_5$ (as 10T39)	trivial	$2$	$5$	$19.459542359$

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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.