Global number field 40.0.100383397447978918530459891214693626269465146363712847232818603515625.3

• Label: 40.0.10038339744...5625.3
• Degree: $40$
• Signature: $[0, 20]$
• Discriminant: $3^{20}\cdot 5^{30}\cdot 11^{36}$
• Root discriminant: $50.12$
• Ramified primes: $3, 5, 11$
• Class number: $18524$ (GRH)
• Class group: $[2, 9262]$ (GRH)
• Galois group: $C_2\times C_{20}$ (as 40T2)

Normalized defining polynomial

\( x^{40} - x^{39} + 21 x^{38} - 18 x^{37} + 249 x^{36} - 191 x^{35} + 2008 x^{34} - 1359 x^{33} + 12160 x^{32} - 7243 x^{31} + 57502 x^{30} - 29496 x^{29} + 218159 x^{28} - 94389 x^{27} + 671689 x^{26} - 234893 x^{25} + 1690845 x^{24} - 452550 x^{23} + 3479454 x^{22} - 637891 x^{21} + 5838877 x^{20} - 598784 x^{19} + 7925819 x^{18} - 188442 x^{17} + 8602640 x^{16} + 356201 x^{15} + 7327393 x^{14} + 688488 x^{13} + 4739830 x^{12} + 533615 x^{11} + 2212864 x^{10} + 232193 x^{9} + 670719 x^{8} + 9081 x^{7} + 109203 x^{6} - 5639 x^{5} + 12525 x^{4} - 994 x^{3} + 252 x^{2} + 12 x + 1 \)

Invariants

Degree:		$40$
Signature:		$[0, 20]$
Discriminant:		\(100383397447978918530459891214693626269465146363712847232818603515625=3^{20}\cdot 5^{30}\cdot 11^{36}\)
Root discriminant:		$50.12$
Ramified primes:		$3, 5, 11$
This field is Galois and abelian over $\Q$.
Conductor:		\(165=3\cdot 5\cdot 11\)
Dirichlet character group:		$\lbrace$$\chi_{165}(128,·)$, $\chi_{165}(1,·)$, $\chi_{165}(2,·)$, $\chi_{165}(4,·)$, $\chi_{165}(7,·)$, $\chi_{165}(8,·)$, $\chi_{165}(13,·)$, $\chi_{165}(14,·)$, $\chi_{165}(16,·)$, $\chi_{165}(17,·)$, $\chi_{165}(146,·)$, $\chi_{165}(26,·)$, $\chi_{165}(28,·)$, $\chi_{165}(31,·)$, $\chi_{165}(32,·)$, $\chi_{165}(34,·)$, $\chi_{165}(43,·)$, $\chi_{165}(49,·)$, $\chi_{165}(52,·)$, $\chi_{165}(56,·)$, $\chi_{165}(59,·)$, $\chi_{165}(62,·)$, $\chi_{165}(64,·)$, $\chi_{165}(68,·)$, $\chi_{165}(71,·)$, $\chi_{165}(73,·)$, $\chi_{165}(83,·)$, $\chi_{165}(142,·)$, $\chi_{165}(86,·)$, $\chi_{165}(89,·)$, $\chi_{165}(91,·)$, $\chi_{165}(98,·)$, $\chi_{165}(104,·)$, $\chi_{165}(107,·)$, $\chi_{165}(112,·)$, $\chi_{165}(118,·)$, $\chi_{165}(119,·)$, $\chi_{165}(136,·)$, $\chi_{165}(124,·)$, $\chi_{165}(127,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $\frac{1}{109} a^{34} + \frac{31}{109} a^{33} + \frac{19}{109} a^{32} + \frac{50}{109} a^{31} - \frac{53}{109} a^{30} - \frac{9}{109} a^{29} + \frac{39}{109} a^{28} - \frac{50}{109} a^{27} + \frac{1}{109} a^{26} - \frac{26}{109} a^{25} + \frac{8}{109} a^{24} - \frac{6}{109} a^{23} + \frac{38}{109} a^{22} - \frac{17}{109} a^{21} - \frac{46}{109} a^{20} - \frac{48}{109} a^{19} - \frac{39}{109} a^{18} - \frac{34}{109} a^{17} + \frac{14}{109} a^{16} + \frac{18}{109} a^{15} + \frac{51}{109} a^{14} + \frac{44}{109} a^{13} + \frac{43}{109} a^{12} + \frac{4}{109} a^{11} - \frac{24}{109} a^{10} - \frac{8}{109} a^{9} + \frac{46}{109} a^{8} - \frac{3}{109} a^{7} - \frac{8}{109} a^{5} - \frac{33}{109} a^{4} + \frac{21}{109} a^{3} - \frac{18}{109} a^{2} - \frac{52}{109} a + \frac{46}{109}$, $\frac{1}{109} a^{35} + \frac{39}{109} a^{33} + \frac{6}{109} a^{32} + \frac{32}{109} a^{31} - \frac{1}{109} a^{30} - \frac{9}{109} a^{29} + \frac{49}{109} a^{28} + \frac{25}{109} a^{27} + \frac{52}{109} a^{26} + \frac{51}{109} a^{25} - \frac{36}{109} a^{24} + \frac{6}{109} a^{23} + \frac{4}{109} a^{22} + \frac{45}{109} a^{21} - \frac{39}{109} a^{20} + \frac{32}{109} a^{19} - \frac{24}{109} a^{18} - \frac{22}{109} a^{17} + \frac{20}{109} a^{16} + \frac{38}{109} a^{15} - \frac{11}{109} a^{14} - \frac{13}{109} a^{13} - \frac{21}{109} a^{12} - \frac{39}{109} a^{11} - \frac{27}{109} a^{10} - \frac{33}{109} a^{9} - \frac{12}{109} a^{8} - \frac{16}{109} a^{7} - \frac{8}{109} a^{6} - \frac{3}{109} a^{5} - \frac{46}{109} a^{4} - \frac{15}{109} a^{3} - \frac{39}{109} a^{2} + \frac{23}{109} a - \frac{9}{109}$, $\frac{1}{109} a^{36} - \frac{4}{109} a^{33} + \frac{54}{109} a^{32} + \frac{11}{109} a^{31} - \frac{13}{109} a^{30} - \frac{36}{109} a^{29} + \frac{30}{109} a^{28} + \frac{40}{109} a^{27} + \frac{12}{109} a^{26} - \frac{3}{109} a^{25} + \frac{21}{109} a^{24} + \frac{20}{109} a^{23} - \frac{20}{109} a^{22} - \frac{30}{109} a^{21} - \frac{27}{109} a^{20} - \frac{5}{109} a^{19} - \frac{27}{109} a^{18} + \frac{38}{109} a^{17} + \frac{37}{109} a^{16} + \frac{50}{109} a^{15} - \frac{40}{109} a^{14} + \frac{7}{109} a^{13} + \frac{28}{109} a^{12} + \frac{35}{109} a^{11} + \frac{31}{109} a^{10} - \frac{27}{109} a^{9} + \frac{43}{109} a^{8} - \frac{3}{109} a^{6} + \frac{48}{109} a^{5} - \frac{36}{109} a^{4} + \frac{14}{109} a^{3} - \frac{38}{109} a^{2} - \frac{52}{109} a - \frac{50}{109}$, $\frac{1}{109} a^{37} - \frac{40}{109} a^{33} - \frac{22}{109} a^{32} - \frac{31}{109} a^{31} - \frac{30}{109} a^{30} - \frac{6}{109} a^{29} - \frac{22}{109} a^{28} + \frac{30}{109} a^{27} + \frac{1}{109} a^{26} + \frac{26}{109} a^{25} + \frac{52}{109} a^{24} - \frac{44}{109} a^{23} + \frac{13}{109} a^{22} + \frac{14}{109} a^{21} + \frac{29}{109} a^{20} - \frac{1}{109} a^{19} - \frac{9}{109} a^{18} + \frac{10}{109} a^{17} - \frac{3}{109} a^{16} + \frac{32}{109} a^{15} - \frac{7}{109} a^{14} - \frac{14}{109} a^{13} - \frac{11}{109} a^{12} + \frac{47}{109} a^{11} - \frac{14}{109} a^{10} + \frac{11}{109} a^{9} - \frac{34}{109} a^{8} - \frac{15}{109} a^{7} + \frac{48}{109} a^{6} + \frac{41}{109} a^{5} - \frac{9}{109} a^{4} + \frac{46}{109} a^{3} - \frac{15}{109} a^{2} - \frac{40}{109} a - \frac{34}{109}$, $\frac{1}{109} a^{38} + \frac{19}{109} a^{33} - \frac{34}{109} a^{32} + \frac{8}{109} a^{31} + \frac{54}{109} a^{30} + \frac{54}{109} a^{29} - \frac{45}{109} a^{28} - \frac{37}{109} a^{27} - \frac{43}{109} a^{26} - \frac{7}{109} a^{25} - \frac{51}{109} a^{24} - \frac{9}{109} a^{23} + \frac{8}{109} a^{22} + \frac{3}{109} a^{21} + \frac{12}{109} a^{20} + \frac{33}{109} a^{19} - \frac{24}{109} a^{18} + \frac{54}{109} a^{17} + \frac{47}{109} a^{16} - \frac{50}{109} a^{15} - \frac{45}{109} a^{14} + \frac{5}{109} a^{13} + \frac{23}{109} a^{12} + \frac{37}{109} a^{11} + \frac{32}{109} a^{10} - \frac{27}{109} a^{9} - \frac{28}{109} a^{8} + \frac{37}{109} a^{7} + \frac{41}{109} a^{6} - \frac{2}{109} a^{5} + \frac{34}{109} a^{4} - \frac{47}{109} a^{3} + \frac{3}{109} a^{2} - \frac{43}{109} a - \frac{13}{109}$, $\frac{1}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{39} - \frac{1716703409260155045804789348407167630414963622780038568434125071751197251719875}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{38} - \frac{823996640306346582348299142889584549366492842050089529856816472125799390117242}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{37} - \frac{2157302124460406557556930245339816632675399976135254556551606197849226561958805}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{36} - \frac{778814884340085413831412025757583813540479046541454722183689347994213764929349}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{35} + \frac{316282511053440341057362998218567489221110770166498533233168362408862094086917}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{34} + \frac{41789800116436329670319562379586539576048911840465777066788105191500862563921345}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{33} + \frac{152273443186792185486193571006807025919342437425952047703059482199970862881841786}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{32} - \frac{160440963127627947345923636896478277741512526964522185293712173897479743291005817}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{31} - \frac{26549791337755426764396542388407482541902978170473545790245377184136048597451567}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{30} + \frac{46424590746140634981424232329170152223598654339262114541114538339483298216908619}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{29} + \frac{24218022129160275514984158238529431796084415471772699004454569314715403794900478}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{28} - \frac{22543965958910782411467793922420593686947683802709553129316660602502788560095381}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{27} + \frac{176847087581919274699149996704954955788770288182989502708193216935207943161619276}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{26} + \frac{237309416245283567616883197010045472957390206705413989172471052090520514321126545}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{25} + \frac{37340012055541694698332801318094233506226975616342395783984583701626474634399301}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{24} - \frac{108201580251440312392695456652839194963512746595905908253216502595081238807376521}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{23} - \frac{222674981221471259070044770982224074848061339713860578882724260137196131038251772}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{22} + \frac{173040899244642868225599634942941910855858555740357329810103399748642927932858849}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{21} - \frac{139569120470423655663548109380560212392454155778490819942712135274097359596128754}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{20} + \frac{113536868770915286576710682030625507123956234440443486009748717953676959907738205}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{19} + \frac{71711736244172963256130504441128085201365916060026811287497086439825504601030545}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{18} + \frac{114057744483676012395365688883069807900014281231105348915613948789059911608149839}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{17} - \frac{124555201514468572818242465220144449902148512527220051421726459800881082025636267}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{16} - \frac{48327983494808401248104497156024425745350002013975148597070027212566672195218604}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{15} - \frac{14478052892187932975134316086328667282643269439125014397058546623515822022183288}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{14} + \frac{210196093837158624699690077231701781190565572380648522078829342409763025965613127}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{13} + \frac{217547781394619734513881078628479985401309613945707166180680000019940153165007617}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{12} - \frac{103989277066864738206479103930750552663851441348130096680975238809490164362482853}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{11} + \frac{34039903500686177644056920014214238947448178589324120221537877192800200406902584}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{10} - \frac{22000553034575837697299362180243403669299605030422577285802907460873207575809163}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{9} + \frac{193974238901163765651944104505689092330406588926067370188415899892031061096743552}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{8} + \frac{106553492293947001358529238513399407643546341372798510559788669935991811562642987}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{7} - \frac{180967422389414199621658482578571876014191518335980551222933180530991840437911522}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{6} - \frac{234069386165477517782363386132316355028479140249347957203114123284703172534314660}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{5} - \frac{228664586178388828568296995338231061054009803921313284465727722541541882618597688}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{4} - \frac{304999390865129538375263681020548904931001713183329709853449085941092543487265}{1454099433870901730639760652006540566616914944088210541317156482260900090629941} a^{3} + \frac{18931656859117244197563852188783474276913508879481184975573402969284635783223053}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{2} + \frac{236373780404466707830801963592544001085675745854558328517634710770605532324121018}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a + \frac{163640994680934822742279452618792776170830896058702043381287340425243348101538757}{481306912611268472841760775814164927550198846493197689175978795628357929998510471}$

Class group and class number

$C_{2}\times C_{9262}$, which has order $18524$ (assuming GRH)

Unit group

Rank:		$19$
Torsion generator:		\( \frac{26281091788646389710510413001060931990337284849179492587352630174565543118306577}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{39} - \frac{28191838206702436115071206864593211495557662405737192671535310756070322999185815}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{38} + \frac{553530774918741203155061821921596781505547770228316063714225382953157618710700775}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{37} - \frac{512861206840069341910611165784833834327423771140132820934010040191208610990794843}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{36} + \frac{6572421852155275578212342572163776511341464200998616305535239765795149413440716032}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{35} - \frac{5489535195768518730344695930199393237149228436896344222566318715773554631192077720}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{34} + \frac{53066597676019725724158738201429444259260499487311294293648097185297595389577262041}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{33} - \frac{39489022677069587940889518690225633053458730948177620792944152083405446286653657633}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{32} + \frac{321603563030319538124771062045217638515104532718139439540475602650991080789709850489}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{31} - \frac{213127303686564493414032469132153690592633271946958754460694144157394818391061468132}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{30} + \frac{1521596896838623255256631508943894470630850947652871095866375427150098203006646960929}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{29} - \frac{882552448617677337055791221632616433929656995454099000295355473984845330514656043504}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{28} + \frac{5773480474136659925336623607329973792286699616748933462542868787610945062136931381811}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{27} - \frac{2887023480164321999980050330572695765919200524293031216214020394010340710005594810112}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{26} + \frac{17771224849974749780939168644149657517041564139345819859411851817685017420684535022317}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{25} - \frac{7422041199873113562758260478990607859689109309319782784293143104426282248204660376523}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{24} + \frac{44696242420947962152154445958929818378212450656271455916368893106004686162413472303319}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{23} - \frac{15033830806129978162714936328453172765311781071788316459671561724016701711024702534627}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{22} + \frac{91832951141033064885127613110989081608809450385283652826401694158553446781927274439522}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{21} - \frac{23225331321836529701668660051575260088699106055172094207769143606847084552355062630406}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{20} + \frac{153698626916630279250667913426645520737811179379855596836037229551646450196560927244500}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{19} - \frac{26592427619805872820697217135062371481347537304419344151381669248007065713386823732299}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{18} + \frac{207826854550735515722097607609679974782355749751268675128073408256722629735074659490676}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{17} - \frac{19728185952563327519086714540077342439345036601866864397769827999967205143503923206231}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{16} + \frac{224279882195865677915577540168702749826879172065526263599681502065804604147708099510960}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{15} - \frac{6756747791223244311676690777428213080311047366619149653609033433255122399099081668344}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{14} + \frac{189579205010169420509303594648260512507849152678196770413831561655981516799156662158910}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{13} + \frac{4276914723075017763490085547747757009045958079557089107548005065939680562401453975801}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{12} + \frac{121325496855133725395412209222278961890287794884495049742831568553563655113124155558058}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{11} + \frac{5008843691213336135558957882548651626210291138051190268674264428974783397780021800029}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{10} + \frac{55926775036531069719434688627395053672945215142030116054832518668951668338210393680875}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{9} + \frac{1870010615639541315517151775822738624320897699057617453644002464291237670070274570249}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{8} + \frac{16637871378949282474502532225298829036666017743671783140962468181252697426065325950698}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{7} - \frac{1046746838637205453752493394911220988046929279711493035498847579846836362829710889699}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{6} + \frac{2695739928797475796640664959613889931372382943502219801443536500225109639688093572503}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{5} - \frac{344916366740290421726946267800867614340496972873203725962369813260732216435305591018}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{4} + \frac{951263246209682559036212236186901557697434956120345523991315088693175415354859854}{1454099433870901730639760652006540566616914944088210541317156482260900090629941} a^{3} - \frac{48940211131362596503019300047876770314739494837452870611858312199096147967851869533}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a^{2} + \frac{5724689276670063775400749057471533927627025753492628873171742541336710223594264361}{481306912611268472841760775814164927550198846493197689175978795628357929998510471} a + \frac{2428368981139489205271122647392097059898538377170735756246700232513697907776590}{4415659748727233695795970420313439702295402261405483386935585280994109449527619} \) (order $6$)
Fundamental units:		Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
Regulator:		\( 60790762850097.93 \) (assuming GRH)

Galois group

$C_2\times C_{20}$ (as 40T2):

An abelian group of order 40

The 40 conjugacy class representatives for $C_2\times C_{20}$

Character table for $C_2\times C_{20}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 4.4.15125.1, 4.0.136125.2, \(\Q(\zeta_{11})^+\), 8.0.18530015625.2, 10.0.52089208083.1, 10.10.669871503125.1, 10.0.162778775259375.1, 20.0.26496929674942114598525390625.1, \(\Q(\zeta_{55})^+\), 20.0.10019151533337487082567413330078125.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$	2	3	5	7	11	13	17	19	23	29	31	37	41	43	47	53	59
Cycle type	$20^{2}$	R	R	$20^{2}$	R	$20^{2}$	$20^{2}$	${\href{/LocalNumberField/19.5.0.1}{5} }^{8}$	${\href{/LocalNumberField/23.4.0.1}{4} }^{10}$	${\href{/LocalNumberField/29.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/31.5.0.1}{5} }^{8}$	$20^{2}$	${\href{/LocalNumberField/41.10.0.1}{10} }^{4}$	${\href{/LocalNumberField/43.4.0.1}{4} }^{10}$	$20^{2}$	$20^{2}$	${\href{/LocalNumberField/59.10.0.1}{10} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Local algebras for ramified primes

$p$	Label	Polynomial	$e$	$f$	$c$	Galois group	Slope content
3	Data not computed
5	Data not computed
11	Data not computed