Normalized defining polynomial
\( x^{32} - 16 x^{31} + 160 x^{30} - 1160 x^{29} + 6758 x^{28} - 32900 x^{27} + 137922 x^{26} - 506220 x^{25} + 1645758 x^{24} - 4772316 x^{23} + 12374350 x^{22} - 28669532 x^{21} + 59055416 x^{20} - 107169676 x^{19} + 168599986 x^{18} - 223250556 x^{17} + 233244785 x^{16} - 155507284 x^{15} - 27458046 x^{14} + 274978956 x^{13} - 481999064 x^{12} + 526514548 x^{11} - 347796254 x^{10} + 4847964 x^{9} + 351155703 x^{8} - 567444664 x^{7} + 586114096 x^{6} - 455539424 x^{5} + 280011493 x^{4} - 136583632 x^{3} + 53725576 x^{2} - 15682728 x + 3489481 \)
Invariants
| Degree: | $32$ | magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
| |
| Signature: | $[0, 16]$ | magma: Signature(K);
sage: K.signature()
gp: K.sign
| |
| Discriminant: | \(1572926909253345615680132503044096000000000000000000000000=2^{64}\cdot 3^{16}\cdot 5^{24}\cdot 7^{16}\) | magma: Discriminant(Integers(K));
sage: K.disc()
gp: K.disc
| |
| Root discriminant: | $61.29$ | magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
| |
| Ramified primes: | $2, 3, 5, 7$ | magma: PrimeDivisors(Discriminant(Integers(K)));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
| |
| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(840=2^{3}\cdot 3\cdot 5\cdot 7\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{840}(1,·)$, $\chi_{840}(517,·)$, $\chi_{840}(769,·)$, $\chi_{840}(13,·)$, $\chi_{840}(659,·)$, $\chi_{840}(407,·)$, $\chi_{840}(671,·)$, $\chi_{840}(673,·)$, $\chi_{840}(419,·)$, $\chi_{840}(421,·)$, $\chi_{840}(167,·)$, $\chi_{840}(169,·)$, $\chi_{840}(71,·)$, $\chi_{840}(433,·)$, $\chi_{840}(181,·)$, $\chi_{840}(827,·)$, $\chi_{840}(323,·)$, $\chi_{840}(839,·)$, $\chi_{840}(587,·)$, $\chi_{840}(589,·)$, $\chi_{840}(337,·)$, $\chi_{840}(83,·)$, $\chi_{840}(601,·)$, $\chi_{840}(349,·)$, $\chi_{840}(97,·)$, $\chi_{840}(743,·)$, $\chi_{840}(491,·)$, $\chi_{840}(239,·)$, $\chi_{840}(757,·)$, $\chi_{840}(503,·)$, $\chi_{840}(251,·)$, $\chi_{840}(253,·)$$\rbrace$ | ||
| This is a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{19} a^{18} - \frac{9}{19} a^{17} - \frac{2}{19} a^{16} - \frac{8}{19} a^{15} + \frac{2}{19} a^{14} + \frac{1}{19} a^{13} + \frac{8}{19} a^{12} + \frac{4}{19} a^{11} - \frac{7}{19} a^{10} - \frac{9}{19} a^{8} + \frac{4}{19} a^{7} + \frac{3}{19} a^{6} + \frac{2}{19} a^{5} + \frac{9}{19} a^{4} + \frac{2}{19} a^{3} + \frac{2}{19} a^{2} - \frac{3}{19} a - \frac{1}{19}$, $\frac{1}{19} a^{19} - \frac{7}{19} a^{17} - \frac{7}{19} a^{16} + \frac{6}{19} a^{15} - \frac{2}{19} a^{13} - \frac{9}{19} a^{11} - \frac{6}{19} a^{10} - \frac{9}{19} a^{9} - \frac{1}{19} a^{8} + \frac{1}{19} a^{7} - \frac{9}{19} a^{6} + \frac{8}{19} a^{5} + \frac{7}{19} a^{4} + \frac{1}{19} a^{3} - \frac{4}{19} a^{2} - \frac{9}{19} a - \frac{9}{19}$, $\frac{1}{209} a^{20} + \frac{1}{209} a^{19} - \frac{1}{209} a^{18} + \frac{8}{209} a^{17} - \frac{89}{209} a^{16} - \frac{61}{209} a^{15} + \frac{29}{209} a^{14} - \frac{91}{209} a^{13} - \frac{94}{209} a^{12} + \frac{6}{19} a^{11} - \frac{1}{11} a^{10} - \frac{10}{209} a^{9} - \frac{16}{209} a^{8} + \frac{92}{209} a^{7} + \frac{36}{209} a^{6} - \frac{49}{209} a^{5} - \frac{3}{19} a^{4} - \frac{67}{209} a^{3} - \frac{1}{209} a^{2} + \frac{78}{209} a + \frac{4}{209}$, $\frac{1}{209} a^{21} - \frac{2}{209} a^{19} - \frac{2}{209} a^{18} + \frac{2}{209} a^{17} + \frac{50}{209} a^{16} - \frac{31}{209} a^{15} + \frac{67}{209} a^{14} - \frac{14}{209} a^{13} + \frac{72}{209} a^{12} + \frac{80}{209} a^{11} + \frac{86}{209} a^{10} - \frac{6}{209} a^{9} - \frac{2}{209} a^{8} - \frac{100}{209} a^{7} + \frac{91}{209} a^{6} - \frac{6}{209} a^{5} + \frac{4}{11} a^{4} + \frac{4}{19} a^{3} + \frac{3}{11} a^{2} - \frac{41}{209} a + \frac{7}{209}$, $\frac{1}{209} a^{22} + \frac{6}{19} a^{17} - \frac{5}{19} a^{15} + \frac{4}{19} a^{14} + \frac{9}{19} a^{13} + \frac{101}{209} a^{12} + \frac{9}{209} a^{11} - \frac{4}{19} a^{10} - \frac{2}{19} a^{9} + \frac{7}{19} a^{8} + \frac{6}{19} a^{7} + \frac{6}{19} a^{6} - \frac{2}{19} a^{5} - \frac{2}{19} a^{4} - \frac{7}{19} a^{3} - \frac{43}{209} a^{2} - \frac{46}{209} a + \frac{8}{209}$, $\frac{1}{209} a^{23} - \frac{3}{19} a^{17} + \frac{7}{19} a^{16} - \frac{5}{19} a^{15} - \frac{3}{19} a^{14} + \frac{35}{209} a^{13} - \frac{101}{209} a^{12} - \frac{9}{19} a^{11} + \frac{2}{19} a^{10} + \frac{7}{19} a^{9} + \frac{3}{19} a^{8} + \frac{1}{19} a^{7} - \frac{1}{19} a^{6} + \frac{5}{19} a^{5} - \frac{4}{19} a^{4} + \frac{34}{209} a^{3} + \frac{31}{209} a^{2} - \frac{3}{209} a + \frac{6}{19}$, $\frac{1}{209} a^{24} - \frac{1}{19} a^{17} + \frac{8}{19} a^{16} - \frac{8}{19} a^{15} + \frac{101}{209} a^{14} - \frac{68}{209} a^{13} - \frac{4}{19} a^{12} - \frac{5}{19} a^{11} + \frac{5}{19} a^{10} + \frac{3}{19} a^{9} - \frac{7}{19} a^{8} - \frac{8}{19} a^{7} - \frac{5}{19} a^{6} + \frac{2}{19} a^{5} - \frac{87}{209} a^{4} + \frac{97}{209} a^{3} + \frac{63}{209} a^{2} - \frac{3}{19} a - \frac{3}{19}$, $\frac{1}{209} a^{25} - \frac{1}{19} a^{17} + \frac{9}{19} a^{16} + \frac{13}{209} a^{15} - \frac{46}{209} a^{14} - \frac{3}{19} a^{13} + \frac{3}{19} a^{12} + \frac{9}{19} a^{11} - \frac{4}{19} a^{10} - \frac{7}{19} a^{9} + \frac{2}{19} a^{8} - \frac{1}{19} a^{7} + \frac{5}{19} a^{6} - \frac{65}{209} a^{5} - \frac{13}{209} a^{4} + \frac{85}{209} a^{3} - \frac{1}{19} a^{2} - \frac{6}{19} a - \frac{1}{19}$, $\frac{1}{209} a^{26} - \frac{9}{209} a^{16} + \frac{75}{209} a^{15} - \frac{1}{19} a^{14} + \frac{4}{19} a^{13} - \frac{2}{19} a^{12} + \frac{5}{19} a^{10} + \frac{2}{19} a^{9} + \frac{9}{19} a^{8} + \frac{9}{19} a^{7} - \frac{32}{209} a^{6} + \frac{9}{209} a^{5} - \frac{25}{209} a^{4} + \frac{1}{19} a^{3} - \frac{4}{19} a^{2} - \frac{4}{19} a - \frac{1}{19}$, $\frac{1}{27379} a^{27} + \frac{52}{27379} a^{26} + \frac{5}{2489} a^{25} + \frac{1}{2489} a^{24} - \frac{8}{27379} a^{23} - \frac{2}{27379} a^{22} + \frac{35}{27379} a^{21} + \frac{30}{27379} a^{20} + \frac{169}{27379} a^{19} + \frac{109}{27379} a^{18} + \frac{11411}{27379} a^{17} + \frac{467}{1441} a^{16} - \frac{12523}{27379} a^{15} + \frac{516}{1441} a^{14} + \frac{3529}{27379} a^{13} - \frac{8912}{27379} a^{12} - \frac{4599}{27379} a^{11} - \frac{7867}{27379} a^{10} - \frac{994}{27379} a^{9} - \frac{15}{2489} a^{8} - \frac{5777}{27379} a^{7} - \frac{4419}{27379} a^{6} - \frac{4548}{27379} a^{5} + \frac{1404}{27379} a^{4} - \frac{10620}{27379} a^{3} + \frac{12308}{27379} a^{2} + \frac{8523}{27379} a - \frac{13005}{27379}$, $\frac{1}{520201} a^{28} + \frac{5}{520201} a^{27} - \frac{555}{520201} a^{26} - \frac{478}{520201} a^{25} - \frac{656}{520201} a^{24} + \frac{1029}{520201} a^{23} - \frac{1181}{520201} a^{22} - \frac{567}{520201} a^{21} - \frac{89}{47291} a^{20} - \frac{12550}{520201} a^{19} - \frac{91}{3971} a^{18} + \frac{249124}{520201} a^{17} - \frac{228207}{520201} a^{16} - \frac{59104}{520201} a^{15} - \frac{182159}{520201} a^{14} - \frac{20688}{47291} a^{13} + \frac{119646}{520201} a^{12} - \frac{137554}{520201} a^{11} + \frac{162823}{520201} a^{10} + \frac{154366}{520201} a^{9} + \frac{244983}{520201} a^{8} + \frac{74006}{520201} a^{7} - \frac{2353}{27379} a^{6} - \frac{12897}{27379} a^{5} - \frac{148003}{520201} a^{4} - \frac{171455}{520201} a^{3} + \frac{118452}{520201} a^{2} - \frac{190493}{520201} a + \frac{80292}{520201}$, $\frac{1}{520201} a^{29} + \frac{9}{520201} a^{27} + \frac{568}{520201} a^{26} - \frac{717}{520201} a^{25} + \frac{832}{520201} a^{24} - \frac{1082}{520201} a^{23} - \frac{818}{520201} a^{22} + \frac{70}{520201} a^{21} + \frac{59}{520201} a^{20} - \frac{6437}{520201} a^{19} - \frac{10376}{520201} a^{18} - \frac{161345}{520201} a^{17} - \frac{178206}{520201} a^{16} - \frac{161569}{520201} a^{15} + \frac{36163}{520201} a^{14} + \frac{77966}{520201} a^{13} - \frac{20944}{47291} a^{12} + \frac{55823}{520201} a^{11} - \frac{49089}{520201} a^{10} + \frac{244192}{520201} a^{9} - \frac{167868}{520201} a^{8} + \frac{162521}{520201} a^{7} - \frac{9479}{27379} a^{6} + \frac{63581}{520201} a^{5} + \frac{178395}{520201} a^{4} + \frac{176435}{520201} a^{3} + \frac{30105}{520201} a^{2} - \frac{40268}{520201} a - \frac{248434}{520201}$, $\frac{1}{145263049978041847766326996433981085658917169} a^{30} - \frac{15}{145263049978041847766326996433981085658917169} a^{29} - \frac{74457194063153442823311393712976928275}{145263049978041847766326996433981085658917169} a^{28} + \frac{1042400716884148199526359511981676996865}{145263049978041847766326996433981085658917169} a^{27} + \frac{43597590937536167570572160578440647229727}{145263049978041847766326996433981085658917169} a^{26} + \frac{67289392515655705816140677364181212110512}{145263049978041847766326996433981085658917169} a^{25} - \frac{105066755978055158850514434739299488499540}{145263049978041847766326996433981085658917169} a^{24} - \frac{103787101386199580066198744158510239344563}{145263049978041847766326996433981085658917169} a^{23} - \frac{346386823663455170739840384840196603600188}{145263049978041847766326996433981085658917169} a^{22} + \frac{253675481250038507281525168878083215309762}{145263049978041847766326996433981085658917169} a^{21} - \frac{67521506921633817464927713713412517695880}{145263049978041847766326996433981085658917169} a^{20} + \frac{29956499920039564748596220991494362144069}{2462085592848166911293677905660696367100291} a^{19} - \frac{1857861934328446720497879247355859705752734}{145263049978041847766326996433981085658917169} a^{18} - \frac{4753590640440441494843819414480309419343230}{145263049978041847766326996433981085658917169} a^{17} + \frac{1127559499637126601301775546910473508882342}{2381361475049866356825032728425919437031429} a^{16} + \frac{72361019426990014590671311617580113150189718}{145263049978041847766326996433981085658917169} a^{15} - \frac{69118078308352303036384276252673737151489579}{145263049978041847766326996433981085658917169} a^{14} + \frac{49395759609212798058769476583570174993623980}{145263049978041847766326996433981085658917169} a^{13} - \frac{2911534553604478863231664021734578334418551}{145263049978041847766326996433981085658917169} a^{12} - \frac{59543593972974551264538509093510526722545121}{145263049978041847766326996433981085658917169} a^{11} + \frac{2139458984722011442856657254292568266765682}{4685904838001349927946032143031647924481199} a^{10} - \frac{54441331455635891630848033925081755238633574}{145263049978041847766326996433981085658917169} a^{9} - \frac{31930502839011280217546347906511439525304904}{145263049978041847766326996433981085658917169} a^{8} + \frac{31116480513074695167653489511923821907062813}{145263049978041847766326996433981085658917169} a^{7} - \frac{789756619748017608524139867305319092421054}{2462085592848166911293677905660696367100291} a^{6} + \frac{21291772399584584834590522706615189127529441}{145263049978041847766326996433981085658917169} a^{5} - \frac{36085424670244420913836495419081002844769447}{145263049978041847766326996433981085658917169} a^{4} + \frac{34765392175764378432776010492679992241456469}{145263049978041847766326996433981085658917169} a^{3} - \frac{50725074589852071543107492678330792235512823}{145263049978041847766326996433981085658917169} a^{2} - \frac{33454892413184670430924833401288976600966859}{145263049978041847766326996433981085658917169} a - \frac{1388333847096991233813679539165738355752461}{13205731816185622524211545130361916878083379}$, $\frac{1}{79838170161481557992698746837076778470083124171259} a^{31} + \frac{274790}{79838170161481557992698746837076778470083124171259} a^{30} - \frac{72930809478589918726244786999767453330858818}{79838170161481557992698746837076778470083124171259} a^{29} - \frac{37293837725294705169740275619454408024587937}{79838170161481557992698746837076778470083124171259} a^{28} + \frac{89443843032634533736748078410457324062843501}{79838170161481557992698746837076778470083124171259} a^{27} + \frac{89752980596576183881689046603023681398704157613}{79838170161481557992698746837076778470083124171259} a^{26} - \frac{1530508720555228780214958347471440370620802366}{2575424843918759935248346672163767047422036263589} a^{25} - \frac{105914175966762365829877134864756311629268314883}{79838170161481557992698746837076778470083124171259} a^{24} + \frac{14500642879016786866856087875177971147305032244}{79838170161481557992698746837076778470083124171259} a^{23} - \frac{89460222414464567106302678735426635404530268330}{79838170161481557992698746837076778470083124171259} a^{22} - \frac{59233802776764393484954478342823192549806582859}{79838170161481557992698746837076778470083124171259} a^{21} + \frac{138119932697926223510564205090567878471284093130}{79838170161481557992698746837076778470083124171259} a^{20} + \frac{28476822498310285386122161859039250540859449907}{2575424843918759935248346672163767047422036263589} a^{19} - \frac{942094425092350341086505677103740079950441724583}{79838170161481557992698746837076778470083124171259} a^{18} + \frac{18717814443009941871317511188416246595348415898}{118983860151239281658269369354808909791480065829} a^{17} + \frac{25659614839133694774960081626866695796033152224139}{79838170161481557992698746837076778470083124171259} a^{16} + \frac{1117286924602325343339489218271753514252327626063}{79838170161481557992698746837076778470083124171259} a^{15} - \frac{1611754921236185182257995449784455083141045221633}{7258015469225596181154431530643343497280284015569} a^{14} - \frac{16673294029011831657427210180111656014125032591829}{79838170161481557992698746837076778470083124171259} a^{13} + \frac{6440651544682267696759660622270069940366882271}{71220490777414413909633137232004262685176738779} a^{12} - \frac{447702980737059564589024040165743288208683451042}{7258015469225596181154431530643343497280284015569} a^{11} + \frac{3504589746185061964058983936008361354625658365860}{7258015469225596181154431530643343497280284015569} a^{10} - \frac{15418431005089108076304820907916101069123241762549}{79838170161481557992698746837076778470083124171259} a^{9} + \frac{6695874779435938505972071003675664646027600529624}{79838170161481557992698746837076778470083124171259} a^{8} - \frac{1206483022416427311669670156085731237156561282449}{2753040350395915792851680925416440636899418074871} a^{7} + \frac{691318471810444376410368562264060490724794674246}{79838170161481557992698746837076778470083124171259} a^{6} - \frac{12156752211271926237300214396469978320510551146278}{79838170161481557992698746837076778470083124171259} a^{5} - \frac{19133466974307667867345714758657511452393258130110}{79838170161481557992698746837076778470083124171259} a^{4} + \frac{10712165186896836180898526103841359418576244607584}{79838170161481557992698746837076778470083124171259} a^{3} - \frac{4706470835337517952068171075593732123139974230898}{79838170161481557992698746837076778470083124171259} a^{2} + \frac{13036579039078467310259462335716575321884265449922}{79838170161481557992698746837076778470083124171259} a + \frac{506325908110003209333529105416741727416455660220}{1308822461663632098240963062902898007706280724119}$
Class group and class number
$C_{2}\times C_{4}\times C_{4}\times C_{8}\times C_{8}$, which has order $2048$ (assuming GRH)
Unit group
| Rank: | $15$ | magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
| |
| Torsion generator: | \( -\frac{51045378445827683392533945584408}{41057248582780751311615881250770149429} a^{30} + \frac{765680676687415250888009183766120}{41057248582780751311615881250770149429} a^{29} - \frac{7337038312525863030392855714231898}{41057248582780751311615881250770149429} a^{28} + \frac{50907477252846983782078025231072452}{41057248582780751311615881250770149429} a^{27} - \frac{25889194975019913620621483748712214}{3732477143889159210146898295524559039} a^{26} + \frac{120932441260884159147758597348704124}{3732477143889159210146898295524559039} a^{25} - \frac{5350078382037684444692919358528702977}{41057248582780751311615881250770149429} a^{24} + \frac{18809123630546744830020064652339734744}{41057248582780751311615881250770149429} a^{23} - \frac{58440647587246990236244365221668192592}{41057248582780751311615881250770149429} a^{22} + \frac{14673864197584718775439606457243495680}{3732477143889159210146898295524559039} a^{21} - \frac{396482094827478066162403193991138111225}{41057248582780751311615881250770149429} a^{20} + \frac{14633515388448175658615728642095850576}{695885569199673751044336970352036431} a^{19} - \frac{1651475660640441092954203365733734563944}{41057248582780751311615881250770149429} a^{18} + \frac{2732219239078235688694493707955876576008}{41057248582780751311615881250770149429} a^{17} - \frac{3798334222838631149532621258761919088349}{41057248582780751311615881250770149429} a^{16} + \frac{4171116232571418154734707943053316027128}{41057248582780751311615881250770149429} a^{15} - \frac{2987747337127269006779557367622182652656}{41057248582780751311615881250770149429} a^{14} - \frac{231110301256778545340581269776767037624}{41057248582780751311615881250770149429} a^{13} + \frac{4822757380637689335962833033866805217125}{41057248582780751311615881250770149429} a^{12} - \frac{8799603699553570478820129387433318035856}{41057248582780751311615881250770149429} a^{11} + \frac{9661999740545829848529337282199482454108}{41057248582780751311615881250770149429} a^{10} - \frac{6205456046840281857857520204444559217136}{41057248582780751311615881250770149429} a^{9} - \frac{367622845690366001719220249544463731268}{41057248582780751311615881250770149429} a^{8} + \frac{6847259030980630554812654972881448561052}{41057248582780751311615881250770149429} a^{7} - \frac{175571958793852250966669632561806558206}{695885569199673751044336970352036431} a^{6} + \frac{10009829979081587943972560325203010543852}{41057248582780751311615881250770149429} a^{5} - \frac{7180340217201566408084853847879185126412}{41057248582780751311615881250770149429} a^{4} + \frac{3893424931290530922096883445506147433984}{41057248582780751311615881250770149429} a^{3} - \frac{1646970160266347535360519955919869773220}{41057248582780751311615881250770149429} a^{2} + \frac{504383548357571090895283581707806049672}{41057248582780751311615881250770149429} a - \frac{101471351168309959716475380373666052350}{41057248582780751311615881250770149429} \) (order $10$) | magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
| |
| 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) | magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
| |
| Regulator: | \( 9540601320446.762 \) (assuming GRH) | magma: Regulator(K);
sage: K.regulator()
gp: K.reg
|
Galois group
$C_2^3\times C_4$ (as 32T34):
| An abelian group of order 32 |
| The 32 conjugacy class representatives for $C_2^3\times C_4$ |
| Character table for $C_2^3\times C_4$ is not computed |
Intermediate fields
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 | R | R | R | R | ${\href{/LocalNumberField/11.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/13.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/17.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/19.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/23.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/29.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/31.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/37.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/41.2.0.1}{2} }^{16}$ | ${\href{/LocalNumberField/43.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/47.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/53.4.0.1}{4} }^{8}$ | ${\href{/LocalNumberField/59.2.0.1}{2} }^{16}$ |
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 |
|---|---|---|---|---|---|---|---|
| 2 | Data not computed | ||||||
| $3$ | 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 3.8.4.1 | $x^{8} + 36 x^{4} - 27 x^{2} + 324$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| $5$ | 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| 5.8.6.1 | $x^{8} - 5 x^{4} + 400$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $[\ ]_{4}^{2}$ | |
| $7$ | 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |
| 7.8.4.1 | $x^{8} + 14 x^{6} + 539 x^{4} + 343 x^{2} + 60025$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $[\ ]_{2}^{4}$ | |