Global number field 42.0.29198428620782310880522337720254845955751250559410488348634029682058779274295867292920491.1

• Label: 42.0.29198428620...0491.1
• Degree: $42$
• Signature: $[0, 21]$
• Discriminant: $-\,7^{77}\cdot 13^{21}$
• Root discriminant: $127.74$
• Ramified primes: $7, 13$
• Class number: Not computed
• Class group: Not computed
• Galois group: $C_{42}$ (as 42T1)

Normalized defining polynomial

\( x^{42} + 126 x^{40} + 7371 x^{38} + 265734 x^{36} - 337 x^{35} + 6608385 x^{34} - 35385 x^{33} + 120236886 x^{32} - 1698480 x^{31} + 1656597096 x^{30} - 49362075 x^{29} + 17647946044 x^{28} - 969725925 x^{27} + 147161484876 x^{26} - 13614951987 x^{25} + 966512559600 x^{24} - 140844330900 x^{23} + 5007866589747 x^{22} - 1090864717741 x^{21} + 20438068055202 x^{20} - 6365649711708 x^{19} + 65504728776627 x^{18} - 27967555663866 x^{17} + 164712579358698 x^{16} - 92087076475296 x^{15} + 327078003099919 x^{14} - 226780519461930 x^{13} + 522529550813886 x^{12} - 423863032664799 x^{11} + 688995831550950 x^{10} - 628212651886323 x^{9} + 777300365125620 x^{8} - 764561060990476 x^{7} + 834254335608561 x^{6} - 771769032155877 x^{5} + 864745657173342 x^{4} - 917560666081710 x^{3} + 525223978833708 x^{2} - 936510480777564 x + 1669935825763633 \)

Invariants

Degree:		$42$
Signature:		$[0, 21]$
Discriminant:		\(-29198428620782310880522337720254845955751250559410488348634029682058779274295867292920491=-\,7^{77}\cdot 13^{21}\)
Root discriminant:		$127.74$
Ramified primes:		$7, 13$
This field is Galois and abelian over $\Q$.
Conductor:		\(637=7^{2}\cdot 13\)
Dirichlet character group:		$\lbrace$$\chi_{637}(1,·)$, $\chi_{637}(363,·)$, $\chi_{637}(261,·)$, $\chi_{637}(129,·)$, $\chi_{637}(12,·)$, $\chi_{637}(493,·)$, $\chi_{637}(272,·)$, $\chi_{637}(402,·)$, $\chi_{637}(534,·)$, $\chi_{637}(285,·)$, $\chi_{637}(545,·)$, $\chi_{637}(547,·)$, $\chi_{637}(326,·)$, $\chi_{637}(38,·)$, $\chi_{637}(92,·)$, $\chi_{637}(274,·)$, $\chi_{637}(558,·)$, $\chi_{637}(584,·)$, $\chi_{637}(53,·)$, $\chi_{637}(311,·)$, $\chi_{637}(443,·)$, $\chi_{637}(181,·)$, $\chi_{637}(194,·)$, $\chi_{637}(454,·)$, $\chi_{637}(417,·)$, $\chi_{637}(456,·)$, $\chi_{637}(183,·)$, $\chi_{637}(79,·)$, $\chi_{637}(467,·)$, $\chi_{637}(599,·)$, $\chi_{637}(90,·)$, $\chi_{637}(220,·)$, $\chi_{637}(352,·)$, $\chi_{637}(144,·)$, $\chi_{637}(103,·)$, $\chi_{637}(636,·)$, $\chi_{637}(235,·)$, $\chi_{637}(365,·)$, $\chi_{637}(625,·)$, $\chi_{637}(376,·)$, $\chi_{637}(508,·)$, $\chi_{637}(170,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{97} a^{21} - \frac{34}{97} a^{19} - \frac{45}{97} a^{17} - \frac{1}{97} a^{15} - \frac{23}{97} a^{14} + \frac{5}{97} a^{13} + \frac{4}{97} a^{12} + \frac{5}{97} a^{11} - \frac{31}{97} a^{10} - \frac{14}{97} a^{9} - \frac{42}{97} a^{8} + \frac{27}{97} a^{7} + \frac{37}{97} a^{6} + \frac{36}{97} a^{5} - \frac{23}{97} a^{4} + \frac{39}{97} a^{3} + \frac{7}{97} a^{2} - \frac{45}{97} a + \frac{42}{97}$, $\frac{1}{97} a^{22} - \frac{34}{97} a^{20} - \frac{45}{97} a^{18} - \frac{1}{97} a^{16} - \frac{23}{97} a^{15} + \frac{5}{97} a^{14} + \frac{4}{97} a^{13} + \frac{5}{97} a^{12} - \frac{31}{97} a^{11} - \frac{14}{97} a^{10} - \frac{42}{97} a^{9} + \frac{27}{97} a^{8} + \frac{37}{97} a^{7} + \frac{36}{97} a^{6} - \frac{23}{97} a^{5} + \frac{39}{97} a^{4} + \frac{7}{97} a^{3} - \frac{45}{97} a^{2} + \frac{42}{97} a$, $\frac{1}{97} a^{23} - \frac{37}{97} a^{19} + \frac{21}{97} a^{17} - \frac{23}{97} a^{16} - \frac{29}{97} a^{15} - \frac{2}{97} a^{14} - \frac{19}{97} a^{13} + \frac{8}{97} a^{12} - \frac{38}{97} a^{11} - \frac{29}{97} a^{10} + \frac{36}{97} a^{9} - \frac{33}{97} a^{8} - \frac{16}{97} a^{7} - \frac{26}{97} a^{6} + \frac{2}{97} a^{5} + \frac{1}{97} a^{4} + \frac{20}{97} a^{3} - \frac{11}{97} a^{2} + \frac{22}{97} a - \frac{27}{97}$, $\frac{1}{97} a^{24} - \frac{37}{97} a^{20} + \frac{21}{97} a^{18} - \frac{23}{97} a^{17} - \frac{29}{97} a^{16} - \frac{2}{97} a^{15} - \frac{19}{97} a^{14} + \frac{8}{97} a^{13} - \frac{38}{97} a^{12} - \frac{29}{97} a^{11} + \frac{36}{97} a^{10} - \frac{33}{97} a^{9} - \frac{16}{97} a^{8} - \frac{26}{97} a^{7} + \frac{2}{97} a^{6} + \frac{1}{97} a^{5} + \frac{20}{97} a^{4} - \frac{11}{97} a^{3} + \frac{22}{97} a^{2} - \frac{27}{97} a$, $\frac{1}{156083863544770561} a^{25} - \frac{420984484639620}{156083863544770561} a^{24} + \frac{75}{156083863544770561} a^{23} + \frac{262245016778707}{156083863544770561} a^{22} + \frac{2475}{156083863544770561} a^{21} + \frac{34811065951093335}{156083863544770561} a^{20} + \frac{47250}{156083863544770561} a^{19} - \frac{16258875952749049}{156083863544770561} a^{18} + \frac{70800927794081297}{156083863544770561} a^{17} + \frac{51055012349153040}{156083863544770561} a^{16} - \frac{57928031826339528}{156083863544770561} a^{15} - \frac{32188078236624983}{156083863544770561} a^{14} - \frac{62755367790944307}{156083863544770561} a^{13} - \frac{14779709294440691}{156083863544770561} a^{12} + \frac{6436448077893052}{156083863544770561} a^{11} - \frac{54334760931180231}{156083863544770561} a^{10} - \frac{57928031596493118}{156083863544770561} a^{9} - \frac{17638246247040121}{156083863544770561} a^{8} - \frac{40227799530839200}{156083863544770561} a^{7} - \frac{16514594812168800}{156083863544770561} a^{6} - \frac{12872895666915059}{156083863544770561} a^{5} - \frac{17972465786368688}{156083863544770561} a^{4} + \frac{74019151899263548}{156083863544770561} a^{3} + \frac{46764501442654802}{156083863544770561} a^{2} - \frac{62755367803683582}{156083863544770561} a - \frac{31976923674276799}{156083863544770561}$, $\frac{1}{156083863544770561} a^{26} + \frac{78}{156083863544770561} a^{24} - \frac{346158541388053}{156083863544770561} a^{23} + \frac{2691}{156083863544770561} a^{22} + \frac{251740573828038}{156083863544770561} a^{21} + \frac{54054}{156083863544770561} a^{20} + \frac{71916697027117660}{156083863544770561} a^{19} + \frac{700245}{156083863544770561} a^{18} + \frac{26159746674141062}{156083863544770561} a^{17} - \frac{19309343937560814}{156083863544770561} a^{16} + \frac{10796324657422689}{156083863544770561} a^{15} - \frac{77237375737998972}{156083863544770561} a^{14} - \frac{33845343078911261}{156083863544770561} a^{13} - \frac{45055135717795540}{156083863544770561} a^{12} - \frac{11132893606038227}{156083863544770561} a^{11} + \frac{57928032245743434}{156083863544770561} a^{10} + \frac{25097021884770040}{156083863544770561} a^{9} - \frac{33791351169631233}{156083863544770561} a^{8} - \frac{48153856335214882}{156083863544770561} a^{7} + \frac{35400464665156723}{156083863544770561} a^{6} + \frac{6555951505220160}{156083863544770561} a^{5} - \frac{61146255402532892}{156083863544770561} a^{4} + \frac{73331640181972888}{156083863544770561} a^{3} - \frac{4827335896107210}{156083863544770561} a^{2} - \frac{59484612499802213}{156083863544770561} a + \frac{24136679932792341}{156083863544770561}$, $\frac{1}{156083863544770561} a^{27} + \frac{308391354364047}{156083863544770561} a^{24} - \frac{3159}{156083863544770561} a^{23} + \frac{715085204078761}{156083863544770561} a^{22} - \frac{138996}{156083863544770561} a^{21} + \frac{220031103815314}{1609111995306913} a^{20} - \frac{2985255}{156083863544770561} a^{19} - \frac{109667600055047}{1609111995306913} a^{18} + \frac{37009575853165391}{156083863544770561} a^{17} + \frac{62549133657916460}{156083863544770561} a^{16} - \frac{33791352232040841}{156083863544770561} a^{15} + \frac{71202326394002478}{156083863544770561} a^{14} - \frac{6436449860403028}{156083863544770561} a^{13} - \frac{28139989227790080}{156083863544770561} a^{12} - \frac{3218231115820460}{156083863544770561} a^{11} + \frac{2279810944122434}{156083863544770561} a^{10} - \frac{14482025521296777}{156083863544770561} a^{9} + \frac{66085546613294764}{156083863544770561} a^{8} - \frac{54709834515053155}{156083863544770561} a^{7} - \frac{45695945236271969}{156083863544770561} a^{6} - \frac{38618710520375220}{156083863544770561} a^{5} - \frac{69563543975905928}{156083863544770561} a^{4} + \frac{28964007023985063}{156083863544770561} a^{3} - \frac{49604159694263520}{156083863544770561} a^{2} - \frac{45055136901714868}{156083863544770561} a - \frac{3141770122738654}{156083863544770561}$, $\frac{1}{156083863544770561} a^{28} - \frac{3402}{156083863544770561} a^{24} + \frac{113301561072018}{156083863544770561} a^{23} - \frac{156492}{156083863544770561} a^{22} - \frac{124955144443974}{156083863544770561} a^{21} - \frac{3536379}{156083863544770561} a^{20} - \frac{55066157275749745}{156083863544770561} a^{19} - \frac{48866328}{156083863544770561} a^{18} + \frac{17793187345731125}{156083863544770561} a^{17} - \frac{53100696290160759}{156083863544770561} a^{16} + \frac{57102041862538561}{156083863544770561} a^{15} + \frac{41836909131492650}{156083863544770561} a^{14} + \frac{39011535622579956}{156083863544770561} a^{13} - \frac{59537155336304959}{156083863544770561} a^{12} + \frac{34929387306294145}{156083863544770561} a^{11} - \frac{27354936074996177}{156083863544770561} a^{10} + \frac{22282705968554935}{156083863544770561} a^{9} - \frac{27354961373241149}{156083863544770561} a^{8} + \frac{433077470448741}{156083863544770561} a^{7} - \frac{9654732906866538}{156083863544770561} a^{6} + \frac{45134921276225762}{156083863544770561} a^{5} + \frac{8045526462270782}{156083863544770561} a^{4} - \frac{22804755933468284}{156083863544770561} a^{3} - \frac{17700239180225171}{156083863544770561} a^{2} + \frac{1188247230666114}{156083863544770561} a - \frac{77237376033012150}{156083863544770561}$, $\frac{1}{156083863544770561} a^{29} + \frac{33760640237348}{156083863544770561} a^{24} + \frac{98658}{156083863544770561} a^{23} + \frac{584546536687438}{156083863544770561} a^{22} + \frac{4883571}{156083863544770561} a^{21} + \frac{13947393543790210}{156083863544770561} a^{20} + \frac{111878172}{156083863544770561} a^{19} + \frac{66595394627808192}{156083863544770561} a^{18} - \frac{30573126392484727}{156083863544770561} a^{17} - \frac{33014103479502620}{156083863544770561} a^{16} - \frac{49882458579826711}{156083863544770561} a^{15} - \frac{32342048543072906}{156083863544770561} a^{14} + \frac{56318996863863377}{156083863544770561} a^{13} - \frac{20428922866261168}{156083863544770561} a^{12} + \frac{12873192663367448}{156083863544770561} a^{11} + \frac{20557366979733035}{156083863544770561} a^{10} + \frac{49883212360152175}{156083863544770561} a^{9} - \frac{70286165763454390}{156083863544770561} a^{8} + \frac{40228935885640015}{156083863544770561} a^{7} + \frac{12446446512697297}{156083863544770561} a^{6} + \frac{1610083908956620}{156083863544770561} a^{5} - \frac{17268427501743947}{156083863544770561} a^{4} + \frac{46664632357212449}{156083863544770561} a^{3} + \frac{23646747082110990}{156083863544770561} a^{2} + \frac{12872940903232028}{156083863544770561} a + \frac{4958550815410819}{156083863544770561}$, $\frac{1}{156083863544770561} a^{30} + \frac{109620}{156083863544770561} a^{24} - \frac{338389485806749}{156083863544770561} a^{23} + \frac{5672835}{156083863544770561} a^{22} - \frac{418375245448831}{156083863544770561} a^{21} + \frac{136739988}{156083863544770561} a^{20} - \frac{38557148932683370}{156083863544770561} a^{19} + \frac{1968227100}{156083863544770561} a^{18} + \frac{49678262473094960}{156083863544770561} a^{17} + \frac{3218242427679926}{156083863544770561} a^{16} + \frac{25172296230391520}{156083863544770561} a^{15} + \frac{20918572092506299}{156083863544770561} a^{14} + \frac{5994857415984995}{156083863544770561} a^{13} - \frac{70800433291983932}{156083863544770561} a^{12} + \frac{29512057431064177}{156083863544770561} a^{11} - \frac{53099297112256593}{156083863544770561} a^{10} + \frac{6956491081540556}{156083863544770561} a^{9} - \frac{30570603459793147}{156083863544770561} a^{8} - \frac{6854975825590976}{156083863544770561} a^{7} + \frac{24139379689741770}{156083863544770561} a^{6} + \frac{36740025209287587}{156083863544770561} a^{5} - \frac{33789856649676393}{156083863544770561} a^{4} - \frac{50165016104475141}{156083863544770561} a^{3} - \frac{32181915333861920}{156083863544770561} a^{2} - \frac{27137870714269652}{156083863544770561} a + \frac{32182251557450744}{156083863544770561}$, $\frac{1}{156083863544770561} a^{31} + \frac{257903302379724}{156083863544770561} a^{24} - \frac{2548665}{156083863544770561} a^{23} + \frac{677793625997487}{156083863544770561} a^{22} - \frac{134569512}{156083863544770561} a^{21} - \frac{73219645280321612}{156083863544770561} a^{20} - \frac{3211317900}{156083863544770561} a^{19} - \frac{6159854817327979}{156083863544770561} a^{18} - \frac{53100740672504529}{156083863544770561} a^{17} + \frac{65768146935617932}{156083863544770561} a^{16} - \frac{16091520037403500}{156083863544770561} a^{15} - \frac{52361960592639987}{156083863544770561} a^{14} + \frac{4824977594054979}{156083863544770561} a^{13} + \frac{30741319301188417}{156083863544770561} a^{12} - \frac{57937229559325332}{156083863544770561} a^{11} - \frac{31492909226580026}{156083863544770561} a^{10} + \frac{77214188224455024}{156083863544770561} a^{9} + \frac{19298191836643591}{156083863544770561} a^{8} - \frac{38654556129200337}{156083863544770561} a^{7} + \frac{6208999195148503}{156083863544770561} a^{6} - \frac{49913373724402423}{156083863544770561} a^{5} + \frac{7445271518310129}{156083863544770561} a^{4} - \frac{61158553504577354}{156083863544770561} a^{3} - \frac{68537157695140100}{156083863544770561} a^{2} - \frac{25747236687658624}{156083863544770561} a - \frac{21034314234552558}{156083863544770561}$, $\frac{1}{156083863544770561} a^{32} - \frac{2912760}{156083863544770561} a^{24} + \frac{644389891201143}{156083863544770561} a^{23} - \frac{160784352}{156083863544770561} a^{22} - \frac{302816744482966}{156083863544770561} a^{21} - \frac{4037085360}{156083863544770561} a^{20} + \frac{59687840007548603}{156083863544770561} a^{19} - \frac{59769835200}{156083863544770561} a^{18} + \frac{32392975025411759}{156083863544770561} a^{17} + \frac{65973020258534333}{156083863544770561} a^{16} + \frac{33098779896522816}{156083863544770561} a^{15} + \frac{51487925935906976}{156083863544770561} a^{14} + \frac{65345896123081466}{156083863544770561} a^{13} + \frac{25730024390722384}{156083863544770561} a^{12} - \frac{50540181171982722}{156083863544770561} a^{11} - \frac{49927521952194943}{156083863544770561} a^{10} - \frac{64780912038099139}{156083863544770561} a^{9} + \frac{17618247395611643}{156083863544770561} a^{8} - \frac{52326952814075246}{156083863544770561} a^{7} - \frac{45143426925416764}{156083863544770561} a^{6} - \frac{47646460312167260}{156083863544770561} a^{5} + \frac{22478377202638142}{156083863544770561} a^{4} + \frac{60314668633798907}{156083863544770561} a^{3} - \frac{25756524448181584}{156083863544770561} a^{2} - \frac{36182923416746242}{156083863544770561} a - \frac{62755754806991397}{156083863544770561}$, $\frac{1}{156083863544770561} a^{33} - \frac{717953379991494}{156083863544770561} a^{24} + \frac{57672648}{156083863544770561} a^{23} - \frac{235700556631137}{156083863544770561} a^{22} + \frac{3171995640}{156083863544770561} a^{21} + \frac{32033428011703326}{156083863544770561} a^{20} + \frac{77858074800}{156083863544770561} a^{19} + \frac{15939394260519247}{156083863544770561} a^{18} - \frac{19308234466117056}{156083863544770561} a^{17} - \frac{23115230967329128}{156083863544770561} a^{16} + \frac{10059263264160}{156083863544770561} a^{15} + \frac{37431050099035049}{156083863544770561} a^{14} + \frac{77297413693371600}{156083863544770561} a^{13} + \frac{25296697962553606}{156083863544770561} a^{12} + \frac{46900760876723837}{156083863544770561} a^{11} - \frac{67670012862739566}{156083863544770561} a^{10} - \frac{68590595744591659}{156083863544770561} a^{9} + \frac{54543540679479480}{156083863544770561} a^{8} + \frac{74955667636849798}{156083863544770561} a^{7} - \frac{7158129117054355}{156083863544770561} a^{6} + \frac{16902767025436690}{156083863544770561} a^{5} + \frac{69217332452033629}{156083863544770561} a^{4} + \frac{53425354674075153}{156083863544770561} a^{3} + \frac{1841356600963082}{156083863544770561} a^{2} + \frac{59575455838512991}{156083863544770561} a + \frac{68362492805974778}{156083863544770561}$, $\frac{1}{156083863544770561} a^{34} + \frac{67616208}{156083863544770561} a^{24} + \frac{510107097602784}{156083863544770561} a^{23} + \frac{3887931960}{156083863544770561} a^{22} + \frac{326160765680764}{156083863544770561} a^{21} + \frac{100410068880}{156083863544770561} a^{20} + \frac{1455505840508664}{156083863544770561} a^{19} + \frac{1517561268300}{156083863544770561} a^{18} + \frac{3527085023151817}{156083863544770561} a^{17} - \frac{77222633750982624}{156083863544770561} a^{16} - \frac{43497015348542110}{156083863544770561} a^{15} + \frac{95528313894816}{156083863544770561} a^{14} - \frac{72929425160383932}{156083863544770561} a^{13} - \frac{2802287312889986}{156083863544770561} a^{12} + \frac{21719430212986734}{156083863544770561} a^{11} + \frac{31771421672845267}{156083863544770561} a^{10} + \frac{68885591086356895}{156083863544770561} a^{9} + \frac{37596432103157686}{156083863544770561} a^{8} - \frac{18631849410529949}{156083863544770561} a^{7} - \frac{13712154396129730}{156083863544770561} a^{6} - \frac{37096041272525203}{156083863544770561} a^{5} + \frac{65696700524162584}{156083863544770561} a^{4} + \frac{17073842413174461}{156083863544770561} a^{3} + \frac{56610883790457245}{156083863544770561} a^{2} - \frac{58918316916266740}{156083863544770561} a - \frac{74008582953178078}{156083863544770561}$, $\frac{1}{156083863544770561} a^{35} + \frac{352525692605662}{156083863544770561} a^{24} - \frac{1183283640}{156083863544770561} a^{23} + \frac{244083573036371}{156083863544770561} a^{22} - \frac{66940045920}{156083863544770561} a^{21} + \frac{41511542147843479}{156083863544770561} a^{20} - \frac{1677304559700}{156083863544770561} a^{19} - \frac{57738922741914410}{156083863544770561} a^{18} + \frac{1584831015014113}{156083863544770561} a^{17} - \frac{33647612731831933}{156083863544770561} a^{16} + \frac{15868220553981226}{156083863544770561} a^{15} + \frac{29158272907284768}{156083863544770561} a^{14} + \frac{43711340448255004}{156083863544770561} a^{13} - \frac{63978426911918078}{156083863544770561} a^{12} - \frac{31083645955699888}{156083863544770561} a^{11} + \frac{54009938689070217}{156083863544770561} a^{10} - \frac{15272914168497313}{156083863544770561} a^{9} + \frac{19835338860234157}{156083863544770561} a^{8} - \frac{45547369942058295}{156083863544770561} a^{7} - \frac{18618874983180593}{156083863544770561} a^{6} - \frac{29914873933553287}{156083863544770561} a^{5} - \frac{40009921913971704}{156083863544770561} a^{4} + \frac{39170506359541367}{156083863544770561} a^{3} - \frac{8620756627794243}{156083863544770561} a^{2} - \frac{76516045578378191}{156083863544770561} a + \frac{47083821214142130}{156083863544770561}$, $\frac{1}{156083863544770561} a^{36} - \frac{1419940368}{156083863544770561} a^{24} - \frac{449551447477671}{156083863544770561} a^{23} - \frac{83979330336}{156083863544770561} a^{22} - \frac{687757472802863}{156083863544770561} a^{21} - \frac{2214042018804}{156083863544770561} a^{20} - \frac{8777163083073144}{156083863544770561} a^{19} - \frac{33993372409920}{156083863544770561} a^{18} + \frac{19356088690773427}{156083863544770561} a^{17} - \frac{38953036985997768}{156083863544770561} a^{16} + \frac{9186183435440570}{156083863544770561} a^{15} + \frac{10684429135046792}{156083863544770561} a^{14} + \frac{19759220388071596}{156083863544770561} a^{13} + \frac{61192790538083468}{156083863544770561} a^{12} - \frac{10577228529753257}{156083863544770561} a^{11} - \frac{66492844320674328}{156083863544770561} a^{10} - \frac{47956421621273110}{156083863544770561} a^{9} + \frac{64470407632206696}{156083863544770561} a^{8} - \frac{5255008040997214}{156083863544770561} a^{7} - \frac{57562381894521601}{156083863544770561} a^{6} + \frac{53730989436800833}{156083863544770561} a^{5} + \frac{13581421550285302}{156083863544770561} a^{4} + \frac{59122628586643473}{156083863544770561} a^{3} - \frac{47152497914508409}{156083863544770561} a^{2} + \frac{51558363825555201}{156083863544770561} a + \frac{20666917762619773}{156083863544770561}$, $\frac{1}{156083863544770561} a^{37} - \frac{358565933151935}{156083863544770561} a^{24} + \frac{22516197264}{156083863544770561} a^{23} - \frac{573248422577523}{156083863544770561} a^{22} + \frac{1300310391996}{156083863544770561} a^{21} - \frac{51834375864875411}{156083863544770561} a^{20} + \frac{33098809978080}{156083863544770561} a^{19} + \frac{62098679384920445}{156083863544770561} a^{18} - \frac{59052009840105637}{156083863544770561} a^{17} + \frac{53370199752405281}{156083863544770561} a^{16} + \frac{35071643056059955}{156083863544770561} a^{15} - \frac{41891517105825682}{156083863544770561} a^{14} - \frac{11272451083476216}{156083863544770561} a^{13} + \frac{30318360222791886}{156083863544770561} a^{12} - \frac{8080228441845865}{156083863544770561} a^{11} + \frac{15019919449175613}{156083863544770561} a^{10} - \frac{56243997072826770}{156083863544770561} a^{9} - \frac{24601084026656070}{156083863544770561} a^{8} + \frac{10778370320499715}{156083863544770561} a^{7} + \frac{69597608182599186}{156083863544770561} a^{6} + \frac{26125610981746473}{156083863544770561} a^{5} + \frac{38174787014072884}{156083863544770561} a^{4} + \frac{31064380974787602}{156083863544770561} a^{3} - \frac{66578108968436957}{156083863544770561} a^{2} - \frac{63450886709265459}{156083863544770561} a - \frac{52257064228361364}{156083863544770561}$, $\frac{1}{156083863544770561} a^{38} + \frac{27600499872}{156083863544770561} a^{24} + \frac{573404638906994}{156083863544770561} a^{23} + \frac{1666380179772}{156083863544770561} a^{22} + \frac{487183121875867}{156083863544770561} a^{21} + \frac{44630008293024}{156083863544770561} a^{20} - \frac{5583881575528128}{156083863544770561} a^{19} + \frac{693793765282464}{156083863544770561} a^{18} + \frac{58931773593880356}{156083863544770561} a^{17} - \frac{57471593702652040}{156083863544770561} a^{16} + \frac{40500501332669929}{156083863544770561} a^{15} - \frac{17262319493448039}{156083863544770561} a^{14} + \frac{38394228055751444}{156083863544770561} a^{13} + \frac{8033721572423688}{156083863544770561} a^{12} - \frac{63713996282713150}{156083863544770561} a^{11} + \frac{72047100773690176}{156083863544770561} a^{10} - \frac{42170143705577639}{156083863544770561} a^{9} + \frac{20766546350629833}{156083863544770561} a^{8} + \frac{15084558746345855}{156083863544770561} a^{7} + \frac{26183057810881911}{156083863544770561} a^{6} - \frac{57369917407269664}{156083863544770561} a^{5} - \frac{13723951647175050}{156083863544770561} a^{4} + \frac{73596054609746264}{156083863544770561} a^{3} + \frac{5099186859381991}{156083863544770561} a^{2} - \frac{2477437144276617}{156083863544770561} a - \frac{52524018106452612}{156083863544770561}$, $\frac{1}{156083863544770561} a^{39} + \frac{666078424142389}{156083863544770561} a^{24} - \frac{403657310628}{156083863544770561} a^{23} + \frac{723247677596655}{156083863544770561} a^{22} - \frac{23681228890176}{156083863544770561} a^{21} - \frac{44812095934223783}{156083863544770561} a^{20} - \frac{610329853669536}{156083863544770561} a^{19} - \frac{15695989035169174}{156083863544770561} a^{18} + \frac{47282867456738435}{156083863544770561} a^{17} + \frac{9219100757946213}{156083863544770561} a^{16} + \frac{68378549810473823}{156083863544770561} a^{15} + \frac{72915306276228017}{156083863544770561} a^{14} + \frac{58877966010366502}{156083863544770561} a^{13} + \frac{65485911297440364}{156083863544770561} a^{12} + \frac{14013396834674262}{156083863544770561} a^{11} + \frac{37215864054162462}{156083863544770561} a^{10} + \frac{71859739468298048}{156083863544770561} a^{9} + \frac{74653488054204448}{156083863544770561} a^{8} + \frac{27999139705612779}{156083863544770561} a^{7} - \frac{58367345427263953}{156083863544770561} a^{6} - \frac{17411965720395519}{156083863544770561} a^{5} - \frac{50601963042745170}{156083863544770561} a^{4} + \frac{51684887418066043}{156083863544770561} a^{3} + \frac{66155903127367313}{156083863544770561} a^{2} + \frac{61899537178425575}{156083863544770561} a + \frac{46171653718329572}{156083863544770561}$, $\frac{1}{156083863544770561} a^{40} - \frac{504571638285}{156083863544770561} a^{24} + \frac{649837721431783}{156083863544770561} a^{23} - \frac{30947060481480}{156083863544770561} a^{22} - \frac{570376623764082}{156083863544770561} a^{21} - \frac{839203548795612}{156083863544770561} a^{20} - \frac{47462253461276317}{156083863544770561} a^{19} - \frac{13177576386046800}{156083863544770561} a^{18} - \frac{77764249324590666}{156083863544770561} a^{17} + \frac{46600353754562793}{156083863544770561} a^{16} + \frac{12053855070098959}{156083863544770561} a^{15} - \frac{64764374323639175}{156083863544770561} a^{14} - \frac{53284807686774983}{156083863544770561} a^{13} + \frac{12999945103732209}{156083863544770561} a^{12} - \frac{40273209804989346}{156083863544770561} a^{11} + \frac{6272754797204269}{156083863544770561} a^{10} - \frac{5560966889056621}{156083863544770561} a^{9} - \frac{9653194503660246}{156083863544770561} a^{8} + \frac{11990743601713634}{156083863544770561} a^{7} + \frac{54806083805220723}{156083863544770561} a^{6} - \frac{46468434018088145}{156083863544770561} a^{5} - \frac{26710339808559767}{156083863544770561} a^{4} + \frac{14080541501344602}{156083863544770561} a^{3} + \frac{70283751836561564}{156083863544770561} a^{2} + \frac{4824383172240516}{156083863544770561} a - \frac{5885966704391955}{156083863544770561}$, $\frac{1}{156083863544770561} a^{41} + \frac{313333331137426}{156083863544770561} a^{24} + \frac{6895812389895}{156083863544770561} a^{23} - \frac{662012795354091}{156083863544770561} a^{22} + \frac{409611255959763}{156083863544770561} a^{21} - \frac{70462692807698013}{156083863544770561} a^{20} + \frac{10663433522919450}{156083863544770561} a^{19} + \frac{47005452352864634}{156083863544770561} a^{18} + \frac{51379325335162904}{156083863544770561} a^{17} + \frac{67190869833834455}{156083863544770561} a^{16} - \frac{52350496388412472}{156083863544770561} a^{15} - \frac{54959054990592645}{156083863544770561} a^{14} + \frac{77029689074902236}{156083863544770561} a^{13} + \frac{25639272164340897}{156083863544770561} a^{12} - \frac{34738603353776412}{156083863544770561} a^{11} + \frac{59600780080400137}{156083863544770561} a^{10} + \frac{14475277269381049}{156083863544770561} a^{9} - \frac{62810625515541010}{156083863544770561} a^{8} + \frac{54793771570377123}{156083863544770561} a^{7} - \frac{24663985787980336}{156083863544770561} a^{6} + \frac{42471133712368868}{156083863544770561} a^{5} - \frac{30634901036574131}{156083863544770561} a^{4} - \frac{59473217607317417}{156083863544770561} a^{3} - \frac{24841792741655876}{156083863544770561} a^{2} + \frac{13614205336158568}{156083863544770561} a - \frac{44145743871263363}{156083863544770561}$

Class group and class number

Not computed

Unit group

Rank:		$20$
Torsion generator:		\( -1 \) (order $2$)
Fundamental units:		Not computed
Regulator:		Not computed

Galois group

$C_{42}$ (as 42T1):

A cyclic group of order 42

The 42 conjugacy class representatives for $C_{42}$

Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{-91}) \), \(\Q(\zeta_{7})^+\), 6.0.36924979.1, 7.7.13841287201.1, 14.0.84150067079150835865691353219.1, \(\Q(\zeta_{49})^+\)

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	$42$	$42$	$21^{2}$	R	$42$	R	$42$	${\href{/LocalNumberField/19.3.0.1}{3} }^{14}$	$21^{2}$	${\href{/LocalNumberField/29.7.0.1}{7} }^{6}$	${\href{/LocalNumberField/31.3.0.1}{3} }^{14}$	$42$	${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$	${\href{/LocalNumberField/43.7.0.1}{7} }^{6}$	$21^{2}$	$21^{2}$	$21^{2}$

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
7	Data not computed
$13$	13.14.7.2	$x^{14} - 48268090 x^{2} + 125497034$	$2$	$7$	$7$	$C_{14}$	$[\ ]_{2}^{7}$
	13.14.7.2	$x^{14} - 48268090 x^{2} + 125497034$	$2$	$7$	$7$	$C_{14}$	$[\ ]_{2}^{7}$
	13.14.7.2	$x^{14} - 48268090 x^{2} + 125497034$	$2$	$7$	$7$	$C_{14}$	$[\ ]_{2}^{7}$