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Properties

Label 42.42.5894751766...5625.1
Degree $42$
Signature $[42, 0]$
Discriminant $3^{21}\cdot 5^{21}\cdot 7^{77}$
Root discriminant $137.21$
Ramified primes $3, 5, 7$
Class number Not computed
Class group Not computed
Galois group $C_{42}$ (as 42T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![825289096969, 56422324200256, 100705981042688, -1194722835542432, -2182766657146880, 6282934993253940, 12286663813613056, -12417833356110027, -27904683042811008, 13111961919565824, 35190719180799952, -8619792825140224, -28496927986493624, 3826593592478720, 16034662507433585, -1203976359523840, -6577707309981696, 276732270859408, 2029606276067328, -47339070983428, -481011410885376, 6086526350717, 88777671908352, -589407436800, -12867182042400, 42732039168, 1469852701968, -2282694400, -132209396199, 87147200, 9307848704, -2248960, -506677248, 35140, 20885760, -251, -629888, 0, 13104, 0, -168, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 168*x^40 + 13104*x^38 - 629888*x^36 - 251*x^35 + 20885760*x^34 + 35140*x^33 - 506677248*x^32 - 2248960*x^31 + 9307848704*x^30 + 87147200*x^29 - 132209396199*x^28 - 2282694400*x^27 + 1469852701968*x^26 + 42732039168*x^25 - 12867182042400*x^24 - 589407436800*x^23 + 88777671908352*x^22 + 6086526350717*x^21 - 481011410885376*x^20 - 47339070983428*x^19 + 2029606276067328*x^18 + 276732270859408*x^17 - 6577707309981696*x^16 - 1203976359523840*x^15 + 16034662507433585*x^14 + 3826593592478720*x^13 - 28496927986493624*x^12 - 8619792825140224*x^11 + 35190719180799952*x^10 + 13111961919565824*x^9 - 27904683042811008*x^8 - 12417833356110027*x^7 + 12286663813613056*x^6 + 6282934993253940*x^5 - 2182766657146880*x^4 - 1194722835542432*x^3 + 100705981042688*x^2 + 56422324200256*x + 825289096969)
 
gp: K = bnfinit(x^42 - 168*x^40 + 13104*x^38 - 629888*x^36 - 251*x^35 + 20885760*x^34 + 35140*x^33 - 506677248*x^32 - 2248960*x^31 + 9307848704*x^30 + 87147200*x^29 - 132209396199*x^28 - 2282694400*x^27 + 1469852701968*x^26 + 42732039168*x^25 - 12867182042400*x^24 - 589407436800*x^23 + 88777671908352*x^22 + 6086526350717*x^21 - 481011410885376*x^20 - 47339070983428*x^19 + 2029606276067328*x^18 + 276732270859408*x^17 - 6577707309981696*x^16 - 1203976359523840*x^15 + 16034662507433585*x^14 + 3826593592478720*x^13 - 28496927986493624*x^12 - 8619792825140224*x^11 + 35190719180799952*x^10 + 13111961919565824*x^9 - 27904683042811008*x^8 - 12417833356110027*x^7 + 12286663813613056*x^6 + 6282934993253940*x^5 - 2182766657146880*x^4 - 1194722835542432*x^3 + 100705981042688*x^2 + 56422324200256*x + 825289096969, 1)
 

Normalized defining polynomial

\( x^{42} - 168 x^{40} + 13104 x^{38} - 629888 x^{36} - 251 x^{35} + 20885760 x^{34} + 35140 x^{33} - 506677248 x^{32} - 2248960 x^{31} + 9307848704 x^{30} + 87147200 x^{29} - 132209396199 x^{28} - 2282694400 x^{27} + 1469852701968 x^{26} + 42732039168 x^{25} - 12867182042400 x^{24} - 589407436800 x^{23} + 88777671908352 x^{22} + 6086526350717 x^{21} - 481011410885376 x^{20} - 47339070983428 x^{19} + 2029606276067328 x^{18} + 276732270859408 x^{17} - 6577707309981696 x^{16} - 1203976359523840 x^{15} + 16034662507433585 x^{14} + 3826593592478720 x^{13} - 28496927986493624 x^{12} - 8619792825140224 x^{11} + 35190719180799952 x^{10} + 13111961919565824 x^{9} - 27904683042811008 x^{8} - 12417833356110027 x^{7} + 12286663813613056 x^{6} + 6282934993253940 x^{5} - 2182766657146880 x^{4} - 1194722835542432 x^{3} + 100705981042688 x^{2} + 56422324200256 x + 825289096969 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $42$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[42, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(589475176645907082922286550311127085690444572711075874815443834048428072939395904541015625=3^{21}\cdot 5^{21}\cdot 7^{77}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $137.21$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(735=3\cdot 5\cdot 7^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{735}(256,·)$, $\chi_{735}(1,·)$, $\chi_{735}(646,·)$, $\chi_{735}(524,·)$, $\chi_{735}(269,·)$, $\chi_{735}(526,·)$, $\chi_{735}(16,·)$, $\chi_{735}(404,·)$, $\chi_{735}(151,·)$, $\chi_{735}(541,·)$, $\chi_{735}(419,·)$, $\chi_{735}(676,·)$, $\chi_{735}(421,·)$, $\chi_{735}(299,·)$, $\chi_{735}(46,·)$, $\chi_{735}(689,·)$, $\chi_{735}(436,·)$, $\chi_{735}(314,·)$, $\chi_{735}(59,·)$, $\chi_{735}(316,·)$, $\chi_{735}(194,·)$, $\chi_{735}(374,·)$, $\chi_{735}(584,·)$, $\chi_{735}(331,·)$, $\chi_{735}(719,·)$, $\chi_{735}(209,·)$, $\chi_{735}(466,·)$, $\chi_{735}(211,·)$, $\chi_{735}(164,·)$, $\chi_{735}(734,·)$, $\chi_{735}(479,·)$, $\chi_{735}(226,·)$, $\chi_{735}(571,·)$, $\chi_{735}(614,·)$, $\chi_{735}(104,·)$, $\chi_{735}(361,·)$, $\chi_{735}(106,·)$, $\chi_{735}(629,·)$, $\chi_{735}(89,·)$, $\chi_{735}(631,·)$, $\chi_{735}(121,·)$, $\chi_{735}(509,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{13} a^{21} - \frac{6}{13} a^{19} - \frac{5}{13} a^{17} + \frac{3}{13} a^{15} - \frac{2}{13} a^{14} + \frac{5}{13} a^{13} - \frac{5}{13} a^{12} + \frac{6}{13} a^{10} - \frac{4}{13} a^{8} + \frac{4}{13} a^{7} - \frac{1}{13} a^{6} + \frac{2}{13} a^{5} - \frac{6}{13} a^{4} - \frac{6}{13} a^{3} + \frac{6}{13} a^{2} - \frac{4}{13} a - \frac{5}{13}$, $\frac{1}{13} a^{22} - \frac{6}{13} a^{20} - \frac{5}{13} a^{18} + \frac{3}{13} a^{16} - \frac{2}{13} a^{15} + \frac{5}{13} a^{14} - \frac{5}{13} a^{13} + \frac{6}{13} a^{11} - \frac{4}{13} a^{9} + \frac{4}{13} a^{8} - \frac{1}{13} a^{7} + \frac{2}{13} a^{6} - \frac{6}{13} a^{5} - \frac{6}{13} a^{4} + \frac{6}{13} a^{3} - \frac{4}{13} a^{2} - \frac{5}{13} a$, $\frac{1}{13} a^{23} - \frac{2}{13} a^{19} - \frac{1}{13} a^{17} - \frac{2}{13} a^{16} - \frac{3}{13} a^{15} - \frac{4}{13} a^{14} + \frac{4}{13} a^{13} + \frac{2}{13} a^{12} + \frac{6}{13} a^{10} + \frac{4}{13} a^{9} + \frac{1}{13} a^{8} + \frac{1}{13} a^{6} + \frac{6}{13} a^{5} - \frac{4}{13} a^{4} - \frac{1}{13} a^{3} + \frac{5}{13} a^{2} + \frac{2}{13} a - \frac{4}{13}$, $\frac{1}{13} a^{24} - \frac{2}{13} a^{20} - \frac{1}{13} a^{18} - \frac{2}{13} a^{17} - \frac{3}{13} a^{16} - \frac{4}{13} a^{15} + \frac{4}{13} a^{14} + \frac{2}{13} a^{13} + \frac{6}{13} a^{11} + \frac{4}{13} a^{10} + \frac{1}{13} a^{9} + \frac{1}{13} a^{7} + \frac{6}{13} a^{6} - \frac{4}{13} a^{5} - \frac{1}{13} a^{4} + \frac{5}{13} a^{3} + \frac{2}{13} a^{2} - \frac{4}{13} a$, $\frac{1}{285743531537269} a^{25} - \frac{345010414285}{21980271656713} a^{24} - \frac{100}{285743531537269} a^{23} - \frac{9032436106580}{285743531537269} a^{22} + \frac{4400}{285743531537269} a^{21} - \frac{16282573344474}{285743531537269} a^{20} - \frac{112000}{285743531537269} a^{19} - \frac{1261718155951}{21980271656713} a^{18} - \frac{109901356459565}{285743531537269} a^{17} + \frac{2578671497797}{285743531537269} a^{16} + \frac{65940795125019}{285743531537269} a^{15} + \frac{91941199021458}{285743531537269} a^{14} + \frac{109901504510765}{285743531537269} a^{13} + \frac{10940843443812}{21980271656713} a^{12} + \frac{43959819140626}{285743531537269} a^{11} - \frac{139936193807779}{285743531537269} a^{10} - \frac{87918743714852}{285743531537269} a^{9} + \frac{5018479785556}{21980271656713} a^{8} - \frac{87925772450852}{285743531537269} a^{7} - \frac{110855536904048}{285743531537269} a^{6} + \frac{65946063093019}{285743531537269} a^{5} + \frac{98159070440143}{285743531537269} a^{4} + \frac{131878903642678}{285743531537269} a^{3} - \frac{105345058261809}{285743531537269} a^{2} + \frac{65941234400539}{285743531537269} a - \frac{19791147029398}{285743531537269}$, $\frac{1}{285743531537269} a^{26} - \frac{8}{21980271656713} a^{24} + \frac{4039730113893}{285743531537269} a^{23} + \frac{368}{21980271656713} a^{22} + \frac{2009447685965}{285743531537269} a^{21} - \frac{9856}{21980271656713} a^{20} - \frac{80377907438600}{285743531537269} a^{19} + \frac{170240}{21980271656713} a^{18} + \frac{96174828719753}{285743531537269} a^{17} + \frac{87921060828196}{285743531537269} a^{16} - \frac{34865173157144}{285743531537269} a^{15} - \frac{87920880237604}{285743531537269} a^{14} + \frac{112932323451785}{285743531537269} a^{13} + \frac{109900228573997}{285743531537269} a^{12} - \frac{9184079487414}{21980271656713} a^{11} - \frac{43956401045010}{285743531537269} a^{10} - \frac{115003158771159}{285743531537269} a^{9} - \frac{749731840}{21980271656713} a^{8} + \frac{95769353465735}{285743531537269} a^{7} + \frac{109915003403053}{285743531537269} a^{6} - \frac{7004514245625}{21980271656713} a^{5} - \frac{43970467036690}{285743531537269} a^{4} - \frac{141786330945417}{285743531537269} a^{3} + \frac{218103808}{21980271656713} a^{2} - \frac{102105884012618}{285743531537269} a + \frac{87920952409124}{285743531537269}$, $\frac{1}{285743531537269} a^{27} - \frac{828645208454}{285743531537269} a^{24} - \frac{432}{21980271656713} a^{23} + \frac{7787773840304}{285743531537269} a^{22} + \frac{25344}{21980271656713} a^{21} - \frac{81284617696995}{285743531537269} a^{20} - \frac{725760}{21980271656713} a^{19} + \frac{60832527784189}{285743531537269} a^{18} + \frac{21980435554057}{285743531537269} a^{17} - \frac{30446597266812}{285743531537269} a^{16} - \frac{109903215786797}{285743531537269} a^{15} - \frac{40463050583729}{285743531537269} a^{14} + \frac{21994349575945}{285743531537269} a^{13} + \frac{7669784176671}{21980271656713} a^{12} + \frac{109830186580781}{285743531537269} a^{11} + \frac{36434423560822}{285743531537269} a^{10} + \frac{110135274617645}{285743531537269} a^{9} + \frac{88850081613130}{285743531537269} a^{8} - \frac{88394767203364}{285743531537269} a^{7} - \frac{102372175096505}{285743531537269} a^{6} - \frac{131345748884022}{285743531537269} a^{5} - \frac{66148238915238}{285743531537269} a^{4} - \frac{22260971257609}{285743531537269} a^{3} + \frac{86005786712737}{285743531537269} a^{2} + \frac{87964573170724}{285743531537269} a - \frac{4467274638193}{21980271656713}$, $\frac{1}{285743531537269} a^{28} - \frac{6048}{285743531537269} a^{24} - \frac{702764002689}{21980271656713} a^{23} + \frac{370944}{285743531537269} a^{22} + \frac{3950291113099}{285743531537269} a^{21} - \frac{11176704}{285743531537269} a^{20} - \frac{6046578027831}{21980271656713} a^{19} + \frac{205922304}{285743531537269} a^{18} + \frac{98895050556734}{285743531537269} a^{17} - \frac{87923587111972}{285743531537269} a^{16} + \frac{46079082886306}{285743531537269} a^{15} - \frac{21959696236297}{285743531537269} a^{14} + \frac{78632778887414}{285743531537269} a^{13} + \frac{43845573639698}{285743531537269} a^{12} + \frac{2699101318745}{21980271656713} a^{11} + \frac{88349333453860}{285743531537269} a^{10} + \frac{53123459941626}{285743531537269} a^{9} + \frac{64920579882267}{285743531537269} a^{8} - \frac{118840018380349}{285743531537269} a^{7} - \frac{20537514966793}{285743531537269} a^{6} - \frac{16052663251666}{285743531537269} a^{5} - \frac{81386274816}{21980271656713} a^{4} + \frac{40856420121274}{285743531537269} a^{3} + \frac{44264949120530}{285743531537269} a^{2} - \frac{90016686828930}{285743531537269} a + \frac{65926319455515}{285743531537269}$, $\frac{1}{285743531537269} a^{29} + \frac{800057789366}{21980271656713} a^{24} - \frac{233856}{285743531537269} a^{23} - \frac{3248214550936}{285743531537269} a^{22} + \frac{15434496}{285743531537269} a^{21} + \frac{47889550273963}{285743531537269} a^{20} - \frac{471453696}{285743531537269} a^{19} + \frac{116513015467792}{285743531537269} a^{18} - \frac{131873098873398}{285743531537269} a^{17} + \frac{57871696953045}{285743531537269} a^{16} + \frac{87821638761508}{285743531537269} a^{15} + \frac{16151274169785}{285743531537269} a^{14} - \frac{131112217508406}{285743531537269} a^{13} - \frac{9677708869122}{21980271656713} a^{12} - \frac{91872636894244}{285743531537269} a^{11} + \frac{91363723885012}{285743531537269} a^{10} - \frac{74771389938724}{285743531537269} a^{9} + \frac{18139761459810}{285743531537269} a^{8} - \frac{70857650175506}{285743531537269} a^{7} - \frac{28093785874044}{285743531537269} a^{6} - \frac{35258189364507}{285743531537269} a^{5} - \frac{46163820682131}{285743531537269} a^{4} + \frac{115697387862582}{285743531537269} a^{3} + \frac{101047632143673}{285743531537269} a^{2} + \frac{46482762858002}{285743531537269} a + \frac{29682480316607}{285743531537269}$, $\frac{1}{285743531537269} a^{30} - \frac{259840}{285743531537269} a^{24} + \frac{288780289181}{21980271656713} a^{23} + \frac{17928960}{285743531537269} a^{22} + \frac{3549044501803}{285743531537269} a^{21} - \frac{576221184}{285743531537269} a^{20} + \frac{6939774606786}{21980271656713} a^{19} + \frac{11058790400}{285743531537269} a^{18} + \frac{40234442784215}{285743531537269} a^{17} + \frac{87782964591652}{285743531537269} a^{16} - \frac{73302877679271}{285743531537269} a^{15} - \frac{20820046561033}{285743531537269} a^{14} - \frac{34690853434283}{285743531537269} a^{13} + \frac{125295712827958}{285743531537269} a^{12} + \frac{7500378321177}{21980271656713} a^{11} + \frac{68798859279890}{285743531537269} a^{10} - \frac{41810480147484}{285743531537269} a^{9} + \frac{72110281357878}{285743531537269} a^{8} + \frac{69826425683831}{285743531537269} a^{7} + \frac{6556116582400}{21980271656713} a^{6} + \frac{85948530596892}{285743531537269} a^{5} + \frac{90923182405951}{285743531537269} a^{4} + \frac{18144935598263}{285743531537269} a^{3} + \frac{106137116671012}{285743531537269} a^{2} + \frac{137120988460203}{285743531537269} a + \frac{109029479922477}{285743531537269}$, $\frac{1}{285743531537269} a^{31} + \frac{2159032752126}{285743531537269} a^{24} - \frac{8055040}{285743531537269} a^{23} + \frac{2817799598604}{285743531537269} a^{22} + \frac{567074816}{285743531537269} a^{21} + \frac{11027997539011}{285743531537269} a^{20} - \frac{18043289600}{285743531537269} a^{19} + \frac{31915329432995}{285743531537269} a^{18} - \frac{65604988845339}{285743531537269} a^{17} - \frac{33941530032457}{285743531537269} a^{16} - \frac{113897689168685}{285743531537269} a^{15} - \frac{127037539266707}{285743531537269} a^{14} + \frac{9429486878967}{285743531537269} a^{13} - \frac{8114717514737}{21980271656713} a^{12} - \frac{31449114445258}{285743531537269} a^{11} + \frac{131682698888101}{285743531537269} a^{10} - \frac{88416972547077}{285743531537269} a^{9} - \frac{81550230610334}{285743531537269} a^{8} + \frac{54599676873702}{285743531537269} a^{7} - \frac{108182372893605}{285743531537269} a^{6} + \frac{91818588828945}{285743531537269} a^{5} + \frac{137442270754304}{285743531537269} a^{4} + \frac{123084912958541}{285743531537269} a^{3} - \frac{31115505590067}{285743531537269} a^{2} - \frac{111689799792218}{285743531537269} a - \frac{5307042546127}{285743531537269}$, $\frac{1}{285743531537269} a^{32} - \frac{9205760}{285743531537269} a^{24} - \frac{1081641755926}{285743531537269} a^{23} + \frac{677543936}{285743531537269} a^{22} + \frac{6761243884627}{285743531537269} a^{21} - \frac{22682992640}{285743531537269} a^{20} - \frac{137227101208422}{285743531537269} a^{19} + \frac{447768166400}{285743531537269} a^{18} + \frac{29650510441614}{285743531537269} a^{17} - \frac{137590674061878}{285743531537269} a^{16} - \frac{72468914162626}{285743531537269} a^{15} + \frac{48717176504320}{285743531537269} a^{14} - \frac{7813967632177}{285743531537269} a^{13} - \frac{60192845236646}{285743531537269} a^{12} - \frac{6872016790084}{21980271656713} a^{11} + \frac{99517806357108}{285743531537269} a^{10} - \frac{73736592520122}{285743531537269} a^{9} + \frac{5477786717734}{285743531537269} a^{8} + \frac{74012795836388}{285743531537269} a^{7} - \frac{86205482473749}{285743531537269} a^{6} + \frac{50318726990425}{285743531537269} a^{5} + \frac{52734218699904}{285743531537269} a^{4} - \frac{53795543301009}{285743531537269} a^{3} + \frac{55795288285390}{285743531537269} a^{2} + \frac{48905897847971}{285743531537269} a + \frac{71289602292525}{285743531537269}$, $\frac{1}{285743531537269} a^{33} + \frac{2066591341967}{285743531537269} a^{24} - \frac{243032064}{285743531537269} a^{23} - \frac{9725601975832}{285743531537269} a^{22} + \frac{17822351360}{285743531537269} a^{21} - \frac{16266638143682}{285743531537269} a^{20} - \frac{583276953600}{285743531537269} a^{19} - \frac{3483159582872}{21980271656713} a^{18} - \frac{98819096165165}{285743531537269} a^{17} - \frac{134930604491906}{285743531537269} a^{16} + \frac{85830481180250}{285743531537269} a^{15} + \frac{108576751103848}{285743531537269} a^{14} - \frac{10896364310713}{285743531537269} a^{13} + \frac{1974391673481}{21980271656713} a^{12} - \frac{16922235270418}{285743531537269} a^{11} - \frac{97297364608117}{285743531537269} a^{10} + \frac{55077407915707}{285743531537269} a^{9} - \frac{67161346426355}{285743531537269} a^{8} - \frac{141444181556269}{285743531537269} a^{7} - \frac{1017485002317}{285743531537269} a^{6} + \frac{75037072690243}{285743531537269} a^{5} - \frac{25493269919168}{285743531537269} a^{4} - \frac{50277232207929}{285743531537269} a^{3} + \frac{48320909272873}{285743531537269} a^{2} + \frac{129878204785176}{285743531537269} a + \frac{97722595617341}{285743531537269}$, $\frac{1}{285743531537269} a^{34} - \frac{284934144}{285743531537269} a^{24} - \frac{888912689549}{285743531537269} a^{23} + \frac{21844951040}{285743531537269} a^{22} - \frac{9436076919300}{285743531537269} a^{21} - \frac{752226140160}{285743531537269} a^{20} + \frac{48609767165626}{285743531537269} a^{19} + \frac{15158496460800}{285743531537269} a^{18} - \frac{225345975163}{21980271656713} a^{17} - \frac{64456990885322}{285743531537269} a^{16} + \frac{37819184273977}{285743531537269} a^{15} + \frac{3884766366283}{285743531537269} a^{14} - \frac{31389564674987}{285743531537269} a^{13} + \frac{130923572152822}{285743531537269} a^{12} + \frac{33500391511958}{285743531537269} a^{11} - \frac{130643723861431}{285743531537269} a^{10} - \frac{8874844775590}{21980271656713} a^{9} - \frac{116407144666200}{285743531537269} a^{8} - \frac{80436386943391}{285743531537269} a^{7} + \frac{28935266247551}{285743531537269} a^{6} + \frac{24770622139020}{285743531537269} a^{5} - \frac{142643028054471}{285743531537269} a^{4} - \frac{95398929751715}{285743531537269} a^{3} + \frac{94633876929167}{285743531537269} a^{2} - \frac{89824635719831}{285743531537269} a + \frac{8510634638049}{21980271656713}$, $\frac{1}{285743531537269} a^{35} - \frac{6222388826843}{285743531537269} a^{24} - \frac{6648463360}{285743531537269} a^{23} - \frac{2968615843475}{285743531537269} a^{22} + \frac{501484093440}{285743531537269} a^{21} - \frac{135102511392172}{285743531537269} a^{20} - \frac{16754127667200}{285743531537269} a^{19} - \frac{6683318427164}{285743531537269} a^{18} + \frac{103578541263270}{285743531537269} a^{17} + \frac{18321690951170}{285743531537269} a^{16} + \frac{130143932304522}{285743531537269} a^{15} - \frac{51851267489887}{285743531537269} a^{14} - \frac{76371872368643}{285743531537269} a^{13} + \frac{65596487694888}{285743531537269} a^{12} + \frac{10471343217291}{285743531537269} a^{11} + \frac{13369034786806}{285743531537269} a^{10} - \frac{37671628398584}{285743531537269} a^{9} + \frac{68518006629785}{285743531537269} a^{8} + \frac{3345863652597}{285743531537269} a^{7} - \frac{85128749370479}{285743531537269} a^{6} + \frac{8207617714252}{21980271656713} a^{5} + \frac{49041152471771}{285743531537269} a^{4} - \frac{69720350028091}{285743531537269} a^{3} - \frac{134004190547379}{285743531537269} a^{2} + \frac{113943244323656}{285743531537269} a - \frac{83024414507909}{285743531537269}$, $\frac{1}{285743531537269} a^{36} - \frac{7978156032}{285743531537269} a^{24} - \frac{9759892139811}{285743531537269} a^{23} + \frac{629134589952}{285743531537269} a^{22} + \frac{9851744049621}{285743531537269} a^{21} - \frac{22115448520704}{285743531537269} a^{20} - \frac{117640135384351}{285743531537269} a^{19} - \frac{118753302111978}{285743531537269} a^{18} - \frac{99511079398223}{285743531537269} a^{17} - \frac{46567089767060}{285743531537269} a^{16} - \frac{37652795247697}{285743531537269} a^{15} + \frac{74701411147582}{285743531537269} a^{14} + \frac{94014652253593}{285743531537269} a^{13} + \frac{27640526453809}{285743531537269} a^{12} + \frac{31726757535142}{285743531537269} a^{11} + \frac{48877031943287}{285743531537269} a^{10} + \frac{113854903867154}{285743531537269} a^{9} + \frac{104743268055770}{285743531537269} a^{8} + \frac{131921259348765}{285743531537269} a^{7} - \frac{97528070621719}{285743531537269} a^{6} + \frac{125977898027959}{285743531537269} a^{5} + \frac{133591800022250}{285743531537269} a^{4} + \frac{105777580898481}{285743531537269} a^{3} + \frac{84279363088821}{285743531537269} a^{2} - \frac{57354683092012}{285743531537269} a + \frac{90790206231938}{285743531537269}$, $\frac{1}{285743531537269} a^{37} - \frac{5235322387052}{285743531537269} a^{24} - \frac{168681013248}{285743531537269} a^{23} - \frac{6881790737696}{285743531537269} a^{22} - \frac{691679510509}{21980271656713} a^{21} - \frac{96220095675140}{285743531537269} a^{20} - \frac{23194553143893}{285743531537269} a^{19} + \frac{59145199481073}{285743531537269} a^{18} - \frac{111291139113205}{285743531537269} a^{17} + \frac{75825541415097}{285743531537269} a^{16} - \frac{104707852566223}{285743531537269} a^{15} + \frac{48613493197505}{285743531537269} a^{14} - \frac{17800746088405}{285743531537269} a^{13} - \frac{102655317208329}{285743531537269} a^{12} + \frac{1666903098650}{21980271656713} a^{11} - \frac{83955620398867}{285743531537269} a^{10} + \frac{111310557877718}{285743531537269} a^{9} - \frac{115390323601620}{285743531537269} a^{8} + \frac{21218979674663}{285743531537269} a^{7} - \frac{115749460522036}{285743531537269} a^{6} - \frac{72379398209985}{285743531537269} a^{5} - \frac{114072818246233}{285743531537269} a^{4} + \frac{54257512881901}{285743531537269} a^{3} - \frac{41506733830715}{285743531537269} a^{2} + \frac{73428680760905}{285743531537269} a + \frac{71166004360265}{285743531537269}$, $\frac{1}{285743531537269} a^{38} - \frac{206770274304}{285743531537269} a^{24} - \frac{222116129368}{21980271656713} a^{23} - \frac{5335264575241}{285743531537269} a^{22} - \frac{8205202254712}{285743531537269} a^{21} + \frac{108973077815584}{285743531537269} a^{20} + \frac{92725107446487}{285743531537269} a^{19} - \frac{142614005226735}{285743531537269} a^{18} - \frac{91102549669892}{285743531537269} a^{17} + \frac{22847652463378}{285743531537269} a^{16} - \frac{61559883792142}{285743531537269} a^{15} - \frac{29613222755365}{285743531537269} a^{14} + \frac{4271126581172}{21980271656713} a^{13} - \frac{23260289720897}{285743531537269} a^{12} - \frac{74986595793106}{285743531537269} a^{11} - \frac{20887188189783}{285743531537269} a^{10} + \frac{115476632262054}{285743531537269} a^{9} - \frac{91914602936721}{285743531537269} a^{8} + \frac{59944505795293}{285743531537269} a^{7} - \frac{22567306271362}{285743531537269} a^{6} - \frac{47966011507354}{285743531537269} a^{5} - \frac{32399109236059}{285743531537269} a^{4} - \frac{33600137263983}{285743531537269} a^{3} - \frac{38895319676255}{285743531537269} a^{2} - \frac{14589901483513}{285743531537269} a + \frac{67296532404045}{285743531537269}$, $\frac{1}{285743531537269} a^{39} + \frac{5892717715345}{285743531537269} a^{24} - \frac{310155411456}{21980271656713} a^{23} + \frac{6123562232711}{285743531537269} a^{22} + \frac{7669788544386}{285743531537269} a^{21} + \frac{62404062049259}{285743531537269} a^{20} - \frac{133678401099233}{285743531537269} a^{19} + \frac{53614467428446}{285743531537269} a^{18} - \frac{9633646235702}{285743531537269} a^{17} + \frac{93743039196948}{285743531537269} a^{16} - \frac{133406343369005}{285743531537269} a^{15} - \frac{71252325917615}{285743531537269} a^{14} + \frac{12711591446493}{285743531537269} a^{13} - \frac{111023943362926}{285743531537269} a^{12} + \frac{72056369794984}{285743531537269} a^{11} + \frac{15304338347878}{285743531537269} a^{10} - \frac{40930002850618}{285743531537269} a^{9} + \frac{55935550891644}{285743531537269} a^{8} - \frac{58595691473429}{285743531537269} a^{7} - \frac{68504526087011}{285743531537269} a^{6} - \frac{37766282855707}{285743531537269} a^{5} - \frac{6575579771400}{285743531537269} a^{4} - \frac{84635050025025}{285743531537269} a^{3} + \frac{9341496380931}{21980271656713} a^{2} + \frac{6701808898585}{285743531537269} a - \frac{105247282609592}{285743531537269}$, $\frac{1}{285743531537269} a^{40} - \frac{387694264320}{21980271656713} a^{24} + \frac{1927999035960}{285743531537269} a^{23} - \frac{5463081364907}{285743531537269} a^{22} + \frac{5225854482460}{285743531537269} a^{21} - \frac{109624384670039}{285743531537269} a^{20} - \frac{3964579744589}{21980271656713} a^{19} + \frac{61050194066704}{285743531537269} a^{18} - \frac{68352062336469}{285743531537269} a^{17} - \frac{14746267539287}{285743531537269} a^{16} + \frac{133501512871876}{285743531537269} a^{15} + \frac{9297541094183}{21980271656713} a^{14} - \frac{57980295676936}{285743531537269} a^{13} - \frac{62716751544822}{285743531537269} a^{12} + \frac{107436373218485}{285743531537269} a^{11} - \frac{15540373173246}{285743531537269} a^{10} + \frac{62999209309343}{285743531537269} a^{9} - \frac{25597734571651}{285743531537269} a^{8} + \frac{93210330331295}{285743531537269} a^{7} + \frac{132005956171026}{285743531537269} a^{6} - \frac{24162997833958}{285743531537269} a^{5} - \frac{32331667137522}{285743531537269} a^{4} + \frac{114019205762661}{285743531537269} a^{3} - \frac{58196694939651}{285743531537269} a^{2} + \frac{22703553439051}{285743531537269} a + \frac{115949083125324}{285743531537269}$, $\frac{1}{285743531537269} a^{41} + \frac{9880995059295}{285743531537269} a^{24} - \frac{3919376876508}{285743531537269} a^{23} - \frac{10932527429458}{285743531537269} a^{22} - \frac{1705208905891}{285743531537269} a^{21} + \frac{99585983859787}{285743531537269} a^{20} - \frac{122284413060447}{285743531537269} a^{19} - \frac{8262624410956}{285743531537269} a^{18} + \frac{72732942939158}{285743531537269} a^{17} - \frac{141844927846782}{285743531537269} a^{16} + \frac{74681416475473}{285743531537269} a^{15} - \frac{46203137774057}{285743531537269} a^{14} - \frac{114568625585815}{285743531537269} a^{13} + \frac{141207299957712}{285743531537269} a^{12} - \frac{71707341034861}{285743531537269} a^{11} - \frac{136712745863971}{285743531537269} a^{10} - \frac{5500718377551}{285743531537269} a^{9} - \frac{18245615230595}{285743531537269} a^{8} + \frac{113792195439539}{285743531537269} a^{7} - \frac{7609348258791}{21980271656713} a^{6} - \frac{31274894176164}{285743531537269} a^{5} - \frac{98225469752881}{285743531537269} a^{4} + \frac{30317251014117}{285743531537269} a^{3} + \frac{19398047643528}{285743531537269} a^{2} + \frac{100640762851151}{285743531537269} a + \frac{28269541883329}{285743531537269}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $41$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$ is not computed

Intermediate fields

\(\Q(\sqrt{105}) \), \(\Q(\zeta_{7})^+\), 6.6.56723625.1, 7.7.13841287201.1, 14.14.229134146187897752973821015625.1, \(\Q(\zeta_{49})^+\)

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 $21^{2}$ R R R $42$ ${\href{/LocalNumberField/13.7.0.1}{7} }^{6}$ $42$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{7}$ $21^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{7}$ $42$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$ $42$ $21^{2}$ $21^{2}$

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

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
5Data not computed
7Data not computed