Properties

Label 42.42.8745717994...9677.1
Degree $42$
Signature $[42, 0]$
Discriminant $7^{77}\cdot 11^{21}$
Root discriminant $117.50$
Ramified primes $7, 11$
Class number Not computed
Class group Not computed
Galois group $C_{42}$ (as 42T1)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 126*x^40 + 7371*x^38 - 265734*x^36 - 83*x^35 + 6608385*x^34 + 8715*x^33 - 120236886*x^32 - 418320*x^31 + 1656597096*x^30 + 12157425*x^29 - 17647825586*x^28 - 238834575*x^27 + 147151366404*x^26 + 3353237433*x^25 - 966133116900*x^24 - 34688663100*x^23 + 4999488494931*x^22 + 268659947305*x^21 - 20317109311296*x^20 - 1567171092690*x^19 + 64305659489211*x^18 + 6871145069880*x^17 - 156373597496214*x^16 - 22423210892208*x^15 + 286220369519701*x^14 + 53473028777910*x^13 - 382576512454206*x^12 - 90465142642875*x^11 + 356767740698292*x^10 + 103652112041415*x^9 - 215794576784304*x^8 - 74585905238228*x^7 + 74443140953685*x^6 + 29468446974645*x^5 - 11170399688028*x^4 - 4811855646528*x^3 + 319923920106*x^2 + 140495689782*x - 11523307067)
 
gp: K = bnfinit(x^42 - 126*x^40 + 7371*x^38 - 265734*x^36 - 83*x^35 + 6608385*x^34 + 8715*x^33 - 120236886*x^32 - 418320*x^31 + 1656597096*x^30 + 12157425*x^29 - 17647825586*x^28 - 238834575*x^27 + 147151366404*x^26 + 3353237433*x^25 - 966133116900*x^24 - 34688663100*x^23 + 4999488494931*x^22 + 268659947305*x^21 - 20317109311296*x^20 - 1567171092690*x^19 + 64305659489211*x^18 + 6871145069880*x^17 - 156373597496214*x^16 - 22423210892208*x^15 + 286220369519701*x^14 + 53473028777910*x^13 - 382576512454206*x^12 - 90465142642875*x^11 + 356767740698292*x^10 + 103652112041415*x^9 - 215794576784304*x^8 - 74585905238228*x^7 + 74443140953685*x^6 + 29468446974645*x^5 - 11170399688028*x^4 - 4811855646528*x^3 + 319923920106*x^2 + 140495689782*x - 11523307067, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11523307067, 140495689782, 319923920106, -4811855646528, -11170399688028, 29468446974645, 74443140953685, -74585905238228, -215794576784304, 103652112041415, 356767740698292, -90465142642875, -382576512454206, 53473028777910, 286220369519701, -22423210892208, -156373597496214, 6871145069880, 64305659489211, -1567171092690, -20317109311296, 268659947305, 4999488494931, -34688663100, -966133116900, 3353237433, 147151366404, -238834575, -17647825586, 12157425, 1656597096, -418320, -120236886, 8715, 6608385, -83, -265734, 0, 7371, 0, -126, 0, 1]);
 

Normalized defining polynomial

\( x^{42} - 126 x^{40} + 7371 x^{38} - 265734 x^{36} - 83 x^{35} + 6608385 x^{34} + 8715 x^{33} - 120236886 x^{32} - 418320 x^{31} + 1656597096 x^{30} + 12157425 x^{29} - 17647825586 x^{28} - 238834575 x^{27} + 147151366404 x^{26} + 3353237433 x^{25} - 966133116900 x^{24} - 34688663100 x^{23} + 4999488494931 x^{22} + 268659947305 x^{21} - 20317109311296 x^{20} - 1567171092690 x^{19} + 64305659489211 x^{18} + 6871145069880 x^{17} - 156373597496214 x^{16} - 22423210892208 x^{15} + 286220369519701 x^{14} + 53473028777910 x^{13} - 382576512454206 x^{12} - 90465142642875 x^{11} + 356767740698292 x^{10} + 103652112041415 x^{9} - 215794576784304 x^{8} - 74585905238228 x^{7} + 74443140953685 x^{6} + 29468446974645 x^{5} - 11170399688028 x^{4} - 4811855646528 x^{3} + 319923920106 x^{2} + 140495689782 x - 11523307067 \)

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[42, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(874571799455406731006354153891279294008499270793470105314249037985839659586368645179677=7^{77}\cdot 11^{21}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $117.50$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 11$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $42$
This field is Galois and abelian over $\Q$.
Conductor:  \(539=7^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{539}(384,·)$, $\chi_{539}(1,·)$, $\chi_{539}(386,·)$, $\chi_{539}(131,·)$, $\chi_{539}(516,·)$, $\chi_{539}(10,·)$, $\chi_{539}(395,·)$, $\chi_{539}(144,·)$, $\chi_{539}(529,·)$, $\chi_{539}(23,·)$, $\chi_{539}(408,·)$, $\chi_{539}(153,·)$, $\chi_{539}(538,·)$, $\chi_{539}(155,·)$, $\chi_{539}(285,·)$, $\chi_{539}(164,·)$, $\chi_{539}(298,·)$, $\chi_{539}(177,·)$, $\chi_{539}(307,·)$, $\chi_{539}(309,·)$, $\chi_{539}(54,·)$, $\chi_{539}(439,·)$, $\chi_{539}(318,·)$, $\chi_{539}(67,·)$, $\chi_{539}(452,·)$, $\chi_{539}(331,·)$, $\chi_{539}(76,·)$, $\chi_{539}(461,·)$, $\chi_{539}(78,·)$, $\chi_{539}(463,·)$, $\chi_{539}(208,·)$, $\chi_{539}(87,·)$, $\chi_{539}(472,·)$, $\chi_{539}(221,·)$, $\chi_{539}(100,·)$, $\chi_{539}(485,·)$, $\chi_{539}(230,·)$, $\chi_{539}(232,·)$, $\chi_{539}(362,·)$, $\chi_{539}(241,·)$, $\chi_{539}(375,·)$, $\chi_{539}(254,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{13} a^{21} + \frac{2}{13} a^{19} - \frac{2}{13} a^{17} - \frac{3}{13} a^{15} + \frac{4}{13} a^{14} + \frac{6}{13} a^{13} + \frac{1}{13} a^{12} + \frac{3}{13} a^{10} + \frac{5}{13} a^{8} + \frac{3}{13} a^{7} + \frac{5}{13} a^{6} + \frac{6}{13} a^{5} + \frac{3}{13} a^{4} + \frac{6}{13} a^{3} + \frac{1}{13} a^{2} + \frac{3}{13} a + \frac{1}{13}$, $\frac{1}{13} a^{22} + \frac{2}{13} a^{20} - \frac{2}{13} a^{18} - \frac{3}{13} a^{16} + \frac{4}{13} a^{15} + \frac{6}{13} a^{14} + \frac{1}{13} a^{13} + \frac{3}{13} a^{11} + \frac{5}{13} a^{9} + \frac{3}{13} a^{8} + \frac{5}{13} a^{7} + \frac{6}{13} a^{6} + \frac{3}{13} a^{5} + \frac{6}{13} a^{4} + \frac{1}{13} a^{3} + \frac{3}{13} a^{2} + \frac{1}{13} a$, $\frac{1}{13} a^{23} - \frac{6}{13} a^{19} + \frac{1}{13} a^{17} + \frac{4}{13} a^{16} - \frac{1}{13} a^{15} + \frac{6}{13} a^{14} + \frac{1}{13} a^{13} + \frac{1}{13} a^{12} - \frac{1}{13} a^{10} + \frac{3}{13} a^{9} - \frac{5}{13} a^{8} + \frac{6}{13} a^{6} - \frac{6}{13} a^{5} - \frac{5}{13} a^{4} + \frac{4}{13} a^{3} - \frac{1}{13} a^{2} - \frac{6}{13} a - \frac{2}{13}$, $\frac{1}{13} a^{24} - \frac{6}{13} a^{20} + \frac{1}{13} a^{18} + \frac{4}{13} a^{17} - \frac{1}{13} a^{16} + \frac{6}{13} a^{15} + \frac{1}{13} a^{14} + \frac{1}{13} a^{13} - \frac{1}{13} a^{11} + \frac{3}{13} a^{10} - \frac{5}{13} a^{9} + \frac{6}{13} a^{7} - \frac{6}{13} a^{6} - \frac{5}{13} a^{5} + \frac{4}{13} a^{4} - \frac{1}{13} a^{3} - \frac{6}{13} a^{2} - \frac{2}{13} a$, $\frac{1}{62107231199} a^{25} + \frac{289951305}{62107231199} a^{24} - \frac{75}{62107231199} a^{23} - \frac{1766576668}{62107231199} a^{22} + \frac{2475}{62107231199} a^{21} + \frac{12649851135}{62107231199} a^{20} - \frac{47250}{62107231199} a^{19} + \frac{5876915716}{62107231199} a^{18} + \frac{14333015094}{62107231199} a^{17} + \frac{2791338207}{62107231199} a^{16} + \frac{28660166598}{62107231199} a^{15} - \frac{15875033475}{62107231199} a^{14} + \frac{28690901238}{62107231199} a^{13} + \frac{2192391360}{4777479323} a^{12} + \frac{9458293246}{62107231199} a^{11} + \frac{20417751387}{62107231199} a^{10} + \frac{14566993719}{62107231199} a^{9} - \frac{18658760916}{62107231199} a^{8} - \frac{24239230240}{62107231199} a^{7} - \frac{6719079062}{62107231199} a^{6} - \frac{4481939078}{62107231199} a^{5} + \frac{21381834122}{62107231199} a^{4} - \frac{9670104196}{62107231199} a^{3} + \frac{19607052484}{62107231199} a^{2} - \frac{28651589913}{62107231199} a - \frac{30278083012}{62107231199}$, $\frac{1}{62107231199} a^{26} - \frac{6}{4777479323} a^{24} + \frac{869853915}{62107231199} a^{23} + \frac{207}{4777479323} a^{22} + \frac{2087311064}{62107231199} a^{21} - \frac{4158}{4777479323} a^{20} + \frac{18597816571}{62107231199} a^{19} + \frac{53865}{4777479323} a^{18} - \frac{1421131524}{4777479323} a^{17} - \frac{19116039434}{62107231199} a^{16} - \frac{11849048020}{62107231199} a^{15} - \frac{14295705117}{62107231199} a^{14} + \frac{11281146295}{62107231199} a^{13} + \frac{14181639945}{62107231199} a^{12} - \frac{2014383677}{4777479323} a^{11} + \frac{31899582}{4777479323} a^{10} - \frac{29226138011}{62107231199} a^{9} - \frac{5509293263}{62107231199} a^{8} - \frac{2811711727}{62107231199} a^{7} + \frac{5545883960}{62107231199} a^{6} + \frac{17144149696}{62107231199} a^{5} + \frac{18690787490}{62107231199} a^{4} + \frac{437288975}{4777479323} a^{3} + \frac{23977210144}{62107231199} a^{2} - \frac{558035941}{62107231199} a + \frac{28661687292}{62107231199}$, $\frac{1}{62107231199} a^{27} - \frac{401340910}{62107231199} a^{24} - \frac{243}{4777479323} a^{23} - \frac{1936247996}{62107231199} a^{22} + \frac{10692}{4777479323} a^{21} - \frac{12316890698}{62107231199} a^{20} - \frac{229635}{4777479323} a^{19} + \frac{24284014935}{62107231199} a^{18} + \frac{9593852254}{62107231199} a^{17} + \frac{14776159206}{62107231199} a^{16} + \frac{4446883655}{62107231199} a^{15} - \frac{13491716713}{62107231199} a^{14} + \frac{144551952}{4777479323} a^{13} + \frac{1780366098}{4777479323} a^{12} - \frac{16680165280}{62107231199} a^{11} + \frac{1122731554}{62107231199} a^{10} - \frac{6323862055}{62107231199} a^{9} - \frac{1063869660}{62107231199} a^{8} + \frac{6767737148}{62107231199} a^{7} + \frac{4246270421}{62107231199} a^{6} + \frac{3523092016}{62107231199} a^{5} + \frac{20459972433}{62107231199} a^{4} - \frac{13669018694}{62107231199} a^{3} + \frac{1837390082}{4777479323} a^{2} + \frac{24920517919}{62107231199} a - \frac{124283798}{4777479323}$, $\frac{1}{62107231199} a^{28} - \frac{3402}{62107231199} a^{24} + \frac{1405539015}{62107231199} a^{23} + \frac{156492}{62107231199} a^{22} + \frac{1618600337}{62107231199} a^{21} - \frac{3536379}{62107231199} a^{20} + \frac{17963971099}{62107231199} a^{19} + \frac{48866328}{62107231199} a^{18} + \frac{7341346624}{62107231199} a^{17} - \frac{2239223736}{4777479323} a^{16} - \frac{7478009499}{62107231199} a^{15} + \frac{17078925057}{62107231199} a^{14} + \frac{23353469405}{62107231199} a^{13} + \frac{26709885406}{62107231199} a^{12} + \frac{1864224547}{4777479323} a^{11} - \frac{10842535251}{62107231199} a^{10} + \frac{2924863785}{62107231199} a^{9} + \frac{18986645540}{62107231199} a^{8} - \frac{13680528054}{62107231199} a^{7} + \frac{27492669799}{62107231199} a^{6} + \frac{30741003829}{62107231199} a^{5} + \frac{14260529447}{62107231199} a^{4} + \frac{14110272119}{62107231199} a^{3} - \frac{2323109518}{62107231199} a^{2} - \frac{2141455415}{62107231199} a + \frac{28406595612}{62107231199}$, $\frac{1}{62107231199} a^{29} - \frac{1118341236}{62107231199} a^{24} - \frac{98658}{62107231199} a^{23} + \frac{1793764135}{62107231199} a^{22} + \frac{4883571}{62107231199} a^{21} + \frac{17223790785}{62107231199} a^{20} - \frac{111878172}{62107231199} a^{19} - \frac{26584709560}{62107231199} a^{18} - \frac{3259132703}{62107231199} a^{17} + \frac{24468031852}{62107231199} a^{16} - \frac{18052166915}{62107231199} a^{15} + \frac{30670813293}{62107231199} a^{14} - \frac{8966506392}{62107231199} a^{13} - \frac{2043071259}{4777479323} a^{12} + \frac{9057764528}{62107231199} a^{11} - \frac{19544191353}{62107231199} a^{10} - \frac{19113574485}{62107231199} a^{9} + \frac{1915038384}{62107231199} a^{8} + \frac{1037111684}{62107231199} a^{7} - \frac{24657156416}{62107231199} a^{6} - \frac{1677080729}{4777479323} a^{5} + \frac{22764741811}{62107231199} a^{4} - \frac{2294966132}{62107231199} a^{3} - \frac{11670171219}{62107231199} a^{2} - \frac{21943933798}{62107231199} a + \frac{29858152317}{62107231199}$, $\frac{1}{62107231199} a^{30} - \frac{109620}{62107231199} a^{24} - \frac{864680074}{62107231199} a^{23} + \frac{5672835}{62107231199} a^{22} - \frac{11699648}{4777479323} a^{21} - \frac{136739988}{62107231199} a^{20} + \frac{13710515727}{62107231199} a^{19} + \frac{1968227100}{62107231199} a^{18} - \frac{14190075563}{62107231199} a^{17} - \frac{18437066100}{62107231199} a^{16} - \frac{16011962884}{62107231199} a^{15} + \frac{25381409293}{62107231199} a^{14} - \frac{158187309}{62107231199} a^{13} + \frac{26243725967}{62107231199} a^{12} - \frac{69292206}{4777479323} a^{11} - \frac{20178487395}{62107231199} a^{10} - \frac{711379762}{4777479323} a^{9} + \frac{17167961636}{62107231199} a^{8} - \frac{7912693481}{62107231199} a^{7} + \frac{24371717195}{62107231199} a^{6} - \frac{1811061565}{62107231199} a^{5} + \frac{4876738642}{62107231199} a^{4} + \frac{20303129260}{62107231199} a^{3} - \frac{24183714239}{62107231199} a^{2} - \frac{12969655269}{62107231199} a + \frac{12236084131}{62107231199}$, $\frac{1}{62107231199} a^{31} - \frac{972561893}{62107231199} a^{24} - \frac{2548665}{62107231199} a^{23} - \frac{1939563102}{62107231199} a^{22} + \frac{134569512}{62107231199} a^{21} - \frac{1646442261}{62107231199} a^{20} - \frac{3211317900}{62107231199} a^{19} - \frac{10666076901}{62107231199} a^{18} - \frac{26834813445}{62107231199} a^{17} + \frac{15928964213}{62107231199} a^{16} + \frac{20333846054}{62107231199} a^{15} - \frac{24484314059}{62107231199} a^{14} + \frac{26981956136}{62107231199} a^{13} - \frac{1225752521}{4777479323} a^{12} + \frac{27584296249}{62107231199} a^{11} + \frac{7258470379}{62107231199} a^{10} + \frac{2443122281}{62107231199} a^{9} - \frac{3839228734}{62107231199} a^{8} + \frac{15405360628}{62107231199} a^{7} + \frac{20616785520}{62107231199} a^{6} + \frac{15466064925}{62107231199} a^{5} + \frac{27383884516}{62107231199} a^{4} - \frac{29116013196}{62107231199} a^{3} + \frac{5393371602}{62107231199} a^{2} + \frac{21073909762}{62107231199} a - \frac{10917269681}{62107231199}$, $\frac{1}{62107231199} a^{32} - \frac{2912760}{62107231199} a^{24} + \frac{1557964091}{62107231199} a^{23} + \frac{160784352}{62107231199} a^{22} + \frac{2372143445}{62107231199} a^{21} - \frac{4037085360}{62107231199} a^{20} + \frac{728308006}{4777479323} a^{19} - \frac{2337395999}{62107231199} a^{18} - \frac{5111676755}{62107231199} a^{17} + \frac{11303428306}{62107231199} a^{16} - \frac{22679848486}{62107231199} a^{15} - \frac{6412726501}{62107231199} a^{14} + \frac{27651228313}{62107231199} a^{13} + \frac{12480015645}{62107231199} a^{12} - \frac{1447347366}{4777479323} a^{11} + \frac{12800102719}{62107231199} a^{10} + \frac{2263751683}{4777479323} a^{9} - \frac{7785061043}{62107231199} a^{8} - \frac{12564037769}{62107231199} a^{7} - \frac{11093503809}{62107231199} a^{6} + \frac{26017602622}{62107231199} a^{5} + \frac{26860326906}{62107231199} a^{4} - \frac{27702801596}{62107231199} a^{3} + \frac{11859670164}{62107231199} a^{2} + \frac{8703951141}{62107231199} a + \frac{19095720665}{62107231199}$, $\frac{1}{62107231199} a^{33} + \frac{2103875274}{62107231199} a^{24} - \frac{57672648}{62107231199} a^{23} + \frac{1286372853}{62107231199} a^{22} - \frac{1605483683}{62107231199} a^{21} + \frac{19782381111}{62107231199} a^{20} - \frac{25305802247}{62107231199} a^{19} + \frac{25736273733}{62107231199} a^{18} + \frac{15434800933}{62107231199} a^{17} + \frac{1381516288}{4777479323} a^{16} - \frac{26556685860}{62107231199} a^{15} + \frac{7803577315}{62107231199} a^{14} + \frac{8890946281}{62107231199} a^{13} - \frac{6061734412}{62107231199} a^{12} + \frac{24765937693}{62107231199} a^{11} - \frac{6307853046}{62107231199} a^{10} - \frac{6609276658}{62107231199} a^{9} - \frac{19668261004}{62107231199} a^{8} - \frac{20023399371}{62107231199} a^{7} - \frac{13447210507}{62107231199} a^{6} + \frac{20220100382}{62107231199} a^{5} - \frac{14276262892}{62107231199} a^{4} + \frac{28220802702}{62107231199} a^{3} - \frac{19331958885}{62107231199} a^{2} - \frac{22481930542}{62107231199} a - \frac{28691082851}{62107231199}$, $\frac{1}{62107231199} a^{34} - \frac{67616208}{62107231199} a^{24} + \frac{1420200744}{62107231199} a^{23} - \frac{889547363}{62107231199} a^{22} + \frac{1033622739}{62107231199} a^{21} + \frac{14249434872}{62107231199} a^{20} + \frac{19275737926}{62107231199} a^{19} - \frac{25564553029}{62107231199} a^{18} - \frac{18130210317}{62107231199} a^{17} + \frac{29942317516}{62107231199} a^{16} + \frac{29407722403}{62107231199} a^{15} + \frac{7392310754}{62107231199} a^{14} - \frac{2998724093}{62107231199} a^{13} + \frac{28891971124}{62107231199} a^{12} + \frac{13710115606}{62107231199} a^{11} - \frac{242826122}{4777479323} a^{10} + \frac{24498031331}{62107231199} a^{9} + \frac{14496849319}{62107231199} a^{8} + \frac{27796635758}{62107231199} a^{7} - \frac{28046741680}{62107231199} a^{6} - \frac{24078201275}{62107231199} a^{5} - \frac{25380146837}{62107231199} a^{4} - \frac{12348717366}{62107231199} a^{3} - \frac{164550093}{4777479323} a^{2} - \frac{6160692487}{62107231199} a - \frac{448012059}{4777479323}$, $\frac{1}{62107231199} a^{35} + \frac{386446562}{62107231199} a^{24} - \frac{1183283640}{62107231199} a^{23} - \frac{975745325}{62107231199} a^{22} + \frac{55335398}{62107231199} a^{21} + \frac{6131162774}{62107231199} a^{20} - \frac{9964275973}{62107231199} a^{19} + \frac{2234948015}{62107231199} a^{18} - \frac{12502064655}{62107231199} a^{17} + \frac{24558790115}{62107231199} a^{16} - \frac{6101273339}{62107231199} a^{15} - \frac{10503466797}{62107231199} a^{14} - \frac{4408676234}{62107231199} a^{13} - \frac{28504727143}{62107231199} a^{12} - \frac{24157994610}{62107231199} a^{11} + \frac{3627345024}{62107231199} a^{10} - \frac{22787076427}{62107231199} a^{9} + \frac{20537485079}{62107231199} a^{8} - \frac{19895646291}{62107231199} a^{7} - \frac{4215638409}{62107231199} a^{6} - \frac{7766372830}{62107231199} a^{5} - \frac{11467566857}{62107231199} a^{4} + \frac{16685046447}{62107231199} a^{3} - \frac{1905721781}{62107231199} a^{2} + \frac{1926277482}{62107231199} a + \frac{4202065600}{62107231199}$, $\frac{1}{62107231199} a^{36} - \frac{1419940368}{62107231199} a^{24} - \frac{657129113}{62107231199} a^{23} - \frac{2015297478}{62107231199} a^{22} + \frac{394307101}{62107231199} a^{21} - \frac{1996652990}{4777479323} a^{20} + \frac{26196426624}{62107231199} a^{19} + \frac{6384506098}{62107231199} a^{18} - \frac{28898341079}{62107231199} a^{17} - \frac{6763170347}{62107231199} a^{16} + \frac{2995353247}{62107231199} a^{15} - \frac{24788267943}{62107231199} a^{14} - \frac{5290866510}{62107231199} a^{13} - \frac{10042035037}{62107231199} a^{12} + \frac{25238967392}{62107231199} a^{11} + \frac{2289456468}{4777479323} a^{10} + \frac{11094832946}{62107231199} a^{9} - \frac{20040171025}{62107231199} a^{8} + \frac{17114558775}{62107231199} a^{7} + \frac{19698195417}{62107231199} a^{6} - \frac{2057750764}{62107231199} a^{5} - \frac{629498903}{62107231199} a^{4} + \frac{19060054952}{62107231199} a^{3} - \frac{755225240}{4777479323} a^{2} - \frac{15673236934}{62107231199} a - \frac{27777410761}{62107231199}$, $\frac{1}{62107231199} a^{37} + \frac{1521429680}{62107231199} a^{24} + \frac{1371199351}{62107231199} a^{23} + \frac{1194891664}{62107231199} a^{22} + \frac{836016140}{62107231199} a^{21} + \frac{3366752551}{62107231199} a^{20} - \frac{5210707659}{62107231199} a^{19} + \frac{26126596699}{62107231199} a^{18} - \frac{1097040091}{4777479323} a^{17} - \frac{27170034956}{62107231199} a^{16} + \frac{16715060068}{62107231199} a^{15} - \frac{14788604281}{62107231199} a^{14} + \frac{12043748218}{62107231199} a^{13} - \frac{29991328135}{62107231199} a^{12} + \frac{4377279681}{62107231199} a^{11} - \frac{11548878400}{62107231199} a^{10} + \frac{4391393896}{62107231199} a^{9} + \frac{17551732852}{62107231199} a^{8} - \frac{2222526944}{4777479323} a^{7} - \frac{24472028294}{62107231199} a^{6} - \frac{27271028565}{62107231199} a^{5} + \frac{1052063002}{4777479323} a^{4} - \frac{30150478445}{62107231199} a^{3} + \frac{2018958047}{62107231199} a^{2} - \frac{8342671222}{62107231199} a + \frac{21358667078}{62107231199}$, $\frac{1}{62107231199} a^{38} + \frac{81875082}{4777479323} a^{24} + \frac{642613912}{62107231199} a^{23} - \frac{960103955}{62107231199} a^{22} - \frac{2295478248}{62107231199} a^{21} - \frac{3573936881}{62107231199} a^{20} + \frac{235400531}{4777479323} a^{19} + \frac{3440517081}{62107231199} a^{18} - \frac{684454848}{62107231199} a^{17} - \frac{1049631279}{62107231199} a^{16} + \frac{24015358713}{62107231199} a^{15} + \frac{7883062306}{62107231199} a^{14} - \frac{12097918657}{62107231199} a^{13} + \frac{27004351566}{62107231199} a^{12} - \frac{1590797056}{4777479323} a^{11} - \frac{28791334311}{62107231199} a^{10} + \frac{84033013}{62107231199} a^{9} - \frac{26603872741}{62107231199} a^{8} + \frac{4118261126}{62107231199} a^{7} + \frac{24768395174}{62107231199} a^{6} + \frac{6859901476}{62107231199} a^{5} - \frac{2520908076}{62107231199} a^{4} + \frac{875060003}{4777479323} a^{3} + \frac{6885173094}{62107231199} a^{2} + \frac{22484024228}{62107231199} a - \frac{10985698713}{62107231199}$, $\frac{1}{62107231199} a^{39} + \frac{167792279}{62107231199} a^{24} - \frac{2349047496}{62107231199} a^{23} - \frac{1550675933}{62107231199} a^{22} - \frac{736113935}{62107231199} a^{21} - \frac{22789805681}{62107231199} a^{20} - \frac{25980073578}{62107231199} a^{19} - \frac{10897753691}{62107231199} a^{18} + \frac{708048447}{4777479323} a^{17} + \frac{24284156506}{62107231199} a^{16} + \frac{14024903084}{62107231199} a^{15} - \frac{20672726267}{62107231199} a^{14} + \frac{10412393229}{62107231199} a^{13} + \frac{25324209702}{62107231199} a^{12} + \frac{3488566528}{62107231199} a^{11} - \frac{6189651038}{62107231199} a^{10} + \frac{1034848048}{4777479323} a^{9} - \frac{23182112710}{62107231199} a^{8} - \frac{18595773243}{62107231199} a^{7} - \frac{30509991196}{62107231199} a^{6} + \frac{17852662869}{62107231199} a^{5} + \frac{21066844502}{62107231199} a^{4} - \frac{12344805674}{62107231199} a^{3} - \frac{1704908359}{62107231199} a^{2} + \frac{1898115404}{4777479323} a + \frac{8771276091}{62107231199}$, $\frac{1}{62107231199} a^{40} + \frac{1841169953}{62107231199} a^{24} + \frac{1478786346}{62107231199} a^{23} - \frac{1450572914}{62107231199} a^{22} + \frac{1452401510}{62107231199} a^{21} + \frac{18024041214}{62107231199} a^{20} - \frac{27660685090}{62107231199} a^{19} + \frac{22381192205}{62107231199} a^{18} - \frac{1993860097}{62107231199} a^{17} + \frac{1226312455}{4777479323} a^{16} + \frac{3803311331}{62107231199} a^{15} + \frac{27816024932}{62107231199} a^{14} - \frac{26206492047}{62107231199} a^{13} - \frac{3013896406}{62107231199} a^{12} + \frac{1618392903}{62107231199} a^{11} + \frac{9477760034}{62107231199} a^{10} + \frac{2204096151}{62107231199} a^{9} + \frac{27561496399}{62107231199} a^{8} - \frac{27980730242}{62107231199} a^{7} + \frac{18166921797}{62107231199} a^{6} - \frac{22526255223}{62107231199} a^{5} - \frac{24149279055}{62107231199} a^{4} + \frac{78345636}{62107231199} a^{3} + \frac{26110444258}{62107231199} a^{2} - \frac{1908188655}{62107231199} a - \frac{12401610918}{62107231199}$, $\frac{1}{62107231199} a^{41} - \frac{781808792}{62107231199} a^{24} - \frac{146902062}{4777479323} a^{23} + \frac{1780954464}{62107231199} a^{22} - \frac{266235611}{62107231199} a^{21} - \frac{14122706526}{62107231199} a^{20} - \frac{23234314282}{62107231199} a^{19} - \frac{15372807883}{62107231199} a^{18} + \frac{4241603189}{62107231199} a^{17} + \frac{16629971113}{62107231199} a^{16} - \frac{29205371854}{62107231199} a^{15} - \frac{2337226753}{4777479323} a^{14} + \frac{29730176373}{62107231199} a^{13} + \frac{6281990375}{62107231199} a^{12} - \frac{9086027749}{62107231199} a^{11} - \frac{23954026702}{62107231199} a^{10} - \frac{24700031221}{62107231199} a^{9} - \frac{20055696278}{62107231199} a^{8} + \frac{5787106090}{62107231199} a^{7} + \frac{105564167}{4777479323} a^{6} - \frac{6870663636}{62107231199} a^{5} - \frac{28934974183}{62107231199} a^{4} + \frac{1819728531}{62107231199} a^{3} - \frac{5629088397}{62107231199} a^{2} + \frac{26967255715}{62107231199} a - \frac{3428072003}{62107231199}$

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

Class group and class number

Not computed

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

Unit group

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

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{77}) \), \(\Q(\zeta_{7})^+\), 6.6.22370117.1, 7.7.13841287201.1, 14.14.26133633514125646560024046997.1, \(\Q(\zeta_{49})^+\)

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 $42$ $42$ $42$ R R ${\href{/LocalNumberField/13.7.0.1}{7} }^{6}$ $21^{2}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{14}$ $21^{2}$ ${\href{/LocalNumberField/29.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{7}$ $21^{2}$ ${\href{/LocalNumberField/41.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/43.14.0.1}{14} }^{3}$ $42$ $21^{2}$ $42$

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

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
7Data not computed
11Data not computed