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Properties

Label 42.42.874...677.1
Degree $42$
Signature $[42, 0]$
Discriminant $8.746\times 10^{86}$
Root discriminant \(117.50\)
Ramified primes $7,11$
Class number not computed
Class group not computed
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

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

\( x^{42} - 126 x^{40} + 7371 x^{38} - 265734 x^{36} - 83 x^{35} + 6608385 x^{34} + 8715 x^{33} + \cdots - 11523307067 \) Copy content Toggle raw display

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

Invariants

Degree:  $42$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[42, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(874\!\cdots\!677\) \(\medspace = 7^{77}\cdot 11^{21}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(117.50\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $7^{11/6}11^{1/2}\approx 117.50132103051723$
Ramified primes:   \(7\), \(11\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{77}) \)
$\card{ \Gal(K/\Q) }$:  $42$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
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 \(a\))

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

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

not computed

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

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $41$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:  not computed
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  not computed
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr $ not computed \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
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)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
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);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(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);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(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);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A cyclic group of order 42
The 42 conjugacy class representatives for $C_{42}$
Character table for $C_{42}$

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.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

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

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

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

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(7\) Copy content Toggle raw display Deg $42$$42$$1$$77$
\(11\) Copy content Toggle raw display Deg $42$$2$$21$$21$