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Properties

Label 42.42.565...149.1
Degree $42$
Signature $[42, 0]$
Discriminant $5.653\times 10^{83}$
Root discriminant $98.65$
Ramified primes $3, 29$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{42}$ (as 42T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^42 - 81*x^40 - 4*x^39 + 2925*x^38 + 276*x^37 - 62221*x^36 - 8388*x^35 + 868038*x^34 + 148164*x^33 - 8369604*x^32 - 1690536*x^31 + 57288192*x^30 + 13101552*x^29 - 281708427*x^28 - 70649728*x^27 + 997233012*x^26 + 267447468*x^25 - 2528669823*x^24 - 709905600*x^23 + 4548805662*x^22 + 1311105276*x^21 - 5738345406*x^20 - 1666490856*x^19 + 5023839240*x^18 + 1442807064*x^17 - 3028872840*x^16 - 846440256*x^15 + 1246931667*x^14 + 334448784*x^13 - 345422658*x^12 - 87695253*x^11 + 62698236*x^10 + 14793089*x^9 - 7127757*x^8 - 1516980*x^7 + 472108*x^6 + 85806*x^5 - 16338*x^4 - 2275*x^3 + 252*x^2 + 21*x - 1)
 
gp: K = bnfinit(x^42 - 81*x^40 - 4*x^39 + 2925*x^38 + 276*x^37 - 62221*x^36 - 8388*x^35 + 868038*x^34 + 148164*x^33 - 8369604*x^32 - 1690536*x^31 + 57288192*x^30 + 13101552*x^29 - 281708427*x^28 - 70649728*x^27 + 997233012*x^26 + 267447468*x^25 - 2528669823*x^24 - 709905600*x^23 + 4548805662*x^22 + 1311105276*x^21 - 5738345406*x^20 - 1666490856*x^19 + 5023839240*x^18 + 1442807064*x^17 - 3028872840*x^16 - 846440256*x^15 + 1246931667*x^14 + 334448784*x^13 - 345422658*x^12 - 87695253*x^11 + 62698236*x^10 + 14793089*x^9 - 7127757*x^8 - 1516980*x^7 + 472108*x^6 + 85806*x^5 - 16338*x^4 - 2275*x^3 + 252*x^2 + 21*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 21, 252, -2275, -16338, 85806, 472108, -1516980, -7127757, 14793089, 62698236, -87695253, -345422658, 334448784, 1246931667, -846440256, -3028872840, 1442807064, 5023839240, -1666490856, -5738345406, 1311105276, 4548805662, -709905600, -2528669823, 267447468, 997233012, -70649728, -281708427, 13101552, 57288192, -1690536, -8369604, 148164, 868038, -8388, -62221, 276, 2925, -4, -81, 0, 1]);
 

\( x^{42} - 81 x^{40} - 4 x^{39} + 2925 x^{38} + 276 x^{37} - 62221 x^{36} - 8388 x^{35} + 868038 x^{34} + 148164 x^{33} - 8369604 x^{32} - 1690536 x^{31} + 57288192 x^{30} + 13101552 x^{29} - 281708427 x^{28} - 70649728 x^{27} + 997233012 x^{26} + 267447468 x^{25} - 2528669823 x^{24} - 709905600 x^{23} + 4548805662 x^{22} + 1311105276 x^{21} - 5738345406 x^{20} - 1666490856 x^{19} + 5023839240 x^{18} + 1442807064 x^{17} - 3028872840 x^{16} - 846440256 x^{15} + 1246931667 x^{14} + 334448784 x^{13} - 345422658 x^{12} - 87695253 x^{11} + 62698236 x^{10} + 14793089 x^{9} - 7127757 x^{8} - 1516980 x^{7} + 472108 x^{6} + 85806 x^{5} - 16338 x^{4} - 2275 x^{3} + 252 x^{2} + 21 x - 1 \)

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:  \(565\!\cdots\!149\)\(\medspace = 3^{56}\cdot 29^{39}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $98.65$
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:  $3, 29$
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:  \(261=3^{2}\cdot 29\)
Dirichlet character group:    $\lbrace$$\chi_{261}(256,·)$, $\chi_{261}(1,·)$, $\chi_{261}(4,·)$, $\chi_{261}(7,·)$, $\chi_{261}(136,·)$, $\chi_{261}(139,·)$, $\chi_{261}(13,·)$, $\chi_{261}(16,·)$, $\chi_{261}(22,·)$, $\chi_{261}(151,·)$, $\chi_{261}(25,·)$, $\chi_{261}(154,·)$, $\chi_{261}(28,·)$, $\chi_{261}(34,·)$, $\chi_{261}(169,·)$, $\chi_{261}(175,·)$, $\chi_{261}(49,·)$, $\chi_{261}(178,·)$, $\chi_{261}(52,·)$, $\chi_{261}(181,·)$, $\chi_{261}(187,·)$, $\chi_{261}(190,·)$, $\chi_{261}(64,·)$, $\chi_{261}(67,·)$, $\chi_{261}(196,·)$, $\chi_{261}(199,·)$, $\chi_{261}(202,·)$, $\chi_{261}(208,·)$, $\chi_{261}(82,·)$, $\chi_{261}(88,·)$, $\chi_{261}(91,·)$, $\chi_{261}(94,·)$, $\chi_{261}(223,·)$, $\chi_{261}(226,·)$, $\chi_{261}(100,·)$, $\chi_{261}(103,·)$, $\chi_{261}(109,·)$, $\chi_{261}(238,·)$, $\chi_{261}(112,·)$, $\chi_{261}(241,·)$, $\chi_{261}(115,·)$, $\chi_{261}(121,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $a^{27}$, $a^{28}$, $a^{29}$, $a^{30}$, $a^{31}$, $a^{32}$, $a^{33}$, $a^{34}$, $a^{35}$, $a^{36}$, $a^{37}$, $a^{38}$, $\frac{1}{233} a^{39} - \frac{37}{233} a^{38} + \frac{13}{233} a^{37} - \frac{58}{233} a^{36} + \frac{30}{233} a^{35} - \frac{106}{233} a^{34} + \frac{109}{233} a^{33} - \frac{75}{233} a^{32} + \frac{62}{233} a^{31} + \frac{52}{233} a^{30} + \frac{15}{233} a^{29} + \frac{39}{233} a^{28} - \frac{55}{233} a^{27} - \frac{86}{233} a^{26} - \frac{45}{233} a^{25} - \frac{8}{233} a^{24} + \frac{35}{233} a^{23} + \frac{3}{233} a^{22} - \frac{24}{233} a^{21} + \frac{3}{233} a^{20} + \frac{7}{233} a^{19} - \frac{3}{233} a^{18} + \frac{92}{233} a^{17} + \frac{21}{233} a^{16} - \frac{14}{233} a^{15} - \frac{31}{233} a^{14} - \frac{13}{233} a^{13} - \frac{48}{233} a^{12} - \frac{115}{233} a^{11} - \frac{73}{233} a^{10} - \frac{71}{233} a^{9} + \frac{72}{233} a^{8} - \frac{94}{233} a^{7} - \frac{73}{233} a^{6} + \frac{1}{233} a^{5} + \frac{86}{233} a^{4} + \frac{106}{233} a^{3} + \frac{46}{233} a^{2} - \frac{22}{233} a - \frac{74}{233}$, $\frac{1}{3961} a^{40} - \frac{7}{3961} a^{39} + \frac{1699}{3961} a^{38} - \frac{367}{3961} a^{37} - \frac{79}{3961} a^{36} + \frac{1726}{3961} a^{35} + \frac{1822}{3961} a^{34} + \frac{166}{3961} a^{33} + \frac{1074}{3961} a^{32} - \frac{52}{233} a^{31} + \frac{643}{3961} a^{30} - \frac{443}{3961} a^{29} + \frac{1115}{3961} a^{28} - \frac{1503}{3961} a^{27} + \frac{870}{3961} a^{26} - \frac{193}{3961} a^{25} + \frac{261}{3961} a^{24} - \frac{1044}{3961} a^{23} + \frac{1464}{3961} a^{22} + \frac{1147}{3961} a^{21} - \frac{8}{233} a^{20} - \frac{725}{3961} a^{19} + \frac{1866}{3961} a^{18} - \frac{481}{3961} a^{17} + \frac{1548}{3961} a^{16} - \frac{1849}{3961} a^{15} + \frac{1387}{3961} a^{14} + \frac{1426}{3961} a^{13} - \frac{856}{3961} a^{12} - \frac{1892}{3961} a^{11} - \frac{863}{3961} a^{10} + \frac{1903}{3961} a^{9} - \frac{1662}{3961} a^{8} + \frac{1534}{3961} a^{7} + \frac{1306}{3961} a^{6} - \frac{350}{3961} a^{5} - \frac{1042}{3961} a^{4} - \frac{1900}{3961} a^{3} + \frac{1125}{3961} a^{2} - \frac{268}{3961} a - \frac{822}{3961}$, $\frac{1}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{41} + \frac{72547867839210821541162495879574118608870492393644538866146546651867489959624049651989620431276533685099317931489926}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{40} - \frac{216356684884737413289424024516383982949140914271912914463471928470064761767054558715801838348556772541530667481573707}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{39} + \frac{144450680688745871204116692506674608386805587545433145049165239103769690349592506340103926164982278659908925657962180539}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{38} - \frac{178875423981496271642402616805879712417528844367611226615197913107768573702141046088379404568632495557257100441595807498}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{37} - \frac{139057497365369040258518699380604702788155370469313546134094175840186102887182481926990100190808819253598901952487171680}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{36} + \frac{198337046425646485878704426069557026461114214861034841499830902079108363555046062920028424468495892544103597984359518398}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{35} + \frac{170576904297327458098360725905527846802587125614945665198945719594432800081812513214654909548475250261071082370732474566}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{34} - \frac{23039722514423700583917622756423403511448874401501246060398199313783260580492752783032319420357568478612900390670850133}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{33} - \frac{201473410232017186309935491200641857924979536683220932559938623772030612592614368433304122034725570268401790544894777755}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{32} + \frac{62553357378990726058442132745396551989162175815820190894882588710285754070077512159186797944001822762652605316088880872}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{31} + \frac{9369805994548377190885736551246848387189210121010241026542875875680845409380302039164825833988014336840959699692614633}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{30} + \frac{146031335203469432789615626568633433135595707749730810413681663307717915653988528979003161495548738545071654842476040791}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{29} - \frac{137147589652084240292543383727267761506407858736341923161536446608214892777923891768481081113207206869385634561791188115}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{28} + \frac{7947411382518262852612125207134800363761526990376948971468976420879833943531078965427499133978022786136997663040177053}{33891935580068308307264970363016594794446441050830516453829392287372590632088966660266542080684360269137798825633238799} a^{27} - \frac{111543984402655609798209078417402251775792705681841580449118526060129055914233649228987444390375644476517626502487102720}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{26} + \frac{23871596830441389474931476649222854625884748705901901319519766979643521605741800484653579662462009852234336888555018990}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{25} + \frac{49952297625097013495740230218186881264771131374586739032662974702807131093416720582353758836890161154626143056346882726}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{24} - \frac{12954841478634522894185075009250127605134154888094486774717898712271420134647212912422590811938108117254162177745580751}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{23} + \frac{137098846046718435951712081047272389100345575200861103507460011229371766549153209757809904084808669312472866475499447031}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{22} - \frac{227694929890258363116706079435054778690436467415775620072061855030837772131172397515046088522697136810979309396339061668}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{21} + \frac{250507988409343427401045825890788858030820674636537508963946553614961833695886052948523445388581279169137698634524528460}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{20} + \frac{202819755675866950648907171992053637328658163070725598961781650923362003923722148959399551545306504351702430593474594288}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{19} - \frac{43861828928093975970309265583963371182092923563596628145508702774732736465110851335745014216413898241480936954953581099}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{18} - \frac{241442663775860915900885265661223573551918048026567552110187778384500465030828730757742565960045230320228064489049562804}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{17} + \frac{71077271118153467388211826868860790596988894238029994017969886729900450634994376542987585248934332270674664890337506960}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{16} - \frac{29334114290140034374336314134962631467388859306943248684554440408543294531194422136982164254303193114523062768332575325}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{15} - \frac{233092765158011959499134836445121676908562236111570626434014354301174692451838557775198479962178418390305983736834362961}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{14} - \frac{236423021353031771575389700358586990969027275273350101794858925918738688121578483755625871849664685434666183096004898040}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{13} - \frac{7711802707876825264236483587659099277930820978701269067932771676837333549288400483505927585957144824351226093372617175}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{12} + \frac{205375891453647418554715885479403249415594238735971907868738886792085144508428026857125875067886675175778014426767407111}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{11} - \frac{272111775310877742655637852304150027362848187807513584329469752412335230064909588737268584738991469443375334459408882819}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{10} - \frac{219735356591816270747157758853153997288522827862385189235811794940545623675086050949031648582332565063863455170761983510}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{9} + \frac{94381862914799675675443923696084980797203147504886869513590583556032522866682815192176771749312158482234202696953900030}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{8} + \frac{156598765368328417959487882029888856465309671257444214513044288260172426331114718498446144035716932153834408094967035928}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{7} - \frac{106636393342829275923410392342200207144911217769224699307795734688609792959742089200401297343155722521501670151850696503}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{6} - \frac{128525436594075968601297793109160441026862937505506198887150596590932025002893687650853469376816505600214517906632691404}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{5} + \frac{261370472030198985417309186184389758653078166308798903431549751573611343341092152620359974431450004016338120187087731684}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{4} + \frac{104973379845238432177946616004002489571344371846599529954713282982170297854593786935756045898533934160380754507889993473}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{3} + \frac{114829475446430554839442716706135441782868131600531063918996575773594918297553489833912135889527992867959092076240423043}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a^{2} + \frac{116424211212625858999866795146349928453984622217345856118565533598997796004905245941156825795849717918963822289675067349}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583} a + \frac{280419571379546274881748408629194777573249899214871595659340598313551896938387280265346245996982067009332384210768490247}{576162904861161241223504496171282111505589497864118779715099668885334040745512433224531215371634124575342580035765059583}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

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:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 43619585644836630000000000000 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{42}\cdot(2\pi)^{0}\cdot 43619585644836630000000000000 \cdot 1}{2\sqrt{565343212441678035532894502003808167878401992443661947648452445739810658542578516149}}\approx 0.127571971394323$ (assuming GRH)

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

\(\Q(\sqrt{29}) \), \(\Q(\zeta_{9})^+\), 6.6.160016229.1, 7.7.594823321.1, \(\Q(\zeta_{29})^+\), 21.21.4814587615056751193058435502319478353721.1

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 $42$ R $21^{2}$ $21^{2}$ $42$ $21^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{21}$ ${\href{/LocalNumberField/19.14.0.1}{14} }^{3}$ $21^{2}$ R $42$ ${\href{/LocalNumberField/37.14.0.1}{14} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{7}$ $42$ $42$ ${\href{/LocalNumberField/53.7.0.1}{7} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{14}$

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
3Data not computed
29Data not computed