Properties

Label 29.29.7839491297...4321.1
Degree $29$
Signature $[29, 0]$
Discriminant $29^{56}$
Root discriminant $666.72$
Ramified prime $29$
Class number Not computed
Class group Not computed
Galois group $C_{29}$ (as 29T1)

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Show commands for: SageMath / Pari/GP / Magma

sage: x = polygen(QQ); K.<a> = NumberField(x^29 - 406*x^27 - 261*x^26 + 60784*x^25 - 10237*x^24 - 4881280*x^23 + 6951010*x^22 + 231791925*x^21 - 664329796*x^20 - 6378435885*x^19 + 29829836682*x^18 + 83025301615*x^17 - 713976297106*x^16 + 200377868604*x^15 + 8552568793352*x^14 - 19940770675013*x^13 - 30466827556936*x^12 + 199293371073354*x^11 - 244879309451696*x^10 - 341168817490702*x^9 + 1410119390706929*x^8 - 1755729397271788*x^7 + 798559330062447*x^6 + 263826444292294*x^5 - 372309027818008*x^4 + 45256457625802*x^3 + 48917493171904*x^2 - 7142036935967*x - 2480435158303)
 
gp: K = bnfinit(x^29 - 406*x^27 - 261*x^26 + 60784*x^25 - 10237*x^24 - 4881280*x^23 + 6951010*x^22 + 231791925*x^21 - 664329796*x^20 - 6378435885*x^19 + 29829836682*x^18 + 83025301615*x^17 - 713976297106*x^16 + 200377868604*x^15 + 8552568793352*x^14 - 19940770675013*x^13 - 30466827556936*x^12 + 199293371073354*x^11 - 244879309451696*x^10 - 341168817490702*x^9 + 1410119390706929*x^8 - 1755729397271788*x^7 + 798559330062447*x^6 + 263826444292294*x^5 - 372309027818008*x^4 + 45256457625802*x^3 + 48917493171904*x^2 - 7142036935967*x - 2480435158303, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2480435158303, -7142036935967, 48917493171904, 45256457625802, -372309027818008, 263826444292294, 798559330062447, -1755729397271788, 1410119390706929, -341168817490702, -244879309451696, 199293371073354, -30466827556936, -19940770675013, 8552568793352, 200377868604, -713976297106, 83025301615, 29829836682, -6378435885, -664329796, 231791925, 6951010, -4881280, -10237, 60784, -261, -406, 0, 1]);
 

Normalized defining polynomial

\( x^{29} - 406 x^{27} - 261 x^{26} + 60784 x^{25} - 10237 x^{24} - 4881280 x^{23} + 6951010 x^{22} + 231791925 x^{21} - 664329796 x^{20} - 6378435885 x^{19} + 29829836682 x^{18} + 83025301615 x^{17} - 713976297106 x^{16} + 200377868604 x^{15} + 8552568793352 x^{14} - 19940770675013 x^{13} - 30466827556936 x^{12} + 199293371073354 x^{11} - 244879309451696 x^{10} - 341168817490702 x^{9} + 1410119390706929 x^{8} - 1755729397271788 x^{7} + 798559330062447 x^{6} + 263826444292294 x^{5} - 372309027818008 x^{4} + 45256457625802 x^{3} + 48917493171904 x^{2} - 7142036935967 x - 2480435158303 \)

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

Invariants

Degree:  $29$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[29, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(7839491297426657080705875253942679356383134413463412253884527413516798034573044321=29^{56}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $666.72$
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:  $29$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $29$
This field is Galois and abelian over $\Q$.
Conductor:  \(841=29^{2}\)
Dirichlet character group:    $\lbrace$$\chi_{841}(320,·)$, $\chi_{841}(1,·)$, $\chi_{841}(581,·)$, $\chi_{841}(262,·)$, $\chi_{841}(523,·)$, $\chi_{841}(204,·)$, $\chi_{841}(784,·)$, $\chi_{841}(465,·)$, $\chi_{841}(146,·)$, $\chi_{841}(726,·)$, $\chi_{841}(407,·)$, $\chi_{841}(88,·)$, $\chi_{841}(668,·)$, $\chi_{841}(349,·)$, $\chi_{841}(30,·)$, $\chi_{841}(610,·)$, $\chi_{841}(291,·)$, $\chi_{841}(552,·)$, $\chi_{841}(233,·)$, $\chi_{841}(813,·)$, $\chi_{841}(494,·)$, $\chi_{841}(175,·)$, $\chi_{841}(755,·)$, $\chi_{841}(436,·)$, $\chi_{841}(117,·)$, $\chi_{841}(697,·)$, $\chi_{841}(378,·)$, $\chi_{841}(59,·)$, $\chi_{841}(639,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{41} a^{21} + \frac{1}{41} a^{20} + \frac{20}{41} a^{19} + \frac{2}{41} a^{18} - \frac{16}{41} a^{17} + \frac{9}{41} a^{16} + \frac{18}{41} a^{15} - \frac{1}{41} a^{14} - \frac{7}{41} a^{13} + \frac{2}{41} a^{12} - \frac{7}{41} a^{11} + \frac{8}{41} a^{10} - \frac{3}{41} a^{9} + \frac{20}{41} a^{8} + \frac{8}{41} a^{7} + \frac{17}{41} a^{6} - \frac{16}{41} a^{5} - \frac{10}{41} a^{4} + \frac{12}{41} a^{3} - \frac{4}{41} a^{2} + \frac{9}{41} a + \frac{4}{41}$, $\frac{1}{41} a^{22} + \frac{19}{41} a^{20} - \frac{18}{41} a^{19} - \frac{18}{41} a^{18} - \frac{16}{41} a^{17} + \frac{9}{41} a^{16} - \frac{19}{41} a^{15} - \frac{6}{41} a^{14} + \frac{9}{41} a^{13} - \frac{9}{41} a^{12} + \frac{15}{41} a^{11} - \frac{11}{41} a^{10} - \frac{18}{41} a^{9} - \frac{12}{41} a^{8} + \frac{9}{41} a^{7} + \frac{8}{41} a^{6} + \frac{6}{41} a^{5} - \frac{19}{41} a^{4} - \frac{16}{41} a^{3} + \frac{13}{41} a^{2} - \frac{5}{41} a - \frac{4}{41}$, $\frac{1}{41} a^{23} + \frac{4}{41} a^{20} + \frac{12}{41} a^{19} - \frac{13}{41} a^{18} - \frac{15}{41} a^{17} + \frac{15}{41} a^{16} - \frac{20}{41} a^{15} - \frac{13}{41} a^{14} + \frac{1}{41} a^{13} + \frac{18}{41} a^{12} - \frac{1}{41} a^{11} - \frac{6}{41} a^{10} + \frac{4}{41} a^{9} - \frac{2}{41} a^{8} + \frac{20}{41} a^{7} + \frac{11}{41} a^{6} - \frac{2}{41} a^{5} + \frac{10}{41} a^{4} - \frac{10}{41} a^{3} - \frac{11}{41} a^{2} - \frac{11}{41} a + \frac{6}{41}$, $\frac{1}{41} a^{24} + \frac{8}{41} a^{20} - \frac{11}{41} a^{19} + \frac{18}{41} a^{18} - \frac{3}{41} a^{17} - \frac{15}{41} a^{16} - \frac{3}{41} a^{15} + \frac{5}{41} a^{14} + \frac{5}{41} a^{13} - \frac{9}{41} a^{12} - \frac{19}{41} a^{11} + \frac{13}{41} a^{10} + \frac{10}{41} a^{9} - \frac{19}{41} a^{8} + \frac{20}{41} a^{7} + \frac{12}{41} a^{6} - \frac{8}{41} a^{5} - \frac{11}{41} a^{4} - \frac{18}{41} a^{3} + \frac{5}{41} a^{2} + \frac{11}{41} a - \frac{16}{41}$, $\frac{1}{41} a^{25} - \frac{19}{41} a^{20} - \frac{19}{41} a^{19} - \frac{19}{41} a^{18} - \frac{10}{41} a^{17} + \frac{7}{41} a^{16} - \frac{16}{41} a^{15} + \frac{13}{41} a^{14} + \frac{6}{41} a^{13} + \frac{6}{41} a^{12} - \frac{13}{41} a^{11} - \frac{13}{41} a^{10} + \frac{5}{41} a^{9} - \frac{17}{41} a^{8} - \frac{11}{41} a^{7} + \frac{20}{41} a^{6} - \frac{6}{41} a^{5} - \frac{20}{41} a^{4} - \frac{9}{41} a^{3} + \frac{2}{41} a^{2} - \frac{6}{41} a + \frac{9}{41}$, $\frac{1}{41} a^{26} - \frac{8}{41} a^{19} - \frac{13}{41} a^{18} - \frac{10}{41} a^{17} - \frac{9}{41} a^{16} - \frac{14}{41} a^{15} - \frac{13}{41} a^{14} - \frac{4}{41} a^{13} - \frac{16}{41} a^{12} + \frac{18}{41} a^{11} - \frac{7}{41} a^{10} + \frac{8}{41} a^{9} + \frac{8}{41} a^{7} - \frac{11}{41} a^{6} + \frac{4}{41} a^{5} + \frac{6}{41} a^{4} - \frac{16}{41} a^{3} + \frac{16}{41} a - \frac{6}{41}$, $\frac{1}{230297} a^{27} + \frac{1480}{230297} a^{26} + \frac{1584}{230297} a^{25} + \frac{1425}{230297} a^{24} + \frac{1860}{230297} a^{23} + \frac{2596}{230297} a^{22} + \frac{650}{230297} a^{21} - \frac{6882}{230297} a^{20} - \frac{85760}{230297} a^{19} + \frac{96649}{230297} a^{18} + \frac{93520}{230297} a^{17} - \frac{46949}{230297} a^{16} - \frac{33746}{230297} a^{15} - \frac{17952}{230297} a^{14} + \frac{108447}{230297} a^{13} - \frac{58535}{230297} a^{12} - \frac{100844}{230297} a^{11} + \frac{58312}{230297} a^{10} + \frac{86990}{230297} a^{9} - \frac{427}{230297} a^{8} - \frac{13708}{230297} a^{7} + \frac{76459}{230297} a^{6} + \frac{41008}{230297} a^{5} + \frac{32566}{230297} a^{4} + \frac{106298}{230297} a^{3} + \frac{36249}{230297} a^{2} - \frac{2403}{5617} a - \frac{529}{1681}$, $\frac{1}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{28} - \frac{2840966059402013812381563179754408838004847572876897525739722695438483119413736614915470839616251774223280904125540036906534738243539011343084072169492426564572034790}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{27} - \frac{11298489965236557797816206293303387961212785608866995773317805078470525832538651199372767639736693890176381959897744576937739114950898610793010041231682983785673703592958}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{26} + \frac{1643996817656586554819035902580479597593256319743260098512514546319604086748594005856409113274378629350185765031717332217774594859816402197336571854025936439386869785681}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{25} - \frac{450325183544140926493210442934693116363180779384316142735513798482522023754040239812250247741733747360525596303971409387064697848089291996826180499550223687610152719082}{59856228871717397886640088292858872319375082290162699569715301956468265892720631550068154757670373667960706566245441514170727002278002746476894548623686987011057071234873} a^{24} - \frac{6699122257770638151590666535927223809712148980514868004278673695175576576021786836383748648933117639351853167614728704474038737550008134993403051328548718839295792621041}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{23} - \frac{22098975240968874813442996325316429062341721977988787043560464073392153150773791063045359352497779557093103877417549890380702758326173980834601266618888881421653235108925}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{22} - \frac{5694251669277787143302772254767990578738966917905113897361042150856237986019185597262992046943788999620196774362906366762292128145185677560571074912011729449933754774368}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{21} - \frac{986975402932910474294030676904603183392449552475734423852346700664594616709746770250839463630726040029140074131301415771469276996839987402443718611401453732121808056722301}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{20} - \frac{227261942756856963504221393289362016625900944028293226414121130834294914337327918014931400537925597636396112825157961243712816840228848621665298822255633985112200027369028}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{19} + \frac{406783468888664755037853392516538631456187669659811317178570865797593646905857577538774740804035057512892234958977932648984194586166740691860122829926576511073381045767641}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{18} + \frac{624471950433051530608946759895684041845776628278338618135920676515877111486735850718237407342751010551998201847953753974129199888633755696147592400583036554984821491740619}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{17} - \frac{124098734220471036156183733141977242572773298918403461664528839982827853300081139021055987586135858631781056446385588624777096169047542413655910701889694577376449994476480}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{16} + \frac{637819225345655535298103849222339322111455748464685946502010337192201862597779061118074585492545665656902937603898477746149595559726744998573876733980828622552677127785078}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{15} + \frac{594805371886850427112193509562916460169640908980616702274451116174307700575484436603711893117022091934895123204608863944987822260857592717851072503097591886450573194700379}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{14} + \frac{191249130411698040584958338761223273915418141286754414976486453606405409351458655812905206749405463162730677838943979886151336432283735914932588818391618674468145394644918}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{13} - \frac{467325395437880390941705965649428623409208411180624945699952436153768447556300716232886233876877201611052065210358773661494757263947019696426425994379519970152241466812753}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{12} + \frac{793102606559039506457055318157007352088166730064602259343397586996985429689004469249067773551800554897331836828442954685182472926159563367178263271914469321262064384411239}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{11} + \frac{154840001677130940129445877437601160327142596568162159408576470873155837884540353612859768575474634279580333243387683843387969120173661137553730954168288471956083025522228}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{10} + \frac{806525911821848145372500589633263815659041401846571631207555313713031506406328504103564664417043638363011779547533356937751291211382702567929268892985611633557317415708698}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{9} + \frac{361813315821924910171410337827109657227286876427169546996734050184293642946199600305891606722469723526761119963978697320903600732126944173731842676185614026330715137288456}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{8} + \frac{1015575624872265605439606638986018061767822522418934102685047246306207647473725166483861240864729023751000165588150556983599989895835499040828790379944808850612165073074887}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{7} + \frac{40156191982303886513014247872452500176319051734581748567079215436378697922538938442908625478515600242374573462168198598231260997723232375003819056815660210202106201289071}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{6} - \frac{44727168035722630478910547323928892681909977460207231502275479705932368000860068511536264296480790553973247034248680785264032649771704109315720255199362067060238710714874}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{5} + \frac{782084154345713789314974181187316505961159397623392886777907806267051779313377009807805194333854837937633002993268532811525429967098088208198053827646831552719575600594721}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{4} + \frac{478660109231224917663985065302439822374269155335653858037914634927623969695133296907261618503511064251184021143633115117669943508130280752519896545433816412520015335993677}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{3} - \frac{784697242237135767035849499954062264927862957409184762806155261412995162148361408381816884836257532502745011684685761492256379136013011014364467374382191396173314320637526}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a^{2} - \frac{437114446227110168428156037746371283578474812555115449195539007951303438002298112179501823372891443256366811811270609777689149080721960610184593548736741653516069641689310}{2454105383740413313352243620007213765094378373896670682358327380215198901601545893552794345064485320386388969216063102080999807093398112605552676493571166467453339920629793} a + \frac{4367636711876734314558723489096577819145659901586526412241875447875063847571381306800847197966626187319178476419329248561067035999592117577361833763417924813567470835400}{17913177983506666520819296496403020183170645064939202061009688906680283953295955427392659453025440294791160359241336511540144577324073814639070631339935521660243357084889}$

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:  $28$
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

$C_{29}$ (as 29T1):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A cyclic group of order 29
The 29 conjugacy class representatives for $C_{29}$
Character table for $C_{29}$ is not computed

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ R $29$ $29$ ${\href{/LocalNumberField/41.1.0.1}{1} }^{29}$ $29$ $29$ $29$ $29$

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
29Data not computed