Properties

Label 29.29.1312342786...4481.1
Degree $29$
Signature $[29, 0]$
Discriminant $523^{28}$
Root discriminant $421.46$
Ramified prime $523$
Class number $1$ (GRH)
Class group Trivial (GRH)
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 - x^28 - 252*x^27 + 493*x^26 + 26301*x^25 - 70613*x^24 - 1477005*x^23 + 4757585*x^22 + 49139083*x^21 - 176708296*x^20 - 1019533247*x^19 + 3897692875*x^18 + 13650905238*x^17 - 52791707208*x^16 - 121558760223*x^15 + 441031541121*x^14 + 744433170879*x^13 - 2221117761731*x^12 - 3198643328695*x^11 + 6333895042002*x^10 + 9161654939551*x^9 - 8623697894872*x^8 - 14867816867065*x^7 + 2648025494151*x^6 + 9891108006231*x^5 + 1898151784214*x^4 - 1956687026062*x^3 - 891062263511*x^2 - 130453149826*x - 6306528127)
 
gp: K = bnfinit(x^29 - x^28 - 252*x^27 + 493*x^26 + 26301*x^25 - 70613*x^24 - 1477005*x^23 + 4757585*x^22 + 49139083*x^21 - 176708296*x^20 - 1019533247*x^19 + 3897692875*x^18 + 13650905238*x^17 - 52791707208*x^16 - 121558760223*x^15 + 441031541121*x^14 + 744433170879*x^13 - 2221117761731*x^12 - 3198643328695*x^11 + 6333895042002*x^10 + 9161654939551*x^9 - 8623697894872*x^8 - 14867816867065*x^7 + 2648025494151*x^6 + 9891108006231*x^5 + 1898151784214*x^4 - 1956687026062*x^3 - 891062263511*x^2 - 130453149826*x - 6306528127, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-6306528127, -130453149826, -891062263511, -1956687026062, 1898151784214, 9891108006231, 2648025494151, -14867816867065, -8623697894872, 9161654939551, 6333895042002, -3198643328695, -2221117761731, 744433170879, 441031541121, -121558760223, -52791707208, 13650905238, 3897692875, -1019533247, -176708296, 49139083, 4757585, -1477005, -70613, 26301, 493, -252, -1, 1]);
 

Normalized defining polynomial

\( x^{29} - x^{28} - 252 x^{27} + 493 x^{26} + 26301 x^{25} - 70613 x^{24} - 1477005 x^{23} + 4757585 x^{22} + 49139083 x^{21} - 176708296 x^{20} - 1019533247 x^{19} + 3897692875 x^{18} + 13650905238 x^{17} - 52791707208 x^{16} - 121558760223 x^{15} + 441031541121 x^{14} + 744433170879 x^{13} - 2221117761731 x^{12} - 3198643328695 x^{11} + 6333895042002 x^{10} + 9161654939551 x^{9} - 8623697894872 x^{8} - 14867816867065 x^{7} + 2648025494151 x^{6} + 9891108006231 x^{5} + 1898151784214 x^{4} - 1956687026062 x^{3} - 891062263511 x^{2} - 130453149826 x - 6306528127 \)

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:  \(13123427860740340635045684158089751314114754706664887333885491004189525894481=523^{28}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $421.46$
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:  $523$
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:  \(523\)
Dirichlet character group:    $\lbrace$$\chi_{523}(1,·)$, $\chi_{523}(387,·)$, $\chi_{523}(134,·)$, $\chi_{523}(520,·)$, $\chi_{523}(9,·)$, $\chi_{523}(394,·)$, $\chi_{523}(11,·)$, $\chi_{523}(206,·)$, $\chi_{523}(408,·)$, $\chi_{523}(465,·)$, $\chi_{523}(150,·)$, $\chi_{523}(473,·)$, $\chi_{523}(280,·)$, $\chi_{523}(345,·)$, $\chi_{523}(285,·)$, $\chi_{523}(160,·)$, $\chi_{523}(304,·)$, $\chi_{523}(226,·)$, $\chi_{523}(99,·)$, $\chi_{523}(81,·)$, $\chi_{523}(490,·)$, $\chi_{523}(43,·)$, $\chi_{523}(428,·)$, $\chi_{523}(174,·)$, $\chi_{523}(368,·)$, $\chi_{523}(496,·)$, $\chi_{523}(73,·)$, $\chi_{523}(121,·)$, $\chi_{523}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{19} a^{17} - \frac{2}{19} a^{16} + \frac{7}{19} a^{15} - \frac{1}{19} a^{14} + \frac{4}{19} a^{13} + \frac{8}{19} a^{12} - \frac{4}{19} a^{11} - \frac{6}{19} a^{10} + \frac{1}{19} a^{8} - \frac{2}{19} a^{7} + \frac{7}{19} a^{6} - \frac{1}{19} a^{5} + \frac{4}{19} a^{4} + \frac{8}{19} a^{3} - \frac{4}{19} a^{2} - \frac{6}{19} a$, $\frac{1}{19} a^{18} + \frac{3}{19} a^{16} - \frac{6}{19} a^{15} + \frac{2}{19} a^{14} - \frac{3}{19} a^{13} - \frac{7}{19} a^{12} + \frac{5}{19} a^{11} + \frac{7}{19} a^{10} + \frac{1}{19} a^{9} + \frac{3}{19} a^{7} - \frac{6}{19} a^{6} + \frac{2}{19} a^{5} - \frac{3}{19} a^{4} - \frac{7}{19} a^{3} + \frac{5}{19} a^{2} + \frac{7}{19} a$, $\frac{1}{19} a^{19} - \frac{1}{19} a$, $\frac{1}{19} a^{20} - \frac{1}{19} a^{2}$, $\frac{1}{19} a^{21} - \frac{1}{19} a^{3}$, $\frac{1}{19} a^{22} - \frac{1}{19} a^{4}$, $\frac{1}{19} a^{23} - \frac{1}{19} a^{5}$, $\frac{1}{361} a^{24} - \frac{8}{361} a^{23} - \frac{7}{361} a^{22} + \frac{1}{361} a^{21} - \frac{8}{361} a^{20} + \frac{8}{361} a^{19} + \frac{7}{361} a^{18} - \frac{9}{361} a^{17} + \frac{172}{361} a^{16} - \frac{48}{361} a^{15} + \frac{61}{361} a^{14} - \frac{9}{19} a^{13} - \frac{83}{361} a^{12} + \frac{166}{361} a^{11} + \frac{160}{361} a^{10} - \frac{69}{361} a^{9} + \frac{86}{361} a^{8} - \frac{37}{361} a^{7} + \frac{84}{361} a^{6} - \frac{102}{361} a^{5} + \frac{64}{361} a^{4} - \frac{122}{361} a^{3} + \frac{98}{361} a^{2} + \frac{4}{19} a$, $\frac{1}{361} a^{25} + \frac{5}{361} a^{23} + \frac{2}{361} a^{22} + \frac{1}{361} a^{20} - \frac{5}{361} a^{19} + \frac{9}{361} a^{18} + \frac{5}{361} a^{17} - \frac{40}{361} a^{16} - \frac{2}{19} a^{15} - \frac{25}{361} a^{14} + \frac{88}{361} a^{13} + \frac{91}{361} a^{12} - \frac{127}{361} a^{11} + \frac{71}{361} a^{10} - \frac{143}{361} a^{9} - \frac{166}{361} a^{8} - \frac{136}{361} a^{7} + \frac{7}{19} a^{6} - \frac{87}{361} a^{5} + \frac{67}{361} a^{4} + \frac{72}{361} a^{3} - \frac{90}{361} a^{2} - \frac{5}{19} a$, $\frac{1}{40793} a^{26} + \frac{1}{40793} a^{25} - \frac{35}{40793} a^{24} - \frac{395}{40793} a^{23} + \frac{681}{40793} a^{22} + \frac{968}{40793} a^{21} - \frac{121}{40793} a^{20} - \frac{31}{40793} a^{19} - \frac{16}{2147} a^{18} + \frac{287}{40793} a^{17} - \frac{10967}{40793} a^{16} + \frac{15176}{40793} a^{15} - \frac{6025}{40793} a^{14} + \frac{483}{40793} a^{13} - \frac{10472}{40793} a^{12} - \frac{16842}{40793} a^{11} - \frac{20228}{40793} a^{10} + \frac{32}{2147} a^{9} - \frac{1975}{40793} a^{8} + \frac{4327}{40793} a^{7} - \frac{18514}{40793} a^{6} - \frac{14389}{40793} a^{5} + \frac{4001}{40793} a^{4} + \frac{8871}{40793} a^{3} - \frac{2984}{40793} a^{2} + \frac{1072}{2147} a + \frac{5}{113}$, $\frac{1}{4605623752707937} a^{27} + \frac{19899893856}{4605623752707937} a^{26} + \frac{1895686225114}{4605623752707937} a^{25} - \frac{5571536327734}{4605623752707937} a^{24} + \frac{42321077651837}{4605623752707937} a^{23} + \frac{76647823822703}{4605623752707937} a^{22} - \frac{24910417720468}{4605623752707937} a^{21} - \frac{31702543407817}{4605623752707937} a^{20} + \frac{114401628107544}{4605623752707937} a^{19} + \frac{47550185191851}{4605623752707937} a^{18} - \frac{59831914544399}{4605623752707937} a^{17} + \frac{1285235015899696}{4605623752707937} a^{16} + \frac{914928674201160}{4605623752707937} a^{15} - \frac{1107260636854052}{4605623752707937} a^{14} - \frac{734808000550862}{4605623752707937} a^{13} - \frac{2222719314621634}{4605623752707937} a^{12} + \frac{1140170124459355}{4605623752707937} a^{11} - \frac{1797112557034840}{4605623752707937} a^{10} + \frac{1315807676212619}{4605623752707937} a^{9} + \frac{842932255131992}{4605623752707937} a^{8} + \frac{71513217065338}{4605623752707937} a^{7} + \frac{194121422850161}{4605623752707937} a^{6} - \frac{938921828943813}{4605623752707937} a^{5} - \frac{2117818442858846}{4605623752707937} a^{4} + \frac{2108917104551760}{4605623752707937} a^{3} + \frac{2124388303766513}{4605623752707937} a^{2} + \frac{116778584174148}{242401250142523} a + \frac{53481690844}{209146893997}$, $\frac{1}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{28} - \frac{724256414056129343591688833528829536189846728111303209534662216981416665842546187312167377465966308145669973941178965335}{143642451746108482566437026413946608120363861765073524818322366454453421604902886046674347184167753669577683031731823987248727223324727761} a^{27} - \frac{12733336496597211548987501490064441078861925385727975705155339506779919204127096281174925978400241541039305991073540989482505283765048}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{26} + \frac{1242138935142427898341741019491669763961061000018170250616874029384789748444546358487190578475107686090952065209972977419072863403863915}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{25} + \frac{81714000662472570008249226894443014251919538709168334710015606873670509340253656220984249333818661993446861469471744424770683663598721}{143642451746108482566437026413946608120363861765073524818322366454453421604902886046674347184167753669577683031731823987248727223324727761} a^{24} - \frac{30993368452883841456495427688575501259954036534344965645543720585374096183387434370437029689328271815402259019792852419815616719717940538}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{23} - \frac{7688681749683416660056025772185325280803535595743570294878085585468027179973512555373987540005991221300577634842330643596935051107877132}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{22} - \frac{23406436039562582054180105631555011919896641590292072942436675153678781703139335957429391054629728130119750467555301262876195056978354870}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{21} + \frac{37290965035853448947621004386794681460983943622681410990572468493481249794829039320788726035979358125838942317470927094464783226212170043}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{20} - \frac{3173417666595416156187150034757081045536572395827203689034290482387377846380351274046355482388877137450850445924028280155197004088306708}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{19} - \frac{31833759237235935939067806830205356280280397481367572568526431419237196853079250755864465691624002802799340261437568362189213295151964858}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{18} + \frac{13111129987151456859537223319277095594415144830113727674827573511519894255254419767812035969408877689072041704750188976600265650793768480}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{17} - \frac{339675172195800663290818882585881184935456008787169372074508928034137973722917434201088636461187562720890172936042585257374777255251075505}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{16} + \frac{436547063798924496567094306552997185984326671456223961656697185265103571615007396134352031186906117514254198162258266021257070945122408282}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{15} + \frac{471988354634425166779513041507460796713842627590438291745511517106353659829675419464464673592343663028931212092485567571602331184511767155}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{14} - \frac{1364246630382527019697472108070369443444332374985153073089665235775283487147740358518111397641436692471767382397473113148090120445107923380}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{13} + \frac{227471593150666733312584837697384291270397173390658269573431603916042403980624034913866342164032349631869134395081488534676858400125572}{2354794290919811189613721744490928001973178061722516800300366663187761009916440754863513888265045142124224311995603671922110282349585701} a^{12} - \frac{888416452180049130469401534108983574252975657984996368668836514566765136118730398922781408910536414356324315411200571668070937034280356217}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{11} - \frac{28566503076043574605463026854464715753259385296247539434286696099553238725595569627218368631897236520165361569387226283647548986733223821}{143642451746108482566437026413946608120363861765073524818322366454453421604902886046674347184167753669577683031731823987248727223324727761} a^{10} - \frac{1268277118766492597285361334495430189527492055409509163339581727383679730582979258274085067485895239949046243975562681583302703913986164954}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{9} - \frac{668479358016667333929737456314910305301871194866561231432071396373757595039966086277091590457232977041010744689508924921934031109699220044}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{8} + \frac{1067455951869136814741969774437844794237101441397953776100130913941877751032262548633674428962918228731136072020975529223866349958627944417}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{7} - \frac{458561595509288101052718991730927918053841113084831601916773527434982220064954352498336708179013501468637706609796535514593954516759656574}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{6} - \frac{591921231430411791495340412084639999597644600382105341899209700600686994748092126585506278113924846335381272847756962442805757151864495369}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{5} - \frac{267873717358758502540182256361925854742867148391167312696729021561001853601567117609047784349133730079541161066373722089225267855866924151}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{4} + \frac{625341294829248851609449944846480465166654193071096368337783517815704867521091050586742512153393918289971109329859560176689020649510352119}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{3} - \frac{152735622910892101852460884848159522843798530559223148291675062806843607282363690201032377294166985570055650948628543774454240974531409166}{2729206583176061168762303501864985554286913373536396971548124962634615010493154834886812596499187319721975977602904655757725817243169827459} a^{2} + \frac{21614580005698768427262384932472670847858483425954745980786774529568526032395061913135160438465763487823024855061136706237712788445703535}{143642451746108482566437026413946608120363861765073524818322366454453421604902886046674347184167753669577683031731823987248727223324727761} a - \frac{55255670267215815479629053989359760582302083694225679014491248939837409660075360963152726686791201703793089626654995340810251456303115}{123936541627358483663880091815312000103851476932764042121071929641461105785075829203342836224476060111801279578715982732742646439451879}$

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:  $28$
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:  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:  \( 404654663243202000000000000000 \) (assuming GRH)
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 $29$ $29$ $29$ $29$ $29$ $29$ $29$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{29}$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$

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
523Data not computed