Properties

Label 29.29.1581655714...5201.1
Degree $29$
Signature $[29, 0]$
Discriminant $349^{28}$
Root discriminant $285.19$
Ramified prime $349$
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 - 168*x^27 + 353*x^26 + 11480*x^25 - 34328*x^24 - 410190*x^23 + 1556443*x^22 + 8267052*x^21 - 38827232*x^20 - 94788585*x^19 + 575669857*x^18 + 574148959*x^17 - 5286405734*x^16 - 1118675686*x^15 + 30698615326*x^14 - 6935797323*x^13 - 112817035992*x^12 + 51055206761*x^11 + 256367413593*x^10 - 136472035194*x^9 - 340428884896*x^8 + 170672875682*x^7 + 236550118296*x^6 - 94550319984*x^5 - 69592403677*x^4 + 18732474566*x^3 + 3480965541*x^2 - 875365500*x + 10242329)
 
gp: K = bnfinit(x^29 - x^28 - 168*x^27 + 353*x^26 + 11480*x^25 - 34328*x^24 - 410190*x^23 + 1556443*x^22 + 8267052*x^21 - 38827232*x^20 - 94788585*x^19 + 575669857*x^18 + 574148959*x^17 - 5286405734*x^16 - 1118675686*x^15 + 30698615326*x^14 - 6935797323*x^13 - 112817035992*x^12 + 51055206761*x^11 + 256367413593*x^10 - 136472035194*x^9 - 340428884896*x^8 + 170672875682*x^7 + 236550118296*x^6 - 94550319984*x^5 - 69592403677*x^4 + 18732474566*x^3 + 3480965541*x^2 - 875365500*x + 10242329, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![10242329, -875365500, 3480965541, 18732474566, -69592403677, -94550319984, 236550118296, 170672875682, -340428884896, -136472035194, 256367413593, 51055206761, -112817035992, -6935797323, 30698615326, -1118675686, -5286405734, 574148959, 575669857, -94788585, -38827232, 8267052, 1556443, -410190, -34328, 11480, 353, -168, -1, 1]);
 

Normalized defining polynomial

\( x^{29} - x^{28} - 168 x^{27} + 353 x^{26} + 11480 x^{25} - 34328 x^{24} - 410190 x^{23} + 1556443 x^{22} + 8267052 x^{21} - 38827232 x^{20} - 94788585 x^{19} + 575669857 x^{18} + 574148959 x^{17} - 5286405734 x^{16} - 1118675686 x^{15} + 30698615326 x^{14} - 6935797323 x^{13} - 112817035992 x^{12} + 51055206761 x^{11} + 256367413593 x^{10} - 136472035194 x^{9} - 340428884896 x^{8} + 170672875682 x^{7} + 236550118296 x^{6} - 94550319984 x^{5} - 69592403677 x^{4} + 18732474566 x^{3} + 3480965541 x^{2} - 875365500 x + 10242329 \)

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:  \(158165571476523684791187605131317359442300677347381843010071500056545201=349^{28}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $285.19$
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:  $349$
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:  \(349\)
Dirichlet character group:    $\lbrace$$\chi_{349}(1,·)$, $\chi_{349}(66,·)$, $\chi_{349}(67,·)$, $\chi_{349}(263,·)$, $\chi_{349}(210,·)$, $\chi_{349}(332,·)$, $\chi_{349}(269,·)$, $\chi_{349}(88,·)$, $\chi_{349}(274,·)$, $\chi_{349}(249,·)$, $\chi_{349}(280,·)$, $\chi_{349}(285,·)$, $\chi_{349}(31,·)$, $\chi_{349}(224,·)$, $\chi_{349}(289,·)$, $\chi_{349}(228,·)$, $\chi_{349}(168,·)$, $\chi_{349}(41,·)$, $\chi_{349}(234,·)$, $\chi_{349}(171,·)$, $\chi_{349}(257,·)$, $\chi_{349}(301,·)$, $\chi_{349}(110,·)$, $\chi_{349}(304,·)$, $\chi_{349}(118,·)$, $\chi_{349}(312,·)$, $\chi_{349}(313,·)$, $\chi_{349}(322,·)$, $\chi_{349}(126,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$, $\frac{1}{373} a^{27} - \frac{172}{373} a^{26} - \frac{74}{373} a^{25} + \frac{61}{373} a^{24} + \frac{94}{373} a^{23} + \frac{90}{373} a^{22} - \frac{155}{373} a^{21} - \frac{82}{373} a^{20} + \frac{132}{373} a^{19} + \frac{102}{373} a^{18} + \frac{28}{373} a^{17} - \frac{100}{373} a^{16} + \frac{141}{373} a^{15} - \frac{43}{373} a^{14} - \frac{182}{373} a^{13} + \frac{9}{373} a^{12} + \frac{104}{373} a^{11} + \frac{143}{373} a^{10} - \frac{34}{373} a^{9} + \frac{185}{373} a^{8} - \frac{131}{373} a^{7} + \frac{136}{373} a^{6} - \frac{134}{373} a^{5} - \frac{173}{373} a^{4} + \frac{84}{373} a^{3} - \frac{150}{373} a^{2} - \frac{90}{373} a - \frac{136}{373}$, $\frac{1}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{28} - \frac{824749344327341563933945379127045768083766822944568625539053619427137853607502282480937020077340485885864393757040498099498941941295154}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{27} - \frac{174398217290330438425683836423270248086205955450022330913447686015120418148612761613495953913771155830750157794525923946811510030477614469}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{26} - \frac{40942314921501226678666493686554475408900323365277428245788621196000341753701784540468473091193866586438165030746709793966752372885291702}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{25} - \frac{71340314147326450445357459927976062073510124795642881366130681789437162281749093105927542627672901724798283400918270047275645306085059445}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{24} + \frac{65787732980860605996189546454028774368562965242436038812277569269093557207285437092520434320406323903055157010442555490456423754742403893}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{23} + \frac{295061371995842645107456798339953109529995696086136282957597523292726722926342082473773691628284565792528634803658229192583918886619273297}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{22} + \frac{391434957397450831770927608307417588918294052760559218297381150146127119440834138337325067304210205479907699842442886351588001226038983163}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{21} + \frac{71203801080478606385567834522128334627880566720368444078789911783614980549287326563119400737101292794092313495441961640052256573930273730}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{20} + \frac{11871574290444190826473264841415349368216020783095399129370502183941623656300128710723648836457627539794697840943610942076584373015667990}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{19} - \frac{73688649795546237166159353107793460038865879534334488785875062222547513887202372785260286332528851647472204268146483988804401011779772462}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{18} + \frac{349660468691601946847499674428446176799055954279752187801443910386043996544414762412469272125517062415052774656268945440724764149033189658}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{17} + \frac{199839035402810787244510255342682241810638605732028814335186943308434376310854664515968536498092024941632340890651717653559388631516802001}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{16} - \frac{308122261508799543379577619578640380063889843010204460634371611907626308684666090642646723289667389386119718045690876153892735120555230155}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{15} + \frac{351987866488893572977075525805768259450811360074003033766884709493001588859607605618366616040827722098895753672524216074967349462468351753}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{14} + \frac{174941514016952825922802606525116710153673955985927036908431779536574952087481719694146154878236959966987832006317199174856809082449891051}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{13} + \frac{112845291199318305600414927970373978255756225351398525478079515719182108989295297194962073874766186821292943826508642348530863785040401494}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{12} - \frac{89950032763637397657354765564136720413014141232919980321838958090215772514957174009517957592923063908066167791661796581212732905832922505}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{11} + \frac{761384514459833216469729832240696519535929593472243599276109297874951271528539193907811330343318308180797390489501161712933086510780759}{2268046798206050466233797005507493056246150091504343067752952659295781495065406168397339437026594704134022300427817311323437429234139483} a^{10} - \frac{389524463670149385263728883226804437677565733357369973588735732840912372322054245751142997457339240387614745206085749586013866292446465617}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{9} - \frac{308917749935522482686832972661499594606012609547125284587299544152042788364472115060437940156633768121586996247606052353966021713070016265}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{8} - \frac{118645933276459505139119719486593133732250699159452949485381317175567705196299092740340602928139313235191022214092981137859606053562542873}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{7} + \frac{103619760847551234308305006872131273669811470513744110411357866848456433529336412547114575742831743212759231626517840717684890944149627976}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{6} + \frac{228044547909647808698218820676118084512835598175567895009559550317980703729326291919703517879842692903184571802509678496684520011632767100}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{5} - \frac{361051679899342818931656492513153615998077036660376672230255999523897836062217359054877099485723765028190962196043445566180418731672458628}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{4} + \frac{17695424575389813770337805078535077153872852884579999015162101234949532854863066202843080867805880308491388171871614843461988881109849496}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{3} + \frac{343282795840494665481298994863742611851691140176123571796049481032727072702233425129871234093482972041672319569635312550686903458444136793}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a^{2} - \frac{113829167136492544189988519486927119274109482472263647204356827501757083200821821719834614309818881358612759554385476168244212096506558272}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159} a + \frac{380228793230315778231061751201219562146867313609419431049400194287920750933288584459642846309950615266888599143833941812946650369393941090}{845981455730856823905206283054294909979813984131119964271851341917326497659396500812207610010919824641990318059575857123642161104334027159}$

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:  \( 158942637080452900000000000 \) (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 $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ $29$

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
349Data not computed