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Properties

Label 29.1.142...409.1
Degree $29$
Signature $[1, 14]$
Discriminant $1.424\times 10^{50}$
Root discriminant $53.63$
Ramified prime $3823$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{29}$ (as 29T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^29 - x^28 + 33*x^27 - 64*x^26 + 535*x^25 - 1144*x^24 + 5145*x^23 - 10327*x^22 + 31220*x^21 - 55583*x^20 + 124455*x^19 - 188951*x^18 + 323227*x^17 - 411652*x^16 + 521736*x^15 - 591430*x^14 + 489137*x^13 - 738131*x^12 + 175881*x^11 - 1211717*x^10 + 917449*x^9 - 3034006*x^8 + 2818536*x^7 - 6798367*x^6 + 6818314*x^5 - 8408506*x^4 + 7552950*x^3 - 8000650*x^2 + 5448750*x - 1749375)
 
gp: K = bnfinit(x^29 - x^28 + 33*x^27 - 64*x^26 + 535*x^25 - 1144*x^24 + 5145*x^23 - 10327*x^22 + 31220*x^21 - 55583*x^20 + 124455*x^19 - 188951*x^18 + 323227*x^17 - 411652*x^16 + 521736*x^15 - 591430*x^14 + 489137*x^13 - 738131*x^12 + 175881*x^11 - 1211717*x^10 + 917449*x^9 - 3034006*x^8 + 2818536*x^7 - 6798367*x^6 + 6818314*x^5 - 8408506*x^4 + 7552950*x^3 - 8000650*x^2 + 5448750*x - 1749375, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1749375, 5448750, -8000650, 7552950, -8408506, 6818314, -6798367, 2818536, -3034006, 917449, -1211717, 175881, -738131, 489137, -591430, 521736, -411652, 323227, -188951, 124455, -55583, 31220, -10327, 5145, -1144, 535, -64, 33, -1, 1]);
 

\( x^{29} - x^{28} + 33 x^{27} - 64 x^{26} + 535 x^{25} - 1144 x^{24} + 5145 x^{23} - 10327 x^{22} + 31220 x^{21} - 55583 x^{20} + 124455 x^{19} - 188951 x^{18} + 323227 x^{17} - 411652 x^{16} + 521736 x^{15} - 591430 x^{14} + 489137 x^{13} - 738131 x^{12} + 175881 x^{11} - 1211717 x^{10} + 917449 x^{9} - 3034006 x^{8} + 2818536 x^{7} - 6798367 x^{6} + 6818314 x^{5} - 8408506 x^{4} + 7552950 x^{3} - 8000650 x^{2} + 5448750 x - 1749375 \)

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

Invariants

Degree:  $29$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 14]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(142449251725173555024565249558533387292386826365409\)\(\medspace = 3823^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $53.63$
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:  $3823$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{15} a^{16} - \frac{1}{15} a^{15} - \frac{1}{15} a^{14} - \frac{2}{15} a^{13} - \frac{1}{15} a^{12} + \frac{1}{15} a^{11} + \frac{1}{15} a^{10} - \frac{1}{15} a^{9} + \frac{2}{15} a^{8} + \frac{7}{15} a^{7} + \frac{7}{15} a^{6} - \frac{7}{15} a^{5} + \frac{4}{15} a^{4} - \frac{4}{15} a^{3} + \frac{1}{3} a^{2} + \frac{4}{15} a$, $\frac{1}{15} a^{17} - \frac{2}{15} a^{15} + \frac{2}{15} a^{14} + \frac{2}{15} a^{13} + \frac{2}{15} a^{11} + \frac{1}{15} a^{9} - \frac{2}{5} a^{8} - \frac{1}{15} a^{7} - \frac{1}{3} a^{6} + \frac{7}{15} a^{5} + \frac{1}{15} a^{3} - \frac{2}{5} a^{2} + \frac{4}{15} a$, $\frac{1}{45} a^{18} + \frac{2}{15} a^{13} - \frac{1}{15} a^{11} - \frac{2}{45} a^{10} - \frac{1}{15} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{1}{5} a^{6} + \frac{7}{15} a^{5} + \frac{1}{5} a^{4} + \frac{7}{15} a^{3} + \frac{19}{45} a^{2} + \frac{1}{15} a$, $\frac{1}{45} a^{19} + \frac{2}{15} a^{14} - \frac{1}{15} a^{12} - \frac{2}{45} a^{11} - \frac{1}{15} a^{10} + \frac{1}{15} a^{9} + \frac{1}{5} a^{8} - \frac{1}{5} a^{7} + \frac{7}{15} a^{6} + \frac{1}{5} a^{5} + \frac{7}{15} a^{4} + \frac{19}{45} a^{3} + \frac{1}{15} a^{2} + \frac{1}{3} a$, $\frac{1}{315} a^{20} + \frac{1}{105} a^{19} + \frac{2}{315} a^{18} - \frac{2}{105} a^{17} + \frac{2}{105} a^{16} + \frac{2}{15} a^{15} - \frac{1}{7} a^{14} + \frac{1}{7} a^{13} - \frac{32}{315} a^{12} + \frac{1}{35} a^{11} + \frac{26}{315} a^{10} - \frac{2}{21} a^{9} + \frac{7}{15} a^{8} - \frac{4}{21} a^{7} - \frac{16}{35} a^{6} - \frac{1}{35} a^{5} + \frac{7}{45} a^{4} - \frac{31}{105} a^{3} + \frac{14}{45} a^{2} - \frac{6}{35} a - \frac{3}{7}$, $\frac{1}{315} a^{21} + \frac{2}{315} a^{18} + \frac{1}{105} a^{17} + \frac{1}{105} a^{16} - \frac{1}{105} a^{15} - \frac{1}{35} a^{14} + \frac{22}{315} a^{13} - \frac{1}{21} a^{11} + \frac{32}{315} a^{10} + \frac{2}{105} a^{9} - \frac{34}{105} a^{8} - \frac{44}{105} a^{7} - \frac{2}{35} a^{6} + \frac{13}{315} a^{5} + \frac{6}{35} a^{4} + \frac{44}{105} a^{3} + \frac{13}{63} a^{2} - \frac{11}{35} a + \frac{2}{7}$, $\frac{1}{315} a^{22} + \frac{2}{315} a^{19} + \frac{1}{105} a^{18} + \frac{1}{105} a^{17} - \frac{1}{105} a^{16} - \frac{1}{35} a^{15} + \frac{22}{315} a^{14} - \frac{1}{21} a^{12} + \frac{32}{315} a^{11} + \frac{2}{105} a^{10} + \frac{1}{105} a^{9} - \frac{44}{105} a^{8} - \frac{2}{35} a^{7} + \frac{13}{315} a^{6} + \frac{6}{35} a^{5} + \frac{44}{105} a^{4} + \frac{13}{63} a^{3} - \frac{11}{35} a^{2} - \frac{1}{21} a$, $\frac{1}{1575} a^{23} - \frac{1}{1575} a^{22} - \frac{1}{1575} a^{21} - \frac{1}{315} a^{19} + \frac{1}{105} a^{18} + \frac{8}{525} a^{17} - \frac{2}{75} a^{16} + \frac{34}{1575} a^{15} + \frac{7}{45} a^{14} - \frac{127}{1575} a^{13} - \frac{61}{525} a^{12} - \frac{71}{1575} a^{11} - \frac{1}{35} a^{10} - \frac{34}{525} a^{9} - \frac{166}{525} a^{8} - \frac{116}{1575} a^{7} + \frac{116}{1575} a^{6} - \frac{547}{1575} a^{5} - \frac{248}{525} a^{4} + \frac{436}{1575} a^{3} - \frac{248}{525} a^{2} - \frac{44}{105} a - \frac{2}{7}$, $\frac{1}{4725} a^{24} + \frac{1}{4725} a^{23} + \frac{2}{4725} a^{22} + \frac{1}{1575} a^{21} + \frac{1}{945} a^{20} - \frac{1}{189} a^{19} - \frac{41}{4725} a^{18} - \frac{8}{1575} a^{17} + \frac{2}{945} a^{16} - \frac{377}{4725} a^{15} + \frac{83}{4725} a^{14} - \frac{239}{1575} a^{13} + \frac{428}{4725} a^{12} - \frac{502}{4725} a^{11} - \frac{302}{4725} a^{10} - \frac{74}{1575} a^{9} + \frac{334}{675} a^{8} - \frac{1046}{4725} a^{7} + \frac{442}{945} a^{6} - \frac{146}{1575} a^{5} - \frac{262}{4725} a^{4} + \frac{113}{4725} a^{3} - \frac{8}{4725} a^{2} + \frac{19}{63} a - \frac{1}{21}$, $\frac{1}{70875} a^{25} - \frac{1}{70875} a^{24} + \frac{2}{23625} a^{23} - \frac{16}{10125} a^{22} + \frac{14}{10125} a^{21} - \frac{4}{14175} a^{20} + \frac{8}{23625} a^{19} - \frac{662}{70875} a^{18} - \frac{29}{10125} a^{17} + \frac{1016}{70875} a^{16} + \frac{137}{23625} a^{15} + \frac{2642}{70875} a^{14} + \frac{2266}{14175} a^{13} - \frac{4511}{70875} a^{12} + \frac{3392}{23625} a^{11} - \frac{2858}{70875} a^{10} - \frac{538}{14175} a^{9} - \frac{6908}{14175} a^{8} - \frac{6238}{23625} a^{7} - \frac{6112}{70875} a^{6} + \frac{17777}{70875} a^{5} - \frac{22307}{70875} a^{4} + \frac{167}{3375} a^{3} + \frac{21547}{70875} a^{2} - \frac{29}{945} a + \frac{44}{315}$, $\frac{1}{496125} a^{26} + \frac{2}{496125} a^{25} + \frac{2}{55125} a^{24} + \frac{8}{70875} a^{23} + \frac{107}{496125} a^{22} - \frac{41}{496125} a^{21} - \frac{62}{165375} a^{20} - \frac{913}{99225} a^{19} - \frac{47}{70875} a^{18} - \frac{4138}{496125} a^{17} + \frac{971}{55125} a^{16} - \frac{5783}{99225} a^{15} - \frac{47224}{496125} a^{14} - \frac{8678}{70875} a^{13} - \frac{16514}{165375} a^{12} + \frac{15026}{99225} a^{11} - \frac{32219}{496125} a^{10} + \frac{1693}{99225} a^{9} - \frac{947}{18375} a^{8} + \frac{243896}{496125} a^{7} - \frac{41749}{496125} a^{6} - \frac{21263}{70875} a^{5} - \frac{15503}{165375} a^{4} - \frac{60152}{496125} a^{3} + \frac{54727}{165375} a^{2} + \frac{866}{2205} a - \frac{12}{245}$, $\frac{1}{496125} a^{27} + \frac{34}{496125} a^{24} - \frac{89}{496125} a^{23} - \frac{262}{496125} a^{22} + \frac{11}{55125} a^{21} - \frac{109}{70875} a^{20} + \frac{748}{99225} a^{19} - \frac{512}{496125} a^{18} + \frac{319}{165375} a^{17} + \frac{808}{496125} a^{16} - \frac{51848}{496125} a^{15} + \frac{37664}{496125} a^{14} - \frac{946}{11025} a^{13} - \frac{5182}{496125} a^{12} + \frac{48332}{496125} a^{11} - \frac{1357}{99225} a^{10} + \frac{12562}{165375} a^{9} + \frac{65119}{496125} a^{8} - \frac{47839}{99225} a^{7} - \frac{8704}{19845} a^{6} - \frac{3484}{11025} a^{5} - \frac{152536}{496125} a^{4} + \frac{136162}{496125} a^{3} - \frac{22144}{99225} a^{2} + \frac{997}{6615} a + \frac{109}{441}$, $\frac{1}{1007739047030091998991375958377067080484985334728545672104844366663125} a^{28} - \frac{888533874791622809913029027304155736160291474289623947810181966}{1007739047030091998991375958377067080484985334728545672104844366663125} a^{27} + \frac{39046103351831464089207161844337759497814786151263006382644843}{47987573668099618999589331351288908594523111177549793909754493650625} a^{26} + \frac{4113629742785890689382925899341841216647976023860638907605938236}{1007739047030091998991375958377067080484985334728545672104844366663125} a^{25} - \frac{1455003702465047771807035095449153837115954605237024665537696192}{40309561881203679959655038335082683219399413389141826884193774666525} a^{24} + \frac{259215581576447114280273870923061695806308910050677729791147060696}{1007739047030091998991375958377067080484985334728545672104844366663125} a^{23} - \frac{34486440150372459001971291857459285886238081932299889610649205347}{67182603135339466599425063891804472032332355648569711473656291110875} a^{22} + \frac{1235906255782690102384559374038915210989200140015925004419967417623}{1007739047030091998991375958377067080484985334728545672104844366663125} a^{21} + \frac{99857522783816509125882733831588472843204268770297969605040621249}{201547809406018399798275191675413416096997066945709134420968873332625} a^{20} - \frac{2143140133244882254903733054643839740593439817097394886820923374318}{1007739047030091998991375958377067080484985334728545672104844366663125} a^{19} - \frac{541806319617338686246196519636140702714892327637577219069646768036}{67182603135339466599425063891804472032332355648569711473656291110875} a^{18} + \frac{18292632742187244438808794011459711366839998083265325034387741631779}{1007739047030091998991375958377067080484985334728545672104844366663125} a^{17} + \frac{13915625465910532141282441252740513989421747300681383656246531041197}{1007739047030091998991375958377067080484985334728545672104844366663125} a^{16} + \frac{37251292741839033109786442408022959367959587899993156442695905281228}{1007739047030091998991375958377067080484985334728545672104844366663125} a^{15} - \frac{40543808415064562663411847388771075441061263493781255932412538390078}{335913015676697332997125319459022360161661778242848557368281455554375} a^{14} - \frac{87719385642432792005591378737420395727144061638845770764194383239}{8061912376240735991931007667016536643879882677828365376838754933305} a^{13} - \frac{75349389202726903144493463198835477659091530481266855264882768234233}{1007739047030091998991375958377067080484985334728545672104844366663125} a^{12} - \frac{131990580403613139759509867959368096216222062324569221556930961896726}{1007739047030091998991375958377067080484985334728545672104844366663125} a^{11} + \frac{4679707326958696918134890544326457349823726119717902236325936540557}{335913015676697332997125319459022360161661778242848557368281455554375} a^{10} + \frac{130783770255184507667406558314012051363455779334931288267709739317968}{1007739047030091998991375958377067080484985334728545672104844366663125} a^{9} + \frac{67201831009746257964060990122243832190817266953255819096439436741}{20566103000614122428395427721980960826224190504664197389894782993125} a^{8} + \frac{330322964219531060145549964664651874559846066872197081429387877045274}{1007739047030091998991375958377067080484985334728545672104844366663125} a^{7} + \frac{1646044075377120794655660735599126791766655454167486476802311266687}{3325871442343537950466587321376459011501601764780678785824568866875} a^{6} - \frac{65547445200447812555672743072296214376424534626153464886736338455481}{143962721004298856998767994053866725783569333532649381729263480951875} a^{5} - \frac{114675910839116799007939276950465208061676801263913229430980409406682}{335913015676697332997125319459022360161661778242848557368281455554375} a^{4} - \frac{7583525652516014858137593826926987588726851158916344706108261023402}{335913015676697332997125319459022360161661778242848557368281455554375} a^{3} + \frac{4239758147887216088798979282788001310012351235742929883909268890143}{9597514733619923799917866270257781718904622235509958781950898730125} a^{2} - \frac{84607030794316144891453459447706734026292197167416083252241573316}{639834315574661586661191084683852114593641482367330585463393248675} a + \frac{82164791010092656726281035279879678420851406720462901342105108502}{298589347268175407108555839519130986810366025104754273216250182715}$

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:  $14$
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:  \( 254102374099303.56 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{14}\cdot 254102374099303.56 \cdot 1}{2\sqrt{142449251725173555024565249558533387292386826365409}}\approx 3.18197522288265$ (assuming GRH)

Galois group

$D_{29}$ (as 29T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 58
The 16 conjugacy class representatives for $D_{29}$
Character table for $D_{29}$

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 $29$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ $29$ $29$ $29$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $29$ $29$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $29$ $29$ $29$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $29$ $29$

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
3823Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3823.2t1.a.a$1$ $ 3823 $ \(\Q(\sqrt{-3823}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3823.29t2.a.d$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.h$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.e$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.k$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.g$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.n$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.a$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.i$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.b$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.l$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.m$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.c$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.j$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3823.29t2.a.f$2$ $ 3823 $ 29.1.142449251725173555024565249558533387292386826365409.1 $D_{29}$ (as 29T2) $1$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.