Properties

Label 29.1.771...361.1
Degree $29$
Signature $[1, 14]$
Discriminant $7.710\times 10^{49}$
Root discriminant $52.51$
Ramified prime $3659$
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 - 7*x^28 + 47*x^27 - 186*x^26 + 710*x^25 - 970*x^24 + 1144*x^23 + 2298*x^22 - 8465*x^21 + 16505*x^20 - 20589*x^19 + 3590*x^18 + 30661*x^17 - 81803*x^16 + 117497*x^15 - 94780*x^14 + 10260*x^13 + 127678*x^12 - 249606*x^11 + 245472*x^10 - 99463*x^9 - 191187*x^8 + 582067*x^7 - 785770*x^6 + 915704*x^5 - 793384*x^4 + 541792*x^3 - 324448*x^2 + 144320*x - 23552)
 
gp: K = bnfinit(x^29 - 7*x^28 + 47*x^27 - 186*x^26 + 710*x^25 - 970*x^24 + 1144*x^23 + 2298*x^22 - 8465*x^21 + 16505*x^20 - 20589*x^19 + 3590*x^18 + 30661*x^17 - 81803*x^16 + 117497*x^15 - 94780*x^14 + 10260*x^13 + 127678*x^12 - 249606*x^11 + 245472*x^10 - 99463*x^9 - 191187*x^8 + 582067*x^7 - 785770*x^6 + 915704*x^5 - 793384*x^4 + 541792*x^3 - 324448*x^2 + 144320*x - 23552, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-23552, 144320, -324448, 541792, -793384, 915704, -785770, 582067, -191187, -99463, 245472, -249606, 127678, 10260, -94780, 117497, -81803, 30661, 3590, -20589, 16505, -8465, 2298, 1144, -970, 710, -186, 47, -7, 1]);
 

\( x^{29} - 7 x^{28} + 47 x^{27} - 186 x^{26} + 710 x^{25} - 970 x^{24} + 1144 x^{23} + 2298 x^{22} - 8465 x^{21} + 16505 x^{20} - 20589 x^{19} + 3590 x^{18} + 30661 x^{17} - 81803 x^{16} + 117497 x^{15} - 94780 x^{14} + 10260 x^{13} + 127678 x^{12} - 249606 x^{11} + 245472 x^{10} - 99463 x^{9} - 191187 x^{8} + 582067 x^{7} - 785770 x^{6} + 915704 x^{5} - 793384 x^{4} + 541792 x^{3} - 324448 x^{2} + 144320 x - 23552 \)

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:  \(77103436114042117740511038546742321228145336964361\)\(\medspace = 3659^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $52.51$
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:  $3659$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{16} a^{13} - \frac{1}{16} a^{12} - \frac{1}{8} a^{11} - \frac{1}{16} a^{10} + \frac{1}{16} a^{9} - \frac{1}{8} a^{8} - \frac{1}{16} a^{7} - \frac{3}{16} a^{6} + \frac{1}{8} a^{5} - \frac{3}{16} a^{4} + \frac{3}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{14} - \frac{1}{16} a^{12} + \frac{1}{16} a^{11} + \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{1}{4} a^{7} - \frac{3}{16} a^{6} + \frac{3}{16} a^{5} + \frac{3}{16} a^{3} - \frac{3}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{15} - \frac{1}{8} a^{11} - \frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{7} + \frac{1}{8} a^{5} + \frac{1}{16} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{10} - \frac{1}{4} a^{7} - \frac{3}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{32} a^{17} + \frac{1}{16} a^{11} - \frac{3}{32} a^{5}$, $\frac{1}{64} a^{18} - \frac{1}{64} a^{17} - \frac{1}{32} a^{14} - \frac{1}{32} a^{13} + \frac{1}{32} a^{12} - \frac{3}{32} a^{10} + \frac{1}{32} a^{8} - \frac{3}{32} a^{7} + \frac{13}{64} a^{6} + \frac{9}{64} a^{5} - \frac{1}{32} a^{4} - \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{19} - \frac{1}{64} a^{17} - \frac{1}{32} a^{15} - \frac{1}{32} a^{12} - \frac{1}{32} a^{11} - \frac{3}{32} a^{10} + \frac{3}{32} a^{9} - \frac{9}{64} a^{7} + \frac{5}{32} a^{6} - \frac{13}{64} a^{5} + \frac{7}{32} a^{4} - \frac{3}{16} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{64} a^{20} - \frac{1}{64} a^{17} - \frac{1}{32} a^{14} - \frac{1}{16} a^{12} + \frac{1}{32} a^{11} + \frac{1}{16} a^{9} + \frac{1}{64} a^{8} - \frac{1}{4} a^{7} - \frac{3}{16} a^{6} + \frac{15}{64} a^{5} + \frac{3}{16} a^{3} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{1664} a^{21} - \frac{9}{1664} a^{20} - \frac{3}{832} a^{19} - \frac{3}{1664} a^{18} + \frac{17}{1664} a^{17} - \frac{3}{416} a^{16} - \frac{1}{64} a^{15} + \frac{3}{832} a^{14} - \frac{7}{416} a^{13} - \frac{23}{832} a^{12} + \frac{45}{832} a^{11} + \frac{1}{16} a^{10} + \frac{189}{1664} a^{9} + \frac{3}{1664} a^{8} - \frac{55}{832} a^{7} + \frac{337}{1664} a^{6} - \frac{235}{1664} a^{5} - \frac{47}{416} a^{4} + \frac{111}{416} a^{3} - \frac{3}{13} a^{2} - \frac{41}{104} a - \frac{4}{13}$, $\frac{1}{3328} a^{22} - \frac{9}{3328} a^{20} + \frac{21}{3328} a^{19} + \frac{1}{208} a^{18} - \frac{41}{3328} a^{17} + \frac{11}{1664} a^{16} + \frac{1}{104} a^{15} + \frac{1}{128} a^{14} + \frac{33}{1664} a^{13} - \frac{29}{832} a^{12} + \frac{145}{1664} a^{11} + \frac{397}{3328} a^{10} - \frac{55}{832} a^{9} - \frac{161}{3328} a^{8} + \frac{569}{3328} a^{7} - \frac{35}{832} a^{6} + \frac{375}{3328} a^{5} + \frac{11}{64} a^{4} + \frac{331}{832} a^{3} + \frac{21}{104} a^{2} + \frac{41}{208} a - \frac{5}{13}$, $\frac{1}{3328} a^{23} - \frac{1}{3328} a^{21} + \frac{1}{3328} a^{20} + \frac{5}{832} a^{19} - \frac{1}{256} a^{18} + \frac{1}{1664} a^{17} + \frac{5}{416} a^{16} - \frac{3}{128} a^{15} - \frac{47}{1664} a^{14} - \frac{7}{832} a^{13} + \frac{5}{128} a^{12} - \frac{131}{3328} a^{11} - \frac{55}{832} a^{10} + \frac{207}{3328} a^{9} + \frac{333}{3328} a^{8} + \frac{35}{416} a^{7} - \frac{101}{3328} a^{6} + \frac{2}{13} a^{5} + \frac{137}{832} a^{4} + \frac{95}{208} a^{3} + \frac{57}{208} a^{2} - \frac{6}{13} a - \frac{3}{13}$, $\frac{1}{6656} a^{24} - \frac{1}{6656} a^{23} - \frac{1}{6656} a^{22} - \frac{1}{3328} a^{21} - \frac{49}{6656} a^{20} - \frac{9}{6656} a^{19} + \frac{27}{6656} a^{18} - \frac{15}{3328} a^{17} + \frac{17}{3328} a^{16} + \frac{11}{832} a^{15} + \frac{21}{3328} a^{14} + \frac{31}{3328} a^{13} + \frac{339}{6656} a^{12} - \frac{657}{6656} a^{11} - \frac{405}{6656} a^{10} + \frac{309}{3328} a^{9} - \frac{61}{512} a^{8} - \frac{1397}{6656} a^{7} - \frac{1151}{6656} a^{6} + \frac{11}{208} a^{5} + \frac{301}{1664} a^{4} + \frac{189}{416} a^{3} + \frac{47}{416} a^{2} + \frac{5}{13} a - \frac{1}{13}$, $\frac{1}{306176} a^{25} + \frac{11}{306176} a^{24} - \frac{17}{306176} a^{23} - \frac{21}{153088} a^{22} - \frac{77}{306176} a^{21} - \frac{3}{13312} a^{20} + \frac{1379}{306176} a^{19} + \frac{53}{11776} a^{18} + \frac{2315}{153088} a^{17} - \frac{1}{1664} a^{16} - \frac{3175}{153088} a^{15} - \frac{3453}{153088} a^{14} - \frac{449}{23552} a^{13} - \frac{17621}{306176} a^{12} - \frac{11197}{306176} a^{11} + \frac{6281}{153088} a^{10} + \frac{1707}{306176} a^{9} - \frac{2373}{23552} a^{8} - \frac{52391}{306176} a^{7} + \frac{8635}{38272} a^{6} + \frac{8159}{38272} a^{5} + \frac{661}{9568} a^{4} - \frac{979}{2392} a^{3} - \frac{49}{104} a^{2} - \frac{53}{4784} a + \frac{1}{13}$, $\frac{1}{23269376} a^{26} + \frac{7}{126464} a^{24} + \frac{191}{23269376} a^{23} + \frac{1719}{23269376} a^{22} - \frac{301}{11634688} a^{21} + \frac{4099}{727168} a^{20} + \frac{162711}{23269376} a^{19} - \frac{2111}{612352} a^{18} - \frac{146399}{11634688} a^{17} + \frac{25943}{11634688} a^{16} - \frac{28749}{1454336} a^{15} - \frac{129419}{23269376} a^{14} + \frac{163731}{11634688} a^{13} - \frac{2635}{45448} a^{12} + \frac{578479}{23269376} a^{11} - \frac{57887}{1224704} a^{10} - \frac{533619}{11634688} a^{9} + \frac{23611}{505856} a^{8} - \frac{1740317}{23269376} a^{7} + \frac{8971}{612352} a^{6} - \frac{72991}{363584} a^{5} - \frac{25531}{223744} a^{4} + \frac{140241}{363584} a^{3} + \frac{212161}{727168} a^{2} - \frac{175367}{363584} a + \frac{73}{988}$, $\frac{1}{77162928724164608} a^{27} + \frac{1003364641}{77162928724164608} a^{26} - \frac{12732044201}{9645366090520576} a^{25} + \frac{14348545361}{3354909944528896} a^{24} - \frac{261437087097}{2967804950929408} a^{23} - \frac{7075224106163}{77162928724164608} a^{22} + \frac{3481158309595}{38581464362082304} a^{21} + \frac{17000181779923}{5935609901858816} a^{20} + \frac{465783261641501}{77162928724164608} a^{19} - \frac{19284418923381}{9645366090520576} a^{18} + \frac{70758806726225}{4822683045260288} a^{17} + \frac{581445909765791}{38581464362082304} a^{16} - \frac{1319854945552891}{77162928724164608} a^{15} - \frac{2740215813125}{77162928724164608} a^{14} + \frac{315087468740011}{38581464362082304} a^{13} + \frac{3991031730294671}{77162928724164608} a^{12} - \frac{2630765209500527}{38581464362082304} a^{11} + \frac{5384823552449725}{77162928724164608} a^{10} + \frac{30260596868299}{326961562390528} a^{9} + \frac{9225470237912381}{77162928724164608} a^{8} - \frac{9732207301119547}{77162928724164608} a^{7} + \frac{8862989339089673}{38581464362082304} a^{6} + \frac{2217476553913125}{9645366090520576} a^{5} + \frac{1910522689713569}{9645366090520576} a^{4} - \frac{619728370329145}{2411341522630144} a^{3} - \frac{63556245260825}{185487809433088} a^{2} + \frac{48655840106577}{1205670761315072} a + \frac{261103363169}{3276279242704}$, $\frac{1}{8181594473904530216746692050944} a^{28} + \frac{2229146955127}{4090797236952265108373346025472} a^{27} - \frac{145414354073906263853067}{8181594473904530216746692050944} a^{26} + \frac{6425519837908305834795999}{8181594473904530216746692050944} a^{25} - \frac{607770813895353108907552079}{8181594473904530216746692050944} a^{24} - \frac{79206506749197482353896805}{8181594473904530216746692050944} a^{23} + \frac{2613239436210306433452815}{23177321455820198914296578048} a^{22} - \frac{292001098392336423017606699}{8181594473904530216746692050944} a^{21} - \frac{6695885046011675839903995999}{1022699309238066277093336506368} a^{20} + \frac{52977417459091256155096281121}{8181594473904530216746692050944} a^{19} - \frac{12368970819467476880421467}{1736331594631691472144883712} a^{18} - \frac{466605655230128073752885427}{49286713698220061546666819584} a^{17} - \frac{40539081002108544773506791477}{8181594473904530216746692050944} a^{16} - \frac{4226709481768994556402631649}{2045398618476132554186673012736} a^{15} + \frac{56631461642532933816965678757}{8181594473904530216746692050944} a^{14} - \frac{69615242631981448898246996147}{8181594473904530216746692050944} a^{13} + \frac{18963397449751163845858211139}{355721498865414357249856176128} a^{12} + \frac{696758728371073847747404517255}{8181594473904530216746692050944} a^{11} + \frac{885346189391873733043599644685}{8181594473904530216746692050944} a^{10} + \frac{340004323456127957223831928113}{8181594473904530216746692050944} a^{9} - \frac{361752595188922427263976917873}{4090797236952265108373346025472} a^{8} - \frac{83007820237493759139775112925}{8181594473904530216746692050944} a^{7} + \frac{306284202111884180994157463}{131961201192008551883011162112} a^{6} - \frac{6494110117431717178283561291}{39334588816848702965128327168} a^{5} + \frac{154542471808164440895629349385}{1022699309238066277093336506368} a^{4} + \frac{1785136913558617700880009629}{4123787537250267246344098816} a^{3} - \frac{4358423265633799904483926203}{19667294408424351482564163584} a^{2} + \frac{2461497756275785857135869995}{5558148419772099332029002752} a + \frac{53933202961865624049220541}{347384276235756208251812672}$

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:  \( 1075559501191057.0 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{14}\cdot 1075559501191057.0 \cdot 1}{2\sqrt{77103436114042117740511038546742321228145336964361}}\approx 18.3069388830337$ (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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $29$ $29$ $29$ $29$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $29$ $29$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
3659Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3659.2t1.a.a$1$ $ 3659 $ \(\Q(\sqrt{-3659}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3659.29t2.a.m$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3659.29t2.a.k$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3659.29t2.a.a$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3659.29t2.a.i$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3659.29t2.a.c$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3659.29t2.a.b$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3659.29t2.a.g$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3659.29t2.a.f$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3659.29t2.a.j$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3659.29t2.a.e$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3659.29t2.a.l$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3659.29t2.a.d$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3659.29t2.a.h$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.3659.29t2.a.n$2$ $ 3659 $ 29.1.77103436114042117740511038546742321228145336964361.1 $D_{29}$ (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 *.