Properties

Label 29.1.123...441.1
Degree $29$
Signature $[1, 14]$
Discriminant $1.239\times 10^{47}$
Root discriminant $42.06$
Ramified prime $2311$
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 - 8*x^28 + 6*x^27 + 45*x^26 + 49*x^25 - 158*x^24 - 252*x^23 - 117*x^22 + 459*x^21 + 1008*x^20 + 2232*x^19 + 2691*x^18 + 1728*x^17 - 2718*x^16 - 7677*x^15 - 13104*x^14 - 15537*x^13 - 15042*x^12 - 6534*x^11 + 6768*x^10 + 22065*x^9 + 32523*x^8 + 38385*x^7 + 37197*x^6 + 29657*x^5 + 17606*x^4 + 7536*x^3 + 1998*x^2 + 377*x - 1)
 
gp: K = bnfinit(x^29 - 8*x^28 + 6*x^27 + 45*x^26 + 49*x^25 - 158*x^24 - 252*x^23 - 117*x^22 + 459*x^21 + 1008*x^20 + 2232*x^19 + 2691*x^18 + 1728*x^17 - 2718*x^16 - 7677*x^15 - 13104*x^14 - 15537*x^13 - 15042*x^12 - 6534*x^11 + 6768*x^10 + 22065*x^9 + 32523*x^8 + 38385*x^7 + 37197*x^6 + 29657*x^5 + 17606*x^4 + 7536*x^3 + 1998*x^2 + 377*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 377, 1998, 7536, 17606, 29657, 37197, 38385, 32523, 22065, 6768, -6534, -15042, -15537, -13104, -7677, -2718, 1728, 2691, 2232, 1008, 459, -117, -252, -158, 49, 45, 6, -8, 1]);
 

\(x^{29} - 8 x^{28} + 6 x^{27} + 45 x^{26} + 49 x^{25} - 158 x^{24} - 252 x^{23} - 117 x^{22} + 459 x^{21} + 1008 x^{20} + 2232 x^{19} + 2691 x^{18} + 1728 x^{17} - 2718 x^{16} - 7677 x^{15} - 13104 x^{14} - 15537 x^{13} - 15042 x^{12} - 6534 x^{11} + 6768 x^{10} + 22065 x^{9} + 32523 x^{8} + 38385 x^{7} + 37197 x^{6} + 29657 x^{5} + 17606 x^{4} + 7536 x^{3} + 1998 x^{2} + 377 x - 1\)  Toggle raw display

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:  \(123936502657789265324679017710546697504158351441\)\(\medspace = 2311^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $42.06$
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:  $2311$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{9} a^{19} + \frac{1}{9} a^{11} - \frac{2}{9} a^{3}$, $\frac{1}{9} a^{20} + \frac{1}{9} a^{12} - \frac{2}{9} a^{4}$, $\frac{1}{9} a^{21} + \frac{1}{9} a^{13} - \frac{2}{9} a^{5}$, $\frac{1}{9} a^{22} + \frac{1}{9} a^{14} - \frac{2}{9} a^{6}$, $\frac{1}{9} a^{23} + \frac{1}{9} a^{15} - \frac{2}{9} a^{7}$, $\frac{1}{999} a^{24} + \frac{8}{333} a^{23} - \frac{1}{333} a^{22} - \frac{16}{333} a^{21} + \frac{11}{333} a^{19} + \frac{1}{37} a^{18} + \frac{10}{333} a^{17} - \frac{5}{111} a^{16} - \frac{55}{333} a^{15} + \frac{29}{333} a^{14} - \frac{13}{333} a^{13} + \frac{11}{111} a^{12} - \frac{55}{333} a^{11} + \frac{13}{111} a^{10} - \frac{50}{333} a^{9} - \frac{1}{333} a^{8} - \frac{106}{333} a^{7} + \frac{62}{333} a^{6} + \frac{92}{333} a^{5} - \frac{23}{111} a^{4} - \frac{1}{9} a^{3} - \frac{10}{111} a^{2} - \frac{32}{333} a - \frac{7}{999}$, $\frac{1}{999} a^{25} - \frac{8}{333} a^{23} + \frac{8}{333} a^{22} + \frac{14}{333} a^{21} + \frac{11}{333} a^{20} + \frac{4}{333} a^{19} + \frac{16}{333} a^{18} + \frac{4}{333} a^{17} + \frac{1}{37} a^{16} - \frac{20}{333} a^{15} - \frac{43}{333} a^{14} - \frac{25}{333} a^{13} + \frac{41}{333} a^{12} - \frac{47}{333} a^{11} + \frac{13}{333} a^{10} + \frac{5}{111} a^{9} - \frac{5}{37} a^{8} + \frac{127}{333} a^{7} - \frac{64}{333} a^{6} + \frac{128}{333} a^{5} + \frac{65}{333} a^{4} + \frac{7}{333} a^{3} + \frac{133}{333} a^{2} + \frac{77}{999} a - \frac{2}{37}$, $\frac{1}{2997} a^{26} + \frac{1}{2997} a^{25} - \frac{1}{2997} a^{24} - \frac{38}{999} a^{23} + \frac{4}{111} a^{22} + \frac{1}{37} a^{21} + \frac{52}{999} a^{20} + \frac{17}{333} a^{19} + \frac{5}{999} a^{18} - \frac{53}{999} a^{17} - \frac{23}{999} a^{16} - \frac{107}{999} a^{15} - \frac{47}{333} a^{14} + \frac{29}{333} a^{13} + \frac{124}{999} a^{12} - \frac{26}{333} a^{11} - \frac{2}{27} a^{10} - \frac{16}{111} a^{9} - \frac{163}{999} a^{8} - \frac{377}{999} a^{7} - \frac{157}{333} a^{6} - \frac{32}{333} a^{5} - \frac{257}{999} a^{4} - \frac{14}{37} a^{3} + \frac{1070}{2997} a^{2} + \frac{590}{2997} a - \frac{548}{2997}$, $\frac{1}{5945895153} a^{27} - \frac{821971}{5945895153} a^{26} - \frac{981727}{1981965051} a^{25} - \frac{2553190}{5945895153} a^{24} + \frac{766835}{37395567} a^{23} + \frac{1151467}{73406113} a^{22} - \frac{101709440}{1981965051} a^{21} + \frac{43792552}{1981965051} a^{20} + \frac{54914366}{1981965051} a^{19} - \frac{1885378}{60059547} a^{18} + \frac{973435}{180178641} a^{17} - \frac{32527093}{1981965051} a^{16} - \frac{166771595}{1981965051} a^{15} + \frac{26282692}{220218339} a^{14} - \frac{131904776}{1981965051} a^{13} - \frac{74091539}{1981965051} a^{12} + \frac{214118263}{1981965051} a^{11} - \frac{7824457}{180178641} a^{10} - \frac{225599566}{1981965051} a^{9} + \frac{7556117}{73406113} a^{8} + \frac{300597247}{1981965051} a^{7} - \frac{400759}{1790393} a^{6} + \frac{569803438}{1981965051} a^{5} + \frac{605270764}{1981965051} a^{4} - \frac{1920541168}{5945895153} a^{3} + \frac{580266025}{5945895153} a^{2} + \frac{129624889}{660655017} a - \frac{1257545330}{5945895153}$, $\frac{1}{45023901421371053540731911} a^{28} - \frac{1784151275979902}{45023901421371053540731911} a^{27} - \frac{3137833376572870760}{49314240330088777153047} a^{26} - \frac{137795543474366956165}{569922802802165234692809} a^{25} - \frac{1923510008271521104717}{4093081947397368503702901} a^{24} + \frac{52941445838978445281194}{1364360649132456167900967} a^{23} + \frac{612851128858011745840825}{15007967140457017846910637} a^{22} + \frac{22742339761622009707061}{714665101926524659376697} a^{21} - \frac{517512193652070593585771}{15007967140457017846910637} a^{20} - \frac{315208381130859171325346}{15007967140457017846910637} a^{19} - \frac{3183550158291612199133}{194908664161779452557281} a^{18} + \frac{590375686922700674432}{15022990130587605452363} a^{17} + \frac{14766504522285443675977}{1364360649132456167900967} a^{16} + \frac{44151157147231029592}{326978085371293881063} a^{15} - \frac{253137978449915827626488}{15007967140457017846910637} a^{14} + \frac{3832383622450685856814}{454786883044152055966989} a^{13} + \frac{2138287087726643501639}{102095014560932094196671} a^{12} - \frac{57094890456843668578210}{555850634831741401737431} a^{11} - \frac{1006847306573395513137620}{15007967140457017846910637} a^{10} + \frac{1038734109487166005249036}{15007967140457017846910637} a^{9} - \frac{1729890477491486217989282}{15007967140457017846910637} a^{8} + \frac{3100407457760906999221697}{15007967140457017846910637} a^{7} + \frac{1354116082320500333362579}{15007967140457017846910637} a^{6} + \frac{129214783323888681101608}{454786883044152055966989} a^{5} - \frac{10071279624841162680922411}{45023901421371053540731911} a^{4} - \frac{6140360594750389010939743}{45023901421371053540731911} a^{3} - \frac{1223134133082725341815227}{4093081947397368503702901} a^{2} - \frac{1382648889213602824878598}{4093081947397368503702901} a - \frac{17659114575663023992831384}{45023901421371053540731911}$  Toggle raw display

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 $2$)  Toggle raw display
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:  \( 4095120529090.4673 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{14}\cdot 4095120529090.4673 \cdot 1}{2\sqrt{123936502657789265324679017710546697504158351441}}\approx 1.73854216153215$ (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 $29$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $29$ $29$ $29$ $29$ $29$ $29$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $29$

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
2311Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2311.2t1.a.a$1$ $ 2311 $ \(\Q(\sqrt{-2311}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2311.29t2.a.d$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2311.29t2.a.h$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2311.29t2.a.e$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2311.29t2.a.k$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2311.29t2.a.g$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2311.29t2.a.n$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2311.29t2.a.a$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2311.29t2.a.i$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2311.29t2.a.b$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2311.29t2.a.l$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2311.29t2.a.m$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2311.29t2.a.c$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2311.29t2.a.j$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2311.29t2.a.f$2$ $ 2311 $ 29.1.123936502657789265324679017710546697504158351441.1 $D_{29}$ (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 *.