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Properties

Label 29.1.358...441.1
Degree $29$
Signature $[1, 14]$
Discriminant $3.588\times 10^{48}$
Root discriminant $47.24$
Ramified prime $2939$
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 - 7*x^27 - 50*x^26 + 119*x^25 + 495*x^24 + 59*x^23 - 530*x^22 + 637*x^21 + 2591*x^20 + 2977*x^19 + 1178*x^18 + 2213*x^17 + 5385*x^16 + 5965*x^15 + 6650*x^14 + 10883*x^13 + 16281*x^12 + 23955*x^11 + 21146*x^10 - 8059*x^9 - 7103*x^8 + 28957*x^7 + 37734*x^6 + 35524*x^5 + 2552*x^4 - 22800*x^3 - 19616*x^2 - 17024*x - 11776)
 
gp: K = bnfinit(x^29 - x^28 - 7*x^27 - 50*x^26 + 119*x^25 + 495*x^24 + 59*x^23 - 530*x^22 + 637*x^21 + 2591*x^20 + 2977*x^19 + 1178*x^18 + 2213*x^17 + 5385*x^16 + 5965*x^15 + 6650*x^14 + 10883*x^13 + 16281*x^12 + 23955*x^11 + 21146*x^10 - 8059*x^9 - 7103*x^8 + 28957*x^7 + 37734*x^6 + 35524*x^5 + 2552*x^4 - 22800*x^3 - 19616*x^2 - 17024*x - 11776, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-11776, -17024, -19616, -22800, 2552, 35524, 37734, 28957, -7103, -8059, 21146, 23955, 16281, 10883, 6650, 5965, 5385, 2213, 1178, 2977, 2591, 637, -530, 59, 495, 119, -50, -7, -1, 1]);
 

\(x^{29} - x^{28} - 7 x^{27} - 50 x^{26} + 119 x^{25} + 495 x^{24} + 59 x^{23} - 530 x^{22} + 637 x^{21} + 2591 x^{20} + 2977 x^{19} + 1178 x^{18} + 2213 x^{17} + 5385 x^{16} + 5965 x^{15} + 6650 x^{14} + 10883 x^{13} + 16281 x^{12} + 23955 x^{11} + 21146 x^{10} - 8059 x^{9} - 7103 x^{8} + 28957 x^{7} + 37734 x^{6} + 35524 x^{5} + 2552 x^{4} - 22800 x^{3} - 19616 x^{2} - 17024 x - 11776\)  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:  \(3587518960220469771354937124331329329937375710441\)\(\medspace = 2939^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $47.24$
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:  $2939$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{9} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{10} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{11} - \frac{1}{4} a^{5}$, $\frac{1}{8} a^{12} - \frac{1}{8} a^{9} - \frac{1}{8} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{13} - \frac{1}{8} a^{10} - \frac{1}{8} a^{7} + \frac{1}{8} a^{4}$, $\frac{1}{8} a^{14} - \frac{1}{8} a^{11} - \frac{1}{8} a^{8} + \frac{1}{8} a^{5}$, $\frac{1}{16} a^{15} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} - \frac{1}{16} a^{12} + \frac{1}{16} a^{11} + \frac{1}{16} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{3}{16} a^{7} - \frac{3}{16} a^{6} + \frac{3}{16} a^{5} + \frac{3}{16} a^{4} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{32} a^{16} + \frac{1}{16} a^{10} - \frac{1}{4} a^{7} - \frac{3}{32} a^{4} + \frac{1}{4} a$, $\frac{1}{32} a^{17} + \frac{1}{16} a^{11} - \frac{3}{32} a^{5}$, $\frac{1}{32} a^{18} - \frac{1}{16} a^{12} - \frac{1}{8} a^{9} + \frac{1}{32} a^{6} + \frac{1}{8} a^{3}$, $\frac{1}{64} a^{19} - \frac{1}{64} a^{18} - \frac{1}{64} a^{17} - \frac{1}{32} a^{15} + \frac{1}{32} a^{14} + \frac{1}{16} a^{11} + \frac{1}{32} a^{10} - \frac{3}{32} a^{9} - \frac{1}{32} a^{8} - \frac{9}{64} a^{7} - \frac{7}{64} a^{6} + \frac{5}{64} a^{5} - \frac{5}{32} a^{4} + \frac{1}{4} a^{3} - \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{64} a^{20} - \frac{1}{64} a^{17} + \frac{1}{32} a^{14} + \frac{3}{32} a^{11} + \frac{5}{64} a^{8} - \frac{5}{64} a^{5} - \frac{1}{8} a^{2}$, $\frac{1}{64} a^{21} - \frac{1}{64} a^{18} - \frac{1}{32} a^{15} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} + \frac{1}{32} a^{12} + \frac{1}{16} a^{11} + \frac{1}{16} a^{10} + \frac{1}{64} a^{9} + \frac{1}{16} a^{8} - \frac{3}{16} a^{7} + \frac{15}{64} a^{6} + \frac{3}{16} a^{5} + \frac{3}{16} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{128} a^{22} - \frac{1}{128} a^{20} - \frac{1}{128} a^{18} - \frac{1}{64} a^{17} + \frac{1}{64} a^{15} + \frac{1}{32} a^{14} - \frac{3}{64} a^{13} + \frac{1}{32} a^{12} + \frac{3}{64} a^{11} - \frac{1}{128} a^{10} + \frac{7}{64} a^{9} + \frac{5}{128} a^{8} + \frac{15}{64} a^{7} - \frac{27}{128} a^{6} + \frac{5}{32} a^{5} - \frac{3}{16} a^{4} - \frac{7}{16} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{128} a^{23} - \frac{1}{128} a^{21} - \frac{1}{128} a^{19} - \frac{1}{64} a^{18} - \frac{1}{64} a^{16} - \frac{1}{32} a^{15} + \frac{1}{64} a^{14} - \frac{1}{32} a^{13} - \frac{1}{64} a^{12} - \frac{9}{128} a^{11} + \frac{7}{64} a^{10} - \frac{3}{128} a^{9} - \frac{5}{64} a^{8} - \frac{19}{128} a^{7} - \frac{1}{32} a^{6} + \frac{1}{8} a^{5} - \frac{5}{32} a^{4} - \frac{3}{8} a^{3} - \frac{1}{2} a^{2} - \frac{1}{4} a$, $\frac{1}{128} a^{24} + \frac{1}{128} a^{18} - \frac{1}{16} a^{14} - \frac{1}{16} a^{13} + \frac{3}{128} a^{12} + \frac{1}{16} a^{11} + \frac{1}{16} a^{10} - \frac{1}{16} a^{9} + \frac{1}{16} a^{8} - \frac{3}{16} a^{7} + \frac{27}{128} a^{6} + \frac{3}{16} a^{5} + \frac{3}{16} a^{4} - \frac{3}{16} a^{3} - \frac{1}{4} a^{2}$, $\frac{1}{359168} a^{25} - \frac{787}{359168} a^{24} - \frac{191}{179584} a^{23} + \frac{873}{359168} a^{22} + \frac{509}{89792} a^{21} - \frac{2261}{359168} a^{20} - \frac{1617}{359168} a^{19} - \frac{363}{179584} a^{18} + \frac{987}{179584} a^{17} + \frac{1321}{89792} a^{16} - \frac{1077}{179584} a^{15} - \frac{2167}{44896} a^{14} - \frac{20347}{359168} a^{13} + \frac{9179}{359168} a^{12} - \frac{10761}{89792} a^{11} - \frac{37317}{359168} a^{10} + \frac{17671}{179584} a^{9} + \frac{1251}{15616} a^{8} - \frac{5165}{359168} a^{7} - \frac{26109}{179584} a^{6} - \frac{22227}{89792} a^{5} - \frac{4781}{22448} a^{4} + \frac{237}{976} a^{3} + \frac{1453}{11224} a^{2} - \frac{1185}{5612} a - \frac{1}{61}$, $\frac{1}{359168} a^{26} + \frac{375}{359168} a^{24} + \frac{481}{359168} a^{23} - \frac{1189}{359168} a^{22} - \frac{2155}{359168} a^{21} - \frac{505}{89792} a^{20} - \frac{2187}{359168} a^{19} - \frac{117}{7808} a^{18} - \frac{657}{179584} a^{17} - \frac{1069}{179584} a^{16} - \frac{141}{7808} a^{15} + \frac{1441}{359168} a^{14} + \frac{757}{179584} a^{13} - \frac{22173}{359168} a^{12} - \frac{36107}{359168} a^{11} + \frac{6399}{359168} a^{10} + \frac{38273}{359168} a^{9} - \frac{5423}{179584} a^{8} - \frac{11}{5888} a^{7} + \frac{2603}{89792} a^{6} + \frac{20115}{89792} a^{5} - \frac{173}{1403} a^{4} - \frac{435}{5612} a^{3} + \frac{1905}{11224} a^{2} + \frac{1711}{5612} a + \frac{6}{61}$, $\frac{1}{412420951496704} a^{27} + \frac{2076281}{6444077367136} a^{26} - \frac{2646223}{12888154734272} a^{25} - \frac{346326983169}{206210475748352} a^{24} + \frac{1463033329189}{412420951496704} a^{23} + \frac{259282335059}{206210475748352} a^{22} + \frac{183644285661}{103105237874176} a^{21} - \frac{140899861469}{103105237874176} a^{20} - \frac{1624817459419}{412420951496704} a^{19} - \frac{177118778683}{51552618937088} a^{18} - \frac{1076908980525}{103105237874176} a^{17} - \frac{42339298785}{206210475748352} a^{16} - \frac{4626194107113}{412420951496704} a^{15} + \frac{5153611907897}{206210475748352} a^{14} + \frac{342950034625}{6444077367136} a^{13} - \frac{1673674349115}{51552618937088} a^{12} + \frac{22648002286811}{412420951496704} a^{11} - \frac{7994206665053}{103105237874176} a^{10} - \frac{11549227505821}{103105237874176} a^{9} + \frac{6836322410973}{206210475748352} a^{8} - \frac{2216804329687}{9591184918528} a^{7} + \frac{247012967465}{3380499602432} a^{6} - \frac{258768458957}{1690249801216} a^{5} + \frac{11353559665101}{51552618937088} a^{4} - \frac{10291156989907}{25776309468544} a^{3} + \frac{5715030153655}{12888154734272} a^{2} - \frac{840033193837}{3222038683568} a - \frac{13727415149}{35022159604}$, $\frac{1}{11144994447581623263273115852158976} a^{28} - \frac{3373261618969304683}{2786248611895405815818278963039744} a^{27} - \frac{9698632740070619444923511}{22469746870124240450147410992256} a^{26} - \frac{1011864625493885596774257317}{5572497223790811631636557926079488} a^{25} + \frac{22315771071283690805350464218485}{11144994447581623263273115852158976} a^{24} - \frac{18907146491876009542174930594115}{5572497223790811631636557926079488} a^{23} + \frac{933642467039186620626294259653}{2786248611895405815818278963039744} a^{22} - \frac{11954190034509046406686079473953}{2786248611895405815818278963039744} a^{21} - \frac{75646364394429446864741583146691}{11144994447581623263273115852158976} a^{20} - \frac{18913258852341881381305278494931}{2786248611895405815818278963039744} a^{19} + \frac{20774650484348326809470729322363}{2786248611895405815818278963039744} a^{18} - \frac{60872197403698336222939067402161}{5572497223790811631636557926079488} a^{17} + \frac{75491199933600334474894458113935}{11144994447581623263273115852158976} a^{16} + \frac{59594925431835565758543417451735}{5572497223790811631636557926079488} a^{15} - \frac{48263358125029974659590372767549}{1393124305947702907909139481519872} a^{14} + \frac{16417434821056948806130642447049}{348281076486925726977284870379968} a^{13} + \frac{10353868119635271432051224023615}{182704827009534807594641243478016} a^{12} - \frac{106963196964385034849908513899531}{1393124305947702907909139481519872} a^{11} - \frac{45992402833673676534214517218371}{2786248611895405815818278963039744} a^{10} - \frac{632863925969831305882886428133555}{5572497223790811631636557926079488} a^{9} + \frac{41901518404871696774350747688925}{359515949921987847202358575876096} a^{8} + \frac{28404532215726691699596279435863}{5572497223790811631636557926079488} a^{7} - \frac{7050716361711239067226647422881}{121141243995452426774707781001728} a^{6} + \frac{111066485706359068863687522171247}{1393124305947702907909139481519872} a^{5} + \frac{116288622698331561098965606056687}{696562152973851453954569740759936} a^{4} - \frac{4894178256617425299005025812081}{15142655499431553346838472625216} a^{3} + \frac{10878495738948014427700965607439}{21767567280432857936080304398748} a^{2} + \frac{2071809838747545712476605612643}{21767567280432857936080304398748} a - \frac{99662773610393508429807022037}{236603992178618021044351134769}$  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:  \( 191758984500558.16 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{14}\cdot 191758984500558.16 \cdot 1}{2\sqrt{3587518960220469771354937124331329329937375710441}}\approx 15.1313185970062$ (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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2939Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2939.2t1.a.a$1$ $ 2939 $ \(\Q(\sqrt{-2939}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2939.29t2.a.k$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2939.29t2.a.g$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2939.29t2.a.n$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2939.29t2.a.c$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2939.29t2.a.j$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2939.29t2.a.l$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2939.29t2.a.b$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2939.29t2.a.d$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2939.29t2.a.e$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2939.29t2.a.f$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2939.29t2.a.i$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2939.29t2.a.h$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2939.29t2.a.a$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2939.29t2.a.m$2$ $ 2939 $ 29.1.3587518960220469771354937124331329329937375710441.1 $D_{29}$ (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 *.