Properties

Label 29.1.107...689.1
Degree $29$
Signature $[1, 14]$
Discriminant $1.071\times 10^{47}$
Root discriminant $41.85$
Ramified prime $2287$
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 - 12*x^28 + 55*x^27 - 97*x^26 + 34*x^25 + 11*x^24 + 138*x^23 - 214*x^22 + 43*x^21 + 224*x^20 - 104*x^19 - 50*x^18 + 748*x^17 - 1490*x^16 + 1965*x^15 - 460*x^14 - 575*x^13 + 2189*x^12 - 1286*x^11 - 53*x^10 + 3178*x^9 - 2804*x^8 + 3117*x^7 - 748*x^6 + 585*x^5 + 965*x^4 + 480*x^3 + 353*x^2 + 369*x - 1)
 
gp: K = bnfinit(x^29 - 12*x^28 + 55*x^27 - 97*x^26 + 34*x^25 + 11*x^24 + 138*x^23 - 214*x^22 + 43*x^21 + 224*x^20 - 104*x^19 - 50*x^18 + 748*x^17 - 1490*x^16 + 1965*x^15 - 460*x^14 - 575*x^13 + 2189*x^12 - 1286*x^11 - 53*x^10 + 3178*x^9 - 2804*x^8 + 3117*x^7 - 748*x^6 + 585*x^5 + 965*x^4 + 480*x^3 + 353*x^2 + 369*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 369, 353, 480, 965, 585, -748, 3117, -2804, 3178, -53, -1286, 2189, -575, -460, 1965, -1490, 748, -50, -104, 224, 43, -214, 138, 11, 34, -97, 55, -12, 1]);
 

\( x^{29} - 12 x^{28} + 55 x^{27} - 97 x^{26} + 34 x^{25} + 11 x^{24} + 138 x^{23} - 214 x^{22} + 43 x^{21} + 224 x^{20} - 104 x^{19} - 50 x^{18} + 748 x^{17} - 1490 x^{16} + 1965 x^{15} - 460 x^{14} - 575 x^{13} + 2189 x^{12} - 1286 x^{11} - 53 x^{10} + 3178 x^{9} - 2804 x^{8} + 3117 x^{7} - 748 x^{6} + 585 x^{5} + 965 x^{4} + 480 x^{3} + 353 x^{2} + 369 x - 1 \)

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:  \(107084423880431831080183695981363790438987742689\)\(\medspace = 2287^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $41.85$
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:  $2287$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{13} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{3} a^{6} - \frac{4}{9} a^{5} + \frac{4}{9} a^{4} - \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{3} a - \frac{2}{9}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{3} a^{7} + \frac{2}{9} a^{6} + \frac{1}{3} a^{4} + \frac{4}{9} a^{3} - \frac{4}{9} a^{2} + \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{12} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{2}{9} a^{7} + \frac{1}{3} a^{5} + \frac{4}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{9} a^{2} - \frac{2}{9} a - \frac{1}{3}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{45} a^{18} - \frac{1}{45} a^{17} + \frac{2}{45} a^{15} - \frac{2}{45} a^{13} - \frac{1}{15} a^{12} - \frac{1}{15} a^{10} + \frac{2}{15} a^{9} + \frac{2}{45} a^{8} + \frac{13}{45} a^{7} - \frac{7}{15} a^{6} + \frac{14}{45} a^{5} + \frac{7}{15} a^{4} - \frac{1}{3} a^{3} - \frac{4}{45} a^{2} - \frac{8}{45} a + \frac{16}{45}$, $\frac{1}{135} a^{19} - \frac{1}{135} a^{18} + \frac{2}{135} a^{16} + \frac{1}{45} a^{14} - \frac{1}{45} a^{13} - \frac{1}{9} a^{12} + \frac{22}{135} a^{11} - \frac{19}{135} a^{10} - \frac{1}{45} a^{9} - \frac{7}{135} a^{8} + \frac{2}{5} a^{7} + \frac{8}{45} a^{6} + \frac{7}{45} a^{5} - \frac{2}{9} a^{4} - \frac{14}{135} a^{3} + \frac{47}{135} a^{2} - \frac{8}{45} a + \frac{10}{27}$, $\frac{1}{405} a^{20} - \frac{1}{405} a^{18} + \frac{2}{405} a^{17} + \frac{17}{405} a^{16} + \frac{2}{45} a^{15} + \frac{1}{27} a^{14} + \frac{4}{135} a^{13} + \frac{7}{405} a^{12} + \frac{11}{135} a^{11} + \frac{53}{405} a^{10} + \frac{10}{81} a^{9} - \frac{28}{405} a^{8} - \frac{8}{45} a^{7} - \frac{1}{27} a^{6} + \frac{62}{135} a^{5} + \frac{136}{405} a^{4} - \frac{44}{135} a^{3} - \frac{7}{405} a^{2} + \frac{146}{405} a + \frac{13}{81}$, $\frac{1}{405} a^{21} - \frac{1}{405} a^{19} + \frac{2}{405} a^{18} + \frac{17}{405} a^{17} + \frac{2}{45} a^{16} + \frac{1}{27} a^{15} + \frac{4}{135} a^{14} + \frac{7}{405} a^{13} + \frac{11}{135} a^{12} + \frac{53}{405} a^{11} + \frac{10}{81} a^{10} - \frac{28}{405} a^{9} + \frac{7}{45} a^{8} - \frac{1}{27} a^{7} + \frac{62}{135} a^{6} + \frac{136}{405} a^{5} - \frac{44}{135} a^{4} - \frac{7}{405} a^{3} + \frac{146}{405} a^{2} + \frac{13}{81} a - \frac{1}{3}$, $\frac{1}{405} a^{22} - \frac{1}{405} a^{19} + \frac{1}{405} a^{18} - \frac{7}{405} a^{17} - \frac{19}{405} a^{16} - \frac{2}{135} a^{15} + \frac{13}{405} a^{14} - \frac{7}{135} a^{12} + \frac{62}{405} a^{11} + \frac{46}{405} a^{10} - \frac{31}{405} a^{9} - \frac{13}{405} a^{8} + \frac{41}{135} a^{7} - \frac{113}{405} a^{6} + \frac{11}{45} a^{5} + \frac{52}{135} a^{4} - \frac{124}{405} a^{3} - \frac{191}{405} a^{2} + \frac{47}{405} a - \frac{103}{405}$, $\frac{1}{1215} a^{23} + \frac{4}{1215} a^{19} - \frac{11}{1215} a^{18} - \frac{17}{1215} a^{17} + \frac{62}{1215} a^{16} + \frac{31}{1215} a^{15} - \frac{22}{405} a^{14} - \frac{7}{135} a^{13} - \frac{22}{405} a^{12} - \frac{16}{243} a^{11} - \frac{34}{243} a^{10} - \frac{152}{1215} a^{9} + \frac{119}{1215} a^{8} + \frac{247}{1215} a^{7} - \frac{98}{405} a^{6} + \frac{13}{27} a^{5} + \frac{139}{405} a^{4} - \frac{1}{243} a^{3} - \frac{359}{1215} a^{2} + \frac{16}{1215} a + \frac{106}{243}$, $\frac{1}{3645} a^{24} + \frac{1}{3645} a^{23} - \frac{1}{1215} a^{21} + \frac{1}{3645} a^{20} - \frac{4}{3645} a^{19} - \frac{4}{3645} a^{18} - \frac{58}{1215} a^{17} - \frac{4}{1215} a^{16} + \frac{11}{729} a^{15} + \frac{4}{243} a^{14} + \frac{2}{243} a^{13} - \frac{347}{3645} a^{12} + \frac{32}{3645} a^{11} + \frac{98}{3645} a^{10} - \frac{53}{405} a^{9} - \frac{5}{81} a^{8} - \frac{238}{729} a^{7} + \frac{52}{1215} a^{6} - \frac{29}{405} a^{5} - \frac{653}{3645} a^{4} - \frac{1702}{3645} a^{3} - \frac{463}{3645} a^{2} + \frac{259}{1215} a - \frac{1528}{3645}$, $\frac{1}{3645} a^{25} - \frac{1}{3645} a^{23} - \frac{1}{1215} a^{22} + \frac{4}{3645} a^{21} + \frac{4}{3645} a^{20} - \frac{17}{3645} a^{18} + \frac{2}{405} a^{17} - \frac{37}{729} a^{16} + \frac{86}{3645} a^{15} + \frac{7}{243} a^{14} - \frac{188}{3645} a^{13} - \frac{44}{3645} a^{12} + \frac{121}{1215} a^{11} + \frac{226}{3645} a^{10} - \frac{13}{135} a^{9} - \frac{488}{3645} a^{8} - \frac{1651}{3645} a^{7} - \frac{508}{1215} a^{6} + \frac{143}{729} a^{5} - \frac{68}{3645} a^{4} - \frac{388}{1215} a^{3} + \frac{934}{3645} a^{2} - \frac{1477}{3645} a + \frac{212}{729}$, $\frac{1}{3645} a^{26} + \frac{1}{3645} a^{23} + \frac{4}{3645} a^{22} + \frac{1}{3645} a^{21} + \frac{1}{3645} a^{20} - \frac{1}{405} a^{19} - \frac{19}{3645} a^{18} - \frac{1}{729} a^{17} - \frac{29}{729} a^{16} - \frac{152}{3645} a^{15} + \frac{79}{3645} a^{14} + \frac{202}{3645} a^{13} - \frac{182}{3645} a^{12} - \frac{43}{405} a^{11} + \frac{452}{3645} a^{10} + \frac{199}{3645} a^{9} - \frac{304}{3645} a^{8} - \frac{353}{3645} a^{7} - \frac{416}{3645} a^{6} - \frac{1409}{3645} a^{5} + \frac{649}{3645} a^{4} - \frac{59}{135} a^{3} - \frac{587}{3645} a^{2} - \frac{271}{729} a + \frac{1682}{3645}$, $\frac{1}{346275} a^{27} - \frac{14}{115425} a^{26} + \frac{37}{346275} a^{25} + \frac{29}{346275} a^{24} + \frac{43}{346275} a^{23} - \frac{341}{346275} a^{22} - \frac{31}{346275} a^{21} + \frac{179}{346275} a^{20} - \frac{194}{346275} a^{19} - \frac{47}{18225} a^{18} + \frac{9188}{346275} a^{17} - \frac{874}{18225} a^{16} - \frac{3791}{346275} a^{15} - \frac{13787}{346275} a^{14} - \frac{913}{18225} a^{13} + \frac{31121}{346275} a^{12} + \frac{52921}{346275} a^{11} - \frac{56}{18225} a^{10} - \frac{12469}{346275} a^{9} + \frac{13169}{346275} a^{8} + \frac{2275}{13851} a^{7} + \frac{141289}{346275} a^{6} + \frac{151982}{346275} a^{5} + \frac{6722}{13851} a^{4} + \frac{12283}{115425} a^{3} - \frac{30193}{69255} a^{2} + \frac{68323}{346275} a + \frac{10436}{115425}$, $\frac{1}{1626427001721958364925} a^{28} - \frac{846713975122591}{1626427001721958364925} a^{27} + \frac{26323939276662373}{325285400344391672985} a^{26} + \frac{50364120216921737}{542142333907319454975} a^{25} + \frac{139088161850022817}{1626427001721958364925} a^{24} + \frac{530343898760202652}{1626427001721958364925} a^{23} + \frac{389108011330849808}{1626427001721958364925} a^{22} + \frac{53923149680633822}{180714111302439818325} a^{21} - \frac{14568380555586946}{325285400344391672985} a^{20} - \frac{14865433777381211}{24275029876447139775} a^{19} - \frac{274307689210619288}{108428466781463890995} a^{18} + \frac{88952611739078633722}{1626427001721958364925} a^{17} - \frac{24153092934563326904}{542142333907319454975} a^{16} - \frac{71384271753318450188}{1626427001721958364925} a^{15} - \frac{86881272790218735979}{1626427001721958364925} a^{14} - \frac{6351971612269213304}{180714111302439818325} a^{13} + \frac{2822017130522643016}{24275029876447139775} a^{12} - \frac{219467508305968619168}{1626427001721958364925} a^{11} + \frac{56939101438483511299}{542142333907319454975} a^{10} + \frac{173268049379867956}{3424056845730438663} a^{9} - \frac{48475091872806068282}{542142333907319454975} a^{8} - \frac{66758935340761069361}{1626427001721958364925} a^{7} - \frac{605149746088217014144}{1626427001721958364925} a^{6} + \frac{34981907385533104108}{180714111302439818325} a^{5} - \frac{206119433252652407972}{542142333907319454975} a^{4} - \frac{48276760410957487501}{1626427001721958364925} a^{3} - \frac{618527947870122193232}{1626427001721958364925} a^{2} - \frac{684969437184629005874}{1626427001721958364925} a - \frac{410629689511917775132}{1626427001721958364925}$

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:  \( 4657227595142.479 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{14}\cdot 4657227595142.479 \cdot 1}{2\sqrt{107084423880431831080183695981363790438987742689}}\approx 2.12707335236917$ (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 $29$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $29$ $29$ $29$ $29$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $29$ $29$ $29$ $29$ $29$ $29$ $29$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
2287Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2287.2t1.a.a$1$ $ 2287 $ \(\Q(\sqrt{-2287}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.2287.29t2.a.c$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2287.29t2.a.j$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2287.29t2.a.l$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2287.29t2.a.h$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2287.29t2.a.a$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2287.29t2.a.f$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2287.29t2.a.e$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2287.29t2.a.k$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2287.29t2.a.n$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2287.29t2.a.m$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2287.29t2.a.d$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2287.29t2.a.g$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2287.29t2.a.b$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.2287.29t2.a.i$2$ $ 2287 $ 29.1.107084423880431831080183695981363790438987742689.1 $D_{29}$ (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 *.