Properties

Label 29.1.186...489.1
Degree $29$
Signature $[1, 14]$
Discriminant $1.866\times 10^{41}$
Root discriminant $26.49$
Ramified prime $887$
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 + 7*x^26 + 29*x^25 + 70*x^24 + 115*x^23 + 217*x^22 + 336*x^21 + 447*x^20 + 628*x^19 + 758*x^18 + 820*x^17 + 951*x^16 + 953*x^15 + 866*x^14 + 839*x^13 + 574*x^12 + 405*x^11 + 413*x^10 + 349*x^9 + 268*x^8 + 247*x^7 - 13*x^6 - 6*x^5 + 265*x^4 + 95*x^3 - 98*x^2 + 32*x + 1)
 
gp: K = bnfinit(x^29 - x^28 + 7*x^27 + 7*x^26 + 29*x^25 + 70*x^24 + 115*x^23 + 217*x^22 + 336*x^21 + 447*x^20 + 628*x^19 + 758*x^18 + 820*x^17 + 951*x^16 + 953*x^15 + 866*x^14 + 839*x^13 + 574*x^12 + 405*x^11 + 413*x^10 + 349*x^9 + 268*x^8 + 247*x^7 - 13*x^6 - 6*x^5 + 265*x^4 + 95*x^3 - 98*x^2 + 32*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 32, -98, 95, 265, -6, -13, 247, 268, 349, 413, 405, 574, 839, 866, 953, 951, 820, 758, 628, 447, 336, 217, 115, 70, 29, 7, 7, -1, 1]);
 

\( x^{29} - x^{28} + 7 x^{27} + 7 x^{26} + 29 x^{25} + 70 x^{24} + 115 x^{23} + 217 x^{22} + 336 x^{21} + 447 x^{20} + 628 x^{19} + 758 x^{18} + 820 x^{17} + 951 x^{16} + 953 x^{15} + 866 x^{14} + 839 x^{13} + 574 x^{12} + 405 x^{11} + 413 x^{10} + 349 x^{9} + 268 x^{8} + 247 x^{7} - 13 x^{6} - 6 x^{5} + 265 x^{4} + 95 x^{3} - 98 x^{2} + 32 x + 1 \)

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:  \(186608180192063027791735955893818125821489\)\(\medspace = 887^{14}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $26.49$
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:  $887$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{5} a^{22} - \frac{2}{5} a^{21} + \frac{1}{5} a^{20} - \frac{1}{5} a^{19} - \frac{1}{5} a^{18} - \frac{2}{5} a^{16} - \frac{1}{5} a^{15} - \frac{2}{5} a^{14} + \frac{2}{5} a^{13} + \frac{2}{5} a^{12} - \frac{1}{5} a^{10} + \frac{2}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{7} + \frac{1}{5} a^{6} + \frac{2}{5} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{23} + \frac{2}{5} a^{21} + \frac{1}{5} a^{20} + \frac{2}{5} a^{19} - \frac{2}{5} a^{18} - \frac{2}{5} a^{17} + \frac{1}{5} a^{15} - \frac{2}{5} a^{14} + \frac{1}{5} a^{13} - \frac{1}{5} a^{12} - \frac{1}{5} a^{11} - \frac{2}{5} a^{9} - \frac{1}{5} a^{8} - \frac{2}{5} a^{7} + \frac{2}{5} a^{6} + \frac{2}{5} a^{5} - \frac{1}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{24} - \frac{1}{5}$, $\frac{1}{3565} a^{25} - \frac{129}{3565} a^{24} + \frac{133}{3565} a^{23} - \frac{231}{3565} a^{22} + \frac{1638}{3565} a^{21} - \frac{988}{3565} a^{20} + \frac{2}{115} a^{19} + \frac{258}{713} a^{18} + \frac{669}{3565} a^{17} - \frac{993}{3565} a^{16} - \frac{156}{3565} a^{15} + \frac{586}{3565} a^{14} + \frac{1206}{3565} a^{13} - \frac{9}{23} a^{12} + \frac{357}{3565} a^{11} + \frac{37}{155} a^{10} + \frac{1652}{3565} a^{9} + \frac{543}{3565} a^{8} - \frac{1007}{3565} a^{7} + \frac{120}{713} a^{6} + \frac{1701}{3565} a^{5} - \frac{1312}{3565} a^{4} + \frac{576}{3565} a^{3} + \frac{844}{3565} a^{2} + \frac{643}{3565} a + \frac{1479}{3565}$, $\frac{1}{5878685} a^{26} + \frac{762}{5878685} a^{25} - \frac{529059}{5878685} a^{24} - \frac{5143}{69161} a^{23} - \frac{60157}{5878685} a^{22} + \frac{15457}{345805} a^{21} + \frac{1075513}{5878685} a^{20} - \frac{2642173}{5878685} a^{19} + \frac{919047}{5878685} a^{18} - \frac{207822}{1175737} a^{17} + \frac{1949969}{5878685} a^{16} + \frac{2163867}{5878685} a^{15} - \frac{1078066}{5878685} a^{14} + \frac{2891301}{5878685} a^{13} - \frac{1771634}{5878685} a^{12} - \frac{88224}{1175737} a^{11} + \frac{1450077}{5878685} a^{10} + \frac{2604006}{5878685} a^{9} + \frac{2551932}{5878685} a^{8} + \frac{2778878}{5878685} a^{7} - \frac{1885047}{5878685} a^{6} + \frac{57156}{1175737} a^{5} + \frac{1226551}{5878685} a^{4} + \frac{464863}{5878685} a^{3} - \frac{374809}{1175737} a^{2} - \frac{2288303}{5878685} a - \frac{626562}{5878685}$, $\frac{1}{5878685} a^{27} + \frac{74}{5878685} a^{25} - \frac{293307}{5878685} a^{24} - \frac{224139}{5878685} a^{23} + \frac{200839}{5878685} a^{22} + \frac{846149}{5878685} a^{21} - \frac{438289}{1175737} a^{20} - \frac{344917}{5878685} a^{19} + \frac{126729}{5878685} a^{18} - \frac{495657}{5878685} a^{17} - \frac{2626883}{5878685} a^{16} + \frac{1673892}{5878685} a^{15} - \frac{168604}{1175737} a^{14} + \frac{2317709}{5878685} a^{13} - \frac{2246848}{5878685} a^{12} - \frac{72176}{189635} a^{11} + \frac{1965791}{5878685} a^{10} - \frac{206174}{5878685} a^{9} + \frac{231157}{1175737} a^{8} - \frac{1307298}{5878685} a^{7} - \frac{835699}{5878685} a^{6} - \frac{1880883}{5878685} a^{5} + \frac{78898}{5878685} a^{4} - \frac{1396453}{5878685} a^{3} - \frac{7353}{51119} a^{2} + \frac{532342}{5878685} a + \frac{259618}{1175737}$, $\frac{1}{281584562291745127357175} a^{28} + \frac{11479488884720458}{281584562291745127357175} a^{27} - \frac{19033360167934471}{281584562291745127357175} a^{26} + \frac{35358306132302069508}{281584562291745127357175} a^{25} - \frac{3281610516165249962879}{281584562291745127357175} a^{24} + \frac{1908329138477231317464}{281584562291745127357175} a^{23} + \frac{24404411634357207229121}{281584562291745127357175} a^{22} - \frac{153418252348501507941}{641422693147483205825} a^{21} + \frac{11592437476734281216638}{56316912458349025471435} a^{20} + \frac{11244682521367161941532}{281584562291745127357175} a^{19} + \frac{3713374888456341543739}{9709812492818797495075} a^{18} - \frac{1626780768893902125203}{9083372977153068624425} a^{17} + \frac{61634017991244879265373}{281584562291745127357175} a^{16} - \frac{4151630375009343142297}{9083372977153068624425} a^{15} + \frac{3883327667842798773514}{11263382491669805094287} a^{14} + \frac{14926908508551393561216}{281584562291745127357175} a^{13} - \frac{82777527378334165497822}{281584562291745127357175} a^{12} + \frac{2665824665323003339808}{16563797781867360432775} a^{11} + \frac{102010997912346846495574}{281584562291745127357175} a^{10} - \frac{117132512431782770481591}{281584562291745127357175} a^{9} + \frac{796001442545231407109}{2448561411232566324845} a^{8} + \frac{104030896883254969064308}{281584562291745127357175} a^{7} - \frac{61288894846440359864496}{281584562291745127357175} a^{6} - \frac{39712180834690456806472}{281584562291745127357175} a^{5} + \frac{5599370890417080989756}{281584562291745127357175} a^{4} + \frac{66765004217076341405159}{281584562291745127357175} a^{3} - \frac{3162735191412940882459}{281584562291745127357175} a^{2} + \frac{6521555215697223342231}{281584562291745127357175} a - \frac{77945159604356487433579}{281584562291745127357175}$

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:  \( 961536136.7336683 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{14}\cdot 961536136.7336683 \cdot 1}{2\sqrt{186608180192063027791735955893818125821489}}\approx 0.332673669671707$ (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 $29$ $29$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ $29$ $29$ $29$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ $29$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $29$ $29$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{14}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ $29$

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
887Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.887.2t1.a.a$1$ $ 887 $ \(\Q(\sqrt{-887}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.887.29t2.a.g$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.887.29t2.a.b$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.887.29t2.a.m$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.887.29t2.a.j$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.887.29t2.a.e$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.887.29t2.a.i$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.887.29t2.a.l$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.887.29t2.a.h$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.887.29t2.a.f$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.887.29t2.a.d$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.887.29t2.a.c$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.887.29t2.a.a$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.887.29t2.a.n$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (as 29T2) $1$ $0$
* 2.887.29t2.a.k$2$ $ 887 $ 29.1.186608180192063027791735955893818125821489.1 $D_{29}$ (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 *.