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Properties

Label 27.1.124...951.1
Degree $27$
Signature $[1, 13]$
Discriminant $-1.250\times 10^{45}$
Root discriminant $46.80$
Ramified primes $7, 199$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 13*x^26 + 64*x^25 - 128*x^24 - 64*x^23 + 1031*x^22 - 3598*x^21 + 9810*x^20 - 21567*x^19 + 36061*x^18 - 52129*x^17 + 97285*x^16 - 181966*x^15 + 235897*x^14 - 374052*x^13 + 634209*x^12 - 258724*x^11 + 64070*x^10 - 1005397*x^9 + 111021*x^8 + 1266741*x^7 + 944398*x^6 - 1133842*x^5 - 1337690*x^4 + 643480*x^3 + 1031258*x^2 - 50650*x + 12587)
 
gp: K = bnfinit(x^27 - 13*x^26 + 64*x^25 - 128*x^24 - 64*x^23 + 1031*x^22 - 3598*x^21 + 9810*x^20 - 21567*x^19 + 36061*x^18 - 52129*x^17 + 97285*x^16 - 181966*x^15 + 235897*x^14 - 374052*x^13 + 634209*x^12 - 258724*x^11 + 64070*x^10 - 1005397*x^9 + 111021*x^8 + 1266741*x^7 + 944398*x^6 - 1133842*x^5 - 1337690*x^4 + 643480*x^3 + 1031258*x^2 - 50650*x + 12587, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![12587, -50650, 1031258, 643480, -1337690, -1133842, 944398, 1266741, 111021, -1005397, 64070, -258724, 634209, -374052, 235897, -181966, 97285, -52129, 36061, -21567, 9810, -3598, 1031, -64, -128, 64, -13, 1]);
 

\( x^{27} - 13 x^{26} + 64 x^{25} - 128 x^{24} - 64 x^{23} + 1031 x^{22} - 3598 x^{21} + 9810 x^{20} - 21567 x^{19} + 36061 x^{18} - 52129 x^{17} + 97285 x^{16} - 181966 x^{15} + 235897 x^{14} - 374052 x^{13} + 634209 x^{12} - 258724 x^{11} + 64070 x^{10} - 1005397 x^{9} + 111021 x^{8} + 1266741 x^{7} + 944398 x^{6} - 1133842 x^{5} - 1337690 x^{4} + 643480 x^{3} + 1031258 x^{2} - 50650 x + 12587 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $27$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 13]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-1249840845592629138161700045087827071680376951\)\(\medspace = -\,7^{18}\cdot 199^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $46.80$
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:  $7, 199$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{6} + \frac{1}{3} a^{5} + \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{8} - \frac{1}{3}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{9} a^{14} - \frac{1}{9} a^{13} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{2}{9} a^{6} - \frac{1}{9} a^{4} - \frac{2}{9} a^{3} - \frac{1}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{13} + \frac{1}{9} a^{12} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{3} + \frac{2}{9} a^{2} + \frac{1}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{8} - \frac{2}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{9} - \frac{2}{9} a$, $\frac{1}{9} a^{18} + \frac{1}{9} a^{10} - \frac{2}{9} a^{2}$, $\frac{1}{297} a^{19} + \frac{1}{99} a^{18} + \frac{14}{297} a^{17} + \frac{10}{297} a^{16} - \frac{14}{297} a^{15} + \frac{8}{297} a^{14} + \frac{2}{33} a^{13} - \frac{35}{297} a^{12} + \frac{13}{99} a^{11} - \frac{5}{99} a^{10} - \frac{4}{33} a^{9} - \frac{20}{297} a^{7} - \frac{35}{99} a^{6} + \frac{38}{297} a^{5} - \frac{116}{297} a^{4} + \frac{115}{297} a^{3} + \frac{7}{27} a^{2} - \frac{46}{99} a + \frac{82}{297}$, $\frac{1}{891} a^{20} + \frac{1}{891} a^{19} - \frac{25}{891} a^{18} - \frac{2}{99} a^{17} - \frac{34}{891} a^{16} + \frac{4}{99} a^{15} + \frac{35}{891} a^{14} - \frac{5}{891} a^{13} + \frac{109}{891} a^{12} + \frac{46}{297} a^{11} - \frac{8}{99} a^{10} - \frac{1}{33} a^{9} + \frac{13}{891} a^{8} + \frac{1}{891} a^{7} - \frac{313}{891} a^{6} + \frac{134}{297} a^{5} + \frac{116}{891} a^{4} + \frac{92}{297} a^{3} + \frac{170}{891} a^{2} - \frac{71}{891} a + \frac{166}{891}$, $\frac{1}{891} a^{21} + \frac{1}{891} a^{19} - \frac{1}{81} a^{18} - \frac{34}{891} a^{17} + \frac{43}{891} a^{16} + \frac{17}{891} a^{15} - \frac{2}{81} a^{14} + \frac{35}{297} a^{13} + \frac{74}{891} a^{12} + \frac{17}{297} a^{11} - \frac{7}{99} a^{10} - \frac{41}{891} a^{9} + \frac{29}{297} a^{8} - \frac{62}{891} a^{7} + \frac{157}{891} a^{6} - \frac{250}{891} a^{5} - \frac{101}{891} a^{4} - \frac{367}{891} a^{3} + \frac{155}{891} a^{2} - \frac{41}{297} a - \frac{130}{891}$, $\frac{1}{891} a^{22} + \frac{1}{33} a^{18} + \frac{31}{891} a^{17} - \frac{1}{33} a^{16} - \frac{28}{891} a^{15} - \frac{32}{891} a^{14} - \frac{2}{891} a^{13} + \frac{17}{891} a^{12} + \frac{23}{297} a^{11} + \frac{148}{891} a^{10} - \frac{7}{297} a^{9} + \frac{41}{297} a^{8} + \frac{38}{297} a^{7} + \frac{32}{99} a^{6} - \frac{245}{891} a^{5} - \frac{64}{297} a^{4} + \frac{71}{891} a^{3} - \frac{161}{891} a^{2} + \frac{364}{891} a + \frac{323}{891}$, $\frac{1}{2673} a^{23} + \frac{1}{2673} a^{22} + \frac{76}{2673} a^{18} + \frac{2}{243} a^{17} + \frac{71}{2673} a^{16} + \frac{7}{891} a^{15} - \frac{52}{2673} a^{14} - \frac{124}{891} a^{13} - \frac{157}{2673} a^{12} + \frac{23}{243} a^{11} - \frac{161}{2673} a^{10} - \frac{104}{891} a^{9} - \frac{119}{891} a^{8} - \frac{49}{891} a^{7} + \frac{403}{2673} a^{6} + \frac{56}{243} a^{5} - \frac{751}{2673} a^{4} - \frac{4}{33} a^{3} + \frac{896}{2673} a^{2} - \frac{146}{891} a - \frac{406}{2673}$, $\frac{1}{8019} a^{24} - \frac{1}{8019} a^{23} - \frac{2}{8019} a^{22} - \frac{1}{2673} a^{21} + \frac{1}{2673} a^{20} - \frac{5}{8019} a^{19} + \frac{179}{8019} a^{18} - \frac{56}{2673} a^{17} + \frac{323}{8019} a^{16} + \frac{206}{8019} a^{15} + \frac{146}{8019} a^{14} + \frac{581}{8019} a^{13} + \frac{278}{2673} a^{12} - \frac{1}{8019} a^{11} + \frac{901}{8019} a^{10} - \frac{113}{2673} a^{9} + \frac{16}{2673} a^{8} + \frac{724}{8019} a^{7} + \frac{668}{8019} a^{6} + \frac{17}{297} a^{5} + \frac{3206}{8019} a^{4} + \frac{1286}{8019} a^{3} - \frac{700}{8019} a^{2} + \frac{1112}{8019} a + \frac{1889}{8019}$, $\frac{1}{796262643} a^{25} + \frac{9943}{265420881} a^{24} - \frac{35924}{265420881} a^{23} + \frac{224548}{796262643} a^{22} - \frac{25055}{88473627} a^{21} + \frac{99754}{796262643} a^{20} - \frac{131125}{265420881} a^{19} - \frac{18814159}{796262643} a^{18} - \frac{40655014}{796262643} a^{17} + \frac{15372040}{796262643} a^{16} + \frac{3058289}{72387513} a^{15} - \frac{3996790}{72387513} a^{14} - \frac{73127674}{796262643} a^{13} + \frac{37665887}{796262643} a^{12} + \frac{24741053}{265420881} a^{11} - \frac{116608952}{796262643} a^{10} - \frac{355940}{24129171} a^{9} + \frac{7389524}{46838979} a^{8} + \frac{14159927}{88473627} a^{7} + \frac{300718223}{796262643} a^{6} - \frac{356496568}{796262643} a^{5} + \frac{299922586}{796262643} a^{4} + \frac{212657569}{796262643} a^{3} + \frac{387826744}{796262643} a^{2} - \frac{158548337}{796262643} a + \frac{349635677}{796262643}$, $\frac{1}{255100998239883840943586201145397255182732430830512237554326453} a^{26} - \frac{142401471298041110375563252321734252651554121135542681}{255100998239883840943586201145397255182732430830512237554326453} a^{25} - \frac{2183031130954372360682272219685516471779600430481261773830}{85033666079961280314528733715132418394244143610170745851442151} a^{24} + \frac{26436626309757943676665681833898032915842929902588414394429}{255100998239883840943586201145397255182732430830512237554326453} a^{23} + \frac{111281189065174969962969637189419506641780649262403582303162}{255100998239883840943586201145397255182732430830512237554326453} a^{22} - \frac{108120595566432712781046835316871009752800462624141655617577}{255100998239883840943586201145397255182732430830512237554326453} a^{21} + \frac{4791131228552065609613978520607687542658596884996852905268}{15005941072934343584916835361493956187219554754736013973783909} a^{20} + \frac{19791425088103491512195755478657999886926187165337722803725}{15005941072934343584916835361493956187219554754736013973783909} a^{19} - \frac{28691181711154993401621794265268395617197923558323751468325}{789786372259702293942991334815471378274713408144000735462311} a^{18} - \frac{194447356045618573240146407052673970226601259140638576206743}{7730333279996480028593521246830219854022194873651885986494741} a^{17} - \frac{134321973466398476386256876317327149636795449049760781010545}{15005941072934343584916835361493956187219554754736013973783909} a^{16} - \frac{214333379288250960782789615224005178939235262324729708694391}{23190999839989440085780563740490659562066584620955657959484223} a^{15} + \frac{3693254460991818035752217898422014163395385056809716812196462}{85033666079961280314528733715132418394244143610170745851442151} a^{14} + \frac{12307367033302643180161574072189108054747452638356205643947175}{255100998239883840943586201145397255182732430830512237554326453} a^{13} - \frac{3637734391538417483099825542006792372176015878355246457298422}{255100998239883840943586201145397255182732430830512237554326453} a^{12} + \frac{37439270357798708699954184049419347582185543711369637413082437}{255100998239883840943586201145397255182732430830512237554326453} a^{11} - \frac{20983365545514577847196504756879104790562247403012541068800549}{255100998239883840943586201145397255182732430830512237554326453} a^{10} + \frac{32115307772983155801356217362103336384865835047123313075902885}{255100998239883840943586201145397255182732430830512237554326453} a^{9} + \frac{7586510365103640422530898538772118382526035340982158358368399}{255100998239883840943586201145397255182732430830512237554326453} a^{8} - \frac{819585677300590763387288725064426710434144969143787921476764}{255100998239883840943586201145397255182732430830512237554326453} a^{7} - \frac{27137089964759235129520393986961207100052254300069742345379310}{255100998239883840943586201145397255182732430830512237554326453} a^{6} - \frac{10775567431112189271305854870103364916767106425228873568794466}{85033666079961280314528733715132418394244143610170745851442151} a^{5} - \frac{372552488090188724757176456193533577947309912817061234097168}{4323745732879387134637054256701648392927668319161224365327567} a^{4} + \frac{124614860926281529742344080270648284205974090729118749886219629}{255100998239883840943586201145397255182732430830512237554326453} a^{3} - \frac{119649915799956260064157452305630061190248085970639225899271710}{255100998239883840943586201145397255182732430830512237554326453} a^{2} - \frac{3143860643689973055777538379557322318845939654840546428660901}{9448185119995697812725414857236935377138238178907860650160239} a - \frac{125515334264813764134304074311762370490237929963458717482458673}{255100998239883840943586201145397255182732430830512237554326453}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  $13$
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:  \( 215328165314.12698 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 215328165314.12698 \cdot 9}{2\sqrt{1249840845592629138161700045087827071680376951}}\approx 1.30393001180370$ (assuming GRH)

Galois group

$D_{27}$ (as 27T8):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 54
The 15 conjugacy class representatives for $D_{27}$
Character table for $D_{27}$

Intermediate fields

3.1.199.1, 9.1.1568239201.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $27$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ $27$ R ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ $27$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ $27$ $27$ $27$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. 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
$7$7.9.6.3$x^{9} - 14 x^{6} + 49 x^{3} - 1372$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
7.9.6.3$x^{9} - 14 x^{6} + 49 x^{3} - 1372$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
7.9.6.3$x^{9} - 14 x^{6} + 49 x^{3} - 1372$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
$199$$\Q_{199}$$x + 2$$1$$1$$0$Trivial$[\ ]$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$
199.2.1.2$x^{2} + 398$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.199.2t1.a.a$1$ $ 199 $ \(\Q(\sqrt{-199}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.199.3t2.a.a$2$ $ 199 $ 3.1.199.1 $S_3$ (as 3T2) $1$ $0$
* 2.199.9t3.a.b$2$ $ 199 $ 9.1.1568239201.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.199.9t3.a.c$2$ $ 199 $ 9.1.1568239201.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.199.9t3.a.a$2$ $ 199 $ 9.1.1568239201.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.9751.27t8.a.g$2$ $ 7^{2} \cdot 199 $ 27.1.1249840845592629138161700045087827071680376951.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.9751.27t8.a.i$2$ $ 7^{2} \cdot 199 $ 27.1.1249840845592629138161700045087827071680376951.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.9751.27t8.a.d$2$ $ 7^{2} \cdot 199 $ 27.1.1249840845592629138161700045087827071680376951.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.9751.27t8.a.f$2$ $ 7^{2} \cdot 199 $ 27.1.1249840845592629138161700045087827071680376951.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.9751.27t8.a.h$2$ $ 7^{2} \cdot 199 $ 27.1.1249840845592629138161700045087827071680376951.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.9751.27t8.a.b$2$ $ 7^{2} \cdot 199 $ 27.1.1249840845592629138161700045087827071680376951.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.9751.27t8.a.c$2$ $ 7^{2} \cdot 199 $ 27.1.1249840845592629138161700045087827071680376951.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.9751.27t8.a.e$2$ $ 7^{2} \cdot 199 $ 27.1.1249840845592629138161700045087827071680376951.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.9751.27t8.a.a$2$ $ 7^{2} \cdot 199 $ 27.1.1249840845592629138161700045087827071680376951.1 $D_{27}$ (as 27T8) $1$ $0$

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 *.