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Properties

Label 27.1.556...943.1
Degree $27$
Signature $[1, 13]$
Discriminant $-5.569\times 10^{46}$
Root discriminant $53.87$
Ramified prime $3943$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 6*x^26 + 27*x^25 - 119*x^24 + 469*x^23 - 1488*x^22 + 4228*x^21 - 11600*x^20 + 26558*x^19 - 54959*x^18 + 133324*x^17 - 272188*x^16 + 522791*x^15 - 983894*x^14 + 1639478*x^13 - 2435791*x^12 + 3557436*x^11 - 4633264*x^10 + 5404652*x^9 - 6079977*x^8 + 6195381*x^7 - 5327698*x^6 + 4651935*x^5 - 3835566*x^4 + 2476602*x^3 - 1723599*x^2 + 1364445*x - 459999)
 
gp: K = bnfinit(x^27 - 6*x^26 + 27*x^25 - 119*x^24 + 469*x^23 - 1488*x^22 + 4228*x^21 - 11600*x^20 + 26558*x^19 - 54959*x^18 + 133324*x^17 - 272188*x^16 + 522791*x^15 - 983894*x^14 + 1639478*x^13 - 2435791*x^12 + 3557436*x^11 - 4633264*x^10 + 5404652*x^9 - 6079977*x^8 + 6195381*x^7 - 5327698*x^6 + 4651935*x^5 - 3835566*x^4 + 2476602*x^3 - 1723599*x^2 + 1364445*x - 459999, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-459999, 1364445, -1723599, 2476602, -3835566, 4651935, -5327698, 6195381, -6079977, 5404652, -4633264, 3557436, -2435791, 1639478, -983894, 522791, -272188, 133324, -54959, 26558, -11600, 4228, -1488, 469, -119, 27, -6, 1]);
 

\( x^{27} - 6 x^{26} + 27 x^{25} - 119 x^{24} + 469 x^{23} - 1488 x^{22} + 4228 x^{21} - 11600 x^{20} + 26558 x^{19} - 54959 x^{18} + 133324 x^{17} - 272188 x^{16} + 522791 x^{15} - 983894 x^{14} + 1639478 x^{13} - 2435791 x^{12} + 3557436 x^{11} - 4633264 x^{10} + 5404652 x^{9} - 6079977 x^{8} + 6195381 x^{7} - 5327698 x^{6} + 4651935 x^{5} - 3835566 x^{4} + 2476602 x^{3} - 1723599 x^{2} + 1364445 x - 459999 \)

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:  \(-55686266560375612156826962909758952173810975943\)\(\medspace = -\,3943^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $53.87$
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:  $3943$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} - \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} - \frac{1}{3} a^{5}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{6}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{7}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{8}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{16} + \frac{1}{9} a^{15} + \frac{1}{9} a^{14} + \frac{1}{9} a^{13} + \frac{1}{9} a^{12} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{4}{9} a^{8} - \frac{4}{9} a^{7} - \frac{4}{9} a^{6} - \frac{4}{9} a^{5} - \frac{4}{9} a^{4} - \frac{4}{9} a^{3} - \frac{4}{9} a^{2} - \frac{1}{3} a$, $\frac{1}{45} a^{18} + \frac{1}{45} a^{17} - \frac{2}{45} a^{16} - \frac{1}{9} a^{15} + \frac{4}{45} a^{14} - \frac{2}{45} a^{13} - \frac{1}{9} a^{12} - \frac{2}{45} a^{11} + \frac{2}{45} a^{10} - \frac{4}{45} a^{9} - \frac{1}{45} a^{8} + \frac{2}{45} a^{7} - \frac{16}{45} a^{6} - \frac{2}{9} a^{5} - \frac{16}{45} a^{4} + \frac{8}{45} a^{3} + \frac{7}{15} a^{2} + \frac{1}{5} a + \frac{1}{5}$, $\frac{1}{45} a^{19} + \frac{2}{45} a^{17} + \frac{2}{45} a^{16} - \frac{1}{45} a^{15} - \frac{1}{45} a^{14} + \frac{2}{45} a^{13} - \frac{7}{45} a^{12} - \frac{2}{15} a^{11} - \frac{1}{45} a^{10} - \frac{2}{45} a^{9} - \frac{17}{45} a^{8} + \frac{22}{45} a^{7} - \frac{14}{45} a^{6} + \frac{19}{45} a^{5} + \frac{19}{45} a^{4} + \frac{8}{45} a^{3} + \frac{13}{45} a^{2} - \frac{1}{3} a - \frac{1}{5}$, $\frac{1}{45} a^{20} + \frac{1}{15} a^{16} - \frac{2}{15} a^{15} - \frac{2}{15} a^{14} - \frac{1}{15} a^{13} + \frac{4}{45} a^{12} + \frac{1}{15} a^{11} - \frac{2}{15} a^{10} + \frac{2}{15} a^{9} - \frac{7}{15} a^{8} - \frac{1}{15} a^{7} + \frac{2}{15} a^{6} - \frac{2}{15} a^{5} - \frac{1}{9} a^{4} - \frac{1}{15} a^{3} - \frac{4}{15} a^{2} + \frac{1}{15} a - \frac{2}{5}$, $\frac{1}{405} a^{21} + \frac{2}{405} a^{20} + \frac{1}{135} a^{19} + \frac{1}{405} a^{18} + \frac{2}{81} a^{17} + \frac{49}{405} a^{16} - \frac{41}{405} a^{15} - \frac{29}{405} a^{14} - \frac{28}{405} a^{13} + \frac{8}{81} a^{11} - \frac{7}{405} a^{10} - \frac{4}{405} a^{9} + \frac{128}{405} a^{8} - \frac{187}{405} a^{7} + \frac{8}{405} a^{6} - \frac{8}{27} a^{5} - \frac{167}{405} a^{4} - \frac{46}{405} a^{3} + \frac{58}{135} a^{2} - \frac{22}{45} a - \frac{2}{15}$, $\frac{1}{1215} a^{22} - \frac{1}{1215} a^{21} - \frac{1}{405} a^{20} + \frac{2}{243} a^{19} - \frac{2}{1215} a^{18} - \frac{44}{1215} a^{17} - \frac{89}{1215} a^{16} + \frac{31}{1215} a^{15} - \frac{17}{243} a^{14} - \frac{29}{405} a^{13} + \frac{4}{1215} a^{12} - \frac{172}{1215} a^{11} + \frac{26}{1215} a^{10} - \frac{35}{243} a^{9} - \frac{238}{1215} a^{8} + \frac{92}{1215} a^{7} + \frac{19}{45} a^{6} - \frac{19}{243} a^{5} - \frac{49}{1215} a^{4} - \frac{202}{405} a^{3} - \frac{1}{27} a^{2} + \frac{17}{45} a - \frac{1}{15}$, $\frac{1}{1215} a^{23} - \frac{1}{1215} a^{21} + \frac{13}{1215} a^{20} - \frac{2}{243} a^{19} + \frac{11}{1215} a^{18} + \frac{32}{1215} a^{17} + \frac{62}{1215} a^{16} + \frac{8}{81} a^{15} + \frac{119}{1215} a^{14} - \frac{194}{1215} a^{13} - \frac{38}{405} a^{12} + \frac{163}{1215} a^{11} + \frac{20}{243} a^{10} + \frac{88}{1215} a^{9} + \frac{103}{1215} a^{8} - \frac{172}{1215} a^{7} - \frac{584}{1215} a^{6} + \frac{37}{135} a^{5} + \frac{572}{1215} a^{4} + \frac{34}{405} a^{3} - \frac{4}{135} a^{2} - \frac{4}{9} a + \frac{2}{5}$, $\frac{1}{8505} a^{24} - \frac{1}{2835} a^{23} + \frac{1}{2835} a^{22} + \frac{1}{2835} a^{21} - \frac{79}{8505} a^{20} - \frac{2}{315} a^{19} + \frac{1}{945} a^{18} - \frac{4}{405} a^{17} - \frac{1133}{8505} a^{16} - \frac{22}{189} a^{15} - \frac{46}{945} a^{14} - \frac{101}{2835} a^{13} - \frac{73}{8505} a^{12} + \frac{37}{405} a^{11} - \frac{122}{945} a^{10} + \frac{166}{2835} a^{9} + \frac{4084}{8505} a^{8} - \frac{74}{405} a^{7} + \frac{262}{567} a^{6} + \frac{262}{2835} a^{5} - \frac{1576}{8505} a^{4} - \frac{65}{567} a^{3} - \frac{40}{189} a^{2} + \frac{41}{315} a - \frac{1}{105}$, $\frac{1}{42525} a^{25} + \frac{1}{42525} a^{24} - \frac{1}{4725} a^{23} + \frac{1}{2835} a^{22} - \frac{4}{42525} a^{21} + \frac{134}{42525} a^{20} - \frac{23}{4725} a^{19} + \frac{68}{14175} a^{18} + \frac{1807}{42525} a^{17} - \frac{2057}{42525} a^{16} - \frac{479}{4725} a^{15} + \frac{2077}{14175} a^{14} + \frac{6401}{42525} a^{13} - \frac{1226}{8505} a^{12} - \frac{328}{2835} a^{11} + \frac{823}{14175} a^{10} - \frac{923}{6075} a^{9} - \frac{7583}{42525} a^{8} + \frac{662}{1575} a^{7} + \frac{8}{45} a^{6} + \frac{1277}{6075} a^{5} + \frac{1543}{8505} a^{4} - \frac{2299}{14175} a^{3} + \frac{41}{945} a^{2} + \frac{47}{225} a - \frac{193}{525}$, $\frac{1}{54298063263936663485108128599183295476574410757011476874973562048025} a^{26} - \frac{2962795100805491297669990813522779101621948615353360686757303}{18099354421312221161702709533061098492191470252337158958324520682675} a^{25} - \frac{24799428491018709633512133723128532761441840047598700857301011}{548465285494309732172809379789730257339135462192035119949227899475} a^{24} + \frac{1728439762641385520734286513266405861559419066097852017369164776}{10859612652787332697021625719836659095314882151402295374994712409605} a^{23} - \frac{3604749594881353268029188718262791104991217494294457667485127149}{54298063263936663485108128599183295476574410757011476874973562048025} a^{22} - \frac{5330588179139908614177751533372448291618209961214012468641996844}{6033118140437407053900903177687032830730490084112386319441506894225} a^{21} - \frac{427474615511288571801712193302003107111127766075737356005534829017}{54298063263936663485108128599183295476574410757011476874973562048025} a^{20} + \frac{68833009340056352159807292446292598397493986631300290796643298974}{54298063263936663485108128599183295476574410757011476874973562048025} a^{19} + \frac{378994337871410049133238820485651061992615994545140193811119984057}{54298063263936663485108128599183295476574410757011476874973562048025} a^{18} + \frac{1602304115483759494869782686561415090940028784195640428430434525308}{54298063263936663485108128599183295476574410757011476874973562048025} a^{17} + \frac{2198740486729600573499778731246587274079876729994925124482318755789}{54298063263936663485108128599183295476574410757011476874973562048025} a^{16} + \frac{7966958895521919745366301170568883951942121994939966335324334168366}{54298063263936663485108128599183295476574410757011476874973562048025} a^{15} - \frac{462724303832488918923042620699905105783913131939879620872733224062}{7756866180562380497872589799883327925224915822430210982139080292575} a^{14} + \frac{78659319964086797579155839576191174749859309804669805637924239346}{987237513889757517911056883621514463210443831945663215908610219055} a^{13} - \frac{1500523273459881573000017063080721872079329038343913729334490946243}{10859612652787332697021625719836659095314882151402295374994712409605} a^{12} + \frac{1199586790264006554654190690361748032611744248588008276753568571787}{7756866180562380497872589799883327925224915822430210982139080292575} a^{11} - \frac{89005849365305630017220646975879065458568681675477856030683639726}{670346460048600783766767019743003647858943342679154035493500766025} a^{10} + \frac{4425884696333324839482141582983462525872554445569668840960496457187}{54298063263936663485108128599183295476574410757011476874973562048025} a^{9} - \frac{715308928910599622264890806126275674846188769484902552129111396423}{7756866180562380497872589799883327925224915822430210982139080292575} a^{8} - \frac{1412308495669024807775915763807690845154862973183094333012083904478}{3619870884262444232340541906612219698438294050467431791664904136535} a^{7} + \frac{6751404893924197728615297195007226829076583264442826958648077163}{18099354421312221161702709533061098492191470252337158958324520682675} a^{6} - \frac{3190067162082803010711018930980984074499084753636094573523527223034}{10859612652787332697021625719836659095314882151402295374994712409605} a^{5} + \frac{2609845549590497244642367251272417384577947563682488239203253246967}{6033118140437407053900903177687032830730490084112386319441506894225} a^{4} + \frac{303279203687535905819925110593542816668561612631387735008648900329}{1206623628087481410780180635537406566146098016822477263888301378845} a^{3} + \frac{166321558289771348015101558711504964128364705703808369711968919849}{2011039380145802351300301059229010943576830028037462106480502298075} a^{2} - \frac{27735565426359736027549475566913972357746800831222502853672921966}{223448820016200261255589006581001215952981114226384678497833588675} a + \frac{20171560483823447022243913983013089971716377558664159242314729234}{44689764003240052251117801316200243190596222845276935699566717735}$

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:  $13$
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:  \( 29482123669434.47 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 29482123669434.47 \cdot 1}{2\sqrt{55686266560375612156826962909758952173810975943}}\approx 2.97182342252825$ (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

3.1.3943.1, 9.1.241716951468001.1

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 $27$ ${\href{/LocalNumberField/3.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $27$ $27$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ $27$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ $27$ $27$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
3943Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3943.2t1.a.a$1$ $ 3943 $ \(\Q(\sqrt{-3943}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.3943.3t2.a.a$2$ $ 3943 $ 3.1.3943.1 $S_3$ (as 3T2) $1$ $0$
* 2.3943.9t3.a.a$2$ $ 3943 $ 9.1.241716951468001.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3943.9t3.a.b$2$ $ 3943 $ 9.1.241716951468001.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3943.9t3.a.c$2$ $ 3943 $ 9.1.241716951468001.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.3943.27t8.a.a$2$ $ 3943 $ 27.1.55686266560375612156826962909758952173810975943.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3943.27t8.a.c$2$ $ 3943 $ 27.1.55686266560375612156826962909758952173810975943.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3943.27t8.a.i$2$ $ 3943 $ 27.1.55686266560375612156826962909758952173810975943.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3943.27t8.a.g$2$ $ 3943 $ 27.1.55686266560375612156826962909758952173810975943.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3943.27t8.a.e$2$ $ 3943 $ 27.1.55686266560375612156826962909758952173810975943.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3943.27t8.a.h$2$ $ 3943 $ 27.1.55686266560375612156826962909758952173810975943.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3943.27t8.a.d$2$ $ 3943 $ 27.1.55686266560375612156826962909758952173810975943.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3943.27t8.a.b$2$ $ 3943 $ 27.1.55686266560375612156826962909758952173810975943.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.3943.27t8.a.f$2$ $ 3943 $ 27.1.55686266560375612156826962909758952173810975943.1 $D_{27}$ (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 *.