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Properties

Label 27.27.1237424251...6729.1
Degree $27$
Signature $[27, 0]$
Discriminant $19^{24}\cdot 43^{18}$
Root discriminant $168.13$
Ramified primes $19, 43$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times C_9$ (as 27T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![2265222521, 20768956575, 8287037006, -115757033393, -142742440584, 129171933167, 280117270166, 13751584729, -199640506816, -77700458908, 61771660232, 40055307468, -8654405011, -9854606544, 256596263, 1422742061, 86373179, -130806305, -13384023, 7956418, 924610, -323554, -35066, 8599, 709, -137, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 6*x^26 - 137*x^25 + 709*x^24 + 8599*x^23 - 35066*x^22 - 323554*x^21 + 924610*x^20 + 7956418*x^19 - 13384023*x^18 - 130806305*x^17 + 86373179*x^16 + 1422742061*x^15 + 256596263*x^14 - 9854606544*x^13 - 8654405011*x^12 + 40055307468*x^11 + 61771660232*x^10 - 77700458908*x^9 - 199640506816*x^8 + 13751584729*x^7 + 280117270166*x^6 + 129171933167*x^5 - 142742440584*x^4 - 115757033393*x^3 + 8287037006*x^2 + 20768956575*x + 2265222521)
 
gp: K = bnfinit(x^27 - 6*x^26 - 137*x^25 + 709*x^24 + 8599*x^23 - 35066*x^22 - 323554*x^21 + 924610*x^20 + 7956418*x^19 - 13384023*x^18 - 130806305*x^17 + 86373179*x^16 + 1422742061*x^15 + 256596263*x^14 - 9854606544*x^13 - 8654405011*x^12 + 40055307468*x^11 + 61771660232*x^10 - 77700458908*x^9 - 199640506816*x^8 + 13751584729*x^7 + 280117270166*x^6 + 129171933167*x^5 - 142742440584*x^4 - 115757033393*x^3 + 8287037006*x^2 + 20768956575*x + 2265222521, 1)
 

Normalized defining polynomial

\( x^{27} - 6 x^{26} - 137 x^{25} + 709 x^{24} + 8599 x^{23} - 35066 x^{22} - 323554 x^{21} + 924610 x^{20} + 7956418 x^{19} - 13384023 x^{18} - 130806305 x^{17} + 86373179 x^{16} + 1422742061 x^{15} + 256596263 x^{14} - 9854606544 x^{13} - 8654405011 x^{12} + 40055307468 x^{11} + 61771660232 x^{10} - 77700458908 x^{9} - 199640506816 x^{8} + 13751584729 x^{7} + 280117270166 x^{6} + 129171933167 x^{5} - 142742440584 x^{4} - 115757033393 x^{3} + 8287037006 x^{2} + 20768956575 x + 2265222521 \)

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

Invariants

Degree:  $27$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[27, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1237424251695265996553186030546217359162421829813695859696729=19^{24}\cdot 43^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $168.13$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $19, 43$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(817=19\cdot 43\)
Dirichlet character group:    $\lbrace$$\chi_{817}(1,·)$, $\chi_{817}(130,·)$, $\chi_{817}(517,·)$, $\chi_{817}(6,·)$, $\chi_{817}(264,·)$, $\chi_{817}(522,·)$, $\chi_{817}(651,·)$, $\chi_{817}(780,·)$, $\chi_{817}(595,·)$, $\chi_{817}(87,·)$, $\chi_{817}(216,·)$, $\chi_{817}(49,·)$, $\chi_{817}(92,·)$, $\chi_{817}(479,·)$, $\chi_{817}(36,·)$, $\chi_{817}(294,·)$, $\chi_{817}(423,·)$, $\chi_{817}(552,·)$, $\chi_{817}(681,·)$, $\chi_{817}(44,·)$, $\chi_{817}(302,·)$, $\chi_{817}(560,·)$, $\chi_{817}(689,·)$, $\chi_{817}(178,·)$, $\chi_{817}(251,·)$, $\chi_{817}(638,·)$, $\chi_{817}(767,·)$$\rbrace$
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{14} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{14} a^{24} + \frac{1}{7} a^{23} - \frac{1}{7} a^{22} + \frac{3}{14} a^{21} - \frac{1}{7} a^{20} + \frac{3}{14} a^{19} - \frac{1}{7} a^{18} - \frac{2}{7} a^{16} - \frac{5}{14} a^{15} + \frac{1}{7} a^{14} - \frac{3}{7} a^{13} - \frac{1}{14} a^{12} + \frac{5}{14} a^{11} + \frac{1}{7} a^{10} + \frac{3}{7} a^{9} - \frac{3}{7} a^{8} - \frac{5}{14} a^{7} - \frac{3}{14} a^{6} - \frac{1}{7} a^{5} + \frac{5}{14} a^{4} + \frac{1}{7} a^{3} - \frac{1}{2} a^{2} + \frac{3}{7} a + \frac{2}{7}$, $\frac{1}{1582} a^{25} - \frac{8}{791} a^{24} - \frac{297}{1582} a^{23} - \frac{241}{1582} a^{22} + \frac{12}{113} a^{21} - \frac{40}{791} a^{20} - \frac{16}{113} a^{19} - \frac{27}{1582} a^{18} - \frac{254}{791} a^{17} - \frac{143}{1582} a^{16} + \frac{18}{791} a^{15} + \frac{3}{226} a^{14} + \frac{667}{1582} a^{13} + \frac{779}{1582} a^{12} + \frac{423}{1582} a^{11} - \frac{499}{1582} a^{10} - \frac{351}{791} a^{9} - \frac{149}{1582} a^{8} - \frac{277}{1582} a^{7} + \frac{73}{1582} a^{6} + \frac{59}{791} a^{5} - \frac{107}{791} a^{4} - \frac{193}{791} a^{3} + \frac{304}{791} a^{2} - \frac{713}{1582} a - \frac{1}{7}$, $\frac{1}{744901861390773662224916928538136443063332597872041668949451615824767389249614616256464707541154065541347928148714316926402} a^{26} - \frac{69897262767242700083577176122259560804723767659899219723175795928847639421563692554607186579697783762050398779473997981}{372450930695386831112458464269068221531666298936020834474725807912383694624807308128232353770577032770673964074357158463201} a^{25} + \frac{44173421177834778838644464925762696944642228007414082004089839354485497665656808428672761501062781258214436268657314239}{53207275813626690158922637752724031647380899848002976353532258273197670660686758304033193395796718967239137724908165494743} a^{24} + \frac{52462447677028479179128384576246753948539664852331961054563518133606675944826167159055170788971782613250733068593657763226}{372450930695386831112458464269068221531666298936020834474725807912383694624807308128232353770577032770673964074357158463201} a^{23} + \frac{63552891321560337264649565149848542203697583603192993794431273755061966915103121748254143791839614297153652152589439473665}{372450930695386831112458464269068221531666298936020834474725807912383694624807308128232353770577032770673964074357158463201} a^{22} - \frac{106737466424962376920702913751248532531623905685121949594394272477865151653346705818172207339557688933453167929511692750699}{744901861390773662224916928538136443063332597872041668949451615824767389249614616256464707541154065541347928148714316926402} a^{21} - \frac{39223694885525325382624363080096191737265148707483312229201469607889888898030534225311411769506883606281670411336202112029}{744901861390773662224916928538136443063332597872041668949451615824767389249614616256464707541154065541347928148714316926402} a^{20} + \frac{150183938305462974011876735050163990816547660179222584826168683950114530102087421090337628663012362921905628166789018991169}{744901861390773662224916928538136443063332597872041668949451615824767389249614616256464707541154065541347928148714316926402} a^{19} - \frac{61913452213418776001908201180339805351881284256574537553104288516200261069188573768465257907992876179875264635672194072330}{372450930695386831112458464269068221531666298936020834474725807912383694624807308128232353770577032770673964074357158463201} a^{18} + \frac{10529559985258888457601409212788726432915551941460839899904959789126194240905874212412750949527512285087198239662164887159}{106414551627253380317845275505448063294761799696005952707064516546395341321373516608066386791593437934478275449816330989486} a^{17} + \frac{29613095103487207195572947970447454151543270195325934706125937734032373899807443285564442582293575752564832244018136387517}{372450930695386831112458464269068221531666298936020834474725807912383694624807308128232353770577032770673964074357158463201} a^{16} + \frac{150885307878136414757907331684650849096983044553649112472730001690180852701089916659233246798001429949861434626681426638661}{744901861390773662224916928538136443063332597872041668949451615824767389249614616256464707541154065541347928148714316926402} a^{15} + \frac{311934081787868023985008932245180530391958298851488905923384598278287800034383401602777832543911749838462648438656311650519}{744901861390773662224916928538136443063332597872041668949451615824767389249614616256464707541154065541347928148714316926402} a^{14} - \frac{63533288033668411718270459363660170869728065129718658703611961723427361140023150947941825410241675872862334733535965371769}{744901861390773662224916928538136443063332597872041668949451615824767389249614616256464707541154065541347928148714316926402} a^{13} + \frac{4740787577656749078327488006813630035177843844687173558505632078426548076376887038321750164003068277088593302232155112229}{106414551627253380317845275505448063294761799696005952707064516546395341321373516608066386791593437934478275449816330989486} a^{12} - \frac{180240162294628659756912118318697054135769940033005058084404548889947615103407224537733475804870059622799404199243999322301}{744901861390773662224916928538136443063332597872041668949451615824767389249614616256464707541154065541347928148714316926402} a^{11} + \frac{51384517555615589586469389983789662909969590051127937138498193149377297218327857863096917815179150496649419119256849109154}{372450930695386831112458464269068221531666298936020834474725807912383694624807308128232353770577032770673964074357158463201} a^{10} + \frac{82934334559305975679919734654893860124375241734005910860982197910622859342934050682259785610796368064260965202842247274556}{372450930695386831112458464269068221531666298936020834474725807912383694624807308128232353770577032770673964074357158463201} a^{9} + \frac{116102085049334147173572914092567520722309395774431521875897076714737973020832810392469256788704218948478915118749855405657}{372450930695386831112458464269068221531666298936020834474725807912383694624807308128232353770577032770673964074357158463201} a^{8} + \frac{51607799294541291083999001323165376989988921785277619581958909951603523321948533792878795304592130296443318310139573999649}{744901861390773662224916928538136443063332597872041668949451615824767389249614616256464707541154065541347928148714316926402} a^{7} - \frac{35735256361149441795218591007325853050981102519627916566624453765989643302430294275718382553770572981842633228850726928748}{372450930695386831112458464269068221531666298936020834474725807912383694624807308128232353770577032770673964074357158463201} a^{6} - \frac{150333136690563689006234645960296263557226484181584954305990243405094486641863050849142538026331285602365816001737959649645}{744901861390773662224916928538136443063332597872041668949451615824767389249614616256464707541154065541347928148714316926402} a^{5} - \frac{164508510348390371335540064241808553663946998834467215342944559337331175183517487150019747229054949458774029956034989006832}{372450930695386831112458464269068221531666298936020834474725807912383694624807308128232353770577032770673964074357158463201} a^{4} + \frac{167258827104331391753468083644583550524281653940881134711382137781544190912113921637028257932561652598303858916170111455569}{744901861390773662224916928538136443063332597872041668949451615824767389249614616256464707541154065541347928148714316926402} a^{3} + \frac{5561076212834644140112367717850752192907143285497330823556299271254194361302321030508036333441510287589331671050998943805}{53207275813626690158922637752724031647380899848002976353532258273197670660686758304033193395796718967239137724908165494743} a^{2} + \frac{29311864405011596794747284601204022785891203424169419276135673696491117877973019723028196891806184061415442630278150985392}{372450930695386831112458464269068221531666298936020834474725807912383694624807308128232353770577032770673964074357158463201} a + \frac{300476262756636416027534319449900957956495775330216449094971366025153918715686415236325521028298442323477387420257627}{7866410346915049129035809328343257683309740827001094778438460893243155735839806283993333342568209871178194269422712283}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $26$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
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)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 2724694690612499700000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_9$ (as 27T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 27
The 27 conjugacy class representatives for $C_3\times C_9$
Character table for $C_3\times C_9$ is not computed

Intermediate fields

3.3.667489.1, 3.3.361.1, 3.3.1849.1, 3.3.667489.2, 9.9.297394093761051169.1, \(\Q(\zeta_{19})^+\), 9.9.107359267847739472009.1, 9.9.107359267847739472009.2

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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];
 
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]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
19Data not computed
43Data not computed