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Properties

Label 27.1.155...607.1
Degree $27$
Signature $[1, 13]$
Discriminant $-1.558\times 10^{49}$
Root discriminant $66.37$
Ramified primes $3, 23$
Class number $9$ (GRH)
Class group $[9]$ (GRH)
Galois group $D_{27}$ (as 27T8)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 + 18*x^25 - 54*x^24 + 81*x^23 - 729*x^22 + 711*x^21 - 2322*x^20 + 7389*x^19 - 646*x^18 + 36288*x^17 - 20349*x^16 + 36072*x^15 - 256311*x^14 - 65700*x^13 - 942516*x^12 + 444744*x^11 - 558486*x^10 + 4867412*x^9 + 1228689*x^8 + 12312063*x^7 - 3080772*x^6 + 6516828*x^5 - 31451760*x^4 - 20430663*x^3 - 52842915*x^2 - 27379269*x - 39209779)
 
gp: K = bnfinit(x^27 + 18*x^25 - 54*x^24 + 81*x^23 - 729*x^22 + 711*x^21 - 2322*x^20 + 7389*x^19 - 646*x^18 + 36288*x^17 - 20349*x^16 + 36072*x^15 - 256311*x^14 - 65700*x^13 - 942516*x^12 + 444744*x^11 - 558486*x^10 + 4867412*x^9 + 1228689*x^8 + 12312063*x^7 - 3080772*x^6 + 6516828*x^5 - 31451760*x^4 - 20430663*x^3 - 52842915*x^2 - 27379269*x - 39209779, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-39209779, -27379269, -52842915, -20430663, -31451760, 6516828, -3080772, 12312063, 1228689, 4867412, -558486, 444744, -942516, -65700, -256311, 36072, -20349, 36288, -646, 7389, -2322, 711, -729, 81, -54, 18, 0, 1]);
 

\( x^{27} + 18 x^{25} - 54 x^{24} + 81 x^{23} - 729 x^{22} + 711 x^{21} - 2322 x^{20} + 7389 x^{19} - 646 x^{18} + 36288 x^{17} - 20349 x^{16} + 36072 x^{15} - 256311 x^{14} - 65700 x^{13} - 942516 x^{12} + 444744 x^{11} - 558486 x^{10} + 4867412 x^{9} + 1228689 x^{8} + 12312063 x^{7} - 3080772 x^{6} + 6516828 x^{5} - 31451760 x^{4} - 20430663 x^{3} - 52842915 x^{2} - 27379269 x - 39209779 \)

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:  \(-15576313507383139645975038414456506401400202794607\)\(\medspace = -\,3^{66}\cdot 23^{13}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $66.37$
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:  $3, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5} a^{15} + \frac{2}{5} a^{14} - \frac{1}{5} a^{13} - \frac{2}{5} a^{12} + \frac{1}{5} a^{11} - \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{1}{5} a^{8} + \frac{1}{5} a^{6} - \frac{1}{5} a^{4} - \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{11} + \frac{2}{5} a^{9} + \frac{2}{5} a^{8} + \frac{1}{5} a^{7} - \frac{2}{5} a^{6} - \frac{1}{5} a^{5} + \frac{2}{5} a^{4} - \frac{2}{5} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{12} + \frac{2}{5} a^{10} + \frac{2}{5} a^{9} + \frac{1}{5} a^{8} - \frac{2}{5} a^{7} - \frac{1}{5} a^{6} + \frac{2}{5} a^{5} - \frac{2}{5} a^{3} + \frac{2}{5} a^{2} - \frac{1}{5} a$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{13} + \frac{2}{5} a^{11} + \frac{2}{5} a^{10} + \frac{1}{5} a^{9} - \frac{2}{5} a^{8} - \frac{1}{5} a^{7} + \frac{2}{5} a^{6} - \frac{2}{5} a^{4} + \frac{2}{5} a^{3} - \frac{1}{5} a^{2}$, $\frac{1}{35} a^{19} - \frac{3}{35} a^{16} + \frac{3}{35} a^{15} + \frac{1}{5} a^{14} - \frac{3}{35} a^{13} + \frac{6}{35} a^{12} + \frac{2}{35} a^{11} - \frac{3}{7} a^{9} + \frac{3}{7} a^{8} - \frac{11}{35} a^{7} - \frac{16}{35} a^{6} - \frac{9}{35} a^{5} + \frac{8}{35} a^{4} + \frac{9}{35} a^{3} - \frac{2}{5} a^{2} + \frac{3}{35} a + \frac{1}{5}$, $\frac{1}{805} a^{20} - \frac{1}{805} a^{19} - \frac{4}{115} a^{18} - \frac{9}{161} a^{17} - \frac{1}{805} a^{16} + \frac{39}{805} a^{15} + \frac{47}{161} a^{14} + \frac{156}{805} a^{13} - \frac{11}{805} a^{12} + \frac{57}{161} a^{11} - \frac{66}{161} a^{10} - \frac{306}{805} a^{9} - \frac{131}{805} a^{8} - \frac{71}{161} a^{7} - \frac{29}{115} a^{6} - \frac{26}{161} a^{5} + \frac{43}{805} a^{4} - \frac{20}{161} a^{3} - \frac{12}{161} a^{2} + \frac{9}{35} a$, $\frac{1}{4025} a^{21} - \frac{1}{4025} a^{20} + \frac{18}{4025} a^{19} - \frac{206}{4025} a^{18} - \frac{1}{4025} a^{17} - \frac{52}{805} a^{16} + \frac{373}{4025} a^{15} - \frac{1132}{4025} a^{14} - \frac{384}{805} a^{13} - \frac{1049}{4025} a^{12} - \frac{218}{575} a^{11} - \frac{1433}{4025} a^{10} + \frac{1916}{4025} a^{9} + \frac{389}{805} a^{8} + \frac{901}{4025} a^{7} - \frac{1671}{4025} a^{6} - \frac{29}{115} a^{5} + \frac{1073}{4025} a^{4} + \frac{837}{4025} a^{3} - \frac{33}{175} a^{2} - \frac{43}{175} a - \frac{7}{25}$, $\frac{1}{4025} a^{22} + \frac{2}{4025} a^{20} + \frac{57}{4025} a^{19} + \frac{213}{4025} a^{18} - \frac{17}{175} a^{17} + \frac{243}{4025} a^{16} + \frac{151}{4025} a^{15} + \frac{668}{4025} a^{14} + \frac{178}{575} a^{13} + \frac{116}{805} a^{12} - \frac{1139}{4025} a^{11} - \frac{1812}{4025} a^{10} + \frac{1781}{4025} a^{9} + \frac{211}{4025} a^{8} + \frac{83}{805} a^{7} + \frac{704}{4025} a^{6} + \frac{1548}{4025} a^{5} - \frac{1}{35} a^{4} + \frac{1233}{4025} a^{3} + \frac{762}{4025} a^{2} + \frac{13}{175} a - \frac{12}{25}$, $\frac{1}{4025} a^{23} - \frac{1}{4025} a^{20} + \frac{1}{575} a^{19} + \frac{13}{575} a^{18} - \frac{11}{161} a^{17} - \frac{27}{575} a^{16} + \frac{16}{575} a^{15} - \frac{347}{805} a^{14} - \frac{206}{805} a^{13} - \frac{1371}{4025} a^{12} - \frac{366}{805} a^{11} + \frac{297}{4025} a^{10} - \frac{1936}{4025} a^{9} - \frac{296}{805} a^{8} + \frac{997}{4025} a^{7} - \frac{197}{805} a^{6} - \frac{219}{805} a^{5} - \frac{503}{4025} a^{4} - \frac{1812}{4025} a^{3} - \frac{1023}{4025} a^{2} + \frac{27}{175} a - \frac{1}{25}$, $\frac{1}{20125} a^{24} - \frac{2}{20125} a^{23} - \frac{1}{20125} a^{22} + \frac{2}{20125} a^{21} - \frac{11}{20125} a^{20} + \frac{89}{20125} a^{19} + \frac{221}{2875} a^{18} + \frac{1424}{20125} a^{17} + \frac{156}{2875} a^{16} - \frac{771}{20125} a^{15} + \frac{4511}{20125} a^{14} + \frac{1003}{20125} a^{13} + \frac{758}{4025} a^{12} + \frac{7513}{20125} a^{11} + \frac{3153}{20125} a^{10} - \frac{6761}{20125} a^{9} - \frac{303}{875} a^{8} - \frac{28}{2875} a^{7} + \frac{4643}{20125} a^{6} + \frac{1459}{20125} a^{5} + \frac{499}{2875} a^{4} - \frac{5891}{20125} a^{3} - \frac{5107}{20125} a^{2} + \frac{257}{875} a - \frac{52}{125}$, $\frac{1}{35641375} a^{25} - \frac{382}{35641375} a^{24} + \frac{204}{35641375} a^{23} - \frac{118}{3240125} a^{22} + \frac{2804}{35641375} a^{21} - \frac{16011}{35641375} a^{20} - \frac{203743}{35641375} a^{19} - \frac{1072356}{35641375} a^{18} - \frac{1604098}{35641375} a^{17} + \frac{1938949}{35641375} a^{16} + \frac{918501}{35641375} a^{15} - \frac{2745892}{35641375} a^{14} - \frac{2105086}{7128275} a^{13} + \frac{340282}{727375} a^{12} + \frac{14542483}{35641375} a^{11} + \frac{15761274}{35641375} a^{10} + \frac{16301136}{35641375} a^{9} + \frac{660304}{3240125} a^{8} + \frac{11931463}{35641375} a^{7} + \frac{5242274}{35641375} a^{6} + \frac{1231278}{3240125} a^{5} + \frac{90059}{35641375} a^{4} + \frac{2272574}{5091625} a^{3} - \frac{5765049}{35641375} a^{2} + \frac{562676}{1549625} a - \frac{344}{1771}$, $\frac{1}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{26} - \frac{2994285130949080481298470102209709934590020311503722038228780652661}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{25} - \frac{54693232359571587890583308646871226550064120426990111968655839072758}{4142682125235872414160375523120185467046560117717701689807969772783589375} a^{24} - \frac{6069818662340573582241939653912048889853028210882165834946038484671023}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{23} - \frac{1056924767225413733137557501801803594599601490269448499961318309373684}{15809827702430778396897759649458666986483811061494086040695721377765943125} a^{22} + \frac{61724237326466643377483696667012268360761120817806193906727770202775497}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{21} + \frac{42651538813247639660106686585121499332611201046388059925772046316948074}{70425596129009831040726383893043152939791522001200928726735486137321019375} a^{20} - \frac{10277944853140501648909099217988758129816775641239894305189164075523186271}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{19} + \frac{14535207156975044543274627682816003095904070153740840332570114502032894184}{154936311483821628289598044564694936467541348402642043198818069502106242625} a^{18} + \frac{21161285923115163272639822763891946764545415959015943094494607973080883879}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{17} + \frac{6790261254344923834683194573753328616352356006434092600703660686640706144}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{16} - \frac{49634653480941327199453898604241935471250841269787920283275897294941564363}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{15} + \frac{10223185690621623351392461374810652878213222956341409401962129358824524}{70909067040650630796154711471256263829538374554984916795797743479224825} a^{14} - \frac{358990294257423086107477677900842920067654693030156482838870743877628242561}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{13} + \frac{242563093005337504186378878525589441648820473077182945233753751490869664086}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{12} + \frac{34073971608002648881876636486077930928006938131243237749057791653030800844}{110668793917015448778284317546210668905386677430458602284870049644361601875} a^{11} - \frac{276510733038849105962671339214867617549947350643171321481314114854274258184}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{10} - \frac{197774010280680555438326929727884494032031156659915135759733456383380095862}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{9} + \frac{44434851096930235280254554584209356359211674095153411676563039084919619404}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{8} + \frac{9228324011827677096119159832297208085705917226671117335706161754714527912}{154936311483821628289598044564694936467541348402642043198818069502106242625} a^{7} - \frac{387228908679598525449520581803349660559441543228652355190533374727147017482}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{6} - \frac{9916111488259249354377716558352773501093785146568136603667468312789012679}{154936311483821628289598044564694936467541348402642043198818069502106242625} a^{5} - \frac{357019502787736569263865078984598578339372569699934730537402597384610367437}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{4} + \frac{362394337897976044212789185388910023082625978124663616572751673551678380722}{774681557419108141447990222823474682337706742013210215994090347510531213125} a^{3} - \frac{1023640819120965782134872557576881958878030622728189275770996091643766587}{3161965540486155679379551929891733397296762212298817208139144275553188625} a^{2} - \frac{2892620801138110439194092482120523774220116845161473463140040830662272838}{6736361368861809925634697589769345063806145582723567095600785630526358375} a + \frac{119344576972247464178178882707676454952432857771934919668326421177576816}{4811686692044149946881926849835246474147246844802547925429132593233113125}$

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

Class group and class number

$C_{9}$, 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:  \( 19566630112992.652 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{1}\cdot(2\pi)^{13}\cdot 19566630112992.652 \cdot 9}{2\sqrt{15576313507383139645975038414456506401400202794607}}\approx 1.06136434349342$ (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

3.1.23.1, 9.1.148718980881.2

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 $27$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ R $27$ $27$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ $27$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{13}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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

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
3Data not computed
$23$$\Q_{23}$$x + 2$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 46$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.23.2t1.a.a$1$ $ 23 $ \(\Q(\sqrt{-23}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.23.3t2.b.a$2$ $ 23 $ 3.1.23.1 $S_3$ (as 3T2) $1$ $0$
* 2.1863.9t3.a.a$2$ $ 3^{4} \cdot 23 $ 9.1.148718980881.2 $D_{9}$ (as 9T3) $1$ $0$
* 2.1863.9t3.a.b$2$ $ 3^{4} \cdot 23 $ 9.1.148718980881.2 $D_{9}$ (as 9T3) $1$ $0$
* 2.1863.9t3.a.c$2$ $ 3^{4} \cdot 23 $ 9.1.148718980881.2 $D_{9}$ (as 9T3) $1$ $0$
* 2.16767.27t8.a.i$2$ $ 3^{6} \cdot 23 $ 27.1.15576313507383139645975038414456506401400202794607.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16767.27t8.a.e$2$ $ 3^{6} \cdot 23 $ 27.1.15576313507383139645975038414456506401400202794607.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16767.27t8.a.h$2$ $ 3^{6} \cdot 23 $ 27.1.15576313507383139645975038414456506401400202794607.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16767.27t8.a.d$2$ $ 3^{6} \cdot 23 $ 27.1.15576313507383139645975038414456506401400202794607.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16767.27t8.a.a$2$ $ 3^{6} \cdot 23 $ 27.1.15576313507383139645975038414456506401400202794607.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16767.27t8.a.g$2$ $ 3^{6} \cdot 23 $ 27.1.15576313507383139645975038414456506401400202794607.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16767.27t8.a.b$2$ $ 3^{6} \cdot 23 $ 27.1.15576313507383139645975038414456506401400202794607.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16767.27t8.a.f$2$ $ 3^{6} \cdot 23 $ 27.1.15576313507383139645975038414456506401400202794607.1 $D_{27}$ (as 27T8) $1$ $0$
* 2.16767.27t8.a.c$2$ $ 3^{6} \cdot 23 $ 27.1.15576313507383139645975038414456506401400202794607.1 $D_{27}$ (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 *.