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Properties

Label 27.27.5659106580...6209.8
Degree $27$
Signature $[27, 0]$
Discriminant $3^{66}\cdot 7^{18}\cdot 13^{18}$
Root discriminant $296.70$
Ramified primes $3, 7, 13$
Class number Not computed
Class group Not computed
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![878893626230101, -6490810357686540, 19464015249722088, -30918692573230389, 27636990777084987, -12365151517166049, 52405731960495, 2846666722896927, -1202353337008818, -18670398026360, 155084845070178, -35021761608051, -6416274457389, 3671235722367, -143798389803, -174741571434, 25557050754, 4183764714, -1158540616, -30624669, 27904176, -888774, -385335, 25515, 2877, -261, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 9*x^26 - 261*x^25 + 2877*x^24 + 25515*x^23 - 385335*x^22 - 888774*x^21 + 27904176*x^20 - 30624669*x^19 - 1158540616*x^18 + 4183764714*x^17 + 25557050754*x^16 - 174741571434*x^15 - 143798389803*x^14 + 3671235722367*x^13 - 6416274457389*x^12 - 35021761608051*x^11 + 155084845070178*x^10 - 18670398026360*x^9 - 1202353337008818*x^8 + 2846666722896927*x^7 + 52405731960495*x^6 - 12365151517166049*x^5 + 27636990777084987*x^4 - 30918692573230389*x^3 + 19464015249722088*x^2 - 6490810357686540*x + 878893626230101)
 
gp: K = bnfinit(x^27 - 9*x^26 - 261*x^25 + 2877*x^24 + 25515*x^23 - 385335*x^22 - 888774*x^21 + 27904176*x^20 - 30624669*x^19 - 1158540616*x^18 + 4183764714*x^17 + 25557050754*x^16 - 174741571434*x^15 - 143798389803*x^14 + 3671235722367*x^13 - 6416274457389*x^12 - 35021761608051*x^11 + 155084845070178*x^10 - 18670398026360*x^9 - 1202353337008818*x^8 + 2846666722896927*x^7 + 52405731960495*x^6 - 12365151517166049*x^5 + 27636990777084987*x^4 - 30918692573230389*x^3 + 19464015249722088*x^2 - 6490810357686540*x + 878893626230101, 1)
 

Normalized defining polynomial

\( x^{27} - 9 x^{26} - 261 x^{25} + 2877 x^{24} + 25515 x^{23} - 385335 x^{22} - 888774 x^{21} + 27904176 x^{20} - 30624669 x^{19} - 1158540616 x^{18} + 4183764714 x^{17} + 25557050754 x^{16} - 174741571434 x^{15} - 143798389803 x^{14} + 3671235722367 x^{13} - 6416274457389 x^{12} - 35021761608051 x^{11} + 155084845070178 x^{10} - 18670398026360 x^{9} - 1202353337008818 x^{8} + 2846666722896927 x^{7} + 52405731960495 x^{6} - 12365151517166049 x^{5} + 27636990777084987 x^{4} - 30918692573230389 x^{3} + 19464015249722088 x^{2} - 6490810357686540 x + 878893626230101 \)

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:  \(5659106580522258442740055894009895600932750567734671570320232266209=3^{66}\cdot 7^{18}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $296.70$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2457=3^{3}\cdot 7\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{2457}(256,·)$, $\chi_{2457}(1,·)$, $\chi_{2457}(835,·)$, $\chi_{2457}(1348,·)$, $\chi_{2457}(1093,·)$, $\chi_{2457}(1927,·)$, $\chi_{2457}(2440,·)$, $\chi_{2457}(2185,·)$, $\chi_{2457}(16,·)$, $\chi_{2457}(529,·)$, $\chi_{2457}(274,·)$, $\chi_{2457}(1108,·)$, $\chi_{2457}(1621,·)$, $\chi_{2457}(1366,·)$, $\chi_{2457}(2200,·)$, $\chi_{2457}(289,·)$, $\chi_{2457}(802,·)$, $\chi_{2457}(547,·)$, $\chi_{2457}(1381,·)$, $\chi_{2457}(1894,·)$, $\chi_{2457}(1639,·)$, $\chi_{2457}(562,·)$, $\chi_{2457}(1075,·)$, $\chi_{2457}(820,·)$, $\chi_{2457}(1654,·)$, $\chi_{2457}(2167,·)$, $\chi_{2457}(1912,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{17} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{17} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{17} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{17} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{17} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{9} - \frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{23} - \frac{1}{2} a^{17} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{10} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{24} - \frac{1}{2} a^{17} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{25} - \frac{1}{2} a^{17} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{26859579401923559969902435304755824096091242659723035816212831924484984688665329212092091310276004667048172179008785709978931238581110861598851456479796418} a^{26} + \frac{459081398351962924235299889006940343625563132038218071035865282131256310867651625450913189983774622267152476179210538796901084036543443800607116042151925}{26859579401923559969902435304755824096091242659723035816212831924484984688665329212092091310276004667048172179008785709978931238581110861598851456479796418} a^{25} - \frac{2578086934257387036487113713337603534867041594493825236589547192548835488096173098006996695717220839287949649489630330176079294691829999531515905535839171}{26859579401923559969902435304755824096091242659723035816212831924484984688665329212092091310276004667048172179008785709978931238581110861598851456479796418} a^{24} + \frac{2732499611608384231215070759715379468778763076690194116083175464393819175141576733748152942739392253303957735262858733886004288740416345463405482777942931}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{23} + \frac{2301240325823830638371264489255454131344001433782106268450730274391962390654676804250418405212386042010445544042233271148677638260083632108877577321187191}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{22} + \frac{839512751281436115667354812232870224292892742998429600049389837308862978512144545245214240615429319337198942699499484107202221027661068658745910494758297}{26859579401923559969902435304755824096091242659723035816212831924484984688665329212092091310276004667048172179008785709978931238581110861598851456479796418} a^{21} - \frac{1667852755675870249003961729471929361858826007645244355937417500978622330654644233963198906516837453651590480899233237019028065661784619410420319419699844}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{20} + \frac{1097361978581095117964729438556715555824588093777912327893213984639396551595011850456126555639862552726350382089650309686696827333823654020505298846149031}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{19} - \frac{1152217542574209735122460635292202967050101213300574233137383156701791343690315850554491343647436310077790428591052960454076495953894246497917891788524381}{26859579401923559969902435304755824096091242659723035816212831924484984688665329212092091310276004667048172179008785709978931238581110861598851456479796418} a^{18} + \frac{11284427838359613019282706527228071401967011246752948866746547109500337217316216853756296281048599441155218757536849723185781113385625668997247398594929717}{26859579401923559969902435304755824096091242659723035816212831924484984688665329212092091310276004667048172179008785709978931238581110861598851456479796418} a^{17} + \frac{12930308257443973950018616381014086606275316245718459911878799080548875218138296621066909666341811871603357132917963154678257166121074095155267081957337905}{26859579401923559969902435304755824096091242659723035816212831924484984688665329212092091310276004667048172179008785709978931238581110861598851456479796418} a^{16} - \frac{6648317344188613301733593734429444091283398261376503768242263529096088160630409472225365314314830886137216065482476707052023544159926873524836978881067423}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{15} + \frac{1784760833505465895564768662722367790621497468327987899546804890037145436997548231245214258128358591783844738749533084334643426166376928889154934619000861}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{14} - \frac{3847496929065508831840569234815197824170121552649346324357030446906379627223847776583148498484048999117670641186165224549952652198930941406472401866929122}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{13} + \frac{790793936311753933452343885988341912304782655419274764732273349438263122914664483902836229976764720046391624637223784276592151935304971825499543621961309}{26859579401923559969902435304755824096091242659723035816212831924484984688665329212092091310276004667048172179008785709978931238581110861598851456479796418} a^{12} - \frac{2922532696318481702850018562724296109990715517674489940303102045631226889205325455181493384676569738434372157047978061718727600983793475473447901382755185}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{11} + \frac{2232966441063595795193662323842277214986254502741651985166256587263874950252650535974946596354070092064899670085501727460983148034293758735580257550822248}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{10} + \frac{4297588407538560569888498933007652524613326807397738079577830622706697664568996476392617687083056417767194571178466288519273346909491653214506832789299509}{26859579401923559969902435304755824096091242659723035816212831924484984688665329212092091310276004667048172179008785709978931238581110861598851456479796418} a^{9} + \frac{4497414187835120506023605028766775232111532466757329868030831391935857979569336298808144978984002438600228586059698243892977479130629401287886825723597476}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{8} + \frac{935229620709445922048823033468737920982088320632998551904659672278149146509619064654891464506372946241217094224937233194582212338915480570301229423564354}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{7} - \frac{6485178818837158845049450614598129209891925395600537609236269917945285978701165696694353270376839049628390241783691853522270578042307671388964381569835127}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{6} - \frac{13314887037573275884221910792507843141968559984946176809153645962573606640618939268156691611715011645172648760035748504750529616989333473419416181900205079}{26859579401923559969902435304755824096091242659723035816212831924484984688665329212092091310276004667048172179008785709978931238581110861598851456479796418} a^{5} + \frac{504111565069973606373909743048672939944565545200817762281525687507003912941367722535179724523985841179645360229637219390270757647821996816996741680029788}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{4} + \frac{224908001678931821253540341643166270378225526460482154045589048643496379007645657303325786007827487152680926520411298067142168562603462564821523748442137}{13429789700961779984951217652377912048045621329861517908106415962242492344332664606046045655138002333524086089504392854989465619290555430799425728239898209} a^{3} + \frac{1532507579469142063281259822936903843112042701064777435096882659733837319161762482727460206629775564315457771523001166771187102091209665940124950587365905}{26859579401923559969902435304755824096091242659723035816212831924484984688665329212092091310276004667048172179008785709978931238581110861598851456479796418} a^{2} + \frac{21706961160396203070769950480415958945761855785957683341261935844775984927459515363289503153646650432843731713366696052459258543384310470786643435188903}{99112839121489151180451790792456915483731522729605298214807497876328356784742912221742034355262009841506170402246441734239598666350962588925651130921758} a + \frac{8581277193716215253598747415460143934837878551971032717955556162736591017516647834815358785426056783220900603947712025276680678435823314372233662214096955}{26859579401923559969902435304755824096091242659723035816212831924484984688665329212092091310276004667048172179008785709978931238581110861598851456479796418}$

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

Class group and class number

Not computed

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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
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

3.3.670761.3, \(\Q(\zeta_{9})^+\), 3.3.670761.1, 3.3.8281.1, 9.9.301789003173921081.12, 9.9.17820338848416865911969.2, 9.9.17820338848416865911969.1, \(\Q(\zeta_{27})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
3Data not computed
7Data not computed
13Data not computed