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Properties

Label 27.27.3463254175...9849.4
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 19^{24}$
Root discriminant $627.71$
Ramified primes $3, 19$
Class number Not computed
Class group Not computed
Galois group $C_{27}$ (as 27T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![18638104434069437, -467536783979896065, -1057165556573924154, 1695530573188686819, 370935283008394440, -853099334603306505, -50108801388853284, 189278999768580696, 3390821146614132, -23533044355733718, -128890862298198, 1824136571981646, 2877945888636, -93545465229828, -37451580390, 3282297025608, 262818108, -80025043581, -768474, 1354686795, 0, -15617943, 0, 116964, 0, -513, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 513*x^25 + 116964*x^23 - 15617943*x^21 + 1354686795*x^19 - 768474*x^18 - 80025043581*x^17 + 262818108*x^16 + 3282297025608*x^15 - 37451580390*x^14 - 93545465229828*x^13 + 2877945888636*x^12 + 1824136571981646*x^11 - 128890862298198*x^10 - 23533044355733718*x^9 + 3390821146614132*x^8 + 189278999768580696*x^7 - 50108801388853284*x^6 - 853099334603306505*x^5 + 370935283008394440*x^4 + 1695530573188686819*x^3 - 1057165556573924154*x^2 - 467536783979896065*x + 18638104434069437)
 
gp: K = bnfinit(x^27 - 513*x^25 + 116964*x^23 - 15617943*x^21 + 1354686795*x^19 - 768474*x^18 - 80025043581*x^17 + 262818108*x^16 + 3282297025608*x^15 - 37451580390*x^14 - 93545465229828*x^13 + 2877945888636*x^12 + 1824136571981646*x^11 - 128890862298198*x^10 - 23533044355733718*x^9 + 3390821146614132*x^8 + 189278999768580696*x^7 - 50108801388853284*x^6 - 853099334603306505*x^5 + 370935283008394440*x^4 + 1695530573188686819*x^3 - 1057165556573924154*x^2 - 467536783979896065*x + 18638104434069437, 1)
 

Normalized defining polynomial

\( x^{27} - 513 x^{25} + 116964 x^{23} - 15617943 x^{21} + 1354686795 x^{19} - 768474 x^{18} - 80025043581 x^{17} + 262818108 x^{16} + 3282297025608 x^{15} - 37451580390 x^{14} - 93545465229828 x^{13} + 2877945888636 x^{12} + 1824136571981646 x^{11} - 128890862298198 x^{10} - 23533044355733718 x^{9} + 3390821146614132 x^{8} + 189278999768580696 x^{7} - 50108801388853284 x^{6} - 853099334603306505 x^{5} + 370935283008394440 x^{4} + 1695530573188686819 x^{3} - 1057165556573924154 x^{2} - 467536783979896065 x + 18638104434069437 \)

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:  \(3463254175600113063839837232871102966170246194081956872095753304432278349849=3^{94}\cdot 19^{24}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $627.71$
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:  $3, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1539=3^{4}\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{1539}(256,·)$, $\chi_{1539}(1,·)$, $\chi_{1539}(514,·)$, $\chi_{1539}(1027,·)$, $\chi_{1539}(4,·)$, $\chi_{1539}(517,·)$, $\chi_{1539}(1030,·)$, $\chi_{1539}(385,·)$, $\chi_{1539}(64,·)$, $\chi_{1539}(898,·)$, $\chi_{1539}(16,·)$, $\chi_{1539}(529,·)$, $\chi_{1539}(1042,·)$, $\chi_{1539}(1411,·)$, $\chi_{1539}(1537,·)$, $\chi_{1539}(481,·)$, $\chi_{1539}(994,·)$, $\chi_{1539}(1507,·)$, $\chi_{1539}(1090,·)$, $\chi_{1539}(1024,·)$, $\chi_{1539}(1282,·)$, $\chi_{1539}(577,·)$, $\chi_{1539}(505,·)$, $\chi_{1539}(1018,·)$, $\chi_{1539}(1531,·)$, $\chi_{1539}(769,·)$, $\chi_{1539}(511,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9}$, $\frac{1}{19} a^{10}$, $\frac{1}{19} a^{11}$, $\frac{1}{19} a^{12}$, $\frac{1}{19} a^{13}$, $\frac{1}{45847} a^{14} - \frac{1}{127} a^{13} - \frac{14}{2413} a^{12} - \frac{6}{2413} a^{11} - \frac{61}{2413} a^{10} + \frac{62}{2413} a^{9} - \frac{53}{127} a^{8} + \frac{29}{127} a^{7} - \frac{38}{127} a^{6} + \frac{689}{2413} a^{5} - \frac{7}{127} a^{4} - \frac{19}{127} a^{3} + \frac{29}{127} a^{2} + \frac{58}{127} a - \frac{27}{127}$, $\frac{1}{45847} a^{15} - \frac{15}{2413} a^{13} + \frac{20}{2413} a^{12} + \frac{59}{2413} a^{11} + \frac{12}{2413} a^{10} + \frac{39}{2413} a^{9} - \frac{54}{127} a^{8} + \frac{17}{127} a^{7} + \frac{651}{2413} a^{6} + \frac{3}{127} a^{5} - \frac{6}{127} a^{4} + \frac{28}{127} a^{3} - \frac{14}{127} a^{2} - \frac{44}{127} a + \frac{32}{127}$, $\frac{1}{45847} a^{16} - \frac{61}{2413} a^{13} + \frac{6}{2413} a^{12} - \frac{47}{2413} a^{11} + \frac{53}{2413} a^{10} + \frac{7}{2413} a^{9} + \frac{25}{127} a^{8} + \frac{841}{2413} a^{7} - \frac{32}{127} a^{6} + \frac{42}{127} a^{5} - \frac{62}{127} a^{4} + \frac{32}{127} a^{3} - \frac{34}{127} a^{2} + \frac{52}{127} a + \frac{52}{127}$, $\frac{1}{45847} a^{17} - \frac{44}{2413} a^{13} - \frac{17}{2413} a^{12} - \frac{43}{2413} a^{11} + \frac{47}{2413} a^{10} - \frac{3}{127} a^{9} - \frac{793}{2413} a^{8} + \frac{51}{127} a^{7} - \frac{58}{127} a^{6} + \frac{57}{127} a^{5} + \frac{47}{127} a^{4} + \frac{43}{127} a^{3} + \frac{8}{127} a^{2} - \frac{36}{127} a - \frac{51}{127}$, $\frac{1}{4027962508901263} a^{18} - \frac{18}{211998026784277} a^{16} + \frac{135}{11157790883383} a^{14} - \frac{546}{587252151757} a^{12} + \frac{24453}{587252151757} a^{10} + \frac{146702853891}{12470472163781} a^{9} - \frac{643302}{587252151757} a^{8} - \frac{7644404621}{656340640199} a^{7} + \frac{9506574}{587252151757} a^{6} - \frac{11611030358}{34544244221} a^{5} - \frac{70373340}{587252151757} a^{4} + \frac{10988541171}{34544244221} a^{3} + \frac{200564019}{587252151757} a^{2} - \frac{3909469499}{34544244221} a - \frac{262697078623}{587252151757}$, $\frac{1}{4027962508901263} a^{19} - \frac{18}{211998026784277} a^{17} + \frac{135}{11157790883383} a^{15} - \frac{546}{587252151757} a^{13} + \frac{24453}{587252151757} a^{11} + \frac{146702853891}{12470472163781} a^{10} - \frac{643302}{587252151757} a^{9} - \frac{7644404621}{656340640199} a^{8} + \frac{9506574}{587252151757} a^{7} - \frac{11611030358}{34544244221} a^{6} - \frac{70373340}{587252151757} a^{5} + \frac{10988541171}{34544244221} a^{4} + \frac{200564019}{587252151757} a^{3} - \frac{3909469499}{34544244221} a^{2} - \frac{262697078623}{587252151757} a$, $\frac{1}{4027962508901263} a^{20} - \frac{189}{11157790883383} a^{16} + \frac{1884}{587252151757} a^{14} - \frac{162279}{587252151757} a^{12} + \frac{146702853891}{12470472163781} a^{11} + \frac{7719624}{587252151757} a^{10} + \frac{7644404621}{656340640199} a^{9} - \frac{210502710}{587252151757} a^{8} - \frac{11033336652}{34544244221} a^{7} + \frac{3180874968}{587252151757} a^{6} + \frac{12604244150}{34544244221} a^{5} - \frac{1403948133}{34544244221} a^{4} - \frac{11151009106}{34544244221} a^{3} - \frac{194104184125}{587252151757} a^{2} + \frac{10186955961}{34544244221} a + \frac{7178329755}{587252151757}$, $\frac{1}{76531287669123997} a^{21} - \frac{189}{211998026784277} a^{17} + \frac{1884}{11157790883383} a^{15} - \frac{8541}{587252151757} a^{13} + \frac{6053768615682}{236938971111839} a^{12} + \frac{406296}{587252151757} a^{11} + \frac{7644404621}{12470472163781} a^{10} - \frac{11079090}{587252151757} a^{9} - \frac{114666069315}{656340640199} a^{8} + \frac{167414472}{587252151757} a^{7} - \frac{15699681781}{34544244221} a^{6} - \frac{73892007}{34544244221} a^{5} - \frac{2405013333}{34544244221} a^{4} - \frac{1955860639396}{11157790883383} a^{3} - \frac{1281962540}{34544244221} a^{2} - \frac{30530201158}{587252151757} a$, $\frac{1}{76531287669123997} a^{22} - \frac{1518}{11157790883383} a^{16} + \frac{16974}{587252151757} a^{14} + \frac{6053768615682}{236938971111839} a^{13} - \frac{1554390}{587252151757} a^{12} + \frac{7644404621}{12470472163781} a^{11} + \frac{76731633}{587252151757} a^{10} + \frac{595904395}{34544244221} a^{9} - \frac{2142683010}{587252151757} a^{8} - \frac{9633897868}{34544244221} a^{7} + \frac{32881943115}{587252151757} a^{6} - \frac{21356332}{272001923} a^{5} + \frac{4400427629127}{11157790883383} a^{4} + \frac{9042482139}{34544244221} a^{3} + \frac{102443039314}{587252151757} a^{2} - \frac{13941817183}{34544244221} a - \frac{218253613451}{587252151757}$, $\frac{1}{2092353236399110687264477} a^{23} + \frac{708132}{110123854547321615119183} a^{22} - \frac{23}{110123854547321615119183} a^{21} + \frac{249425}{5795992344595874479957} a^{20} + \frac{230}{5795992344595874479957} a^{19} + \frac{305366}{5795992344595874479957} a^{18} - \frac{69}{16055380455944250637} a^{17} - \frac{84960657}{1215347524553548853} a^{16} + \frac{276}{944434144467308861} a^{15} - \frac{994930275126887150}{110123854547321615119183} a^{14} + \frac{87992041105013266288}{5795992344595874479957} a^{13} + \frac{13943986718972461783}{5795992344595874479957} a^{12} + \frac{3399754716756007352}{305052228662940762103} a^{11} + \frac{3562231353586504843}{305052228662940762103} a^{10} + \frac{2271222023447959422}{305052228662940762103} a^{9} + \frac{527453251375004561}{16055380455944250637} a^{8} - \frac{878008617176754190}{16055380455944250637} a^{7} + \frac{257908433002833053}{845020023997065823} a^{6} - \frac{131852269564879256851}{305052228662940762103} a^{5} - \frac{1576283323625552751}{16055380455944250637} a^{4} + \frac{831592957818022646}{16055380455944250637} a^{3} - \frac{397360472415011542}{845020023997065823} a^{2} - \frac{203274456029173075}{845020023997065823} a + \frac{151402549605682586}{845020023997065823}$, $\frac{1}{2092353236399110687264477} a^{24} - \frac{24}{110123854547321615119183} a^{22} + \frac{504057}{110123854547321615119183} a^{21} + \frac{252}{5795992344595874479957} a^{20} - \frac{512624}{5795992344595874479957} a^{19} - \frac{80}{16055380455944250637} a^{18} - \frac{86039541}{305052228662940762103} a^{17} + \frac{18}{49707060235121519} a^{16} - \frac{996552195567354236}{110123854547321615119183} a^{15} - \frac{864}{49707060235121519} a^{14} + \frac{14948345908246521273}{5795992344595874479957} a^{13} + \frac{34012977350595023175}{5795992344595874479957} a^{12} + \frac{6642111036987670902}{305052228662940762103} a^{11} + \frac{2293881257784444747}{305052228662940762103} a^{10} + \frac{266926872609374590}{16055380455944250637} a^{9} + \frac{5078424781842917408}{16055380455944250637} a^{8} + \frac{254506730818809141}{845020023997065823} a^{7} + \frac{31896851075746367369}{305052228662940762103} a^{6} - \frac{38960183205624457}{845020023997065823} a^{5} + \frac{1379579673847801945}{16055380455944250637} a^{4} - \frac{4046077953846405896}{16055380455944250637} a^{3} - \frac{67561602348405099}{845020023997065823} a^{2} - \frac{77227262640017427}{845020023997065823} a - \frac{415208894066654311}{845020023997065823}$, $\frac{1}{39754711491583103058025063} a^{25} + \frac{436032}{110123854547321615119183} a^{22} - \frac{300}{110123854547321615119183} a^{21} - \frac{250938}{5795992344595874479957} a^{20} + \frac{4000}{5795992344595874479957} a^{19} + \frac{430888}{5795992344595874479957} a^{18} - \frac{1350}{16055380455944250637} a^{17} + \frac{11013291896369874447}{2092353236399110687264477} a^{16} + \frac{5760}{944434144467308861} a^{15} + \frac{639823419574487860}{110123854547321615119183} a^{14} - \frac{55238963463271119003}{5795992344595874479957} a^{13} - \frac{110344981161292598578}{5795992344595874479957} a^{12} + \frac{5392652958188637433}{305052228662940762103} a^{11} - \frac{4640321992722031223}{305052228662940762103} a^{10} + \frac{2994467674298436579}{305052228662940762103} a^{9} - \frac{7286800842311282124}{16055380455944250637} a^{8} + \frac{109830373834968556556}{5795992344595874479957} a^{7} + \frac{375362486713561852}{845020023997065823} a^{6} - \frac{74281730299832951130}{305052228662940762103} a^{5} - \frac{4067251182501160551}{16055380455944250637} a^{4} - \frac{7616341146514994365}{16055380455944250637} a^{3} + \frac{255853574536277187}{845020023997065823} a^{2} + \frac{111672844092851909}{845020023997065823} a - \frac{14340953980483158}{49707060235121519}$, $\frac{1}{39754711491583103058025063} a^{26} - \frac{325}{110123854547321615119183} a^{22} + \frac{155031}{110123854547321615119183} a^{21} + \frac{4550}{5795992344595874479957} a^{20} + \frac{126284}{5795992344595874479957} a^{19} - \frac{1625}{16055380455944250637} a^{18} + \frac{11013371811255164988}{2092353236399110687264477} a^{17} + \frac{390}{49707060235121519} a^{16} + \frac{638922219661505740}{110123854547321615119183} a^{15} - \frac{19500}{49707060235121519} a^{14} + \frac{17858736485553061118}{5795992344595874479957} a^{13} - \frac{138589963112015061855}{5795992344595874479957} a^{12} - \frac{1483742892784184106}{305052228662940762103} a^{11} + \frac{1767385150253878482}{305052228662940762103} a^{10} - \frac{294230848597057186}{16055380455944250637} a^{9} + \frac{2516356902950716991466}{5795992344595874479957} a^{8} + \frac{351199629824821621}{845020023997065823} a^{7} + \frac{64231439262231945974}{305052228662940762103} a^{6} + \frac{193380949458551702}{845020023997065823} a^{5} + \frac{3388896060729875192}{16055380455944250637} a^{4} - \frac{4851485990733935973}{16055380455944250637} a^{3} + \frac{2857852502502470}{11901690478831913} a^{2} - \frac{18774842757936924}{845020023997065823} a - \frac{100254343198116070}{845020023997065823}$

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

$C_{27}$ (as 27T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 27
The 27 conjugacy class representatives for $C_{27}$
Character table for $C_{27}$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 9.9.1476349596018920529.1

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 $27$ R $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ R $27$ $27$ $27$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ $27$

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
$19$19.9.8.7$x^{9} - 1245184$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.7$x^{9} - 1245184$$9$$1$$8$$C_9$$[\ ]_{9}$
19.9.8.7$x^{9} - 1245184$$9$$1$$8$$C_9$$[\ ]_{9}$