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Properties

Label 27.27.9667962870...5089.2
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 61^{18}$
Root discriminant $710.08$
Ramified primes $3, 61$
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![-1146875791564075382836381, -4371712760668124402754687, 0, 2173911809949723063664899, 0, -320741086713893566770231, 0, 22033735855443117779688, 0, -852854074550393903130, 0, 20590515659785963086, 0, -328894323275269692, 0, 3594473478418248, 0, -27296507852019, 0, 143927517195, 0, -516835737, 0, 1205604, 0, -1647, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 1647*x^25 + 1205604*x^23 - 516835737*x^21 + 143927517195*x^19 - 27296507852019*x^17 + 3594473478418248*x^15 - 328894323275269692*x^13 + 20590515659785963086*x^11 - 852854074550393903130*x^9 + 22033735855443117779688*x^7 - 320741086713893566770231*x^5 + 2173911809949723063664899*x^3 - 4371712760668124402754687*x - 1146875791564075382836381)
 
gp: K = bnfinit(x^27 - 1647*x^25 + 1205604*x^23 - 516835737*x^21 + 143927517195*x^19 - 27296507852019*x^17 + 3594473478418248*x^15 - 328894323275269692*x^13 + 20590515659785963086*x^11 - 852854074550393903130*x^9 + 22033735855443117779688*x^7 - 320741086713893566770231*x^5 + 2173911809949723063664899*x^3 - 4371712760668124402754687*x - 1146875791564075382836381, 1)
 

Normalized defining polynomial

\( x^{27} - 1647 x^{25} + 1205604 x^{23} - 516835737 x^{21} + 143927517195 x^{19} - 27296507852019 x^{17} + 3594473478418248 x^{15} - 328894323275269692 x^{13} + 20590515659785963086 x^{11} - 852854074550393903130 x^{9} + 22033735855443117779688 x^{7} - 320741086713893566770231 x^{5} + 2173911809949723063664899 x^{3} - 4371712760668124402754687 x - 1146875791564075382836381 \)

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:  \(96679628704383224900411211844182831619287468040811424837360737394634888225089=3^{94}\cdot 61^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $710.08$
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:  $3, 61$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4941=3^{4}\cdot 61\)
Dirichlet character group:    $\lbrace$$\chi_{4941}(1,·)$, $\chi_{4941}(3844,·)$, $\chi_{4941}(2182,·)$, $\chi_{4941}(4039,·)$, $\chi_{4941}(1099,·)$, $\chi_{4941}(1294,·)$, $\chi_{4941}(3280,·)$, $\chi_{4941}(2197,·)$, $\chi_{4941}(535,·)$, $\chi_{4941}(2392,·)$, $\chi_{4941}(196,·)$, $\chi_{4941}(4378,·)$, $\chi_{4941}(3295,·)$, $\chi_{4941}(1633,·)$, $\chi_{4941}(3490,·)$, $\chi_{4941}(550,·)$, $\chi_{4941}(4393,·)$, $\chi_{4941}(2731,·)$, $\chi_{4941}(4588,·)$, $\chi_{4941}(1648,·)$, $\chi_{4941}(1843,·)$, $\chi_{4941}(3829,·)$, $\chi_{4941}(745,·)$, $\chi_{4941}(2746,·)$, $\chi_{4941}(1084,·)$, $\chi_{4941}(2941,·)$, $\chi_{4941}(4927,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{61} a^{3}$, $\frac{1}{61} a^{4}$, $\frac{1}{61} a^{5}$, $\frac{1}{3721} a^{6}$, $\frac{1}{3721} a^{7}$, $\frac{1}{3721} a^{8}$, $\frac{1}{226981} a^{9}$, $\frac{1}{226981} a^{10}$, $\frac{1}{226981} a^{11}$, $\frac{1}{13845841} a^{12}$, $\frac{1}{13845841} a^{13}$, $\frac{1}{1540944060899879} a^{14} + \frac{49872332}{1540944060899879} a^{13} - \frac{14}{25261378047539} a^{12} + \frac{19417198}{25261378047539} a^{11} + \frac{77}{414120951599} a^{10} - \frac{23720683}{25261378047539} a^{9} - \frac{210}{6788868059} a^{8} - \frac{32104696}{414120951599} a^{7} + \frac{294}{111292919} a^{6} + \frac{357839}{6788868059} a^{5} - \frac{11956}{111292919} a^{4} + \frac{44732370}{6788868059} a^{3} + \frac{182329}{111292919} a^{2} - \frac{22289327}{111292919} a - \frac{453962}{111292919}$, $\frac{1}{93997587714892619} a^{15} - \frac{15}{1540944060899879} a^{13} + \frac{49872332}{1540944060899879} a^{12} + \frac{90}{25261378047539} a^{11} - \frac{42003389}{25261378047539} a^{10} - \frac{275}{414120951599} a^{9} + \frac{22075872}{414120951599} a^{8} + \frac{450}{6788868059} a^{7} + \frac{51145754}{414120951599} a^{6} - \frac{378}{111292919} a^{5} + \frac{21608930}{6788868059} a^{4} + \frac{8540}{111292919} a^{3} - \frac{7408776}{111292919} a^{2} - \frac{55815}{111292919} a - \frac{11990121}{111292919}$, $\frac{1}{93997587714892619} a^{16} + \frac{18906879}{1540944060899879} a^{13} - \frac{120}{25261378047539} a^{12} + \frac{26668743}{25261378047539} a^{11} + \frac{880}{414120951599} a^{10} - \frac{10818324}{25261378047539} a^{9} - \frac{2700}{6788868059} a^{8} + \frac{14746990}{414120951599} a^{7} + \frac{4032}{111292919} a^{6} + \frac{26976515}{6788868059} a^{5} - \frac{170800}{111292919} a^{4} - \frac{3535624}{6788868059} a^{3} + \frac{2679120}{111292919} a^{2} - \frac{12451269}{111292919} a - \frac{6809430}{111292919}$, $\frac{1}{93997587714892619} a^{17} - \frac{136}{25261378047539} a^{13} - \frac{33599051}{1540944060899879} a^{12} + \frac{1088}{414120951599} a^{11} - \frac{4679625}{25261378047539} a^{10} - \frac{3740}{6788868059} a^{9} + \frac{38475236}{414120951599} a^{8} + \frac{6528}{111292919} a^{7} + \frac{19829715}{414120951599} a^{6} - \frac{348432}{111292919} a^{5} + \frac{4457959}{6788868059} a^{4} + \frac{48738980}{6788868059} a^{3} + \frac{13373565}{111292919} a^{2} - \frac{54021478}{111292919} a - \frac{16601601}{111292919}$, $\frac{1}{5733852850608449759} a^{18} - \frac{48744615}{1540944060899879} a^{13} - \frac{816}{25261378047539} a^{12} - \frac{26718895}{25261378047539} a^{11} + \frac{6732}{414120951599} a^{10} + \frac{39956999}{25261378047539} a^{9} - \frac{22032}{6788868059} a^{8} + \frac{34717907}{414120951599} a^{7} + \frac{16233193}{414120951599} a^{6} - \frac{2346089}{6788868059} a^{5} + \frac{20202839}{6788868059} a^{4} - \frac{24134660}{6788868059} a^{3} + \frac{23911146}{111292919} a^{2} + \frac{4304243}{111292919} a + \frac{49554087}{111292919}$, $\frac{1}{5733852850608449759} a^{19} - \frac{969}{25261378047539} a^{13} + \frac{35191686}{1540944060899879} a^{12} + \frac{8721}{414120951599} a^{11} - \frac{47413648}{25261378047539} a^{10} - \frac{7693498}{25261378047539} a^{9} - \frac{30524653}{414120951599} a^{8} - \frac{6246898}{414120951599} a^{7} - \frac{24749836}{414120951599} a^{6} + \frac{27880792}{6788868059} a^{5} + \frac{45348170}{6788868059} a^{4} - \frac{55466229}{6788868059} a^{3} + \frac{42579995}{111292919} a^{2} + \frac{43260554}{111292919} a - \frac{54415698}{111292919}$, $\frac{1}{5733852850608449759} a^{20} + \frac{12025402}{1540944060899879} a^{13} - \frac{4845}{414120951599} a^{12} + \frac{31162206}{25261378047539} a^{11} + \frac{47355637}{25261378047539} a^{10} - \frac{7682095}{25261378047539} a^{9} + \frac{15617245}{414120951599} a^{8} - \frac{45663831}{414120951599} a^{7} + \frac{33857754}{414120951599} a^{6} + \frac{51199011}{6788868059} a^{5} + \frac{17046899}{6788868059} a^{4} + \frac{26131286}{6788868059} a^{3} + \frac{25132272}{111292919} a^{2} + \frac{42622700}{111292919} a - \frac{11646379}{111292919}$, $\frac{1}{349765023887115435299} a^{21} - \frac{5985}{25261378047539} a^{13} - \frac{378164}{25261378047539} a^{12} + \frac{57456}{414120951599} a^{11} - \frac{13844849}{25261378047539} a^{10} - \frac{37523017}{25261378047539} a^{9} - \frac{4135245}{414120951599} a^{8} - \frac{31002466}{414120951599} a^{7} - \frac{37976354}{414120951599} a^{6} + \frac{43786292}{6788868059} a^{5} - \frac{30736921}{6788868059} a^{4} - \frac{21957616}{6788868059} a^{3} + \frac{53645792}{111292919} a^{2} - \frac{19788352}{111292919} a + \frac{13534808}{111292919}$, $\frac{1}{349765023887115435299} a^{22} - \frac{15581103}{1540944060899879} a^{13} + \frac{13304105}{1540944060899879} a^{12} + \frac{118357}{25261378047539} a^{11} + \frac{7887443}{25261378047539} a^{10} + \frac{51981904}{25261378047539} a^{9} - \frac{33438718}{414120951599} a^{8} + \frac{55434809}{414120951599} a^{7} - \frac{36854745}{414120951599} a^{6} - \frac{46972512}{6788868059} a^{5} + \frac{39665691}{6788868059} a^{4} + \frac{39266051}{6788868059} a^{3} - \frac{7370949}{111292919} a^{2} + \frac{30238455}{111292919} a - \frac{19560379}{111292919}$, $\frac{1}{349765023887115435299} a^{23} - \frac{13915010}{1540944060899879} a^{13} - \frac{55184824}{1540944060899879} a^{12} + \frac{27857181}{25261378047539} a^{11} + \frac{5681993}{25261378047539} a^{10} - \frac{6148801}{25261378047539} a^{9} + \frac{9709146}{414120951599} a^{8} + \frac{1621082}{414120951599} a^{7} - \frac{51022218}{414120951599} a^{6} + \frac{13326046}{6788868059} a^{5} + \frac{55044998}{6788868059} a^{4} - \frac{35793933}{6788868059} a^{3} + \frac{54116948}{111292919} a^{2} + \frac{16714415}{111292919} a - \frac{7213041}{111292919}$, $\frac{1}{21335666457114041553239} a^{24} - \frac{44845737}{1540944060899879} a^{13} + \frac{55632879}{1540944060899879} a^{12} - \frac{12504678}{25261378047539} a^{11} + \frac{47825010}{25261378047539} a^{10} - \frac{43206264}{25261378047539} a^{9} + \frac{13453273}{414120951599} a^{8} + \frac{55006065}{414120951599} a^{7} + \frac{760508}{6788868059} a^{6} - \frac{14074163}{6788868059} a^{5} - \frac{570381}{111292919} a^{4} + \frac{10083283}{6788868059} a^{3} + \frac{39841680}{111292919} a^{2} + \frac{34842584}{111292919} a - \frac{53089750}{111292919}$, $\frac{1}{21335666457114041553239} a^{25} + \frac{8768414}{1540944060899879} a^{13} + \frac{2769813}{1540944060899879} a^{12} - \frac{175319}{414120951599} a^{11} + \frac{31017677}{25261378047539} a^{10} - \frac{28122694}{25261378047539} a^{9} - \frac{36129946}{414120951599} a^{8} - \frac{53379424}{414120951599} a^{7} - \frac{42445333}{414120951599} a^{6} - \frac{29687346}{6788868059} a^{5} - \frac{51699808}{6788868059} a^{4} - \frac{6892157}{6788868059} a^{3} + \frac{22465127}{111292919} a^{2} + \frac{8904835}{111292919} a - \frac{3251919}{111292919}$, $\frac{1}{21335666457114041553239} a^{26} + \frac{12002523}{1540944060899879} a^{13} + \frac{770418}{25261378047539} a^{12} + \frac{22285366}{25261378047539} a^{11} - \frac{34983222}{25261378047539} a^{10} + \frac{14256473}{25261378047539} a^{9} - \frac{24551355}{414120951599} a^{8} + \frac{20492317}{414120951599} a^{7} + \frac{802891}{414120951599} a^{6} - \frac{50931787}{6788868059} a^{5} + \frac{46606927}{6788868059} a^{4} - \frac{44213381}{6788868059} a^{3} - \frac{4469936}{111292919} a^{2} - \frac{7855036}{111292919} a + \frac{24215314}{111292919}$

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

\(\Q(\zeta_{9})^+\), \(\Q(\zeta_{27})^+\)

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 $27$ R $27$ $27$ $27$ $27$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/19.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{3}$ $27$ $27$ $27$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ $27$

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
61Data not computed