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Properties

Label 27.27.7950204228...7601.2
Degree $27$
Signature $[27, 0]$
Discriminant $3^{94}\cdot 13^{18}$
Root discriminant $253.35$
Ramified primes $3, 13$
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![-920226642090821, -8177627877990831, 0, 19081131715311939, 0, -13210014264446727, 0, 4258173096231912, 0, -773386139272890, 0, 87614373819726, 0, -6566757801084, 0, 336756810312, 0, -11999818467, 0, 296891595, 0, -5002569, 0, 54756, 0, -351, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 351*x^25 + 54756*x^23 - 5002569*x^21 + 296891595*x^19 - 11999818467*x^17 + 336756810312*x^15 - 6566757801084*x^13 + 87614373819726*x^11 - 773386139272890*x^9 + 4258173096231912*x^7 - 13210014264446727*x^5 + 19081131715311939*x^3 - 8177627877990831*x - 920226642090821)
 
gp: K = bnfinit(x^27 - 351*x^25 + 54756*x^23 - 5002569*x^21 + 296891595*x^19 - 11999818467*x^17 + 336756810312*x^15 - 6566757801084*x^13 + 87614373819726*x^11 - 773386139272890*x^9 + 4258173096231912*x^7 - 13210014264446727*x^5 + 19081131715311939*x^3 - 8177627877990831*x - 920226642090821, 1)
 

Normalized defining polynomial

\( x^{27} - 351 x^{25} + 54756 x^{23} - 5002569 x^{21} + 296891595 x^{19} - 11999818467 x^{17} + 336756810312 x^{15} - 6566757801084 x^{13} + 87614373819726 x^{11} - 773386139272890 x^{9} + 4258173096231912 x^{7} - 13210014264446727 x^{5} + 19081131715311939 x^{3} - 8177627877990831 x - 920226642090821 \)

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:  \(79502042287804104995388608594472718525612115183651404436672117601=3^{94}\cdot 13^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $253.35$
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:  $3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(1053=3^{4}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{1053}(256,·)$, $\chi_{1053}(1,·)$, $\chi_{1053}(835,·)$, $\chi_{1053}(133,·)$, $\chi_{1053}(841,·)$, $\chi_{1053}(586,·)$, $\chi_{1053}(139,·)$, $\chi_{1053}(718,·)$, $\chi_{1053}(16,·)$, $\chi_{1053}(724,·)$, $\chi_{1053}(469,·)$, $\chi_{1053}(22,·)$, $\chi_{1053}(601,·)$, $\chi_{1053}(607,·)$, $\chi_{1053}(352,·)$, $\chi_{1053}(484,·)$, $\chi_{1053}(937,·)$, $\chi_{1053}(490,·)$, $\chi_{1053}(235,·)$, $\chi_{1053}(367,·)$, $\chi_{1053}(820,·)$, $\chi_{1053}(373,·)$, $\chi_{1053}(118,·)$, $\chi_{1053}(952,·)$, $\chi_{1053}(250,·)$, $\chi_{1053}(958,·)$, $\chi_{1053}(703,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{13} a^{3}$, $\frac{1}{13} a^{4}$, $\frac{1}{13} a^{5}$, $\frac{1}{169} a^{6}$, $\frac{1}{169} a^{7}$, $\frac{1}{169} a^{8}$, $\frac{1}{2197} a^{9}$, $\frac{1}{2197} a^{10}$, $\frac{1}{2197} a^{11}$, $\frac{1}{28561} a^{12}$, $\frac{1}{28561} a^{13}$, $\frac{1}{3079989679} a^{14} - \frac{28801}{3079989679} a^{13} - \frac{14}{236922283} a^{12} + \frac{50896}{236922283} a^{11} + \frac{77}{18224791} a^{10} + \frac{34769}{236922283} a^{9} - \frac{210}{1401907} a^{8} - \frac{40310}{18224791} a^{7} + \frac{294}{107839} a^{6} + \frac{11082}{1401907} a^{5} - \frac{2548}{107839} a^{4} + \frac{35806}{1401907} a^{3} + \frac{8281}{107839} a^{2} + \frac{25696}{107839} a - \frac{4394}{107839}$, $\frac{1}{40039865827} a^{15} - \frac{15}{3079989679} a^{13} - \frac{28801}{3079989679} a^{12} + \frac{90}{236922283} a^{11} + \frac{22095}{236922283} a^{10} - \frac{275}{18224791} a^{9} - \frac{45508}{18224791} a^{8} + \frac{450}{1401907} a^{7} - \frac{15115}{18224791} a^{6} - \frac{378}{107839} a^{5} + \frac{47870}{1401907} a^{4} + \frac{1820}{107839} a^{3} - \frac{1007}{107839} a^{2} - \frac{2535}{107839} a - \frac{29228}{107839}$, $\frac{1}{40039865827} a^{16} - \frac{29460}{3079989679} a^{13} - \frac{120}{236922283} a^{12} + \frac{30662}{236922283} a^{11} + \frac{880}{18224791} a^{10} + \frac{37770}{236922283} a^{9} - \frac{2700}{1401907} a^{8} + \frac{27269}{18224791} a^{7} + \frac{34374}{18224791} a^{6} - \frac{1578}{1401907} a^{5} - \frac{41844}{1401907} a^{4} - \frac{15196}{1401907} a^{3} + \frac{13841}{107839} a^{2} + \frac{32695}{107839} a + \frac{41929}{107839}$, $\frac{1}{40039865827} a^{17} - \frac{136}{236922283} a^{13} - \frac{2520}{3079989679} a^{12} + \frac{1088}{18224791} a^{11} - \frac{20656}{236922283} a^{10} + \frac{14974}{236922283} a^{9} + \frac{49363}{18224791} a^{8} + \frac{24842}{18224791} a^{7} + \frac{30299}{18224791} a^{6} + \frac{5223}{1401907} a^{5} - \frac{13125}{1401907} a^{4} + \frac{35756}{1401907} a^{3} - \frac{48702}{107839} a^{2} + \frac{16309}{107839} a - \frac{40440}{107839}$, $\frac{1}{520518255751} a^{18} + \frac{1142}{236922283} a^{13} - \frac{816}{236922283} a^{12} + \frac{43457}{236922283} a^{11} - \frac{20323}{236922283} a^{10} + \frac{33031}{236922283} a^{9} + \frac{37101}{18224791} a^{8} + \frac{28255}{18224791} a^{7} - \frac{31338}{18224791} a^{6} + \frac{999}{107839} a^{5} - \frac{39238}{1401907} a^{4} - \frac{31841}{1401907} a^{3} + \frac{7604}{107839} a^{2} + \frac{7516}{107839} a + \frac{49450}{107839}$, $\frac{1}{520518255751} a^{19} - \frac{969}{236922283} a^{13} + \frac{31743}{3079989679} a^{12} + \frac{5534}{236922283} a^{11} - \frac{4144}{18224791} a^{10} - \frac{12163}{236922283} a^{9} + \frac{10371}{18224791} a^{8} + \frac{947}{1401907} a^{7} + \frac{41396}{18224791} a^{6} - \frac{296}{1401907} a^{5} - \frac{18777}{1401907} a^{4} - \frac{46432}{1401907} a^{3} + \frac{4250}{107839} a^{2} - \frac{6823}{107839} a - \frac{9271}{107839}$, $\frac{1}{520518255751} a^{20} - \frac{4058}{3079989679} a^{13} + \frac{43907}{3079989679} a^{12} - \frac{19815}{236922283} a^{11} - \frac{19729}{236922283} a^{10} - \frac{29941}{236922283} a^{9} + \frac{23142}{18224791} a^{8} - \frac{37662}{18224791} a^{7} - \frac{6862}{18224791} a^{6} + \frac{37511}{1401907} a^{5} + \frac{27470}{1401907} a^{4} + \frac{12895}{1401907} a^{3} + \frac{28621}{107839} a^{2} - \frac{49437}{107839} a - \frac{29811}{107839}$, $\frac{1}{6766737324763} a^{21} + \frac{30034}{3079989679} a^{13} + \frac{31212}{3079989679} a^{12} - \frac{7945}{236922283} a^{11} + \frac{44713}{236922283} a^{10} + \frac{9566}{236922283} a^{9} - \frac{1327}{1401907} a^{8} + \frac{17123}{18224791} a^{7} + \frac{18371}{18224791} a^{6} + \frac{10554}{1401907} a^{5} - \frac{11134}{1401907} a^{4} - \frac{26245}{1401907} a^{3} + \frac{42779}{107839} a^{2} - \frac{44227}{107839} a + \frac{30303}{107839}$, $\frac{1}{6766737324763} a^{22} - \frac{44012}{3079989679} a^{13} - \frac{29047}{3079989679} a^{12} + \frac{52074}{236922283} a^{11} + \frac{32613}{236922283} a^{10} + \frac{52145}{236922283} a^{9} + \frac{52303}{18224791} a^{8} - \frac{19542}{18224791} a^{7} + \frac{36121}{18224791} a^{6} + \frac{51071}{1401907} a^{5} + \frac{5196}{1401907} a^{4} - \frac{9964}{1401907} a^{3} + \frac{28792}{107839} a^{2} - \frac{27477}{107839} a - \frac{25540}{107839}$, $\frac{1}{6766737324763} a^{23} + \frac{28786}{3079989679} a^{13} - \frac{170}{3079989679} a^{12} + \frac{35657}{236922283} a^{11} + \frac{2006}{236922283} a^{10} + \frac{50723}{236922283} a^{9} - \frac{39656}{18224791} a^{8} - \frac{2170}{1401907} a^{7} + \frac{37879}{18224791} a^{6} - \frac{9617}{1401907} a^{5} + \frac{11989}{1401907} a^{4} - \frac{14695}{1401907} a^{3} + \frac{47914}{107839} a^{2} - \frac{781}{107839} a - \frac{33401}{107839}$, $\frac{1}{87967585221919} a^{24} - \frac{33244}{3079989679} a^{13} + \frac{7305}{3079989679} a^{12} - \frac{7495}{236922283} a^{11} - \frac{1743}{18224791} a^{10} + \frac{26193}{236922283} a^{9} + \frac{3906}{18224791} a^{8} + \frac{29032}{18224791} a^{7} - \frac{33929}{18224791} a^{6} - \frac{41992}{1401907} a^{5} - \frac{1513}{1401907} a^{4} - \frac{50580}{1401907} a^{3} - \frac{28998}{107839} a^{2} + \frac{4391}{107839} a + \frac{24158}{107839}$, $\frac{1}{87967585221919} a^{25} + \frac{49342}{3079989679} a^{13} - \frac{1020}{3079989679} a^{12} - \frac{29945}{236922283} a^{11} - \frac{18814}{236922283} a^{10} - \frac{14827}{236922283} a^{9} - \frac{2653}{1401907} a^{8} + \frac{15684}{18224791} a^{7} - \frac{15280}{18224791} a^{6} + \frac{30471}{1401907} a^{5} + \frac{27032}{1401907} a^{4} - \frac{45675}{1401907} a^{3} - \frac{15012}{107839} a^{2} - \frac{38576}{107839} a + \frac{47709}{107839}$, $\frac{1}{87967585221919} a^{26} - \frac{340}{236922283} a^{13} - \frac{36161}{3079989679} a^{12} + \frac{25386}{236922283} a^{11} - \frac{15907}{236922283} a^{10} + \frac{21652}{236922283} a^{9} + \frac{28433}{18224791} a^{8} - \frac{21776}{18224791} a^{7} - \frac{30019}{18224791} a^{6} - \frac{37282}{1401907} a^{5} - \frac{49151}{1401907} a^{4} + \frac{7207}{1401907} a^{3} - \frac{37707}{107839} a^{2} + \frac{18800}{107839} a + \frac{52358}{107839}$

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

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
13Data not computed