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Properties

Label 27.27.7950204228...7601.1
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![-2175481232872831, -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 - 2175481232872831)
 
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 - 2175481232872831, 1)
 

Normalized defining polynomial

\( x^{27} - 351 x^{25} + 54756 x^{23} - 5002569 x^{21} + 296891595 x^{19} - 11999818467 x^{17} + 336756810312 x^{15} - 6566757801084 x^{13} + 87614373819726 x^{11} - 773386139272890 x^{9} + 4258173096231912 x^{7} - 13210014264446727 x^{5} + 19081131715311939 x^{3} - 8177627877990831 x - 2175481232872831 \)

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$$\chi_{1053}(1,·)$, $\chi_{1053}(646,·)$, $\chi_{1053}(328,·)$, $\chi_{1053}(586,·)$, $\chi_{1053}(529,·)$, $\chi_{1053}(211,·)$, $\chi_{1053}(469,·)$, $\chi_{1053}(412,·)$, $\chi_{1053}(94,·)$, $\chi_{1053}(997,·)$, $\chi_{1053}(352,·)$, $\chi_{1053}(913,·)$, $\chi_{1053}(679,·)$, $\chi_{1053}(1030,·)$, $\chi_{1053}(796,·)$, $\chi_{1053}(295,·)$, $\chi_{1053}(937,·)$, $\chi_{1053}(235,·)$, $\chi_{1053}(178,·)$, $\chi_{1053}(61,·)$, $\chi_{1053}(880,·)$, $\chi_{1053}(562,·)$, $\chi_{1053}(820,·)$, $\chi_{1053}(118,·)$, $\chi_{1053}(763,·)$, $\chi_{1053}(445,·)$, $\chi_{1053}(703,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{13} a^{3}$, $\frac{1}{13} a^{4}$, $\frac{1}{13} a^{5}$, $\frac{1}{169} a^{6}$, $\frac{1}{169} a^{7}$, $\frac{1}{169} a^{8}$, $\frac{1}{2197} a^{9}$, $\frac{1}{2197} a^{10}$, $\frac{1}{2197} a^{11}$, $\frac{1}{28561} a^{12}$, $\frac{1}{28561} a^{13}$, $\frac{1}{300775891} a^{14} - \frac{2816}{300775891} a^{13} - \frac{14}{23136607} a^{12} + \frac{5015}{23136607} a^{11} + \frac{77}{1779739} a^{10} + \frac{486}{23136607} a^{9} - \frac{210}{136903} a^{8} + \frac{3046}{1779739} a^{7} - \frac{2969}{1779739} a^{6} + \frac{3467}{136903} a^{5} - \frac{1531}{136903} a^{4} + \frac{3792}{136903} a^{3} - \frac{2250}{10531} a^{2} - \frac{5055}{10531} a - \frac{4394}{10531}$, $\frac{1}{3910086583} a^{15} - \frac{15}{300775891} a^{13} - \frac{2816}{300775891} a^{12} + \frac{90}{23136607} a^{11} + \frac{2199}{23136607} a^{10} - \frac{275}{1779739} a^{9} - \frac{4630}{1779739} a^{8} - \frac{4681}{1779739} a^{7} + \frac{3537}{1779739} a^{6} - \frac{378}{10531} a^{5} - \frac{25}{136903} a^{4} + \frac{2598}{136903} a^{3} + \frac{1513}{10531} a^{2} - \frac{2535}{10531} a - \frac{4018}{10531}$, $\frac{1}{3910086583} a^{16} - \frac{2932}{300775891} a^{13} - \frac{120}{23136607} a^{12} + \frac{3707}{23136607} a^{11} + \frac{909}{23136607} a^{10} - \frac{245}{23136607} a^{9} - \frac{3507}{1779739} a^{8} - \frac{3428}{1779739} a^{7} - \frac{239}{136903} a^{6} - \frac{675}{136903} a^{5} + \frac{695}{136903} a^{4} + \frac{2832}{136903} a^{3} - \frac{4692}{10531} a^{2} + \frac{4405}{10531} a - \frac{2724}{10531}$, $\frac{1}{3910086583} a^{17} - \frac{136}{23136607} a^{13} - \frac{1007}{300775891} a^{12} + \frac{3613}{23136607} a^{11} - \frac{3462}{23136607} a^{10} - \frac{200}{23136607} a^{9} - \frac{4228}{1779739} a^{8} - \frac{2523}{1779739} a^{7} - \frac{4746}{1779739} a^{6} + \frac{3524}{136903} a^{5} + \frac{146}{136903} a^{4} - \frac{402}{136903} a^{3} - \frac{189}{10531} a^{2} + \frac{3664}{10531} a - \frac{3795}{10531}$, $\frac{1}{50831125579} a^{18} + \frac{71}{23136607} a^{13} - \frac{77}{300775891} a^{12} + \frac{499}{23136607} a^{11} + \frac{3268}{23136607} a^{10} - \frac{1318}{23136607} a^{9} - \frac{2079}{1779739} a^{8} + \frac{3992}{1779739} a^{7} - \frac{82}{1779739} a^{6} - \frac{4802}{136903} a^{5} + \frac{1563}{136903} a^{4} - \frac{496}{136903} a^{3} + \frac{1301}{10531} a^{2} - \frac{4067}{10531} a + \frac{2683}{10531}$, $\frac{1}{50831125579} a^{19} - \frac{2066}{300775891} a^{13} - \frac{4554}{300775891} a^{12} - \frac{2468}{23136607} a^{11} + \frac{1487}{23136607} a^{10} - \frac{1710}{23136607} a^{9} - \frac{3658}{1779739} a^{8} + \frac{237}{1779739} a^{7} + \frac{3087}{1779739} a^{6} + \frac{2946}{136903} a^{5} + \frac{1463}{136903} a^{4} + \frac{2658}{136903} a^{3} - \frac{1924}{10531} a^{2} + \frac{3215}{10531} a + \frac{1227}{10531}$, $\frac{1}{50831125579} a^{20} + \frac{1233}{300775891} a^{13} + \frac{201}{23136607} a^{12} - \frac{27}{23136607} a^{11} + \frac{2280}{23136607} a^{10} - \frac{1799}{23136607} a^{9} + \frac{4673}{1779739} a^{8} - \frac{1415}{1779739} a^{7} + \frac{1793}{1779739} a^{6} + \frac{3205}{136903} a^{5} - \frac{1088}{136903} a^{4} - \frac{4742}{136903} a^{3} - \frac{1114}{10531} a^{2} + \frac{4349}{10531} a - \frac{282}{10531}$, $\frac{1}{660804632527} a^{21} - \frac{4088}{300775891} a^{13} - \frac{3827}{300775891} a^{12} - \frac{773}{23136607} a^{11} + \frac{3750}{23136607} a^{10} + \frac{3132}{23136607} a^{9} + \frac{4457}{1779739} a^{8} + \frac{434}{1779739} a^{7} - \frac{1964}{1779739} a^{6} + \frac{786}{136903} a^{5} + \frac{651}{136903} a^{4} + \frac{3764}{136903} a^{3} - \frac{932}{10531} a^{2} - \frac{950}{10531} a - \frac{4486}{10531}$, $\frac{1}{660804632527} a^{22} - \frac{404}{23136607} a^{13} + \frac{4167}{300775891} a^{12} + \frac{1213}{23136607} a^{11} - \frac{103}{1779739} a^{10} + \frac{1695}{23136607} a^{9} + \frac{3054}{1779739} a^{8} + \frac{2442}{1779739} a^{7} + \frac{4658}{1779739} a^{6} - \frac{979}{136903} a^{5} + \frac{450}{136903} a^{4} - \frac{117}{10531} a^{3} + \frac{5144}{10531} a^{2} + \frac{3027}{10531} a + \frac{3214}{10531}$, $\frac{1}{660804632527} a^{23} + \frac{59}{300775891} a^{13} - \frac{2836}{300775891} a^{12} - \frac{590}{23136607} a^{11} + \frac{306}{1779739} a^{10} + \frac{1548}{23136607} a^{9} - \frac{2827}{1779739} a^{8} - \frac{4870}{1779739} a^{7} + \frac{79}{136903} a^{6} + \frac{1035}{136903} a^{5} + \frac{3351}{136903} a^{4} + \frac{5149}{136903} a^{3} + \frac{1809}{10531} a^{2} + \frac{3005}{10531} a - \frac{3867}{10531}$, $\frac{1}{8590460222851} a^{24} + \frac{411}{300775891} a^{13} + \frac{236}{300775891} a^{12} + \frac{2658}{23136607} a^{11} - \frac{5234}{23136607} a^{10} + \frac{2258}{23136607} a^{9} - \frac{2566}{1779739} a^{8} - \frac{1594}{1779739} a^{7} - \frac{3312}{1779739} a^{6} - \frac{4136}{136903} a^{5} + \frac{2484}{136903} a^{4} + \frac{4041}{136903} a^{3} + \frac{3152}{10531} a^{2} - \frac{3278}{10531} a - \frac{1120}{10531}$, $\frac{1}{8590460222851} a^{25} - \frac{798}{300775891} a^{13} + \frac{4046}{300775891} a^{12} - \frac{2323}{23136607} a^{11} + \frac{1556}{23136607} a^{10} - \frac{1422}{23136607} a^{9} + \frac{4150}{1779739} a^{8} - \frac{2029}{1779739} a^{7} - \frac{2450}{1779739} a^{6} - \frac{768}{136903} a^{5} + \frac{1422}{136903} a^{4} - \frac{1072}{136903} a^{3} - \frac{5256}{10531} a^{2} + \frac{1878}{10531} a + \frac{5133}{10531}$, $\frac{1}{8590460222851} a^{26} - \frac{19}{300775891} a^{13} + \frac{3592}{300775891} a^{12} + \frac{1746}{23136607} a^{11} - \frac{2980}{23136607} a^{10} - \frac{524}{23136607} a^{9} - \frac{652}{1779739} a^{8} - \frac{4403}{1779739} a^{7} + \frac{760}{1779739} a^{6} - \frac{1565}{136903} a^{5} - \frac{1214}{136903} a^{4} - \frac{1523}{136903} a^{3} - \frac{3352}{10531} a^{2} + \frac{4616}{10531} a + \frac{411}{10531}$

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