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Properties

Label 27.27.550...809.1
Degree $27$
Signature $[27, 0]$
Discriminant $5.509\times 10^{50}$
Root discriminant $75.74$
Ramified primes $13, 19$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_3\times C_9$ (as 27T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 6*x^26 - 47*x^25 + 302*x^24 + 943*x^23 - 6448*x^22 - 10567*x^21 + 76481*x^20 + 71695*x^19 - 556066*x^18 - 291044*x^17 + 2587104*x^16 + 603975*x^15 - 7811439*x^14 - 35156*x^13 + 15161145*x^12 - 2848142*x^11 - 18230502*x^10 + 6439289*x^9 + 12558384*x^8 - 6282221*x^7 - 4210683*x^6 + 2778704*x^5 + 421825*x^4 - 457387*x^3 + 20126*x^2 + 21522*x - 2699)
 
gp: K = bnfinit(x^27 - 6*x^26 - 47*x^25 + 302*x^24 + 943*x^23 - 6448*x^22 - 10567*x^21 + 76481*x^20 + 71695*x^19 - 556066*x^18 - 291044*x^17 + 2587104*x^16 + 603975*x^15 - 7811439*x^14 - 35156*x^13 + 15161145*x^12 - 2848142*x^11 - 18230502*x^10 + 6439289*x^9 + 12558384*x^8 - 6282221*x^7 - 4210683*x^6 + 2778704*x^5 + 421825*x^4 - 457387*x^3 + 20126*x^2 + 21522*x - 2699, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-2699, 21522, 20126, -457387, 421825, 2778704, -4210683, -6282221, 12558384, 6439289, -18230502, -2848142, 15161145, -35156, -7811439, 603975, 2587104, -291044, -556066, 71695, 76481, -10567, -6448, 943, 302, -47, -6, 1]);
 

\( x^{27} - 6 x^{26} - 47 x^{25} + 302 x^{24} + 943 x^{23} - 6448 x^{22} - 10567 x^{21} + 76481 x^{20} + 71695 x^{19} - 556066 x^{18} - 291044 x^{17} + 2587104 x^{16} + 603975 x^{15} - 7811439 x^{14} - 35156 x^{13} + 15161145 x^{12} - 2848142 x^{11} - 18230502 x^{10} + 6439289 x^{9} + 12558384 x^{8} - 6282221 x^{7} - 4210683 x^{6} + 2778704 x^{5} + 421825 x^{4} - 457387 x^{3} + 20126 x^{2} + 21522 x - 2699 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $27$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[27, 0]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(550892378962365588304561118053988796799287710804809\)\(\medspace = 13^{18}\cdot 19^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $75.74$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $13, 19$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $27$
This field is Galois and abelian over $\Q$.
Conductor:  \(247=13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{247}(1,·)$, $\chi_{247}(66,·)$, $\chi_{247}(131,·)$, $\chi_{247}(196,·)$, $\chi_{247}(9,·)$, $\chi_{247}(74,·)$, $\chi_{247}(139,·)$, $\chi_{247}(16,·)$, $\chi_{247}(81,·)$, $\chi_{247}(87,·)$, $\chi_{247}(68,·)$, $\chi_{247}(92,·)$, $\chi_{247}(157,·)$, $\chi_{247}(159,·)$, $\chi_{247}(144,·)$, $\chi_{247}(35,·)$, $\chi_{247}(100,·)$, $\chi_{247}(42,·)$, $\chi_{247}(235,·)$, $\chi_{247}(172,·)$, $\chi_{247}(237,·)$, $\chi_{247}(178,·)$, $\chi_{247}(118,·)$, $\chi_{247}(55,·)$, $\chi_{247}(120,·)$, $\chi_{247}(61,·)$, $\chi_{247}(191,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $\frac{1}{51733786095913174054700521748046212109398285809635601845975070558361} a^{26} - \frac{23136196471736133623394495990255629477644556796805128598123797068170}{51733786095913174054700521748046212109398285809635601845975070558361} a^{25} - \frac{17878508898838244508381554483693896339784487584005586378141186652304}{51733786095913174054700521748046212109398285809635601845975070558361} a^{24} - \frac{18489942620427680598281697992439671428668195439867272065872428705087}{51733786095913174054700521748046212109398285809635601845975070558361} a^{23} + \frac{3330816577363812980152029648152980628649330651057579296229853861896}{51733786095913174054700521748046212109398285809635601845975070558361} a^{22} + \frac{4675935436202677327685959725202563255611998810562963806316588645741}{51733786095913174054700521748046212109398285809635601845975070558361} a^{21} + \frac{6107822779644381847296778576455026447848287256805419831894207769883}{51733786095913174054700521748046212109398285809635601845975070558361} a^{20} + \frac{11376424817425625558135098543515457997197429163987298177137347128817}{51733786095913174054700521748046212109398285809635601845975070558361} a^{19} - \frac{12900395787513687517271596811102297927269232320412823285416543395229}{51733786095913174054700521748046212109398285809635601845975070558361} a^{18} + \frac{11659032230594255920309901657363485727459857196943750997719619869925}{51733786095913174054700521748046212109398285809635601845975070558361} a^{17} + \frac{5208850043330506872317715405245213628302812638479627207601555334428}{51733786095913174054700521748046212109398285809635601845975070558361} a^{16} + \frac{18984929931277340034128164720167141689659022760605648147927232735581}{51733786095913174054700521748046212109398285809635601845975070558361} a^{15} - \frac{9841379589573940613720653149114193980847707360047024808827367883633}{51733786095913174054700521748046212109398285809635601845975070558361} a^{14} + \frac{21011273263900748900320715971382944912568228832359575731354055665178}{51733786095913174054700521748046212109398285809635601845975070558361} a^{13} - \frac{18511390326035274044975306790582461098641827160335984642528465748364}{51733786095913174054700521748046212109398285809635601845975070558361} a^{12} + \frac{976192916707436212289414592509462489666609874464428178226942606344}{51733786095913174054700521748046212109398285809635601845975070558361} a^{11} + \frac{18799225300461857864776555732933435452448656185164497017907760510551}{51733786095913174054700521748046212109398285809635601845975070558361} a^{10} - \frac{24159221902258457803294142411936396190304731334639757530542375848973}{51733786095913174054700521748046212109398285809635601845975070558361} a^{9} - \frac{799336047749843647820510816516408321544877219748817058816138704059}{51733786095913174054700521748046212109398285809635601845975070558361} a^{8} - \frac{11022535822339638557842632870514779158179011064485110680828738314759}{51733786095913174054700521748046212109398285809635601845975070558361} a^{7} - \frac{21498100628276247044024958133733263672411715134437999150794101164213}{51733786095913174054700521748046212109398285809635601845975070558361} a^{6} - \frac{20951763693092196257673044117818159470930177749414768225324866754296}{51733786095913174054700521748046212109398285809635601845975070558361} a^{5} + \frac{14802114185340069105582736930980536650800977132701923584524267317977}{51733786095913174054700521748046212109398285809635601845975070558361} a^{4} - \frac{23792610842095672193140370895124145540448946858543066672795666628958}{51733786095913174054700521748046212109398285809635601845975070558361} a^{3} - \frac{21748761245917518966346031480835184025120609952940256590044748869611}{51733786095913174054700521748046212109398285809635601845975070558361} a^{2} + \frac{14218918498251839727319784899491623411355717432490884003735650645488}{51733786095913174054700521748046212109398285809635601845975070558361} a + \frac{20583614622405125161846035718845216903232568143112296086220591058383}{51733786095913174054700521748046212109398285809635601845975070558361}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $26$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 43333147066107930 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{27}\cdot(2\pi)^{0}\cdot 43333147066107930 \cdot 1}{2\sqrt{550892378962365588304561118053988796799287710804809}}\approx 0.123898696079419$ (assuming GRH)

Galois group

$C_3\times C_9$ (as 27T2):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

3.3.169.1, 3.3.361.1, 3.3.61009.2, 3.3.61009.1, 9.9.227081481823729.1, 9.9.81976414938366169.2, 9.9.81976414938366169.1, \(\Q(\zeta_{19})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
13Data not computed
19Data not computed