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Properties

Label 27.27.503...521.1
Degree $27$
Signature $[27, 0]$
Discriminant $5.032\times 10^{46}$
Root discriminant $53.67$
Ramified primes $3, 7$
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 - 45*x^25 + 837*x^23 - 8430*x^21 + 50652*x^19 - 193*x^18 - 188811*x^17 + 2934*x^16 + 441720*x^15 - 15822*x^14 - 646731*x^13 + 37506*x^12 + 585063*x^11 - 42453*x^10 - 319331*x^9 + 24462*x^8 + 99963*x^7 - 7245*x^6 - 16119*x^5 + 1080*x^4 + 1059*x^3 - 81*x^2 - 18*x + 1)
 
gp: K = bnfinit(x^27 - 45*x^25 + 837*x^23 - 8430*x^21 + 50652*x^19 - 193*x^18 - 188811*x^17 + 2934*x^16 + 441720*x^15 - 15822*x^14 - 646731*x^13 + 37506*x^12 + 585063*x^11 - 42453*x^10 - 319331*x^9 + 24462*x^8 + 99963*x^7 - 7245*x^6 - 16119*x^5 + 1080*x^4 + 1059*x^3 - 81*x^2 - 18*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -18, -81, 1059, 1080, -16119, -7245, 99963, 24462, -319331, -42453, 585063, 37506, -646731, -15822, 441720, 2934, -188811, -193, 50652, 0, -8430, 0, 837, 0, -45, 0, 1]);
 

\( x^{27} - 45 x^{25} + 837 x^{23} - 8430 x^{21} + 50652 x^{19} - 193 x^{18} - 188811 x^{17} + 2934 x^{16} + 441720 x^{15} - 15822 x^{14} - 646731 x^{13} + 37506 x^{12} + 585063 x^{11} - 42453 x^{10} - 319331 x^{9} + 24462 x^{8} + 99963 x^{7} - 7245 x^{6} - 16119 x^{5} + 1080 x^{4} + 1059 x^{3} - 81 x^{2} - 18 x + 1 \)

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:  \(50323116815004832630295337440131512593194174521\)\(\medspace = 3^{66}\cdot 7^{18}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $53.67$
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:  $3, 7$
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:  \(189=3^{3}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{189}(64,·)$, $\chi_{189}(1,·)$, $\chi_{189}(130,·)$, $\chi_{189}(67,·)$, $\chi_{189}(4,·)$, $\chi_{189}(142,·)$, $\chi_{189}(79,·)$, $\chi_{189}(16,·)$, $\chi_{189}(148,·)$, $\chi_{189}(85,·)$, $\chi_{189}(22,·)$, $\chi_{189}(151,·)$, $\chi_{189}(88,·)$, $\chi_{189}(25,·)$, $\chi_{189}(163,·)$, $\chi_{189}(100,·)$, $\chi_{189}(37,·)$, $\chi_{189}(169,·)$, $\chi_{189}(106,·)$, $\chi_{189}(43,·)$, $\chi_{189}(172,·)$, $\chi_{189}(109,·)$, $\chi_{189}(46,·)$, $\chi_{189}(184,·)$, $\chi_{189}(121,·)$, $\chi_{189}(58,·)$, $\chi_{189}(127,·)$$\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}{6572051699031783636148040372400866530872797} a^{26} - \frac{3128161542052233584117625196754683002257122}{6572051699031783636148040372400866530872797} a^{25} + \frac{2569691265818507607056039605209773585422534}{6572051699031783636148040372400866530872797} a^{24} + \frac{2774681163185792832662405554343946928493372}{6572051699031783636148040372400866530872797} a^{23} + \frac{1454777682648645288668097496809188348046066}{6572051699031783636148040372400866530872797} a^{22} + \frac{1694389064104381340725743221609596918381309}{6572051699031783636148040372400866530872797} a^{21} + \frac{1978027081260875059152436659686064584709554}{6572051699031783636148040372400866530872797} a^{20} + \frac{1563192202820656498208988469452317902549175}{6572051699031783636148040372400866530872797} a^{19} - \frac{1365959405605485581057798212081872268857849}{6572051699031783636148040372400866530872797} a^{18} - \frac{2705917195863283295942289316564958167920850}{6572051699031783636148040372400866530872797} a^{17} - \frac{1739799140668879090576627029453840469415116}{6572051699031783636148040372400866530872797} a^{16} + \frac{594331372401851110066806380016232586911282}{6572051699031783636148040372400866530872797} a^{15} - \frac{2337347656597575200150116187211243713948911}{6572051699031783636148040372400866530872797} a^{14} - \frac{2630134618704916679327749160767056611937588}{6572051699031783636148040372400866530872797} a^{13} + \frac{1467534671506952570156185074642065836617494}{6572051699031783636148040372400866530872797} a^{12} + \frac{1967064295859597746570152556470827151367232}{6572051699031783636148040372400866530872797} a^{11} - \frac{2176561166100209722329943228621379902286173}{6572051699031783636148040372400866530872797} a^{10} + \frac{2502256983355177077064058537443154872705816}{6572051699031783636148040372400866530872797} a^{9} + \frac{3022239167622824187757096378197236468805472}{6572051699031783636148040372400866530872797} a^{8} + \frac{1852917314222978828288216900482185401091718}{6572051699031783636148040372400866530872797} a^{7} + \frac{23280823669944550123817670601448927019797}{6572051699031783636148040372400866530872797} a^{6} + \frac{3282202200872936776644303109046507838580716}{6572051699031783636148040372400866530872797} a^{5} - \frac{3050372896729224631797033839455054423822405}{6572051699031783636148040372400866530872797} a^{4} + \frac{3008149247626434564389768126125792597694940}{6572051699031783636148040372400866530872797} a^{3} + \frac{1711534243682350508413812638647972427747685}{6572051699031783636148040372400866530872797} a^{2} + \frac{6386274974164382911762239736463906360227}{6572051699031783636148040372400866530872797} a + \frac{3078692040347290382348439189636650455637611}{6572051699031783636148040372400866530872797}$

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:  \( 347763474421246.5 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{27}\cdot(2\pi)^{0}\cdot 347763474421246.5 \cdot 1}{2\sqrt{50323116815004832630295337440131512593194174521}}\approx 0.104035147566184$ (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

3.3.3969.1, \(\Q(\zeta_{9})^+\), 3.3.3969.2, \(\Q(\zeta_{7})^+\), 9.9.62523502209.1, \(\Q(\zeta_{27})^+\), 9.9.3691950281939241.2, 9.9.3691950281939241.1

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/11.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/23.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{3}$

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
3Data not computed
7Data not computed