Properties

Label 27.27.706...369.1
Degree $27$
Signature $[27, 0]$
Discriminant $7.070\times 10^{44}$
Root discriminant $45.82$
Ramified prime $3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{27}$ (as 27T1)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 27*x^25 + 324*x^23 - 2277*x^21 + 10395*x^19 - 32319*x^17 + 69768*x^15 - 104652*x^13 + 107406*x^11 - 72930*x^9 + 30888*x^7 - 7371*x^5 + 819*x^3 - 27*x - 1)
 
gp: K = bnfinit(x^27 - 27*x^25 + 324*x^23 - 2277*x^21 + 10395*x^19 - 32319*x^17 + 69768*x^15 - 104652*x^13 + 107406*x^11 - 72930*x^9 + 30888*x^7 - 7371*x^5 + 819*x^3 - 27*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, -27, 0, 819, 0, -7371, 0, 30888, 0, -72930, 0, 107406, 0, -104652, 0, 69768, 0, -32319, 0, 10395, 0, -2277, 0, 324, 0, -27, 0, 1]);
 

\( x^{27} - 27 x^{25} + 324 x^{23} - 2277 x^{21} + 10395 x^{19} - 32319 x^{17} + 69768 x^{15} - 104652 x^{13} + 107406 x^{11} - 72930 x^{9} + 30888 x^{7} - 7371 x^{5} + 819 x^{3} - 27 x - 1 \)

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:  \(706965049015104706497203195837614914543357369\)\(\medspace = 3^{94}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $45.82$
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:  $3$
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:  \(81=3^{4}\)
Dirichlet character group:    $\lbrace$$\chi_{81}(64,·)$, $\chi_{81}(1,·)$, $\chi_{81}(67,·)$, $\chi_{81}(4,·)$, $\chi_{81}(70,·)$, $\chi_{81}(7,·)$, $\chi_{81}(73,·)$, $\chi_{81}(10,·)$, $\chi_{81}(76,·)$, $\chi_{81}(13,·)$, $\chi_{81}(79,·)$, $\chi_{81}(16,·)$, $\chi_{81}(19,·)$, $\chi_{81}(22,·)$, $\chi_{81}(25,·)$, $\chi_{81}(28,·)$, $\chi_{81}(31,·)$, $\chi_{81}(34,·)$, $\chi_{81}(37,·)$, $\chi_{81}(40,·)$, $\chi_{81}(43,·)$, $\chi_{81}(46,·)$, $\chi_{81}(49,·)$, $\chi_{81}(52,·)$, $\chi_{81}(55,·)$, $\chi_{81}(58,·)$, $\chi_{81}(61,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $a^{25}$, $a^{26}$

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:  \( 68981171346332.37 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{27}\cdot(2\pi)^{0}\cdot 68981171346332.37 \cdot 1}{2\sqrt{706965049015104706497203195837614914543357369}}\approx 0.174105094869797$ (assuming GRH)

Galois group

$C_{27}$ (as 27T1):

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

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