Properties

Label 27.27.797...529.1
Degree $27$
Signature $[27, 0]$
Discriminant $7.977\times 10^{45}$
Root discriminant $50.13$
Ramified primes $7, 19$
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 - x^26 - 42*x^25 + 37*x^24 + 728*x^23 - 564*x^22 - 6817*x^21 + 4664*x^20 + 37948*x^19 - 23103*x^18 - 130429*x^17 + 71289*x^16 + 279661*x^15 - 138143*x^14 - 372684*x^13 + 166778*x^12 + 305327*x^11 - 124486*x^10 - 150120*x^9 + 56020*x^8 + 42107*x^7 - 14253*x^6 - 6122*x^5 + 1790*x^4 + 395*x^3 - 85*x^2 - 10*x + 1)
 
gp: K = bnfinit(x^27 - x^26 - 42*x^25 + 37*x^24 + 728*x^23 - 564*x^22 - 6817*x^21 + 4664*x^20 + 37948*x^19 - 23103*x^18 - 130429*x^17 + 71289*x^16 + 279661*x^15 - 138143*x^14 - 372684*x^13 + 166778*x^12 + 305327*x^11 - 124486*x^10 - 150120*x^9 + 56020*x^8 + 42107*x^7 - 14253*x^6 - 6122*x^5 + 1790*x^4 + 395*x^3 - 85*x^2 - 10*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -10, -85, 395, 1790, -6122, -14253, 42107, 56020, -150120, -124486, 305327, 166778, -372684, -138143, 279661, 71289, -130429, -23103, 37948, 4664, -6817, -564, 728, 37, -42, -1, 1]);
 

\( x^{27} - x^{26} - 42 x^{25} + 37 x^{24} + 728 x^{23} - 564 x^{22} - 6817 x^{21} + 4664 x^{20} + 37948 x^{19} - 23103 x^{18} - 130429 x^{17} + 71289 x^{16} + 279661 x^{15} - 138143 x^{14} - 372684 x^{13} + 166778 x^{12} + 305327 x^{11} - 124486 x^{10} - 150120 x^{9} + 56020 x^{8} + 42107 x^{7} - 14253 x^{6} - 6122 x^{5} + 1790 x^{4} + 395 x^{3} - 85 x^{2} - 10 x + 1 \)

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:  \(7977212169716289044333767743376433611324896529\)\(\medspace = 7^{18}\cdot 19^{24}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $50.13$
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:  $7, 19$
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:  \(133=7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{133}(64,·)$, $\chi_{133}(1,·)$, $\chi_{133}(130,·)$, $\chi_{133}(4,·)$, $\chi_{133}(100,·)$, $\chi_{133}(9,·)$, $\chi_{133}(74,·)$, $\chi_{133}(11,·)$, $\chi_{133}(16,·)$, $\chi_{133}(81,·)$, $\chi_{133}(85,·)$, $\chi_{133}(23,·)$, $\chi_{133}(25,·)$, $\chi_{133}(92,·)$, $\chi_{133}(93,·)$, $\chi_{133}(30,·)$, $\chi_{133}(99,·)$, $\chi_{133}(36,·)$, $\chi_{133}(102,·)$, $\chi_{133}(39,·)$, $\chi_{133}(106,·)$, $\chi_{133}(43,·)$, $\chi_{133}(44,·)$, $\chi_{133}(120,·)$, $\chi_{133}(121,·)$, $\chi_{133}(58,·)$, $\chi_{133}(123,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{113} a^{24} + \frac{27}{113} a^{23} + \frac{32}{113} a^{22} + \frac{6}{113} a^{21} + \frac{12}{113} a^{20} - \frac{18}{113} a^{19} + \frac{34}{113} a^{18} + \frac{41}{113} a^{17} + \frac{27}{113} a^{16} + \frac{41}{113} a^{15} - \frac{22}{113} a^{14} + \frac{32}{113} a^{13} + \frac{48}{113} a^{12} - \frac{33}{113} a^{11} + \frac{12}{113} a^{10} + \frac{18}{113} a^{9} + \frac{25}{113} a^{8} - \frac{7}{113} a^{7} + \frac{48}{113} a^{6} - \frac{34}{113} a^{5} + \frac{27}{113} a^{4} - \frac{15}{113} a^{3} - \frac{48}{113} a^{2} - \frac{24}{113} a - \frac{4}{113}$, $\frac{1}{113} a^{25} - \frac{19}{113} a^{23} + \frac{46}{113} a^{22} - \frac{37}{113} a^{21} - \frac{3}{113} a^{20} - \frac{45}{113} a^{19} + \frac{27}{113} a^{18} + \frac{50}{113} a^{17} - \frac{10}{113} a^{16} + \frac{1}{113} a^{15} - \frac{52}{113} a^{14} - \frac{25}{113} a^{13} + \frac{27}{113} a^{12} - \frac{1}{113} a^{11} + \frac{33}{113} a^{10} - \frac{9}{113} a^{9} - \frac{4}{113} a^{8} + \frac{11}{113} a^{7} + \frac{26}{113} a^{6} + \frac{41}{113} a^{5} + \frac{47}{113} a^{4} + \frac{18}{113} a^{3} + \frac{29}{113} a^{2} - \frac{34}{113} a - \frac{5}{113}$, $\frac{1}{8206544682495678472680089967879773} a^{26} + \frac{14669387966349457650803639027094}{8206544682495678472680089967879773} a^{25} - \frac{27945535847640849390740210846844}{8206544682495678472680089967879773} a^{24} + \frac{3742464948774644485192036085284255}{8206544682495678472680089967879773} a^{23} - \frac{1327541569494088178244529599129901}{8206544682495678472680089967879773} a^{22} - \frac{658886807479998098622240062336646}{8206544682495678472680089967879773} a^{21} - \frac{694961768360920198946127230809891}{8206544682495678472680089967879773} a^{20} - \frac{2309153772045912282863070608472196}{8206544682495678472680089967879773} a^{19} - \frac{1574994466038791749158137346267}{8206544682495678472680089967879773} a^{18} + \frac{3301909933939954784265101539864587}{8206544682495678472680089967879773} a^{17} + \frac{459648920194588690607000112177860}{8206544682495678472680089967879773} a^{16} + \frac{2757068035596579369764651155621985}{8206544682495678472680089967879773} a^{15} + \frac{3257553843522999448324846275435568}{8206544682495678472680089967879773} a^{14} + \frac{2342967951202671702013075889463633}{8206544682495678472680089967879773} a^{13} - \frac{3430477967030387439023911928846622}{8206544682495678472680089967879773} a^{12} + \frac{188613811539485153419623697366638}{8206544682495678472680089967879773} a^{11} + \frac{4009723612964788006229743716129247}{8206544682495678472680089967879773} a^{10} + \frac{1063182427617136563970509047153085}{8206544682495678472680089967879773} a^{9} - \frac{3914265650992212190793577659924791}{8206544682495678472680089967879773} a^{8} + \frac{3463845486354937268700293132362653}{8206544682495678472680089967879773} a^{7} - \frac{1066498817418124604979083074586013}{8206544682495678472680089967879773} a^{6} + \frac{3486686275783114067746518820906586}{8206544682495678472680089967879773} a^{5} - \frac{1775750959556524252412577928499423}{8206544682495678472680089967879773} a^{4} - \frac{573928709447212093576965780811753}{8206544682495678472680089967879773} a^{3} + \frac{958037281550647932241699355858838}{8206544682495678472680089967879773} a^{2} - \frac{1468986731164605974315582480867614}{8206544682495678472680089967879773} a - \frac{2548044515053912648408539789703558}{8206544682495678472680089967879773}$

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:  \( 173879598132270.75 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{27}\cdot(2\pi)^{0}\cdot 173879598132270.75 \cdot 1}{2\sqrt{7977212169716289044333767743376433611324896529}}\approx 0.130648053453212$ (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

3.3.17689.1, 3.3.361.1, 3.3.17689.2, \(\Q(\zeta_{7})^+\), 9.9.5534900853769.1, 9.9.1998099208210609.1, 9.9.1998099208210609.2, \(\Q(\zeta_{19})^+\)

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 ${\href{/LocalNumberField/2.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/3.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/5.9.0.1}{9} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{9}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }^{3}$ 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
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
19Data not computed