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Properties

Label 27.9.675...923.1
Degree $27$
Signature $[9, 9]$
Discriminant $-6.759\times 10^{34}$
Root discriminant $19.50$
Ramified prime $3$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_9\times S_3$ (as 27T12)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^27 - 18*x^23 - 18*x^22 - 18*x^21 + 18*x^20 + 135*x^19 + 219*x^18 - 162*x^17 + 324*x^16 - 153*x^15 - 81*x^14 - 675*x^13 - 630*x^12 - 540*x^11 - 513*x^10 - 405*x^9 + 27*x^8 + 243*x^7 + 387*x^6 + 162*x^5 - 81*x^4 - 135*x^3 + 27*x - 3)
 
gp: K = bnfinit(x^27 - 18*x^23 - 18*x^22 - 18*x^21 + 18*x^20 + 135*x^19 + 219*x^18 - 162*x^17 + 324*x^16 - 153*x^15 - 81*x^14 - 675*x^13 - 630*x^12 - 540*x^11 - 513*x^10 - 405*x^9 + 27*x^8 + 243*x^7 + 387*x^6 + 162*x^5 - 81*x^4 - 135*x^3 + 27*x - 3, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-3, 27, 0, -135, -81, 162, 387, 243, 27, -405, -513, -540, -630, -675, -81, -153, 324, -162, 219, 135, 18, -18, -18, -18, 0, 0, 0, 1]);
 

\( x^{27} - 18 x^{23} - 18 x^{22} - 18 x^{21} + 18 x^{20} + 135 x^{19} + 219 x^{18} - 162 x^{17} + 324 x^{16} - 153 x^{15} - 81 x^{14} - 675 x^{13} - 630 x^{12} - 540 x^{11} - 513 x^{10} - 405 x^{9} + 27 x^{8} + 243 x^{7} + 387 x^{6} + 162 x^{5} - 81 x^{4} - 135 x^{3} + 27 x - 3 \)

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

Invariants

Degree:  $27$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[9, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-67585198634817523235520443624317923\)\(\medspace = -\,3^{73}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $19.50$
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)));
 
$|\Aut(K/\Q)|$:  $9$
This field is not Galois over $\Q$.
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}{2067989529675373872884142445660740956329891519} a^{26} - \frac{722974281903480422799051858697221070742482387}{2067989529675373872884142445660740956329891519} a^{25} - \frac{918901109894091580648639856802626304305232628}{2067989529675373872884142445660740956329891519} a^{24} - \frac{295987832165049516451713142671550340473251091}{2067989529675373872884142445660740956329891519} a^{23} - \frac{642777361828979540235103178402636871106262830}{2067989529675373872884142445660740956329891519} a^{22} + \frac{417072548283108759086905711277752467903911114}{2067989529675373872884142445660740956329891519} a^{21} - \frac{324789928860912273563572044304887478851174346}{2067989529675373872884142445660740956329891519} a^{20} + \frac{695884600294162644250792752766551951776614362}{2067989529675373872884142445660740956329891519} a^{19} + \frac{669096430000185971564537158048054086135371489}{2067989529675373872884142445660740956329891519} a^{18} - \frac{390090589481473694433135220033384417982096277}{2067989529675373872884142445660740956329891519} a^{17} + \frac{236998120770844126550684541769708975802059520}{2067989529675373872884142445660740956329891519} a^{16} - \frac{666876739092276732929631066605921154801864686}{2067989529675373872884142445660740956329891519} a^{15} - \frac{410874737164245420506206270481420947413439594}{2067989529675373872884142445660740956329891519} a^{14} + \frac{618139372204512679612844946670604040657041499}{2067989529675373872884142445660740956329891519} a^{13} + \frac{566345675009369576985715326577590916703595742}{2067989529675373872884142445660740956329891519} a^{12} + \frac{959221259000895114077985293216290119837643174}{2067989529675373872884142445660740956329891519} a^{11} - \frac{190014449903910784685826210382469719913539507}{2067989529675373872884142445660740956329891519} a^{10} - \frac{448259172440178594540466386319143354158218435}{2067989529675373872884142445660740956329891519} a^{9} - \frac{13075459083820824895251889955627525817297204}{2067989529675373872884142445660740956329891519} a^{8} + \frac{281559977539359769251539520365739911501054175}{2067989529675373872884142445660740956329891519} a^{7} - \frac{250093779461215400345076533857602512248203790}{2067989529675373872884142445660740956329891519} a^{6} + \frac{410599330461173958194533282523002377218650008}{2067989529675373872884142445660740956329891519} a^{5} - \frac{937902165296727869075064491784868463451609918}{2067989529675373872884142445660740956329891519} a^{4} - \frac{371297873541147134549144758359820500631716035}{2067989529675373872884142445660740956329891519} a^{3} + \frac{873675768297034425964017081646149327941872348}{2067989529675373872884142445660740956329891519} a^{2} + \frac{340260045433520020513952473001492281215732106}{2067989529675373872884142445660740956329891519} a - \frac{524393845738372758729930025949274252823095372}{2067989529675373872884142445660740956329891519}$

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:  $17$
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:  \( 13605448.556703938 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{9}\cdot(2\pi)^{9}\cdot 13605448.556703938 \cdot 1}{2\sqrt{67585198634817523235520443624317923}}\approx 0.204477651195874$ (assuming GRH)

Galois group

$C_9\times S_3$ (as 27T12):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 54
The 27 conjugacy class representatives for $C_9\times S_3$
Character table for $C_9\times S_3$ is not computed

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.1.243.1, \(\Q(\zeta_{27})^+\), 9.3.1162261467.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 18 sibling: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $18{,}\,{\href{/LocalNumberField/2.9.0.1}{9} }$ R $18{,}\,{\href{/LocalNumberField/5.9.0.1}{9} }$ ${\href{/LocalNumberField/7.9.0.1}{9} }^{3}$ $18{,}\,{\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{9}$ $18{,}\,{\href{/LocalNumberField/23.9.0.1}{9} }$ $18{,}\,{\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{9}$ $18{,}\,{\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{3}$ $18{,}\,{\href{/LocalNumberField/47.9.0.1}{9} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{9}$ $18{,}\,{\href{/LocalNumberField/59.9.0.1}{9} }$

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

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.9.6t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})\) $C_6$ (as 6T1) $0$ $-1$
* 1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.9.6t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})\) $C_6$ (as 6T1) $0$ $-1$
* 1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.27.18t1.a.a$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
1.27.18t1.a.b$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
* 1.27.9t1.a.a$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.27.18t1.a.c$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
* 1.27.9t1.a.b$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.27.18t1.a.d$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
1.27.18t1.a.e$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
* 1.27.9t1.a.c$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.27.18t1.a.f$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
* 1.27.9t1.a.d$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
* 1.27.9t1.a.e$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
* 1.27.9t1.a.f$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
* 2.243.3t2.b.a$2$ $ 3^{5}$ 3.1.243.1 $S_3$ (as 3T2) $1$ $0$
* 2.243.6t5.b.a$2$ $ 3^{5}$ 6.0.177147.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.243.6t5.b.b$2$ $ 3^{5}$ 6.0.177147.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.729.18t16.b.a$2$ $ 3^{6}$ 27.9.67585198634817523235520443624317923.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.729.18t16.b.b$2$ $ 3^{6}$ 27.9.67585198634817523235520443624317923.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.729.18t16.b.c$2$ $ 3^{6}$ 27.9.67585198634817523235520443624317923.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.729.18t16.b.d$2$ $ 3^{6}$ 27.9.67585198634817523235520443624317923.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.729.18t16.b.e$2$ $ 3^{6}$ 27.9.67585198634817523235520443624317923.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.729.18t16.b.f$2$ $ 3^{6}$ 27.9.67585198634817523235520443624317923.1 $C_9\times S_3$ (as 27T12) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.