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Properties

Label 27.9.218...752.1
Degree $27$
Signature $[9, 9]$
Discriminant $-2.187\times 10^{38}$
Root discriminant $26.30$
Ramified primes $2, 3$
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^25 - 9*x^24 + 135*x^23 + 63*x^22 - 642*x^21 - 135*x^20 + 2133*x^19 + 336*x^18 - 6156*x^17 - 270*x^16 + 15525*x^15 - 4347*x^14 - 26667*x^13 + 16194*x^12 + 26730*x^11 - 24912*x^10 - 13179*x^9 + 19197*x^8 + 693*x^7 - 6957*x^6 + 1917*x^5 + 648*x^4 - 507*x^3 + 135*x^2 - 18*x + 1)
 
gp: K = bnfinit(x^27 - 18*x^25 - 9*x^24 + 135*x^23 + 63*x^22 - 642*x^21 - 135*x^20 + 2133*x^19 + 336*x^18 - 6156*x^17 - 270*x^16 + 15525*x^15 - 4347*x^14 - 26667*x^13 + 16194*x^12 + 26730*x^11 - 24912*x^10 - 13179*x^9 + 19197*x^8 + 693*x^7 - 6957*x^6 + 1917*x^5 + 648*x^4 - 507*x^3 + 135*x^2 - 18*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -18, 135, -507, 648, 1917, -6957, 693, 19197, -13179, -24912, 26730, 16194, -26667, -4347, 15525, -270, -6156, 336, 2133, -135, -642, 63, 135, -9, -18, 0, 1]);
 

\( x^{27} - 18 x^{25} - 9 x^{24} + 135 x^{23} + 63 x^{22} - 642 x^{21} - 135 x^{20} + 2133 x^{19} + 336 x^{18} - 6156 x^{17} - 270 x^{16} + 15525 x^{15} - 4347 x^{14} - 26667 x^{13} + 16194 x^{12} + 26730 x^{11} - 24912 x^{10} - 13179 x^{9} + 19197 x^{8} + 693 x^{7} - 6957 x^{6} + 1917 x^{5} + 648 x^{4} - 507 x^{3} + 135 x^{2} - 18 x + 1 \)

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

Invariants

Degree:  $27$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[9, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(-218729065566982775445089767573496266752\)\(\medspace = -\,2^{18}\cdot 3^{69}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $26.30$
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:  $2, 3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{53} a^{25} - \frac{19}{53} a^{24} + \frac{10}{53} a^{23} - \frac{20}{53} a^{22} - \frac{6}{53} a^{21} - \frac{22}{53} a^{19} + \frac{18}{53} a^{18} + \frac{1}{53} a^{17} - \frac{6}{53} a^{16} - \frac{15}{53} a^{15} - \frac{1}{53} a^{14} - \frac{25}{53} a^{13} + \frac{12}{53} a^{12} - \frac{20}{53} a^{11} + \frac{17}{53} a^{10} - \frac{5}{53} a^{9} - \frac{3}{53} a^{8} - \frac{9}{53} a^{7} + \frac{15}{53} a^{6} + \frac{13}{53} a^{5} - \frac{9}{53} a^{4} - \frac{15}{53} a^{3} + \frac{8}{53} a^{2} - \frac{10}{53} a - \frac{7}{53}$, $\frac{1}{2852468315136422218788511661435728003394789} a^{26} + \frac{5182105344944835593657101376591251615283}{2852468315136422218788511661435728003394789} a^{25} + \frac{1005998949987806990487381506647986161658786}{2852468315136422218788511661435728003394789} a^{24} + \frac{1073154836042779104984267379506506316695508}{2852468315136422218788511661435728003394789} a^{23} + \frac{1311113602917772319032920710680785354355201}{2852468315136422218788511661435728003394789} a^{22} + \frac{874992971872720327892903302830656790577228}{2852468315136422218788511661435728003394789} a^{21} - \frac{208771419596483366353386530283725941372035}{2852468315136422218788511661435728003394789} a^{20} - \frac{231754735186413685535703360121346974749671}{2852468315136422218788511661435728003394789} a^{19} - \frac{385878629697495464215196868433632692980447}{2852468315136422218788511661435728003394789} a^{18} - \frac{205749050034918326325351193481724360301767}{2852468315136422218788511661435728003394789} a^{17} - \frac{586739861087500565661176488052431487552114}{2852468315136422218788511661435728003394789} a^{16} - \frac{90832374734768949085718066206037114548007}{2852468315136422218788511661435728003394789} a^{15} + \frac{847671336260582855096496813028862209653071}{2852468315136422218788511661435728003394789} a^{14} + \frac{1104714419644441020345414585384176989542851}{2852468315136422218788511661435728003394789} a^{13} + \frac{1379930473388021444875037160719485388679137}{2852468315136422218788511661435728003394789} a^{12} + \frac{108300386159691395940215086580249457354525}{2852468315136422218788511661435728003394789} a^{11} + \frac{97871363767983647305334810005073740369484}{2852468315136422218788511661435728003394789} a^{10} + \frac{772805190921319707156702884491804778466951}{2852468315136422218788511661435728003394789} a^{9} - \frac{682701025388252762801038886288415742422059}{2852468315136422218788511661435728003394789} a^{8} + \frac{10630344126660540981193428743735410616367}{2852468315136422218788511661435728003394789} a^{7} + \frac{1063058768461405712745985675649591848318964}{2852468315136422218788511661435728003394789} a^{6} + \frac{1102715873090825185940595144319718357704266}{2852468315136422218788511661435728003394789} a^{5} + \frac{1116213759299017630816496132079633357218619}{2852468315136422218788511661435728003394789} a^{4} + \frac{829403287961859005297536211909072301075466}{2852468315136422218788511661435728003394789} a^{3} - \frac{1134577424934260901563671176128173843908480}{2852468315136422218788511661435728003394789} a^{2} + \frac{447179909271429046069062868686940534428118}{2852468315136422218788511661435728003394789} a - \frac{293565525383250620684079298808389716228543}{2852468315136422218788511661435728003394789}$

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:  \( 932907706.847076 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{9}\cdot(2\pi)^{9}\cdot 932907706.847076 \cdot 1}{2\sqrt{218729065566982775445089767573496266752}}\approx 0.246458733596961$ (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

\(\Q(\zeta_{9})^+\), 3.1.108.1, \(\Q(\zeta_{27})^+\), 9.3.918330048.1

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 R 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
2Data not computed
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) $1$ $-1$
* 1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.9.6t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})\) $C_6$ (as 6T1) $0$ $-1$
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) $0$ $-1$
* 1.27.9t1.a.a$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
* 1.27.9t1.a.b$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
* 1.27.9t1.a.c$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
* 1.27.9t1.a.d$1$ $ 3^{3}$ \(\Q(\zeta_{27})^+\) $C_9$ (as 9T1) $0$ $1$
1.27.18t1.a.b$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
1.27.18t1.a.c$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
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.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) $0$ $1$
1.27.18t1.a.f$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
* 2.108.3t2.b.a$2$ $ 2^{2} \cdot 3^{3}$ 3.1.108.1 $S_3$ (as 3T2) $1$ $0$
* 2.324.6t5.c.a$2$ $ 2^{2} \cdot 3^{4}$ 6.0.314928.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.324.6t5.c.b$2$ $ 2^{2} \cdot 3^{4}$ 6.0.314928.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2916.18t16.b.a$2$ $ 2^{2} \cdot 3^{6}$ 27.9.218729065566982775445089767573496266752.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.b.b$2$ $ 2^{2} \cdot 3^{6}$ 27.9.218729065566982775445089767573496266752.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.b.c$2$ $ 2^{2} \cdot 3^{6}$ 27.9.218729065566982775445089767573496266752.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.b.d$2$ $ 2^{2} \cdot 3^{6}$ 27.9.218729065566982775445089767573496266752.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.b.e$2$ $ 2^{2} \cdot 3^{6}$ 27.9.218729065566982775445089767573496266752.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.b.f$2$ $ 2^{2} \cdot 3^{6}$ 27.9.218729065566982775445089767573496266752.1 $C_9\times S_3$ (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 *.