Properties

Label 27.9.177...912.2
Degree $27$
Signature $[9, 9]$
Discriminant $-1.772\times 10^{40}$
Root discriminant $30.95$
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 - 9*x^26 + 36*x^25 - 54*x^24 - 180*x^23 + 1296*x^22 - 3699*x^21 + 5184*x^20 + 1737*x^19 - 28278*x^18 + 81630*x^17 - 152073*x^16 + 206037*x^15 - 206640*x^14 + 143748*x^13 - 36954*x^12 - 70929*x^11 + 125325*x^10 - 106329*x^9 + 53154*x^8 - 10548*x^7 - 9594*x^6 + 13185*x^5 - 9180*x^4 + 4068*x^3 - 1053*x^2 - 207*x + 163)
 
gp: K = bnfinit(x^27 - 9*x^26 + 36*x^25 - 54*x^24 - 180*x^23 + 1296*x^22 - 3699*x^21 + 5184*x^20 + 1737*x^19 - 28278*x^18 + 81630*x^17 - 152073*x^16 + 206037*x^15 - 206640*x^14 + 143748*x^13 - 36954*x^12 - 70929*x^11 + 125325*x^10 - 106329*x^9 + 53154*x^8 - 10548*x^7 - 9594*x^6 + 13185*x^5 - 9180*x^4 + 4068*x^3 - 1053*x^2 - 207*x + 163, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![163, -207, -1053, 4068, -9180, 13185, -9594, -10548, 53154, -106329, 125325, -70929, -36954, 143748, -206640, 206037, -152073, 81630, -28278, 1737, 5184, -3699, 1296, -180, -54, 36, -9, 1]);
 

\( x^{27} - 9 x^{26} + 36 x^{25} - 54 x^{24} - 180 x^{23} + 1296 x^{22} - 3699 x^{21} + 5184 x^{20} + 1737 x^{19} - 28278 x^{18} + 81630 x^{17} - 152073 x^{16} + 206037 x^{15} - 206640 x^{14} + 143748 x^{13} - 36954 x^{12} - 70929 x^{11} + 125325 x^{10} - 106329 x^{9} + 53154 x^{8} - 10548 x^{7} - 9594 x^{6} + 13185 x^{5} - 9180 x^{4} + 4068 x^{3} - 1053 x^{2} - 207 x + 163 \)

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:  \(-17717054310925604811052271173453197606912\)\(\medspace = -\,2^{18}\cdot 3^{73}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $30.95$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{163} a^{25} + \frac{29}{163} a^{24} + \frac{13}{163} a^{23} - \frac{74}{163} a^{22} - \frac{13}{163} a^{21} - \frac{56}{163} a^{20} - \frac{4}{163} a^{19} + \frac{61}{163} a^{18} + \frac{79}{163} a^{17} - \frac{13}{163} a^{16} - \frac{78}{163} a^{15} - \frac{69}{163} a^{14} + \frac{47}{163} a^{13} + \frac{74}{163} a^{12} - \frac{40}{163} a^{11} + \frac{37}{163} a^{10} - \frac{73}{163} a^{9} + \frac{78}{163} a^{8} - \frac{50}{163} a^{7} + \frac{16}{163} a^{6} + \frac{18}{163} a^{5} - \frac{15}{163} a^{4} + \frac{26}{163} a^{3} + \frac{44}{163} a^{2} - \frac{38}{163} a$, $\frac{1}{1524136820005435899121894905321137646356922434503210356308437} a^{26} - \frac{1282330206265927842874535649482941320621152635346372092447}{1524136820005435899121894905321137646356922434503210356308437} a^{25} - \frac{567436243839472746766601501309532256354593823118537593794718}{1524136820005435899121894905321137646356922434503210356308437} a^{24} + \frac{686258393391893155539267212800818687592151803914015835427599}{1524136820005435899121894905321137646356922434503210356308437} a^{23} + \frac{159878869136843396316775338923093070352900229221699482719140}{1524136820005435899121894905321137646356922434503210356308437} a^{22} + \frac{447572022868076099307092915638205734839176703245227474880739}{1524136820005435899121894905321137646356922434503210356308437} a^{21} + \frac{510799391117163993311641239750122059600519377756178296056467}{1524136820005435899121894905321137646356922434503210356308437} a^{20} - \frac{202985305370684644612057126679189585449254792396172529306253}{1524136820005435899121894905321137646356922434503210356308437} a^{19} - \frac{235272658508703100223291220299370330886088778110150248022978}{1524136820005435899121894905321137646356922434503210356308437} a^{18} + \frac{726853260580140219145296117611833390017002385777509359367157}{1524136820005435899121894905321137646356922434503210356308437} a^{17} - \frac{199834349575901744366745917661841326460886638939595200374188}{1524136820005435899121894905321137646356922434503210356308437} a^{16} - \frac{32902225929891826224217255113351512108493894611782273869504}{1524136820005435899121894905321137646356922434503210356308437} a^{15} - \frac{100197917441804833462265116803375045849705965644530824005195}{1524136820005435899121894905321137646356922434503210356308437} a^{14} - \frac{573682730601917231436720822464028505789284516623145794822832}{1524136820005435899121894905321137646356922434503210356308437} a^{13} + \frac{561456940645008088819404788537832911210627808218216388047797}{1524136820005435899121894905321137646356922434503210356308437} a^{12} - \frac{713820731322308435371223789306030673368783215844298131341147}{1524136820005435899121894905321137646356922434503210356308437} a^{11} + \frac{679777987703164358027707848444585043887632688817979757576140}{1524136820005435899121894905321137646356922434503210356308437} a^{10} + \frac{281527794991461582998240636906124492592671303322359688191434}{1524136820005435899121894905321137646356922434503210356308437} a^{9} + \frac{360251220136554046830154754865881442948345319747926112331949}{1524136820005435899121894905321137646356922434503210356308437} a^{8} + \frac{122859419247500506123758625065890996816780116220191616641912}{1524136820005435899121894905321137646356922434503210356308437} a^{7} + \frac{661377277672705005901470433149585356044367103695286814833682}{1524136820005435899121894905321137646356922434503210356308437} a^{6} + \frac{19737500023298518252466419379105770999554262744730089223519}{1524136820005435899121894905321137646356922434503210356308437} a^{5} - \frac{507199956282057594825480326906484298540136527313722711045466}{1524136820005435899121894905321137646356922434503210356308437} a^{4} - \frac{159892784030657483029764234496039983057354216952763395380316}{1524136820005435899121894905321137646356922434503210356308437} a^{3} - \frac{730279703387151631445942060781857611244101326126486230469251}{1524136820005435899121894905321137646356922434503210356308437} a^{2} + \frac{507653776608159136499535168897495558644560909150879344626770}{1524136820005435899121894905321137646356922434503210356308437} a + \frac{1274048676279607916414748098656726539676142608149991011904}{9350532638070158890318373652276918075809340088976750652199}$

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:  \( 8061331492.668233 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{9}\cdot(2\pi)^{9}\cdot 8061331492.668233 \cdot 1}{2\sqrt{17717054310925604811052271173453197606912}}\approx 0.236629999374393$ (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

\(\Q(\zeta_{9})^+\), 3.1.972.2, \(\Q(\zeta_{27})^+\), 9.3.74384733888.4

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) $1$ $-1$
1.9.6t1.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) $0$ $-1$
1.27.18t1.a.c$1$ $ 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.d$1$ $ 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) $0$ $1$
* 1.27.9t1.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) $0$ $1$
* 1.27.9t1.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.e$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
1.27.18t1.a.f$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
* 2.972.3t2.d.a$2$ $ 2^{2} \cdot 3^{5}$ 3.1.972.2 $S_3$ (as 3T2) $1$ $0$
* 2.972.6t5.d.a$2$ $ 2^{2} \cdot 3^{5}$ 6.0.2834352.4 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.972.6t5.d.b$2$ $ 2^{2} \cdot 3^{5}$ 6.0.2834352.4 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2916.18t16.d.a$2$ $ 2^{2} \cdot 3^{6}$ 27.9.17717054310925604811052271173453197606912.2 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.d.b$2$ $ 2^{2} \cdot 3^{6}$ 27.9.17717054310925604811052271173453197606912.2 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.d.c$2$ $ 2^{2} \cdot 3^{6}$ 27.9.17717054310925604811052271173453197606912.2 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.d.d$2$ $ 2^{2} \cdot 3^{6}$ 27.9.17717054310925604811052271173453197606912.2 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.d.e$2$ $ 2^{2} \cdot 3^{6}$ 27.9.17717054310925604811052271173453197606912.2 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.d.f$2$ $ 2^{2} \cdot 3^{6}$ 27.9.17717054310925604811052271173453197606912.2 $C_9\times S_3$ (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 *.