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Properties

Label 27.9.177...912.1
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 - 18*x^25 + 189*x^23 - 63*x^22 - 1062*x^21 + 648*x^20 + 4464*x^19 - 2679*x^18 - 14499*x^17 + 7092*x^16 + 36963*x^15 - 21411*x^14 - 74187*x^13 + 24255*x^12 + 61398*x^11 - 36351*x^10 - 17520*x^9 + 66663*x^8 + 42507*x^7 - 11583*x^6 - 18603*x^5 - 10539*x^4 - 6327*x^3 - 2835*x^2 - 639*x - 53)
 
gp: K = bnfinit(x^27 - 18*x^25 + 189*x^23 - 63*x^22 - 1062*x^21 + 648*x^20 + 4464*x^19 - 2679*x^18 - 14499*x^17 + 7092*x^16 + 36963*x^15 - 21411*x^14 - 74187*x^13 + 24255*x^12 + 61398*x^11 - 36351*x^10 - 17520*x^9 + 66663*x^8 + 42507*x^7 - 11583*x^6 - 18603*x^5 - 10539*x^4 - 6327*x^3 - 2835*x^2 - 639*x - 53, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-53, -639, -2835, -6327, -10539, -18603, -11583, 42507, 66663, -17520, -36351, 61398, 24255, -74187, -21411, 36963, 7092, -14499, -2679, 4464, 648, -1062, -63, 189, 0, -18, 0, 1]);
 

\( x^{27} - 18 x^{25} + 189 x^{23} - 63 x^{22} - 1062 x^{21} + 648 x^{20} + 4464 x^{19} - 2679 x^{18} - 14499 x^{17} + 7092 x^{16} + 36963 x^{15} - 21411 x^{14} - 74187 x^{13} + 24255 x^{12} + 61398 x^{11} - 36351 x^{10} - 17520 x^{9} + 66663 x^{8} + 42507 x^{7} - 11583 x^{6} - 18603 x^{5} - 10539 x^{4} - 6327 x^{3} - 2835 x^{2} - 639 x - 53 \)

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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{12} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{16} - \frac{1}{3} a^{13} + \frac{1}{3} a^{10} + \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{17} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{18} - \frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{19} - \frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{3} a^{20} - \frac{1}{3} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{21} - \frac{1}{3} a^{12} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{22} - \frac{1}{3} a^{13} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{23} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{24} - \frac{1}{3} a^{12} + \frac{1}{3} a^{9} - \frac{1}{3} a^{6} - \frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{25} - \frac{1}{3} a^{13} + \frac{1}{3} a^{10} - \frac{1}{3} a^{7} - \frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{11635504228846358448927348648104456129976298362080203870293} a^{26} - \frac{243671552971932894966189505516475546636191078626474118814}{3878501409615452816309116216034818709992099454026734623431} a^{25} - \frac{402496355682565522865890882062676263760597275779512889769}{3878501409615452816309116216034818709992099454026734623431} a^{24} - \frac{732113475165636202202006766957359866193410049208005703953}{11635504228846358448927348648104456129976298362080203870293} a^{23} - \frac{444615295729171802380101902949602842609811436545241769002}{3878501409615452816309116216034818709992099454026734623431} a^{22} + \frac{1930138224547042928127337708620447392391031412519046808457}{11635504228846358448927348648104456129976298362080203870293} a^{21} + \frac{63957912991709353558655667472164195740015747186240816000}{3878501409615452816309116216034818709992099454026734623431} a^{20} - \frac{788023945844073307283080035487410611470263178658546990413}{11635504228846358448927348648104456129976298362080203870293} a^{19} + \frac{363312292676128639534226300361200324658971513590440525216}{3878501409615452816309116216034818709992099454026734623431} a^{18} - \frac{214703717668866498046849305967448255473405777892150136337}{11635504228846358448927348648104456129976298362080203870293} a^{17} + \frac{1717224856312352334272441001456070842865724747077479616681}{11635504228846358448927348648104456129976298362080203870293} a^{16} + \frac{109090457486133487227852370635314340738907184827619957501}{11635504228846358448927348648104456129976298362080203870293} a^{15} + \frac{348319907864351190901024064766597369705671888074085652371}{11635504228846358448927348648104456129976298362080203870293} a^{14} - \frac{83052603271863379943808483456993636890209671785399575402}{11635504228846358448927348648104456129976298362080203870293} a^{13} - \frac{5726739346432612054774755763335009788879480465245125545839}{11635504228846358448927348648104456129976298362080203870293} a^{12} - \frac{756324911303443926242673298150075085000843967713200570388}{3878501409615452816309116216034818709992099454026734623431} a^{11} - \frac{5440959265665826375071538496712318639813418141808176087951}{11635504228846358448927348648104456129976298362080203870293} a^{10} - \frac{5765026625579127968539086721555067912153213671321484799814}{11635504228846358448927348648104456129976298362080203870293} a^{9} - \frac{15623389345379607521447892059842740103862704340783960780}{11635504228846358448927348648104456129976298362080203870293} a^{8} - \frac{4124364837579362572532153580576427163443986984052139640083}{11635504228846358448927348648104456129976298362080203870293} a^{7} + \frac{2201277476446152687916791974388944087245088413101960947105}{11635504228846358448927348648104456129976298362080203870293} a^{6} - \frac{2273161887676426629511981478418464549354578657659554141789}{11635504228846358448927348648104456129976298362080203870293} a^{5} + \frac{4471798604251571855348836735035906671680245641860071335351}{11635504228846358448927348648104456129976298362080203870293} a^{4} - \frac{1040228896385672335313638267007509609336761729624314407446}{3878501409615452816309116216034818709992099454026734623431} a^{3} + \frac{700795977678828098278146549292110198156613041581628139018}{3878501409615452816309116216034818709992099454026734623431} a^{2} - \frac{1918515476391826337162876822938636307631771211203646312283}{3878501409615452816309116216034818709992099454026734623431} a + \frac{1510500826948893899066693640611073486249676124872193205569}{11635504228846358448927348648104456129976298362080203870293}$

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:  \( 6360080771.530314 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{9}\cdot(2\pi)^{9}\cdot 6360080771.530314 \cdot 1}{2\sqrt{17717054310925604811052271173453197606912}}\approx 0.186691976425618$ (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

\(\Q(\zeta_{9})^+\), 3.1.972.1, \(\Q(\zeta_{27})^+\), 9.3.74384733888.5

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) $0$ $1$
* 1.27.9t1.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) $0$ $1$
1.27.18t1.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.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) $0$ $1$
1.27.18t1.a.d$1$ $ 3^{3}$ \(\Q(\zeta_{27})\) $C_{18}$ (as 18T1) $0$ $-1$
* 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.c.a$2$ $ 2^{2} \cdot 3^{5}$ 3.1.972.1 $S_3$ (as 3T2) $1$ $0$
* 2.972.6t5.c.a$2$ $ 2^{2} \cdot 3^{5}$ 6.0.2834352.3 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.972.6t5.c.b$2$ $ 2^{2} \cdot 3^{5}$ 6.0.2834352.3 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.2916.18t16.c.a$2$ $ 2^{2} \cdot 3^{6}$ 27.9.17717054310925604811052271173453197606912.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.c.b$2$ $ 2^{2} \cdot 3^{6}$ 27.9.17717054310925604811052271173453197606912.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.c.c$2$ $ 2^{2} \cdot 3^{6}$ 27.9.17717054310925604811052271173453197606912.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.c.d$2$ $ 2^{2} \cdot 3^{6}$ 27.9.17717054310925604811052271173453197606912.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.c.e$2$ $ 2^{2} \cdot 3^{6}$ 27.9.17717054310925604811052271173453197606912.1 $C_9\times S_3$ (as 27T12) $0$ $0$
* 2.2916.18t16.c.f$2$ $ 2^{2} \cdot 3^{6}$ 27.9.17717054310925604811052271173453197606912.1 $C_9\times S_3$ (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 *.