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Properties

Label 18.0.508...703.2
Degree $18$
Signature $[0, 9]$
Discriminant $-5.087\times 10^{23}$
Root discriminant \(20.75\)
Ramified primes $3,23$
Class number $3$
Class group [3]
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 3*x^15 + 45*x^14 - 60*x^13 + 254*x^12 - 198*x^11 + 450*x^10 - 400*x^9 + 468*x^8 - 534*x^7 + 956*x^6 - 774*x^5 + 405*x^4 - 138*x^3 + 45*x^2 - 9*x + 1)
 
gp: K = bnfinit(y^18 + 3*y^16 - 3*y^15 + 45*y^14 - 60*y^13 + 254*y^12 - 198*y^11 + 450*y^10 - 400*y^9 + 468*y^8 - 534*y^7 + 956*y^6 - 774*y^5 + 405*y^4 - 138*y^3 + 45*y^2 - 9*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 3*x^16 - 3*x^15 + 45*x^14 - 60*x^13 + 254*x^12 - 198*x^11 + 450*x^10 - 400*x^9 + 468*x^8 - 534*x^7 + 956*x^6 - 774*x^5 + 405*x^4 - 138*x^3 + 45*x^2 - 9*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 3*x^16 - 3*x^15 + 45*x^14 - 60*x^13 + 254*x^12 - 198*x^11 + 450*x^10 - 400*x^9 + 468*x^8 - 534*x^7 + 956*x^6 - 774*x^5 + 405*x^4 - 138*x^3 + 45*x^2 - 9*x + 1)
 

\( x^{18} + 3 x^{16} - 3 x^{15} + 45 x^{14} - 60 x^{13} + 254 x^{12} - 198 x^{11} + 450 x^{10} - 400 x^{9} + \cdots + 1 \) Copy content Toggle raw display

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

Invariants

Degree:  $18$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[0, 9]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-508698711308514601331703\) \(\medspace = -\,3^{24}\cdot 23^{9}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(20.75\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{4/3}23^{1/2}\approx 20.75035786129348$
Ramified primes:   \(3\), \(23\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{-23}) \)
$\card{ \Gal(K/\Q) }$:  $18$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{4}$, $\frac{1}{26}a^{12}-\frac{2}{13}a^{11}+\frac{3}{26}a^{10}-\frac{1}{13}a^{9}-\frac{3}{26}a^{8}+\frac{3}{13}a^{7}+\frac{2}{13}a^{6}+\frac{7}{26}a^{5}-\frac{5}{13}a^{4}-\frac{5}{26}a^{3}+\frac{5}{13}a^{2}+\frac{7}{26}a-\frac{6}{13}$, $\frac{1}{104}a^{13}-\frac{1}{104}a^{12}+\frac{17}{104}a^{11}-\frac{19}{104}a^{10}+\frac{17}{104}a^{9}-\frac{3}{104}a^{8}+\frac{9}{104}a^{7}+\frac{19}{104}a^{6}+\frac{37}{104}a^{5}+\frac{43}{104}a^{4}-\frac{5}{104}a^{3}+\frac{11}{104}a^{2}+\frac{35}{104}a-\frac{49}{104}$, $\frac{1}{208}a^{14}-\frac{21}{104}a^{11}+\frac{1}{104}a^{10}+\frac{23}{104}a^{9}+\frac{1}{104}a^{8}+\frac{9}{52}a^{7}+\frac{6}{13}a^{6}-\frac{2}{13}a^{5}-\frac{5}{104}a^{4}+\frac{17}{104}a^{3}+\frac{47}{104}a^{2}-\frac{37}{104}a+\frac{3}{16}$, $\frac{1}{208}a^{15}-\frac{1}{104}a^{12}+\frac{25}{104}a^{11}-\frac{21}{104}a^{10}+\frac{1}{8}a^{9}+\frac{5}{52}a^{8}+\frac{3}{26}a^{7}-\frac{5}{13}a^{6}+\frac{31}{104}a^{5}+\frac{25}{104}a^{4}+\frac{51}{104}a^{3}+\frac{7}{104}a^{2}+\frac{7}{208}a+\frac{5}{26}$, $\frac{1}{191152}a^{16}+\frac{379}{191152}a^{15}+\frac{1}{11947}a^{14}+\frac{331}{95576}a^{13}-\frac{123}{47788}a^{12}+\frac{8009}{47788}a^{11}+\frac{7217}{47788}a^{10}+\frac{4661}{95576}a^{9}+\frac{6445}{47788}a^{8}+\frac{2463}{23894}a^{7}+\frac{10451}{95576}a^{6}-\frac{20411}{47788}a^{5}-\frac{19719}{47788}a^{4}-\frac{9869}{23894}a^{3}-\frac{81911}{191152}a^{2}-\frac{22235}{191152}a+\frac{6973}{23894}$, $\frac{1}{2780879296}a^{17}+\frac{3833}{2780879296}a^{16}+\frac{166905}{695219824}a^{15}-\frac{2071559}{2780879296}a^{14}-\frac{4605317}{1390439648}a^{13}+\frac{6298693}{1390439648}a^{12}+\frac{6180341}{347609912}a^{11}-\frac{252042831}{1390439648}a^{10}+\frac{159223221}{695219824}a^{9}-\frac{162112095}{695219824}a^{8}+\frac{64818553}{347609912}a^{7}-\frac{398590551}{1390439648}a^{6}-\frac{543304225}{1390439648}a^{5}+\frac{121440327}{347609912}a^{4}-\frac{44352703}{213913792}a^{3}-\frac{1364399973}{2780879296}a^{2}+\frac{15734112}{43451239}a-\frac{819081329}{2780879296}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

$C_{3}$, which has order $3$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $8$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{305901}{764608}a^{17}+\frac{98037}{764608}a^{16}+\frac{228765}{191152}a^{15}-\frac{637335}{764608}a^{14}+\frac{6728799}{382304}a^{13}-\frac{6988831}{382304}a^{12}+\frac{2241279}{23894}a^{11}-\frac{18057519}{382304}a^{10}+\frac{29534367}{191152}a^{9}-\frac{20235177}{191152}a^{8}+\frac{1000851}{7352}a^{7}-\frac{60950731}{382304}a^{6}+\frac{121020147}{382304}a^{5}-\frac{9153009}{47788}a^{4}+\frac{50793137}{764608}a^{3}-\frac{10632249}{764608}a^{2}+\frac{50103}{7352}a-\frac{79265}{764608}$, $\frac{1052302221}{2780879296}a^{17}+\frac{380687001}{2780879296}a^{16}+\frac{412416517}{347609912}a^{15}-\frac{1967801787}{2780879296}a^{14}+\frac{23317796655}{1390439648}a^{13}-\frac{23156016163}{1390439648}a^{12}+\frac{15667997499}{173804956}a^{11}-\frac{59069295087}{1390439648}a^{10}+\frac{8295882981}{53478448}a^{9}-\frac{66618986805}{695219824}a^{8}+\frac{24666799329}{173804956}a^{7}-\frac{211463530003}{1390439648}a^{6}+\frac{424790728551}{1390439648}a^{5}-\frac{31854065487}{173804956}a^{4}+\frac{243463776337}{2780879296}a^{3}-\frac{58702644261}{2780879296}a^{2}+\frac{4566921465}{695219824}a-\frac{199497605}{2780879296}$, $\frac{352003}{213913792}a^{17}+\frac{12803499}{2780879296}a^{16}+\frac{1619697}{173804956}a^{15}+\frac{34695959}{2780879296}a^{14}+\frac{102829905}{1390439648}a^{13}+\frac{140058651}{1390439648}a^{12}+\frac{112930319}{347609912}a^{11}+\frac{1004538663}{1390439648}a^{10}+\frac{523830117}{695219824}a^{9}+\frac{829640031}{695219824}a^{8}+\frac{151838847}{347609912}a^{7}-\frac{205917261}{1390439648}a^{6}-\frac{64681551}{1390439648}a^{5}+\frac{83632509}{347609912}a^{4}-\frac{180907125}{2780879296}a^{3}+\frac{101096873}{2780879296}a^{2}-\frac{6696081}{695219824}a-\frac{5222070719}{2780879296}$, $\frac{3196371}{2780879296}a^{17}+\frac{5707371}{2780879296}a^{16}+\frac{4332591}{695219824}a^{15}+\frac{12346287}{2780879296}a^{14}+\frac{74375929}{1390439648}a^{13}+\frac{25825239}{1390439648}a^{12}+\frac{24761049}{86902478}a^{11}+\frac{278195583}{1390439648}a^{10}+\frac{478880433}{695219824}a^{9}+\frac{16339437}{53478448}a^{8}+\frac{13482711}{26739224}a^{7}-\frac{262497069}{1390439648}a^{6}-\frac{8001387}{1390439648}a^{5}+\frac{23200167}{173804956}a^{4}-\frac{106139313}{2780879296}a^{3}+\frac{59857593}{2780879296}a^{2}-\frac{1976289}{347609912}a+\frac{968269977}{2780879296}$, $\frac{11751703}{2780879296}a^{17}-\frac{877968297}{2780879296}a^{16}+\frac{16433091}{695219824}a^{15}-\frac{2576476081}{2780879296}a^{14}+\frac{1619291457}{1390439648}a^{13}-\frac{20015499729}{1390439648}a^{12}+\frac{1772463579}{86902478}a^{11}-\frac{8574921049}{106956896}a^{10}+\frac{45163814361}{695219824}a^{9}-\frac{95381217411}{695219824}a^{8}+\frac{21797000489}{173804956}a^{7}-\frac{14841266649}{106956896}a^{6}+\frac{17441354337}{106956896}a^{5}-\frac{12773974542}{43451239}a^{4}+\frac{663248500995}{2780879296}a^{3}-\frac{287515183907}{2780879296}a^{2}+\frac{6806039371}{347609912}a-\frac{4064003695}{2780879296}$, $\frac{901980845}{2780879296}a^{17}+\frac{125556877}{2780879296}a^{16}+\frac{52062575}{53478448}a^{15}-\frac{2348187191}{2780879296}a^{14}+\frac{20100521979}{1390439648}a^{13}-\frac{24260547747}{1390439648}a^{12}+\frac{27704858589}{347609912}a^{11}-\frac{73791127595}{1390439648}a^{10}+\frac{95448221221}{695219824}a^{9}-\frac{77022037303}{695219824}a^{8}+\frac{11620993081}{86902478}a^{7}-\frac{213881000935}{1390439648}a^{6}+\frac{398251146919}{1390439648}a^{5}-\frac{72803840511}{347609912}a^{4}+\frac{272183681033}{2780879296}a^{3}-\frac{84313030441}{2780879296}a^{2}+\frac{3179911699}{347609912}a-\frac{1764925289}{2780879296}$, $\frac{23207747}{43451239}a^{17}+\frac{286552365}{695219824}a^{16}+\frac{1266693273}{695219824}a^{15}-\frac{204494831}{695219824}a^{14}+\frac{8152737511}{347609912}a^{13}-\frac{1213483229}{86902478}a^{12}+\frac{41914095407}{347609912}a^{11}-\frac{3848963883}{347609912}a^{10}+\frac{2808877649}{13369612}a^{9}-\frac{18819490481}{347609912}a^{8}+\frac{29447967637}{173804956}a^{7}-\frac{52782061313}{347609912}a^{6}+\frac{31629217829}{86902478}a^{5}-\frac{38022593801}{347609912}a^{4}+\frac{24691065751}{347609912}a^{3}-\frac{8237946305}{695219824}a^{2}+\frac{2488842349}{695219824}a+\frac{138903335}{695219824}$, $\frac{5451415}{347609912}a^{17}+\frac{23708563}{695219824}a^{16}+\frac{37978335}{695219824}a^{15}+\frac{1109121}{26739224}a^{14}+\frac{107083395}{173804956}a^{13}+\frac{184529003}{347609912}a^{12}+\frac{795356287}{347609912}a^{11}+\frac{1571432673}{347609912}a^{10}+\frac{451933073}{173804956}a^{9}+\frac{126787307}{26739224}a^{8}-\frac{992809671}{347609912}a^{7}+\frac{81275629}{86902478}a^{6}+\frac{15517921}{26739224}a^{5}+\frac{5368535953}{347609912}a^{4}-\frac{1860030677}{173804956}a^{3}-\frac{90286087}{53478448}a^{2}+\frac{857007847}{695219824}a+\frac{139161493}{173804956}$ Copy content Toggle raw display
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 8598.44783956 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 8598.44783956 \cdot 3}{2\cdot\sqrt{508698711308514601331703}}\cr\approx \mathstrut & 0.275994111752 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 3*x^15 + 45*x^14 - 60*x^13 + 254*x^12 - 198*x^11 + 450*x^10 - 400*x^9 + 468*x^8 - 534*x^7 + 956*x^6 - 774*x^5 + 405*x^4 - 138*x^3 + 45*x^2 - 9*x + 1)
 
DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent()
 
hK = K.class_number(); wK = K.unit_group().torsion_generator().order();
 
2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^18 + 3*x^16 - 3*x^15 + 45*x^14 - 60*x^13 + 254*x^12 - 198*x^11 + 450*x^10 - 400*x^9 + 468*x^8 - 534*x^7 + 956*x^6 - 774*x^5 + 405*x^4 - 138*x^3 + 45*x^2 - 9*x + 1, 1);
 
[polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^18 + 3*x^16 - 3*x^15 + 45*x^14 - 60*x^13 + 254*x^12 - 198*x^11 + 450*x^10 - 400*x^9 + 468*x^8 - 534*x^7 + 956*x^6 - 774*x^5 + 405*x^4 - 138*x^3 + 45*x^2 - 9*x + 1);
 
OK := Integers(K); DK := Discriminant(OK);
 
UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK);
 
r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK);
 
hK := #clK; wK := #TorsionSubgroup(UK);
 
2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 + 3*x^16 - 3*x^15 + 45*x^14 - 60*x^13 + 254*x^12 - 198*x^11 + 450*x^10 - 400*x^9 + 468*x^8 - 534*x^7 + 956*x^6 - 774*x^5 + 405*x^4 - 138*x^3 + 45*x^2 - 9*x + 1);
 
OK = ring_of_integers(K); DK = discriminant(OK);
 
UK, fUK = unit_group(OK); clK, fclK = class_group(OK);
 
r1,r2 = signature(K); RK = regulator(K); RR = parent(RK);
 
hK = order(clK); wK = torsion_units_order(K);
 
2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$C_3\times S_3$ (as 18T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-23}) \), 3.1.23.1 x3, \(\Q(\zeta_{9})^+\), 6.0.12167.1, 6.0.79827687.1, 6.0.79827687.2 x2, 9.3.6466042647.5 x3

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 6 sibling: 6.0.79827687.2
Degree 9 sibling: 9.3.6466042647.5
Minimal sibling: 6.0.79827687.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{6}$ R ${\href{/padicField/5.6.0.1}{6} }^{3}$ ${\href{/padicField/7.6.0.1}{6} }^{3}$ ${\href{/padicField/11.6.0.1}{6} }^{3}$ ${\href{/padicField/13.3.0.1}{3} }^{6}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.2.0.1}{2} }^{9}$ R ${\href{/padicField/29.3.0.1}{3} }^{6}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.2.0.1}{2} }^{9}$ ${\href{/padicField/41.3.0.1}{3} }^{6}$ ${\href{/padicField/43.6.0.1}{6} }^{3}$ ${\href{/padicField/47.3.0.1}{3} }^{6}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.3.0.1}{3} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Oscar:
 
p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(3\) Copy content Toggle raw display 3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{8} + 108 x^{7} + 225 x^{6} + 108 x^{5} + 324 x^{4} + 675 x^{3} + 4050 x^{2} - 3861$$3$$3$$12$$C_3^2$$[2]^{3}$
\(23\) Copy content Toggle raw display 23.6.3.2$x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
23.6.3.2$x^{6} + 73 x^{4} + 36 x^{3} + 1591 x^{2} - 2412 x + 10467$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.23.2t1.a.a$1$ $ 23 $ \(\Q(\sqrt{-23}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.207.6t1.b.a$1$ $ 3^{2} \cdot 23 $ 6.0.79827687.1 $C_6$ (as 6T1) $0$ $-1$
* 1.207.6t1.b.b$1$ $ 3^{2} \cdot 23 $ 6.0.79827687.1 $C_6$ (as 6T1) $0$ $-1$
* 1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
*2 2.23.3t2.b.a$2$ $ 23 $ 3.1.23.1 $S_3$ (as 3T2) $1$ $0$
*2 2.1863.6t5.c.a$2$ $ 3^{4} \cdot 23 $ 18.0.508698711308514601331703.2 $S_3 \times C_3$ (as 18T3) $0$ $0$
*2 2.1863.6t5.c.b$2$ $ 3^{4} \cdot 23 $ 18.0.508698711308514601331703.2 $S_3 \times C_3$ (as 18T3) $0$ $0$

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 *.