Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 3*x^16 - 3*x^15 + 18*x^14 + 24*x^13 + 39*x^12 + 63*x^11 + 66*x^10 + 92*x^9 + 108*x^8 + 21*x^7 - 75*x^6 - 36*x^5 + 33*x^4 + 30*x^3 + 9*x^2 + 3*x + 1)
gp: K = bnfinit(y^18 + 3*y^16 - 3*y^15 + 18*y^14 + 24*y^13 + 39*y^12 + 63*y^11 + 66*y^10 + 92*y^9 + 108*y^8 + 21*y^7 - 75*y^6 - 36*y^5 + 33*y^4 + 30*y^3 + 9*y^2 + 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 + 3*x^16 - 3*x^15 + 18*x^14 + 24*x^13 + 39*x^12 + 63*x^11 + 66*x^10 + 92*x^9 + 108*x^8 + 21*x^7 - 75*x^6 - 36*x^5 + 33*x^4 + 30*x^3 + 9*x^2 + 3*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 + 3*x^16 - 3*x^15 + 18*x^14 + 24*x^13 + 39*x^12 + 63*x^11 + 66*x^10 + 92*x^9 + 108*x^8 + 21*x^7 - 75*x^6 - 36*x^5 + 33*x^4 + 30*x^3 + 9*x^2 + 3*x + 1)
\( x^{18} + 3 x^{16} - 3 x^{15} + 18 x^{14} + 24 x^{13} + 39 x^{12} + 63 x^{11} + 66 x^{10} + 92 x^{9} + \cdots + 1 \)
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: |
\(-2981391508243921935123\)
\(\medspace = -\,3^{31}\cdot 13^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(15.60\) | 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: | $3^{31/18}13^{1/2}\approx 23.91555719607668$ | ||
| Ramified primes: |
\(3\), \(13\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-3}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}$, $\frac{1}{3}a^{10}+\frac{1}{3}a$, $\frac{1}{9}a^{11}-\frac{1}{9}a^{10}+\frac{1}{9}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{4}{9}a^{2}-\frac{4}{9}a+\frac{4}{9}$, $\frac{1}{9}a^{12}+\frac{1}{9}a^{9}-\frac{2}{9}a^{3}-\frac{2}{9}$, $\frac{1}{9}a^{13}+\frac{1}{9}a^{10}-\frac{2}{9}a^{4}-\frac{2}{9}a$, $\frac{1}{9}a^{14}+\frac{1}{9}a^{10}-\frac{1}{9}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}+\frac{4}{9}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{4}{9}a-\frac{4}{9}$, $\frac{1}{9}a^{15}-\frac{1}{9}a^{9}-\frac{2}{9}a^{6}+\frac{2}{9}$, $\frac{1}{9}a^{16}-\frac{1}{9}a^{10}-\frac{2}{9}a^{7}+\frac{2}{9}a$, $\frac{1}{505770651}a^{17}-\frac{26800795}{505770651}a^{16}+\frac{3135059}{505770651}a^{15}-\frac{25026709}{505770651}a^{14}+\frac{24869824}{505770651}a^{13}-\frac{22158187}{505770651}a^{12}+\frac{11058808}{505770651}a^{11}-\frac{41452174}{505770651}a^{10}-\frac{4037029}{56196739}a^{9}-\frac{179414669}{505770651}a^{8}+\frac{111463877}{505770651}a^{7}+\frac{233138468}{505770651}a^{6}+\frac{123235982}{505770651}a^{5}-\frac{177885833}{505770651}a^{4}-\frac{248191789}{505770651}a^{3}+\frac{130524265}{505770651}a^{2}+\frac{209339984}{505770651}a-\frac{14539693}{56196739}$
sage: K.integral_basis()
gp: K.zk
magma: IntegralBasis(K);
oscar: basis(OK)
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$
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: |
\( -\frac{1476827171}{505770651} a^{17} + \frac{425343308}{168590217} a^{16} - \frac{5404238693}{505770651} a^{15} + \frac{8981335211}{505770651} a^{14} - \frac{33863679400}{505770651} a^{13} - \frac{2335731508}{168590217} a^{12} - \frac{48529484567}{505770651} a^{11} - \frac{50718098032}{505770651} a^{10} - \frac{49355626366}{505770651} a^{9} - \frac{89277384656}{505770651} a^{8} - \frac{26118987181}{168590217} a^{7} + \frac{43947231568}{505770651} a^{6} + \frac{78592698215}{505770651} a^{5} - \frac{19359031537}{505770651} a^{4} - \frac{13204647298}{168590217} a^{3} - \frac{9100516115}{505770651} a^{2} - \frac{2227238122}{505770651} a - \frac{1570535296}{505770651} \)
(order $18$)
| 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: |
$\frac{573511403}{505770651}a^{17}-\frac{108179090}{505770651}a^{16}+\frac{1698264025}{505770651}a^{15}-\frac{2005598563}{505770651}a^{14}+\frac{10573527878}{505770651}a^{13}+\frac{3992337350}{168590217}a^{12}+\frac{6422554619}{168590217}a^{11}+\frac{31989260425}{505770651}a^{10}+\frac{31218976322}{505770651}a^{9}+\frac{44838117032}{505770651}a^{8}+\frac{53109168982}{505770651}a^{7}-\frac{712756832}{505770651}a^{6}-\frac{44654512351}{505770651}a^{5}-\frac{10255091704}{505770651}a^{4}+\frac{7874201219}{168590217}a^{3}+\frac{3497552132}{168590217}a^{2}+\frac{1704517255}{505770651}a+\frac{2063067365}{505770651}$, $\frac{679001176}{168590217}a^{17}-\frac{1286736407}{505770651}a^{16}+\frac{2349796280}{168590217}a^{15}-\frac{10592301425}{505770651}a^{14}+\frac{43738083452}{505770651}a^{13}+\frac{20794844428}{505770651}a^{12}+\frac{68667099232}{505770651}a^{11}+\frac{87439870573}{505770651}a^{10}+\frac{83511029200}{505770651}a^{9}+\frac{47171761222}{168590217}a^{8}+\frac{137200228756}{505770651}a^{7}-\frac{11315993677}{168590217}a^{6}-\frac{120284213084}{505770651}a^{5}+\frac{2521518503}{505770651}a^{4}+\frac{55905736423}{505770651}a^{3}+\frac{23035531654}{505770651}a^{2}+\frac{8509888093}{505770651}a+\frac{3318455746}{505770651}$, $\frac{1403329513}{505770651}a^{17}-\frac{893563322}{505770651}a^{16}+\frac{4847032712}{505770651}a^{15}-\frac{2438136194}{168590217}a^{14}+\frac{30125853782}{505770651}a^{13}+\frac{14241595103}{505770651}a^{12}+\frac{46944524201}{505770651}a^{11}+\frac{59828918083}{505770651}a^{10}+\frac{18927822838}{168590217}a^{9}+\frac{96727351564}{505770651}a^{8}+\frac{93368359765}{505770651}a^{7}-\frac{24663828244}{505770651}a^{6}-\frac{27848241011}{168590217}a^{5}+\frac{2158565087}{505770651}a^{4}+\frac{40142560223}{505770651}a^{3}+\frac{15018811091}{505770651}a^{2}+\frac{5052913717}{505770651}a+\frac{834042952}{168590217}$, $\frac{203077331}{56196739}a^{17}-\frac{1581883313}{505770651}a^{16}+\frac{6816278230}{505770651}a^{15}-\frac{3770205283}{168590217}a^{14}+\frac{4720958218}{56196739}a^{13}+\frac{833068573}{56196739}a^{12}+\frac{63590461250}{505770651}a^{11}+\frac{20369226257}{168590217}a^{10}+\frac{65944082839}{505770651}a^{9}+\frac{37089329014}{168590217}a^{8}+\frac{100287536227}{505770651}a^{7}-\frac{50006445815}{505770651}a^{6}-\frac{31371736517}{168590217}a^{5}+\frac{5768051956}{168590217}a^{4}+\frac{14858861522}{168590217}a^{3}+\frac{15235246172}{505770651}a^{2}+\frac{1401230911}{168590217}a+\frac{1777285813}{505770651}$, $\frac{3648485194}{505770651}a^{17}-\frac{271433797}{56196739}a^{16}+\frac{12617683433}{505770651}a^{15}-\frac{19427862134}{505770651}a^{14}+\frac{26281265558}{168590217}a^{13}+\frac{11499235523}{168590217}a^{12}+\frac{120163919009}{505770651}a^{11}+\frac{149273010625}{505770651}a^{10}+\frac{47601310315}{168590217}a^{9}+\frac{240841611547}{505770651}a^{8}+\frac{78266660060}{168590217}a^{7}-\frac{78280762513}{505770651}a^{6}-\frac{218879679032}{505770651}a^{5}+\frac{1658507588}{56196739}a^{4}+\frac{36216531856}{168590217}a^{3}+\frac{36015032324}{505770651}a^{2}+\frac{9062021266}{505770651}a+\frac{2003280365}{168590217}$, $\frac{4542997889}{505770651}a^{17}-\frac{3564140458}{505770651}a^{16}+\frac{16332747955}{505770651}a^{15}-\frac{26354601977}{505770651}a^{14}+\frac{102120198071}{505770651}a^{13}+\frac{9830667899}{168590217}a^{12}+\frac{50641979359}{168590217}a^{11}+\frac{166666753073}{505770651}a^{10}+\frac{55479529945}{168590217}a^{9}+\frac{284508289163}{505770651}a^{8}+\frac{265081809944}{505770651}a^{7}-\frac{117448448840}{505770651}a^{6}-\frac{252595938182}{505770651}a^{5}+\frac{38419127924}{505770651}a^{4}+\frac{13869066633}{56196739}a^{3}+\frac{4005422474}{56196739}a^{2}+\frac{9078249446}{505770651}a+\frac{661889405}{56196739}$, $\frac{2783325611}{505770651}a^{17}-\frac{227580728}{56196739}a^{16}+\frac{3268699717}{168590217}a^{15}-\frac{15519672928}{505770651}a^{14}+\frac{61318387360}{505770651}a^{13}+\frac{22010022994}{505770651}a^{12}+\frac{91125536690}{505770651}a^{11}+\frac{108183852533}{505770651}a^{10}+\frac{34052324786}{168590217}a^{9}+\frac{179144682317}{505770651}a^{8}+\frac{55672153025}{168590217}a^{7}-\frac{22612828495}{168590217}a^{6}-\frac{161004397924}{505770651}a^{5}+\frac{19898279164}{505770651}a^{4}+\frac{80170816390}{505770651}a^{3}+\frac{25911466577}{505770651}a^{2}+\frac{6799956221}{505770651}a+\frac{428224696}{56196739}$, $\frac{575058890}{505770651}a^{17}-\frac{403272082}{505770651}a^{16}+\frac{1977816583}{505770651}a^{15}-\frac{3035517647}{505770651}a^{14}+\frac{12326419054}{505770651}a^{13}+\frac{5516099887}{505770651}a^{12}+\frac{17575236469}{505770651}a^{11}+\frac{2763884883}{56196739}a^{10}+\frac{19848916610}{505770651}a^{9}+\frac{39362668433}{505770651}a^{8}+\frac{35329661693}{505770651}a^{7}-\frac{13121037332}{505770651}a^{6}-\frac{32710982726}{505770651}a^{5}+\frac{5651540488}{505770651}a^{4}+\frac{14136442117}{505770651}a^{3}+\frac{4178513584}{505770651}a^{2}+\frac{745139933}{168590217}a+\frac{1294479590}{505770651}$
| 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: | \( 20791.745081119407 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 20791.745081119407 \cdot 1}{18\cdot\sqrt{2981391508243921935123}}\cr\approx \mathstrut & 0.322870039525109 \end{aligned}\]
# 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 + 18*x^14 + 24*x^13 + 39*x^12 + 63*x^11 + 66*x^10 + 92*x^9 + 108*x^8 + 21*x^7 - 75*x^6 - 36*x^5 + 33*x^4 + 30*x^3 + 9*x^2 + 3*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 + 18*x^14 + 24*x^13 + 39*x^12 + 63*x^11 + 66*x^10 + 92*x^9 + 108*x^8 + 21*x^7 - 75*x^6 - 36*x^5 + 33*x^4 + 30*x^3 + 9*x^2 + 3*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 + 18*x^14 + 24*x^13 + 39*x^12 + 63*x^11 + 66*x^10 + 92*x^9 + 108*x^8 + 21*x^7 - 75*x^6 - 36*x^5 + 33*x^4 + 30*x^3 + 9*x^2 + 3*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 + 18*x^14 + 24*x^13 + 39*x^12 + 63*x^11 + 66*x^10 + 92*x^9 + 108*x^8 + 21*x^7 - 75*x^6 - 36*x^5 + 33*x^4 + 30*x^3 + 9*x^2 + 3*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
$C_6\times S_3$ (as 18T6):
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 36 |
| The 18 conjugacy class representatives for $S_3 \times C_6$ |
| Character table for $S_3 \times C_6$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{9})^+\), 3.1.351.1, \(\Q(\zeta_{9})\), 6.0.369603.1, 9.3.31524548679.1 |
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
| Galois closure: | data not computed |
| Degree 12 sibling: | 12.0.1870004703089601.2 |
| Degree 18 sibling: | deg 18 |
| Minimal sibling: | 12.0.1870004703089601.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/17.2.0.1}{2} }^{9}$ | ${\href{/padicField/19.2.0.1}{2} }^{6}{,}\,{\href{/padicField/19.1.0.1}{1} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.3.0.1}{3} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.2.0.1}{2} }^{9}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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
| $p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
|---|---|---|---|---|---|---|---|
|
\(3\)
| Deg $18$ | $18$ | $1$ | $31$ | |||
|
\(13\)
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 13.3.0.1 | $x^{3} + 2 x + 11$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 13.6.3.1 | $x^{6} + 390 x^{5} + 50743 x^{4} + 2208202 x^{3} + 765301 x^{2} + 5017316 x + 24555184$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 13.6.3.1 | $x^{6} + 390 x^{5} + 50743 x^{4} + 2208202 x^{3} + 765301 x^{2} + 5017316 x + 24555184$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.13.2t1.a.a | $1$ | $ 13 $ | \(\Q(\sqrt{13}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.39.2t1.a.a | $1$ | $ 3 \cdot 13 $ | \(\Q(\sqrt{-39}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 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) | $0$ | $1$ |
| * | 1.9.3t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| 1.117.6t1.h.a | $1$ | $ 3^{2} \cdot 13 $ | 6.0.43243551.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.9.6t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ |
| 1.117.6t1.h.b | $1$ | $ 3^{2} \cdot 13 $ | 6.0.43243551.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.117.6t1.c.a | $1$ | $ 3^{2} \cdot 13 $ | 6.6.14414517.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.117.6t1.c.b | $1$ | $ 3^{2} \cdot 13 $ | 6.6.14414517.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| * | 1.9.6t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ |
| * | 2.351.3t2.b.a | $2$ | $ 3^{3} \cdot 13 $ | 3.1.351.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.351.6t3.a.a | $2$ | $ 3^{3} \cdot 13 $ | 6.2.1601613.2 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.1053.12t18.a.a | $2$ | $ 3^{4} \cdot 13 $ | 18.0.2981391508243921935123.1 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
| * | 2.1053.12t18.a.b | $2$ | $ 3^{4} \cdot 13 $ | 18.0.2981391508243921935123.1 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
| * | 2.1053.6t5.f.a | $2$ | $ 3^{4} \cdot 13 $ | 6.0.43243551.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.1053.6t5.f.b | $2$ | $ 3^{4} \cdot 13 $ | 6.0.43243551.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
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 *.