Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*x^2 - 3*x + 1)
gp: K = bnfinit(y^18 - 3*y^17 - 2*y^16 + 12*y^15 + 11*y^14 - 55*y^13 + 101*y^11 - 8*y^10 - 107*y^9 - 8*y^8 + 101*y^7 - 55*y^5 + 11*y^4 + 12*y^3 - 2*y^2 - 3*y + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 3*x^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*x^2 - 3*x + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 3*x^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*x^2 - 3*x + 1)
\( x^{18} - 3 x^{17} - 2 x^{16} + 12 x^{15} + 11 x^{14} - 55 x^{13} + 101 x^{11} - 8 x^{10} - 107 x^{9} + \cdots + 1 \)
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: |
\(-6131323620948659515767\)
\(\medspace = -\,3^{6}\cdot 7^{15}\cdot 11^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(16.23\) | 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: | $3^{1/2}7^{5/6}11^{1/2}\approx 29.074036854473896$ | ||
| Ramified primes: |
\(3\), \(7\), \(11\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{4}$, $\frac{1}{8}a^{12}-\frac{1}{8}a^{10}-\frac{1}{8}a^{8}-\frac{1}{4}a^{7}-\frac{1}{8}a^{6}+\frac{1}{4}a^{5}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}+\frac{3}{8}a^{2}-\frac{3}{8}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{9}-\frac{1}{4}a^{8}-\frac{1}{8}a^{7}-\frac{1}{4}a^{6}+\frac{3}{8}a^{5}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}+\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{40}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{10}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}-\frac{1}{2}a^{3}+\frac{1}{40}$, $\frac{1}{40}a^{15}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{10}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{40}a$, $\frac{1}{200}a^{16}+\frac{1}{200}a^{15}+\frac{1}{200}a^{14}-\frac{1}{20}a^{13}-\frac{1}{40}a^{12}+\frac{1}{20}a^{11}+\frac{9}{40}a^{10}+\frac{23}{100}a^{9}+\frac{31}{200}a^{8}-\frac{1}{50}a^{7}+\frac{9}{40}a^{6}-\frac{1}{5}a^{5}+\frac{9}{40}a^{4}+\frac{9}{20}a^{3}+\frac{19}{50}a^{2}-\frac{49}{200}a-\frac{37}{100}$, $\frac{1}{200}a^{17}-\frac{1}{200}a^{14}+\frac{1}{40}a^{13}-\frac{1}{20}a^{12}+\frac{7}{40}a^{11}+\frac{13}{100}a^{10}-\frac{3}{40}a^{9}-\frac{1}{20}a^{8}-\frac{41}{200}a^{7}+\frac{1}{5}a^{6}-\frac{13}{40}a^{5}-\frac{3}{20}a^{4}-\frac{7}{100}a^{3}-\frac{1}{2}a^{2}+\frac{3}{8}a+\frac{59}{200}$
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
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: |
\( -\frac{297}{200} a^{17} + \frac{693}{200} a^{16} + \frac{549}{100} a^{15} - \frac{297}{20} a^{14} - \frac{1053}{40} a^{13} + \frac{2651}{40} a^{12} + \frac{1827}{40} a^{11} - \frac{26037}{200} a^{10} - \frac{14067}{200} a^{9} + \frac{25083}{200} a^{8} + \frac{3609}{40} a^{7} - \frac{4059}{40} a^{6} - \frac{2583}{40} a^{5} + \frac{1989}{40} a^{4} + \frac{1089}{100} a^{3} - \frac{1341}{100} a^{2} - \frac{477}{200} a + \frac{117}{40} \)
(order $14$)
| 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: |
$a^{17}-3a^{16}-2a^{15}+12a^{14}+11a^{13}-55a^{12}+101a^{10}-8a^{9}-107a^{8}-8a^{7}+101a^{6}-55a^{4}+11a^{3}+12a^{2}-2a-3$, $\frac{53}{40}a^{17}-\frac{287}{100}a^{16}-\frac{1049}{200}a^{15}+\frac{2391}{200}a^{14}+\frac{1013}{40}a^{13}-\frac{267}{5}a^{12}-\frac{1943}{40}a^{11}+\frac{2021}{20}a^{10}+\frac{16271}{200}a^{9}-\frac{4361}{50}a^{8}-\frac{19739}{200}a^{7}+\frac{641}{10}a^{6}+\frac{2797}{40}a^{5}-\frac{219}{10}a^{4}-\frac{327}{20}a^{3}+\frac{169}{50}a^{2}+\frac{463}{100}a-\frac{59}{200}$, $\frac{139}{100}a^{17}-\frac{79}{25}a^{16}-\frac{1077}{200}a^{15}+\frac{541}{40}a^{14}+\frac{1047}{40}a^{13}-\frac{303}{5}a^{12}-\frac{1973}{40}a^{11}+\frac{5947}{50}a^{10}+\frac{16733}{200}a^{9}-\frac{5773}{50}a^{8}-\frac{4361}{40}a^{7}+\frac{1783}{20}a^{6}+\frac{3317}{40}a^{5}-\frac{743}{20}a^{4}-\frac{4997}{200}a^{3}+\frac{171}{25}a^{2}+\frac{549}{100}a-\frac{83}{40}$, $\frac{467}{200}a^{17}-\frac{601}{100}a^{16}-\frac{184}{25}a^{15}+\frac{5061}{200}a^{14}+\frac{733}{20}a^{13}-\frac{2291}{20}a^{12}-\frac{199}{4}a^{11}+\frac{22251}{100}a^{10}+\frac{3707}{50}a^{9}-\frac{11663}{50}a^{8}-\frac{11167}{100}a^{7}+\frac{4089}{20}a^{6}+\frac{327}{4}a^{5}-\frac{439}{4}a^{4}-\frac{3293}{200}a^{3}+\frac{3099}{100}a^{2}+\frac{207}{50}a-\frac{1419}{200}$, $\frac{757}{200}a^{17}-\frac{1779}{200}a^{16}-\frac{2719}{200}a^{15}+\frac{3707}{100}a^{14}+\frac{267}{4}a^{13}-\frac{667}{4}a^{12}-\frac{2267}{20}a^{11}+\frac{31751}{100}a^{10}+\frac{9309}{50}a^{9}-\frac{30017}{100}a^{8}-\frac{24323}{100}a^{7}+\frac{951}{4}a^{6}+\frac{3473}{20}a^{5}-\frac{2089}{20}a^{4}-\frac{8183}{200}a^{3}+\frac{4171}{200}a^{2}+\frac{1981}{200}a-\frac{127}{25}$, $\frac{299}{100}a^{17}-\frac{351}{50}a^{16}-\frac{1067}{100}a^{15}+\frac{5823}{200}a^{14}+\frac{2101}{40}a^{13}-\frac{1313}{10}a^{12}-\frac{3537}{40}a^{11}+\frac{24909}{100}a^{10}+\frac{29121}{200}a^{9}-\frac{5923}{25}a^{8}-\frac{37777}{200}a^{7}+\frac{1927}{10}a^{6}+\frac{5393}{40}a^{5}-\frac{1717}{20}a^{4}-\frac{6007}{200}a^{3}+\frac{1049}{50}a^{2}+\frac{1891}{200}a-\frac{997}{200}$, $\frac{513}{200}a^{17}-\frac{1373}{200}a^{16}-\frac{1593}{200}a^{15}+\frac{1501}{50}a^{14}+\frac{1559}{40}a^{13}-\frac{5393}{40}a^{12}-\frac{407}{8}a^{11}+\frac{54653}{200}a^{10}+\frac{13647}{200}a^{9}-\frac{59913}{200}a^{8}-\frac{25001}{200}a^{7}+\frac{10667}{40}a^{6}+\frac{831}{8}a^{5}-\frac{1179}{8}a^{4}-\frac{719}{25}a^{3}+\frac{844}{25}a^{2}+\frac{433}{50}a-\frac{1141}{200}$, $\frac{269}{200}a^{17}-\frac{519}{200}a^{16}-\frac{1179}{200}a^{15}+\frac{2067}{200}a^{14}+\frac{144}{5}a^{13}-\frac{927}{20}a^{12}-62a^{11}+\frac{4091}{50}a^{10}+\frac{2827}{25}a^{9}-\frac{2791}{50}a^{8}-\frac{6337}{50}a^{7}+\frac{112}{5}a^{6}+\frac{343}{4}a^{5}+11a^{4}-\frac{4301}{200}a^{3}-\frac{1169}{200}a^{2}+\frac{471}{200}a+\frac{157}{200}$
| 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: | \( 27994.834823771416 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 27994.834823771416 \cdot 1}{14\cdot\sqrt{6131323620948659515767}}\cr\approx \mathstrut & 0.389754886856357 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*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^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*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^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*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^17 - 2*x^16 + 12*x^15 + 11*x^14 - 55*x^13 + 101*x^11 - 8*x^10 - 107*x^9 - 8*x^8 + 101*x^7 - 55*x^5 + 11*x^4 + 12*x^3 - 2*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
| \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), 3.1.231.1, \(\Q(\zeta_{7})\), 6.0.373527.1, 9.3.29595664791.1 |
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: | 12.0.364807736118799281.4 |
| Degree 18 sibling: | deg 18 |
| Minimal sibling: | 12.0.364807736118799281.4 |
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 | ${\href{/padicField/5.6.0.1}{6} }^{3}$ | R | R | ${\href{/padicField/13.6.0.1}{6} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.6.0.1}{6} }^{3}$ | ${\href{/padicField/23.6.0.1}{6} }^{2}{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{6}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}$ | ${\href{/padicField/43.2.0.1}{2} }^{6}{,}\,{\href{/padicField/43.1.0.1}{1} }^{6}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{2}{,}\,{\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.6.0.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(7\)
| 7.6.5.5 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $[\ ]_{6}$ |
| 7.12.10.1 | $x^{12} + 36 x^{11} + 558 x^{10} + 4860 x^{9} + 26055 x^{8} + 88776 x^{7} + 193010 x^{6} + 266580 x^{5} + 237645 x^{4} + 153900 x^{3} + 137808 x^{2} + 210600 x + 184108$ | $6$ | $2$ | $10$ | $C_6\times C_2$ | $[\ ]_{6}^{2}$ | |
|
\(11\)
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 11.3.0.1 | $x^{3} + 2 x + 9$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 11.6.3.1 | $x^{6} + 242 x^{2} - 11979$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 11.6.3.1 | $x^{6} + 242 x^{2} - 11979$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.33.2t1.a.a | $1$ | $ 3 \cdot 11 $ | \(\Q(\sqrt{33}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.231.2t1.a.a | $1$ | $ 3 \cdot 7 \cdot 11 $ | \(\Q(\sqrt{-231}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.7.2t1.a.a | $1$ | $ 7 $ | \(\Q(\sqrt{-7}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| 1.231.6t1.a.a | $1$ | $ 3 \cdot 7 \cdot 11 $ | 6.0.603993159.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.7.6t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})\) | $C_6$ (as 6T1) | $0$ | $-1$ |
| 1.231.6t1.a.b | $1$ | $ 3 \cdot 7 \cdot 11 $ | 6.0.603993159.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.231.6t1.b.a | $1$ | $ 3 \cdot 7 \cdot 11 $ | 6.6.86284737.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.231.6t1.b.b | $1$ | $ 3 \cdot 7 \cdot 11 $ | 6.6.86284737.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| * | 1.7.6t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})\) | $C_6$ (as 6T1) | $0$ | $-1$ |
| * | 2.231.3t2.a.a | $2$ | $ 3 \cdot 7 \cdot 11 $ | 3.1.231.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.231.6t3.d.a | $2$ | $ 3 \cdot 7 \cdot 11 $ | 6.2.1760913.1 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.1617.6t5.a.a | $2$ | $ 3 \cdot 7^{2} \cdot 11 $ | 6.0.603993159.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.1617.12t18.a.a | $2$ | $ 3 \cdot 7^{2} \cdot 11 $ | 18.0.6131323620948659515767.1 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
| * | 2.1617.12t18.a.b | $2$ | $ 3 \cdot 7^{2} \cdot 11 $ | 18.0.6131323620948659515767.1 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
| * | 2.1617.6t5.a.b | $2$ | $ 3 \cdot 7^{2} \cdot 11 $ | 6.0.603993159.2 | $S_3\times C_3$ (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 *.