Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 78*x^15 + 84*x^14 - 111*x^12 + 90*x^11 + 72*x^10 - 156*x^9 + 45*x^8 + 81*x^7 - 57*x^6 - 27*x^5 + 36*x^4 - 9*x^2 + 3)
gp: K = bnfinit(y^18 - 9*y^17 + 36*y^16 - 78*y^15 + 84*y^14 - 111*y^12 + 90*y^11 + 72*y^10 - 156*y^9 + 45*y^8 + 81*y^7 - 57*y^6 - 27*y^5 + 36*y^4 - 9*y^2 + 3, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 9*x^17 + 36*x^16 - 78*x^15 + 84*x^14 - 111*x^12 + 90*x^11 + 72*x^10 - 156*x^9 + 45*x^8 + 81*x^7 - 57*x^6 - 27*x^5 + 36*x^4 - 9*x^2 + 3);
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 9*x^17 + 36*x^16 - 78*x^15 + 84*x^14 - 111*x^12 + 90*x^11 + 72*x^10 - 156*x^9 + 45*x^8 + 81*x^7 - 57*x^6 - 27*x^5 + 36*x^4 - 9*x^2 + 3)
\( x^{18} - 9 x^{17} + 36 x^{16} - 78 x^{15} + 84 x^{14} - 111 x^{12} + 90 x^{11} + 72 x^{10} - 156 x^{9} + \cdots + 3 \)
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: |
\(-9651146816936671875\)
\(\medspace = -\,3^{31}\cdot 5^{6}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(11.34\) | 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: | $3^{31/18}5^{1/2}\approx 14.831798946842026$ | ||
| Ramified primes: |
\(3\), \(5\)
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{78986}a^{17}-\frac{5056}{39493}a^{16}-\frac{6819}{78986}a^{15}+\frac{16487}{78986}a^{14}+\frac{13397}{78986}a^{13}-\frac{3690}{39493}a^{12}-\frac{2755}{78986}a^{11}-\frac{4355}{39493}a^{10}+\frac{3399}{39493}a^{9}-\frac{2023}{78986}a^{8}-\frac{9480}{39493}a^{7}-\frac{13791}{39493}a^{6}-\frac{1719}{78986}a^{5}+\frac{29603}{78986}a^{4}-\frac{38077}{78986}a^{3}+\frac{30111}{78986}a^{2}+\frac{3137}{78986}a-\frac{19725}{78986}$
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: |
\( -\frac{40878}{39493} a^{17} + \frac{301049}{39493} a^{16} - \frac{1924181}{78986} a^{15} + \frac{1493193}{39493} a^{14} - \frac{625023}{39493} a^{13} - \frac{1508121}{39493} a^{12} + \frac{4037487}{78986} a^{11} + \frac{1418225}{78986} a^{10} - \frac{3056857}{39493} a^{9} + \frac{2562749}{78986} a^{8} + \frac{3350415}{78986} a^{7} - \frac{1489395}{39493} a^{6} - \frac{1122827}{78986} a^{5} + \frac{2080287}{78986} a^{4} - \frac{26003}{39493} a^{3} - \frac{236085}{39493} a^{2} + \frac{78471}{39493} a + \frac{177403}{78986} \)
(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: |
$\frac{57180}{39493}a^{17}-\frac{882633}{78986}a^{16}+\frac{1504703}{39493}a^{15}-\frac{5274055}{78986}a^{14}+\frac{3662327}{78986}a^{13}+\frac{1534532}{39493}a^{12}-\frac{3745158}{39493}a^{11}+\frac{1035241}{39493}a^{10}+\frac{3376439}{39493}a^{9}-\frac{3199076}{39493}a^{8}-\frac{523866}{39493}a^{7}+\frac{2028338}{39493}a^{6}-\frac{389273}{39493}a^{5}-\frac{839286}{39493}a^{4}+\frac{525869}{78986}a^{3}+\frac{296755}{78986}a^{2}-\frac{122025}{39493}a-\frac{187291}{78986}$, $\frac{36879}{78986}a^{17}-\frac{132250}{39493}a^{16}+\frac{421438}{39493}a^{15}-\frac{696205}{39493}a^{14}+\frac{498929}{39493}a^{13}+\frac{453159}{78986}a^{12}-\frac{1526383}{78986}a^{11}+\frac{1086283}{78986}a^{10}+\frac{278329}{78986}a^{9}-\frac{1465181}{78986}a^{8}+\frac{1656431}{78986}a^{7}-\frac{54363}{78986}a^{6}-\frac{1548963}{78986}a^{5}+\frac{406949}{39493}a^{4}+\frac{281910}{39493}a^{3}-\frac{335993}{39493}a^{2}-\frac{71773}{39493}a+\frac{180757}{78986}$, $\frac{166339}{78986}a^{17}-\frac{1316367}{78986}a^{16}+\frac{4554507}{78986}a^{15}-\frac{3989953}{39493}a^{14}+\frac{2553081}{39493}a^{13}+\frac{5903049}{78986}a^{12}-\frac{6055762}{39493}a^{11}+\frac{527163}{39493}a^{10}+\frac{6994734}{39493}a^{9}-\frac{10133645}{78986}a^{8}-\frac{5208015}{78986}a^{7}+\frac{7956591}{78986}a^{6}+\frac{450459}{39493}a^{5}-\frac{4277039}{78986}a^{4}+\frac{197351}{39493}a^{3}+\frac{559347}{39493}a^{2}-\frac{47276}{39493}a-\frac{432251}{78986}$, $\frac{87879}{78986}a^{17}-\frac{335918}{39493}a^{16}+\frac{2231489}{78986}a^{15}-\frac{1825987}{39493}a^{14}+\frac{902909}{39493}a^{13}+\frac{1741205}{39493}a^{12}-\frac{5227631}{78986}a^{11}-\frac{815735}{39493}a^{10}+\frac{8521319}{78986}a^{9}-\frac{2182727}{39493}a^{8}-\frac{4359893}{78986}a^{7}+\frac{2233503}{39493}a^{6}+\frac{1615937}{78986}a^{5}-\frac{3239285}{78986}a^{4}+\frac{95843}{39493}a^{3}+\frac{382475}{39493}a^{2}+\frac{15283}{78986}a-\frac{381449}{78986}$, $\frac{39235}{39493}a^{17}-\frac{587679}{78986}a^{16}+\frac{1899391}{78986}a^{15}-\frac{1489136}{39493}a^{14}+\frac{650846}{39493}a^{13}+\frac{2741769}{78986}a^{12}-\frac{1619297}{39493}a^{11}-\frac{2574887}{78986}a^{10}+\frac{6483961}{78986}a^{9}-\frac{741842}{39493}a^{8}-\frac{4394627}{78986}a^{7}+\frac{2265933}{78986}a^{6}+\frac{1233362}{39493}a^{5}-\frac{2281951}{78986}a^{4}-\frac{246849}{39493}a^{3}+\frac{366920}{39493}a^{2}+\frac{79507}{78986}a-\frac{287545}{78986}$, $\frac{39749}{39493}a^{17}-\frac{298078}{39493}a^{16}+\frac{979353}{39493}a^{15}-\frac{1624292}{39493}a^{14}+\frac{2001625}{78986}a^{13}+\frac{954216}{39493}a^{12}-\frac{3503689}{78986}a^{11}-\frac{255110}{39493}a^{10}+\frac{2293190}{39493}a^{9}-\frac{2497017}{78986}a^{8}-\frac{825474}{39493}a^{7}+\frac{877101}{39493}a^{6}+\frac{502127}{78986}a^{5}-\frac{557190}{39493}a^{4}+\frac{7059}{39493}a^{3}+\frac{133041}{78986}a^{2}+\frac{13212}{39493}a-\frac{107471}{78986}$, $\frac{7488}{39493}a^{17}-\frac{50068}{39493}a^{16}+\frac{284005}{78986}a^{15}-\frac{356361}{78986}a^{14}+\frac{4516}{39493}a^{13}+\frac{265718}{39493}a^{12}-\frac{290545}{39493}a^{11}+\frac{241377}{78986}a^{10}-\frac{282557}{78986}a^{9}+\frac{389613}{78986}a^{8}+\frac{123334}{39493}a^{7}-\frac{301570}{39493}a^{6}-\frac{76140}{39493}a^{5}+\frac{657491}{78986}a^{4}-\frac{238683}{78986}a^{3}-\frac{73355}{39493}a^{2}-\frac{8479}{39493}a+\frac{3020}{39493}$, $\frac{67518}{39493}a^{17}-\frac{1040375}{78986}a^{16}+\frac{1741844}{39493}a^{15}-\frac{5766335}{78986}a^{14}+\frac{1452215}{39493}a^{13}+\frac{5806153}{78986}a^{12}-\frac{4897192}{39493}a^{11}+\frac{126962}{39493}a^{10}+\frac{5686710}{39493}a^{9}-\frac{7824361}{78986}a^{8}-\frac{4256107}{78986}a^{7}+\frac{6617209}{78986}a^{6}+\frac{45978}{39493}a^{5}-\frac{1624589}{39493}a^{4}+\frac{657251}{78986}a^{3}+\frac{329788}{39493}a^{2}-\frac{191451}{78986}a-\frac{216673}{78986}$
| 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: | \( 868.676171318 \) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 868.676171318 \cdot 1}{18\cdot\sqrt{9651146816936671875}}\cr\approx \mathstrut & 0.237090830253 \end{aligned}\]
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 36*x^16 - 78*x^15 + 84*x^14 - 111*x^12 + 90*x^11 + 72*x^10 - 156*x^9 + 45*x^8 + 81*x^7 - 57*x^6 - 27*x^5 + 36*x^4 - 9*x^2 + 3)
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 - 9*x^17 + 36*x^16 - 78*x^15 + 84*x^14 - 111*x^12 + 90*x^11 + 72*x^10 - 156*x^9 + 45*x^8 + 81*x^7 - 57*x^6 - 27*x^5 + 36*x^4 - 9*x^2 + 3, 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 - 9*x^17 + 36*x^16 - 78*x^15 + 84*x^14 - 111*x^12 + 90*x^11 + 72*x^10 - 156*x^9 + 45*x^8 + 81*x^7 - 57*x^6 - 27*x^5 + 36*x^4 - 9*x^2 + 3);
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 - 9*x^17 + 36*x^16 - 78*x^15 + 84*x^14 - 111*x^12 + 90*x^11 + 72*x^10 - 156*x^9 + 45*x^8 + 81*x^7 - 57*x^6 - 27*x^5 + 36*x^4 - 9*x^2 + 3);
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
| \(\Q(\sqrt{-3}) \), 3.1.135.1, \(\Q(\zeta_{9})^+\), 6.0.54675.1, \(\Q(\zeta_{9})\), 9.3.1793613375.1 |
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: | 12.0.6053445140625.2 |
| Degree 18 sibling: | 18.6.402131117372361328125.1 |
| Minimal sibling: | 12.0.6053445140625.2 |
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 | R | ${\href{/padicField/7.6.0.1}{6} }^{2}{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }^{2}{,}\,{\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}$ | ${\href{/padicField/19.3.0.1}{3} }^{6}$ | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{6}$ | ${\href{/padicField/37.2.0.1}{2} }^{6}{,}\,{\href{/padicField/37.1.0.1}{1} }^{6}$ | ${\href{/padicField/41.6.0.1}{6} }^{3}$ | ${\href{/padicField/43.6.0.1}{6} }^{2}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }^{3}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 $18$ | $18$ | $1$ | $31$ | |||
|
\(5\)
| 5.6.0.1 | $x^{6} + x^{4} + 4 x^{3} + x^{2} + 2$ | $1$ | $6$ | $0$ | $C_6$ | $[\ ]^{6}$ |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
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) | $1$ | $-1$ |
| 1.15.2t1.a.a | $1$ | $ 3 \cdot 5 $ | \(\Q(\sqrt{-15}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.5.2t1.a.a | $1$ | $ 5 $ | \(\Q(\sqrt{5}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.45.6t1.a.a | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (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) | $0$ | $1$ |
| 1.45.6t1.b.a | $1$ | $ 3^{2} \cdot 5 $ | 6.0.2460375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.9.6t1.a.a | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ |
| 1.45.6t1.b.b | $1$ | $ 3^{2} \cdot 5 $ | 6.0.2460375.1 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| 1.45.6t1.a.b | $1$ | $ 3^{2} \cdot 5 $ | 6.6.820125.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| * | 1.9.6t1.a.b | $1$ | $ 3^{2}$ | \(\Q(\zeta_{9})\) | $C_6$ (as 6T1) | $0$ | $-1$ |
| * | 2.135.3t2.b.a | $2$ | $ 3^{3} \cdot 5 $ | 3.1.135.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.135.6t3.a.a | $2$ | $ 3^{3} \cdot 5 $ | 6.2.91125.1 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.405.12t18.b.a | $2$ | $ 3^{4} \cdot 5 $ | 18.0.9651146816936671875.1 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
| * | 2.405.12t18.b.b | $2$ | $ 3^{4} \cdot 5 $ | 18.0.9651146816936671875.1 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
| * | 2.405.6t5.a.a | $2$ | $ 3^{4} \cdot 5 $ | 6.0.2460375.2 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.405.6t5.a.b | $2$ | $ 3^{4} \cdot 5 $ | 6.0.2460375.2 | $S_3\times C_3$ (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 *.