Normalized defining polynomial
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 11*x^16 + 50*x^14 - 44*x^12 - 395*x^10 + 549*x^8 + 1449*x^6 + 581*x^4 + 35*x^2 + 1)
gp: K = bnfinit(y^18 - 11*y^16 + 50*y^14 - 44*y^12 - 395*y^10 + 549*y^8 + 1449*y^6 + 581*y^4 + 35*y^2 + 1, 1)
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 11*x^16 + 50*x^14 - 44*x^12 - 395*x^10 + 549*x^8 + 1449*x^6 + 581*x^4 + 35*x^2 + 1);
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 11*x^16 + 50*x^14 - 44*x^12 - 395*x^10 + 549*x^8 + 1449*x^6 + 581*x^4 + 35*x^2 + 1)
\( x^{18} - 11x^{16} + 50x^{14} - 44x^{12} - 395x^{10} + 549x^{8} + 1449x^{6} + 581x^{4} + 35x^{2} + 1 \)
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: |
\(-37133262473195501387776000000\)
\(\medspace = -\,2^{30}\cdot 5^{6}\cdot 19^{12}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(38.66\) | 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: | $2^{2}5^{1/2}19^{2/3}\approx 63.686501757102$ | ||
| Ramified primes: |
\(2\), \(5\), \(19\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\card{ \Aut(K/\Q) }$: | $6$ | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a+\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{14}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{828141712}a^{16}+\frac{5138849}{207035428}a^{14}+\frac{2115063}{414070856}a^{12}-\frac{9836385}{414070856}a^{10}-\frac{196198185}{828141712}a^{8}-\frac{87938569}{414070856}a^{6}+\frac{403688355}{828141712}a^{4}+\frac{70114745}{414070856}a^{2}-\frac{108016327}{828141712}$, $\frac{1}{828141712}a^{17}+\frac{5138849}{207035428}a^{15}+\frac{2115063}{414070856}a^{13}-\frac{9836385}{414070856}a^{11}-\frac{196198185}{828141712}a^{9}-\frac{87938569}{414070856}a^{7}+\frac{403688355}{828141712}a^{5}+\frac{70114745}{414070856}a^{3}-\frac{108016327}{828141712}a$
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 $1$ (assuming GRH)
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: |
\( \frac{205675}{2540312} a^{17} - \frac{278255}{317539} a^{15} + \frac{4934709}{1270156} a^{13} - \frac{3534245}{1270156} a^{11} - \frac{83644899}{2540312} a^{9} + \frac{49860575}{1270156} a^{7} + \frac{322287861}{2540312} a^{5} + \frac{81320401}{1270156} a^{3} + \frac{16407311}{2540312} a \)
(order $4$)
| 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: |
$\frac{125536809}{828141712}a^{17}-\frac{338497465}{207035428}a^{15}+\frac{2983754803}{414070856}a^{13}-\frac{2012924793}{414070856}a^{11}-\frac{51508001009}{828141712}a^{9}+\frac{29622903355}{414070856}a^{7}+\frac{201347509579}{828141712}a^{5}+\frac{51285728205}{414070856}a^{3}+\frac{6303467921}{828141712}a$, $\frac{58486759}{828141712}a^{17}-\frac{157075205}{207035428}a^{15}+\frac{1375039669}{414070856}a^{13}-\frac{860760923}{414070856}a^{11}-\frac{24239763935}{828141712}a^{9}+\frac{13368355905}{414070856}a^{7}+\frac{96281666893}{828141712}a^{5}+\frac{24775277479}{414070856}a^{3}+\frac{126542823}{828141712}a$, $\frac{52065231}{828141712}a^{16}-\frac{146465271}{207035428}a^{14}+\frac{1376911061}{414070856}a^{12}-\frac{1505934875}{414070856}a^{10}-\frac{19697796471}{828141712}a^{8}+\frac{16768323929}{414070856}a^{6}+\frac{65956497093}{828141712}a^{4}+\frac{7257980471}{414070856}a^{2}+\frac{442614647}{828141712}$, $\frac{4418851}{103517714}a^{16}-\frac{99093467}{207035428}a^{14}+\frac{464570777}{207035428}a^{12}-\frac{506750407}{207035428}a^{10}-\frac{3285466343}{207035428}a^{8}+\frac{1346680970}{51758857}a^{6}+\frac{5605321863}{103517714}a^{4}+\frac{4017966805}{207035428}a^{2}+\frac{181478941}{207035428}$, $\frac{114570063}{51758857}a^{17}+\frac{66245931}{207035428}a^{16}-\frac{2534982355}{103517714}a^{15}-\frac{363108821}{103517714}a^{14}+\frac{11615737425}{103517714}a^{13}+\frac{1642411657}{103517714}a^{12}-\frac{21604678765}{207035428}a^{11}-\frac{2788518161}{207035428}a^{10}-\frac{89886554385}{103517714}a^{9}-\frac{13143052523}{103517714}a^{8}+\frac{263065460913}{207035428}a^{7}+\frac{35396261229}{207035428}a^{6}+\frac{648246996511}{207035428}a^{5}+\frac{97468219369}{207035428}a^{4}+\frac{223922658829}{207035428}a^{3}+\frac{20934236473}{103517714}a^{2}-\frac{165269167}{51758857}a+\frac{779205938}{51758857}$, $\frac{1215121305}{828141712}a^{17}+\frac{2288512}{51758857}a^{16}-\frac{3354315955}{207035428}a^{15}-\frac{25567983}{51758857}a^{14}+\frac{30659408339}{414070856}a^{13}+\frac{474521495}{207035428}a^{12}-\frac{28021489883}{414070856}a^{11}-\frac{118910501}{51758857}a^{10}-\frac{477567983213}{828141712}a^{9}-\frac{3577197659}{207035428}a^{8}+\frac{343454148013}{414070856}a^{7}+\frac{5696234853}{207035428}a^{6}+\frac{1732103665459}{828141712}a^{5}+\frac{12555501311}{207035428}a^{4}+\frac{317232133193}{414070856}a^{3}+\frac{673788054}{51758857}a^{2}+\frac{14228869413}{828141712}a-\frac{431076541}{103517714}$, $\frac{948227595}{828141712}a^{16}-\frac{2622173803}{207035428}a^{14}+\frac{24024313729}{414070856}a^{12}-\frac{22309691383}{414070856}a^{10}-\frac{372003880187}{828141712}a^{8}+\frac{271637460317}{414070856}a^{6}+\frac{1342394978273}{828141712}a^{4}+\frac{234731197535}{414070856}a^{2}+\frac{2700268107}{828141712}$, $\frac{14947726863}{828141712}a^{17}+\frac{703119575}{103517714}a^{16}-\frac{41062238535}{207035428}a^{15}-\frac{3886067116}{51758857}a^{14}+\frac{372626889591}{414070856}a^{13}+\frac{35537700389}{103517714}a^{12}-\frac{323639647659}{414070856}a^{11}-\frac{16287848135}{51758857}a^{10}-\frac{5917918472471}{828141712}a^{9}-\frac{553673454217}{207035428}a^{8}+\frac{4066723977987}{414070856}a^{7}+\frac{199750493011}{51758857}a^{6}+\frac{21811601030301}{828141712}a^{5}+\frac{2013877939331}{207035428}a^{4}+\frac{4486007640465}{414070856}a^{3}+\frac{738791739247}{207035428}a^{2}+\frac{581577214611}{828141712}a+\frac{18123666849}{207035428}$
| 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: | \( 7210682.545838993 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 7210682.545838993 \cdot 1}{4\cdot\sqrt{37133262473195501387776000000}}\cr\approx \mathstrut & 0.142775485668223 \end{aligned}\] (assuming GRH)
# self-contained SageMath code snippet to compute the analytic class number formula
x = polygen(QQ); K.<a> = NumberField(x^18 - 11*x^16 + 50*x^14 - 44*x^12 - 395*x^10 + 549*x^8 + 1449*x^6 + 581*x^4 + 35*x^2 + 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 - 11*x^16 + 50*x^14 - 44*x^12 - 395*x^10 + 549*x^8 + 1449*x^6 + 581*x^4 + 35*x^2 + 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 - 11*x^16 + 50*x^14 - 44*x^12 - 395*x^10 + 549*x^8 + 1449*x^6 + 581*x^4 + 35*x^2 + 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 - 11*x^16 + 50*x^14 - 44*x^12 - 395*x^10 + 549*x^8 + 1449*x^6 + 581*x^4 + 35*x^2 + 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
| \(\Q(\sqrt{-1}) \), 3.3.361.1, 3.1.14440.1, 6.0.3336217600.4, 6.0.8340544.1, 9.3.3010936384000.1 |
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: | 12.0.34162868224000000.2 |
| Degree 18 sibling: | deg 18 |
| Minimal sibling: | 12.0.34162868224000000.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }^{3}$ | R | ${\href{/padicField/7.6.0.1}{6} }^{3}$ | ${\href{/padicField/11.6.0.1}{6} }^{3}$ | ${\href{/padicField/13.3.0.1}{3} }^{6}$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/23.6.0.1}{6} }^{3}$ | ${\href{/padicField/29.6.0.1}{6} }^{2}{,}\,{\href{/padicField/29.3.0.1}{3} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{9}$ | ${\href{/padicField/37.3.0.1}{3} }^{6}$ | ${\href{/padicField/41.3.0.1}{3} }^{6}$ | ${\href{/padicField/43.6.0.1}{6} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}$ | ${\href{/padicField/53.3.0.1}{3} }^{6}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.6.6.3 | $x^{6} + 6 x^{5} + 20 x^{4} + 42 x^{3} + 55 x^{2} + 36 x + 9$ | $2$ | $3$ | $6$ | $C_6$ | $[2]^{3}$ |
| 2.12.24.315 | $x^{12} - 8 x^{11} + 78 x^{10} + 76 x^{9} - 30 x^{8} + 720 x^{7} + 1200 x^{6} + 1248 x^{5} + 2676 x^{4} + 3712 x^{3} + 3448 x^{2} + 2160 x + 4072$ | $4$ | $3$ | $24$ | $C_6\times C_2$ | $[2, 3]^{3}$ | |
|
\(5\)
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ |
| 5.3.0.1 | $x^{3} + 3 x + 3$ | $1$ | $3$ | $0$ | $C_3$ | $[\ ]^{3}$ | |
| 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
| 5.6.3.2 | $x^{6} + 75 x^{2} - 375$ | $2$ | $3$ | $3$ | $C_6$ | $[\ ]_{2}^{3}$ | |
|
\(19\)
| 19.18.12.1 | $x^{18} + 1938 x^{16} + 165 x^{15} + 1564986 x^{14} + 217074 x^{13} + 673956770 x^{12} + 124220751 x^{11} + 163260541175 x^{10} + 40698676942 x^{9} + 21101810097561 x^{8} + 7872565858164 x^{7} + 1139107203476720 x^{6} + 760256762812749 x^{5} + 441034593923007 x^{4} + 19443033036221259 x^{3} + 22753074450170540 x^{2} + 9283251966890258 x + 2743830934025437$ | $3$ | $6$ | $12$ | $C_6 \times C_3$ | $[\ ]_{3}^{6}$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.40.2t1.a.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{10}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.40.2t1.b.a | $1$ | $ 2^{3} \cdot 5 $ | \(\Q(\sqrt{-10}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 1.4.2t1.a.a | $1$ | $ 2^{2}$ | \(\Q(\sqrt{-1}) \) | $C_2$ (as 2T1) | $1$ | $-1$ |
| * | 1.76.6t1.a.a | $1$ | $ 2^{2} \cdot 19 $ | 6.0.8340544.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| 1.760.6t1.a.a | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.6.8340544000.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.760.6t1.a.b | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.6.8340544000.1 | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.760.6t1.b.a | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.8340544000.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.76.6t1.a.b | $1$ | $ 2^{2} \cdot 19 $ | 6.0.8340544.1 | $C_6$ (as 6T1) | $0$ | $-1$ |
| 1.760.6t1.b.b | $1$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.8340544000.3 | $C_6$ (as 6T1) | $0$ | $-1$ | |
| * | 1.19.3t1.a.a | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 1.19.3t1.a.b | $1$ | $ 19 $ | 3.3.361.1 | $C_3$ (as 3T1) | $0$ | $1$ |
| * | 2.14440.3t2.a.a | $2$ | $ 2^{3} \cdot 5 \cdot 19^{2}$ | 3.1.14440.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| * | 2.57760.6t3.c.a | $2$ | $ 2^{5} \cdot 5 \cdot 19^{2}$ | 6.2.33362176000.17 | $D_{6}$ (as 6T3) | $1$ | $0$ |
| * | 2.760.6t5.a.a | $2$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.23104000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.760.6t5.a.b | $2$ | $ 2^{3} \cdot 5 \cdot 19 $ | 6.0.23104000.1 | $S_3\times C_3$ (as 6T5) | $0$ | $0$ |
| * | 2.3040.12t18.a.a | $2$ | $ 2^{5} \cdot 5 \cdot 19 $ | 18.0.37133262473195501387776000000.1 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
| * | 2.3040.12t18.a.b | $2$ | $ 2^{5} \cdot 5 \cdot 19 $ | 18.0.37133262473195501387776000000.1 | $S_3 \times C_6$ (as 18T6) | $0$ | $0$ |
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 *.