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Properties

Label 18.2.178...688.2
Degree $18$
Signature $[2, 8]$
Discriminant $1.790\times 10^{29}$
Root discriminant \(42.18\)
Ramified primes $2,3,11$
Class number $18$ (GRH)
Class group [3, 6] (GRH)
Galois group $C_6:S_3$ (as 18T12)

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Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^18 - 37*x^15 + 245*x^12 + 3983*x^9 - 490*x^6 - 148*x^3 - 8)
 
Copy content gp:K = bnfinit(y^18 - 37*y^15 + 245*y^12 + 3983*y^9 - 490*y^6 - 148*y^3 - 8, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 37*x^15 + 245*x^12 + 3983*x^9 - 490*x^6 - 148*x^3 - 8);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 37*x^15 + 245*x^12 + 3983*x^9 - 490*x^6 - 148*x^3 - 8)
 

\( x^{18} - 37x^{15} + 245x^{12} + 3983x^{9} - 490x^{6} - 148x^{3} - 8 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $18$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $[2, 8]$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(178976732236484163896033906688\) \(\medspace = 2^{12}\cdot 3^{21}\cdot 11^{15}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(42.18\)
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:(1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2^{2/3}3^{7/6}11^{5/6}\approx 42.18473888411525$
Ramified primes:   \(2\), \(3\), \(11\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q(\sqrt{33}) \)
$\Aut(K/\Q)$:   $C_2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{9}a^{6}+\frac{2}{9}$, $\frac{1}{9}a^{7}+\frac{2}{9}a$, $\frac{1}{9}a^{8}+\frac{2}{9}a^{2}$, $\frac{1}{9}a^{9}+\frac{2}{9}a^{3}$, $\frac{1}{9}a^{10}+\frac{2}{9}a^{4}$, $\frac{1}{27}a^{11}+\frac{1}{27}a^{10}+\frac{1}{27}a^{9}+\frac{1}{27}a^{8}+\frac{1}{27}a^{7}+\frac{1}{27}a^{6}+\frac{2}{27}a^{5}+\frac{2}{27}a^{4}+\frac{2}{27}a^{3}+\frac{2}{27}a^{2}+\frac{2}{27}a+\frac{2}{27}$, $\frac{1}{810}a^{12}+\frac{1}{90}a^{9}+\frac{31}{810}a^{6}-\frac{7}{90}a^{3}-\frac{133}{405}$, $\frac{1}{810}a^{13}+\frac{1}{90}a^{10}+\frac{31}{810}a^{7}-\frac{7}{90}a^{4}-\frac{133}{405}a$, $\frac{1}{4860}a^{14}+\frac{1}{2430}a^{13}-\frac{1}{2430}a^{12}-\frac{1}{60}a^{11}-\frac{1}{30}a^{10}+\frac{1}{30}a^{9}-\frac{239}{4860}a^{8}+\frac{31}{2430}a^{7}-\frac{31}{2430}a^{6}+\frac{9}{20}a^{5}-\frac{1}{10}a^{4}+\frac{1}{10}a^{3}+\frac{407}{2430}a^{2}-\frac{538}{1215}a+\frac{538}{1215}$, $\frac{1}{14580}a^{15}-\frac{1}{2916}a^{12}+\frac{17}{2916}a^{9}+\frac{223}{14580}a^{6}-\frac{2347}{7290}a^{3}-\frac{734}{3645}$, $\frac{1}{29160}a^{16}+\frac{13}{29160}a^{13}+\frac{247}{29160}a^{10}+\frac{781}{29160}a^{7}+\frac{1094}{3645}a^{4}-\frac{1931}{7290}a$, $\frac{1}{87480}a^{17}-\frac{1}{87480}a^{16}-\frac{1}{43740}a^{15}-\frac{1}{17496}a^{14}-\frac{49}{87480}a^{13}+\frac{1}{8748}a^{12}-\frac{307}{17496}a^{11}-\frac{3811}{87480}a^{10}+\frac{307}{8748}a^{9}-\frac{1397}{87480}a^{8}+\frac{4583}{87480}a^{7}+\frac{1397}{43740}a^{6}-\frac{3967}{43740}a^{5}-\frac{10531}{21870}a^{4}+\frac{3967}{21870}a^{3}-\frac{4417}{10935}a^{2}-\frac{1403}{4374}a-\frac{2101}{10935}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

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

Class group and class number

Ideal class group:  $C_{3}\times C_{6}$, which has order $18$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{6}\times C_{6}$, which has order $36$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $9$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{2717}{29160}a^{17}-\frac{100903}{29160}a^{14}+\frac{679523}{29160}a^{11}+\frac{10729433}{29160}a^{8}-\frac{351923}{3645}a^{5}-\frac{36751}{7290}a^{2}+1$, $\frac{1249}{5832}a^{17}+\frac{91}{1215}a^{16}-\frac{97}{4860}a^{15}-\frac{231721}{29160}a^{14}-\frac{1123}{405}a^{13}+\frac{1201}{1620}a^{12}+\frac{1554341}{29160}a^{11}+\frac{22369}{1215}a^{10}-\frac{24283}{4860}a^{9}+\frac{24711479}{29160}a^{8}+\frac{120653}{405}a^{7}-\frac{127631}{1620}a^{6}-\frac{2830691}{14580}a^{5}-\frac{52502}{1215}a^{4}+\frac{25661}{1215}a^{3}-\frac{54098}{3645}a^{2}-\frac{4114}{405}a+\frac{337}{405}$, $\frac{9}{40}a^{17}-\frac{9023}{1080}a^{14}+\frac{2249}{40}a^{11}+\frac{959881}{1080}a^{8}-\frac{4547}{20}a^{5}-\frac{110}{27}a^{2}$, $\frac{203}{2430}a^{17}-\frac{3013}{972}a^{14}+\frac{20213}{972}a^{11}+\frac{1606423}{4860}a^{8}-\frac{369329}{4860}a^{5}-\frac{11473}{2430}a^{2}$, $\frac{937}{43740}a^{17}-\frac{8}{2187}a^{16}-\frac{25}{8748}a^{15}-\frac{17341}{21870}a^{14}+\frac{1469}{10935}a^{13}+\frac{935}{8748}a^{12}+\frac{115031}{21870}a^{11}-\frac{9394}{10935}a^{10}-\frac{6499}{8748}a^{9}+\frac{1864187}{21870}a^{8}-\frac{161956}{10935}a^{7}-\frac{97105}{8748}a^{6}-\frac{508577}{43740}a^{5}-\frac{25432}{10935}a^{4}+\frac{12935}{2187}a^{3}-\frac{7409}{4374}a^{2}-\frac{7084}{10935}a+\frac{935}{2187}$, $\frac{7739}{17496}a^{17}-\frac{7631}{43740}a^{16}+\frac{1783}{21870}a^{15}-\frac{286879}{17496}a^{14}+\frac{282829}{43740}a^{13}-\frac{33037}{10935}a^{12}+\frac{1915907}{17496}a^{11}-\frac{1887449}{43740}a^{10}+\frac{220322}{10935}a^{9}+\frac{30692369}{17496}a^{8}-\frac{30275219}{43740}a^{7}+\frac{707641}{2187}a^{6}-\frac{2961029}{8748}a^{5}+\frac{1412623}{10935}a^{4}-\frac{256105}{4374}a^{3}-\frac{100712}{2187}a^{2}+\frac{199723}{10935}a-\frac{94807}{10935}$, $\frac{3683}{17496}a^{17}-\frac{509}{4374}a^{16}+\frac{2213}{43740}a^{15}-\frac{683411}{87480}a^{14}+\frac{47218}{10935}a^{13}-\frac{82093}{43740}a^{12}+\frac{4587871}{87480}a^{11}-\frac{316778}{10935}a^{10}+\frac{550013}{43740}a^{9}+\frac{72838789}{87480}a^{8}-\frac{5035142}{10935}a^{7}+\frac{8763311}{43740}a^{6}-\frac{8593801}{43740}a^{5}+\frac{2328047}{21870}a^{4}-\frac{971219}{21870}a^{3}-\frac{147508}{10935}a^{2}+\frac{100177}{10935}a-\frac{8270}{2187}$, $\frac{1831}{8748}a^{17}-\frac{476}{10935}a^{16}+\frac{38}{10935}a^{15}-\frac{67925}{8748}a^{14}+\frac{17563}{10935}a^{13}-\frac{1441}{10935}a^{12}+\frac{455173}{8748}a^{11}-\frac{114818}{10935}a^{10}+\frac{10601}{10935}a^{9}+\frac{7249555}{8748}a^{8}-\frac{381505}{2187}a^{7}+\frac{143033}{10935}a^{6}-\frac{402035}{2187}a^{5}+\frac{7136}{2187}a^{4}-\frac{160714}{10935}a^{3}-\frac{53750}{2187}a^{2}+\frac{61018}{10935}a-\frac{18967}{10935}$, $\frac{1825}{17496}a^{17}-\frac{964}{10935}a^{16}-\frac{16}{2187}a^{15}-\frac{338737}{87480}a^{14}+\frac{7147}{2187}a^{13}+\frac{2938}{10935}a^{12}+\frac{2276717}{87480}a^{11}-\frac{47735}{2187}a^{10}-\frac{18788}{10935}a^{9}+\frac{36074423}{87480}a^{8}-\frac{3822577}{10935}a^{7}-\frac{323912}{10935}a^{6}-\frac{4457567}{43740}a^{5}+\frac{734216}{10935}a^{4}-\frac{50864}{10935}a^{3}-\frac{127856}{10935}a^{2}+\frac{47684}{10935}a+\frac{7702}{10935}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 28469096.209551767 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{8}\cdot 28469096.209551767 \cdot 18}{2\cdot\sqrt{178976732236484163896033906688}}\cr\approx \mathstrut & 5.88459855102584 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^18 - 37*x^15 + 245*x^12 + 3983*x^9 - 490*x^6 - 148*x^3 - 8) 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))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^18 - 37*x^15 + 245*x^12 + 3983*x^9 - 490*x^6 - 148*x^3 - 8, 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)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 37*x^15 + 245*x^12 + 3983*x^9 - 490*x^6 - 148*x^3 - 8); 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)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = PolynomialRing(QQ); K, a = NumberField(x^18 - 37*x^15 + 245*x^12 + 3983*x^9 - 490*x^6 - 148*x^3 - 8); 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

$C_6:S_3$ (as 18T12):

Copy content comment:Galois group
 
Copy content sage:K.galois_group(type='pari')
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:G = GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 36
The 12 conjugacy class representatives for $C_6:S_3$
Character table for $C_6:S_3$

Intermediate fields

\(\Q(\sqrt{33}) \), 3.1.3267.1, 3.1.13068.1, 3.1.1452.1, 3.1.108.1, 6.2.352218537.1, 6.2.69574032.1, 6.2.5635496592.1, 6.2.46574352.1, 9.1.6694969951296.1

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

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

Sibling fields

Galois closure: data not computed
Degree 18 sibling: 18.0.59658910745494721298677968896.1
Minimal sibling: 18.0.59658910745494721298677968896.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.2.0.1}{2} }^{9}$ ${\href{/padicField/7.6.0.1}{6} }^{3}$ R ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{8}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{9}$ ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.2.0.1}{2} }^{9}$ ${\href{/padicField/53.2.0.1}{2} }^{9}$ ${\href{/padicField/59.2.0.1}{2} }^{9}$

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

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_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
\(2\) Copy content Toggle raw display 2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
2.1.3.2a1.1$x^{3} + 2$$3$$1$$2$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
2.2.3.4a1.2$x^{6} + 3 x^{5} + 6 x^{4} + 7 x^{3} + 6 x^{2} + 3 x + 3$$3$$2$$4$$S_3$$$[\ ]_{3}^{2}$$
\(3\) Copy content Toggle raw display 3.1.6.7a1.1$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$$[\frac{3}{2}]_{2}$$
3.1.6.7a1.1$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$$[\frac{3}{2}]_{2}$$
3.1.6.7a1.1$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$$[\frac{3}{2}]_{2}$$
\(11\) Copy content Toggle raw display 11.1.6.5a1.2$x^{6} + 22$$6$$1$$5$$D_{6}$$$[\ ]_{6}^{2}$$
11.2.6.10a1.2$x^{12} + 42 x^{11} + 747 x^{10} + 7280 x^{9} + 41955 x^{8} + 143682 x^{7} + 279509 x^{6} + 287364 x^{5} + 167820 x^{4} + 58240 x^{3} + 11952 x^{2} + 1344 x + 75$$6$$2$$10$$D_6$$$[\ ]_{6}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)