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Properties

Label 18.0.596...896.1
Degree $18$
Signature $[0, 9]$
Discriminant $-5.966\times 10^{28}$
Root discriminant \(39.69\)
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 - 4*x^15 + 47*x^12 - 296*x^9 + 1215*x^6 - 1512*x^3 + 729)
 
Copy content gp:K = bnfinit(y^18 - 4*y^15 + 47*y^12 - 296*y^9 + 1215*y^6 - 1512*y^3 + 729, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^18 - 4*x^15 + 47*x^12 - 296*x^9 + 1215*x^6 - 1512*x^3 + 729);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^18 - 4*x^15 + 47*x^12 - 296*x^9 + 1215*x^6 - 1512*x^3 + 729)
 

\( x^{18} - 4x^{15} + 47x^{12} - 296x^{9} + 1215x^{6} - 1512x^{3} + 729 \) 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:  $[0, 9]$
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:   \(-59658910745494721298677968896\) \(\medspace = -\,2^{12}\cdot 3^{20}\cdot 11^{15}\) 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:  \(39.69\)
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:  \(\Q(\sqrt{-11}) \)
$\Aut(K/\Q)$:   $C_2$
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.
Maximal CM subfield:  \(\Q(\sqrt{-11}) \)

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{2}$, $\frac{1}{6}a^{9}-\frac{1}{2}a^{6}-\frac{1}{6}a^{3}-\frac{1}{2}$, $\frac{1}{18}a^{10}-\frac{1}{2}a^{7}-\frac{7}{18}a^{4}-\frac{1}{2}a$, $\frac{1}{18}a^{11}-\frac{1}{6}a^{8}-\frac{7}{18}a^{5}+\frac{1}{6}a^{2}$, $\frac{1}{18}a^{12}+\frac{1}{9}a^{6}-\frac{1}{2}$, $\frac{1}{54}a^{13}+\frac{1}{54}a^{12}+\frac{1}{54}a^{11}-\frac{1}{18}a^{9}-\frac{1}{6}a^{8}+\frac{10}{27}a^{7}+\frac{11}{54}a^{6}+\frac{11}{54}a^{5}+\frac{1}{3}a^{4}-\frac{5}{18}a^{3}-\frac{1}{2}a^{2}-\frac{1}{6}a$, $\frac{1}{54}a^{14}-\frac{1}{54}a^{11}+\frac{1}{18}a^{9}-\frac{7}{54}a^{8}+\frac{1}{3}a^{7}-\frac{1}{2}a^{6}+\frac{7}{54}a^{5}-\frac{7}{18}a^{3}-\frac{1}{3}a-\frac{1}{2}$, $\frac{1}{266220}a^{15}+\frac{7229}{266220}a^{12}+\frac{2377}{133110}a^{9}-\frac{22526}{66555}a^{6}-\frac{9823}{29580}a^{3}-\frac{4519}{9860}$, $\frac{1}{798660}a^{16}+\frac{7229}{798660}a^{13}-\frac{1}{54}a^{12}-\frac{1}{54}a^{11}+\frac{2377}{399330}a^{10}+\frac{1}{18}a^{9}-\frac{1}{6}a^{8}+\frac{44029}{199665}a^{7}-\frac{11}{54}a^{6}-\frac{11}{54}a^{5}-\frac{9823}{88740}a^{4}+\frac{5}{18}a^{3}-\frac{1}{6}a^{2}-\frac{4519}{29580}a$, $\frac{1}{2395980}a^{17}+\frac{22019}{2395980}a^{14}+\frac{2377}{1197990}a^{11}-\frac{1}{18}a^{9}-\frac{81686}{598995}a^{8}-\frac{1}{3}a^{7}-\frac{1}{2}a^{6}+\frac{108497}{266220}a^{5}+\frac{7}{18}a^{3}+\frac{20131}{88740}a^{2}+\frac{1}{3}a-\frac{1}{2}$ 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:  $2$, $3$

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:  $C_{3}\times C_{6}$, which has order $18$ (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:  $8$
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:   $\frac{4499}{1197990}a^{17}-\frac{14729}{1197990}a^{14}+\frac{201571}{1197990}a^{11}-\frac{1174591}{1197990}a^{8}+\frac{28831}{7395}a^{5}-\frac{65153}{22185}a^{2}-1$, $\frac{8329}{798660}a^{16}-\frac{29329}{798660}a^{13}+\frac{186493}{399330}a^{10}-\frac{569924}{199665}a^{7}+\frac{988513}{88740}a^{4}-\frac{284141}{29580}a$, $\frac{167}{79866}a^{16}-\frac{550}{39933}a^{13}+\frac{8569}{79866}a^{10}-\frac{66577}{79866}a^{7}+\frac{16154}{4437}a^{4}-\frac{19117}{2958}a$, $\frac{755}{159732}a^{17}+\frac{1997}{399330}a^{16}+\frac{7}{7830}a^{15}-\frac{2573}{159732}a^{14}-\frac{3061}{199665}a^{13}-\frac{1}{3915}a^{12}+\frac{16877}{79866}a^{11}+\frac{43649}{199665}a^{10}+\frac{109}{3915}a^{9}-\frac{50395}{39933}a^{8}-\frac{252704}{199665}a^{7}-\frac{569}{3915}a^{6}+\frac{263257}{53244}a^{5}+\frac{70643}{14790}a^{4}+\frac{487}{2610}a^{3}-\frac{21977}{5916}a^{2}-\frac{4943}{2465}a+\frac{61}{145}$, $\frac{3413}{598995}a^{17}-\frac{1997}{399330}a^{16}+\frac{601}{133110}a^{15}-\frac{11933}{598995}a^{14}+\frac{3061}{199665}a^{13}-\frac{583}{66555}a^{12}+\frac{304739}{1197990}a^{11}-\frac{43649}{199665}a^{10}+\frac{24869}{133110}a^{9}-\frac{1849109}{1197990}a^{8}+\frac{252704}{199665}a^{7}-\frac{127099}{133110}a^{6}+\frac{797327}{133110}a^{5}-\frac{70643}{14790}a^{4}+\frac{69098}{22185}a^{3}-\frac{199349}{44370}a^{2}+\frac{7408}{2465}a+\frac{511}{4930}$, $\frac{20333}{2395980}a^{17}+\frac{350}{39933}a^{16}+\frac{1033}{133110}a^{15}-\frac{69713}{2395980}a^{14}-\frac{2317}{79866}a^{13}-\frac{643}{22185}a^{12}+\frac{456311}{1197990}a^{11}+\frac{31109}{79866}a^{10}+\frac{22486}{66555}a^{9}-\frac{1372513}{598995}a^{8}-\frac{187199}{79866}a^{7}-\frac{48317}{22185}a^{6}+\frac{2408801}{266220}a^{5}+\frac{80623}{8874}a^{4}+\frac{375953}{44370}a^{3}-\frac{625597}{88740}a^{2}-\frac{3815}{493}a-\frac{18196}{2465}$, $\frac{4499}{598995}a^{17}+\frac{16}{2465}a^{16}+\frac{245}{53244}a^{15}-\frac{14729}{598995}a^{14}-\frac{973}{44370}a^{13}-\frac{737}{53244}a^{12}+\frac{201571}{598995}a^{11}+\frac{6701}{22185}a^{10}+\frac{5555}{26622}a^{9}-\frac{1174591}{598995}a^{8}-\frac{38656}{22185}a^{7}-\frac{15511}{13311}a^{6}+\frac{57662}{7395}a^{5}+\frac{156412}{22185}a^{4}+\frac{9595}{1972}a^{3}-\frac{130306}{22185}a^{2}-\frac{31901}{4930}a-\frac{9737}{1972}$, $\frac{551}{20655}a^{17}+\frac{419}{14790}a^{16}+\frac{1225}{53244}a^{15}-\frac{1961}{20655}a^{14}-\frac{4067}{44370}a^{13}-\frac{3685}{53244}a^{12}+\frac{24574}{20655}a^{11}+\frac{27424}{22185}a^{10}+\frac{27775}{26622}a^{9}-\frac{150784}{20655}a^{8}-\frac{164609}{22185}a^{7}-\frac{77555}{13311}a^{6}+\frac{21529}{765}a^{5}+\frac{1222171}{44370}a^{4}+\frac{47975}{1972}a^{3}-\frac{16129}{765}a^{2}-\frac{106599}{4930}a-\frac{42769}{1972}$ 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:  \( 7436740.6823236905 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{9}\cdot 7436740.6823236905 \cdot 18}{2\cdot\sqrt{59658910745494721298677968896}}\cr\approx \mathstrut & 4.18221449390943 \end{aligned}\] (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 - 4*x^15 + 47*x^12 - 296*x^9 + 1215*x^6 - 1512*x^3 + 729) 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 - 4*x^15 + 47*x^12 - 296*x^9 + 1215*x^6 - 1512*x^3 + 729, 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 - 4*x^15 + 47*x^12 - 296*x^9 + 1215*x^6 - 1512*x^3 + 729); 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 - 4*x^15 + 47*x^12 - 296*x^9 + 1215*x^6 - 1512*x^3 + 729); 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

\(\Q(\sqrt{-11}) \), 3.1.3267.1, 3.1.13068.1, 3.1.1452.1, 3.1.108.1, 6.0.117406179.1, 6.0.23191344.1, 6.0.1878498864.1, 6.0.15524784.3, 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: 18.2.178976732236484163896033906688.2
Minimal sibling: This field is its own minimal sibling

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 ${\href{/padicField/5.2.0.1}{2} }^{8}{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ ${\href{/padicField/7.6.0.1}{6} }^{3}$ R ${\href{/padicField/13.6.0.1}{6} }^{3}$ ${\href{/padicField/17.2.0.1}{2} }^{9}$ ${\href{/padicField/19.6.0.1}{6} }^{3}$ ${\href{/padicField/23.2.0.1}{2} }^{8}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ ${\href{/padicField/29.2.0.1}{2} }^{9}$ ${\href{/padicField/31.3.0.1}{3} }^{6}$ ${\href{/padicField/37.3.0.1}{3} }^{6}$ ${\href{/padicField/41.2.0.1}{2} }^{9}$ ${\href{/padicField/43.2.0.1}{2} }^{9}$ ${\href{/padicField/47.2.0.1}{2} }^{8}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ ${\href{/padicField/59.2.0.1}{2} }^{8}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$

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 2.2.3.4a1.2$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$$$[\ ]_{3}^{2}$$
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 3.1.3.3a1.1$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$$[\frac{3}{2}]_{2}$$
3.1.3.3a1.1$x^{3} + 3 x + 3$$3$$1$$3$$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 11.1.6.5a1.1$x^{6} + 11$$6$$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)