Properties

Label 9.5.74310349154112.1
Degree $9$
Signature $(5, 2)$
Discriminant $7.431\times 10^{13}$
Root discriminant \(34.77\)
Ramified primes $2,3,37$
Class number $3$
Class group [3]
Galois group $S_3 \wr C_3 $ (as 9T28)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^9 - 18*x^7 - 36*x^6 + 144*x^4 + 420*x^3 + 216*x^2 - 432*x - 296)
 
Copy content gp:K = bnfinit(y^9 - 18*y^7 - 36*y^6 + 144*y^4 + 420*y^3 + 216*y^2 - 432*y - 296, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 18*x^7 - 36*x^6 + 144*x^4 + 420*x^3 + 216*x^2 - 432*x - 296);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - 18*x^7 - 36*x^6 + 144*x^4 + 420*x^3 + 216*x^2 - 432*x - 296)
 

\( x^{9} - 18x^{7} - 36x^{6} + 144x^{4} + 420x^{3} + 216x^{2} - 432x - 296 \) 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:  $9$
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:  $(5, 2)$
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:   \(74310349154112\) \(\medspace = 2^{6}\cdot 3^{22}\cdot 37\) 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:  \(34.77\)
Copy content comment:Root discriminant
 
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:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $2^{2/3}3^{22/9}37^{1/2}\approx 141.6067404717884$
Ramified primes:   \(2\), \(3\), \(37\) 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:  \(\Q(\sqrt{37}) \)
$\Aut(K/\Q)$:   $C_1$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2}a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{4}a^{6}$, $\frac{1}{4}a^{7}$, $\frac{1}{7278772}a^{8}+\frac{814663}{7278772}a^{7}-\frac{201929}{1819693}a^{6}-\frac{198350}{1819693}a^{5}+\frac{332350}{1819693}a^{4}+\frac{433539}{3639386}a^{3}-\frac{745441}{1819693}a^{2}-\frac{695825}{1819693}a+\frac{602505}{1819693}$ 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:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  $C_{3}$, which has order $3$
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:  $C_{3}$, which has order $3$
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:  $6$
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:   $\frac{12015}{7278772}a^{8}+\frac{23649}{3639386}a^{7}-\frac{284971}{7278772}a^{6}-\frac{574533}{3639386}a^{5}-\frac{262077}{3639386}a^{4}+\frac{1009719}{3639386}a^{3}+\frac{1875024}{1819693}a^{2}+\frac{2971653}{1819693}a-\frac{1460872}{1819693}$, $\frac{15687}{7278772}a^{8}-\frac{42729}{3639386}a^{7}-\frac{118533}{7278772}a^{6}+\frac{158580}{1819693}a^{5}+\frac{154005}{1819693}a^{4}+\frac{366776}{1819693}a^{3}-\frac{385749}{1819693}a^{2}-\frac{2707854}{1819693}a+\frac{1830186}{1819693}$, $\frac{69771}{7278772}a^{8}-\frac{78375}{7278772}a^{7}-\frac{1080519}{7278772}a^{6}-\frac{312585}{1819693}a^{5}+\frac{43951}{1819693}a^{4}+\frac{1666108}{1819693}a^{3}+\frac{3940701}{1819693}a^{2}-\frac{2636221}{1819693}a-\frac{2991024}{1819693}$, $\frac{44220}{1819693}a^{8}-\frac{64461}{1819693}a^{7}-\frac{2888957}{7278772}a^{6}-\frac{466960}{1819693}a^{5}+\frac{885635}{1819693}a^{4}+\frac{4897036}{1819693}a^{3}+\frac{10449165}{1819693}a^{2}-\frac{6229331}{1819693}a-\frac{8334610}{1819693}$, $\frac{360061}{7278772}a^{8}-\frac{339039}{3639386}a^{7}-\frac{5515109}{7278772}a^{6}-\frac{608179}{1819693}a^{5}+\frac{4703647}{3639386}a^{4}+\frac{21777883}{3639386}a^{3}+\frac{18682529}{1819693}a^{2}-\frac{22310015}{1819693}a-\frac{21623892}{1819693}$, $\frac{13335}{7278772}a^{8}-\frac{36105}{7278772}a^{7}-\frac{129993}{7278772}a^{6}-\frac{146949}{3639386}a^{5}+\frac{49897}{3639386}a^{4}+\frac{1897597}{3639386}a^{3}+\frac{2346817}{1819693}a^{2}-\frac{211768}{1819693}a-\frac{3179806}{1819693}$ Copy content Toggle raw display
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:  \( 1539.48699795 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 5 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{2}\cdot 1539.48699795 \cdot 3}{2\cdot\sqrt{74310349154112}}\cr\approx \mathstrut & 0.33841713813 \end{aligned}\]

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^9 - 18*x^7 - 36*x^6 + 144*x^4 + 420*x^3 + 216*x^2 - 432*x - 296) 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^9 - 18*x^7 - 36*x^6 + 144*x^4 + 420*x^3 + 216*x^2 - 432*x - 296, 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^9 - 18*x^7 - 36*x^6 + 144*x^4 + 420*x^3 + 216*x^2 - 432*x - 296); 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 = polynomial_ring(QQ); K, a = number_field(x^9 - 18*x^7 - 36*x^6 + 144*x^4 + 420*x^3 + 216*x^2 - 432*x - 296); 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

$S_3\wr C_3$ (as 9T28):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 648
The 17 conjugacy class representatives for $S_3 \wr C_3 $
Character table for $S_3 \wr C_3 $

Intermediate fields

\(\Q(\zeta_{9})^+\)

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(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 27 siblings: data not computed
Degree 36 siblings: data not computed
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 ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ ${\href{/padicField/7.9.0.1}{9} }$ ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.3.0.1}{3} }$ R ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{5}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$

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 2.3.3.6a1.1$x^{9} + 3 x^{7} + 3 x^{6} + 3 x^{5} + 6 x^{4} + 4 x^{3} + 3 x^{2} + 3 x + 3$$3$$3$$6$$S_3\times C_3$$$[\ ]_{3}^{6}$$
\(3\) Copy content Toggle raw display 3.1.9.22a3.8$x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$$9$$1$$22$$C_9$$$[2, 3]$$
\(37\) Copy content Toggle raw display $\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{37}$$x + 35$$1$$1$$0$Trivial$$[\ ]$$
37.2.1.0a1.1$x^{2} + 33 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
37.1.2.1a1.2$x^{2} + 74$$2$$1$$1$$C_2$$$[\ ]_{2}$$
37.3.1.0a1.1$x^{3} + 6 x + 35$$1$$3$$0$$C_3$$$[\ ]^{3}$$

Spectrum of ring of integers

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