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Properties

Label 9.1.194...832.1
Degree $9$
Signature $[1, 4]$
Discriminant $1.947\times 10^{35}$
Root discriminant \(8337.74\)
Ramified primes $2,3,2287$
Class number $5832$ (GRH)
Class group [3, 6, 6, 54] (GRH)
Galois group $S_3^2$ (as 9T8)

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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 + 342*x^7 - 20559*x^6 + 38988*x^5 + 2351934*x^4 + 143854347*x^3 - 267184764*x^2 - 32179357980*x - 332006485861)
 
Copy content gp:K = bnfinit(y^9 + 342*y^7 - 20559*y^6 + 38988*y^5 + 2351934*y^4 + 143854347*y^3 - 267184764*y^2 - 32179357980*y - 332006485861, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 + 342*x^7 - 20559*x^6 + 38988*x^5 + 2351934*x^4 + 143854347*x^3 - 267184764*x^2 - 32179357980*x - 332006485861);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 + 342*x^7 - 20559*x^6 + 38988*x^5 + 2351934*x^4 + 143854347*x^3 - 267184764*x^2 - 32179357980*x - 332006485861)
 

\( x^{9} + 342 x^{7} - 20559 x^{6} + 38988 x^{5} + 2351934 x^{4} + 143854347 x^{3} - 267184764 x^{2} + \cdots - 332006485861 \) 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:  $[1, 4]$
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:   \(194731833464812587366907938019256832\) \(\medspace = 2^{9}\cdot 3^{19}\cdot 2287^{7}\) 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:  \(8337.74\)
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:  $2^{3/2}3^{13/6}2287^{5/6}\approx 19261.76954796617$
Ramified primes:   \(2\), \(3\), \(2287\) 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:  \(\Q(\sqrt{13722}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3}a^{3}-\frac{1}{3}$, $\frac{1}{9}a^{4}-\frac{1}{9}a^{3}-\frac{1}{3}a^{2}-\frac{1}{9}a+\frac{4}{9}$, $\frac{1}{9}a^{5}-\frac{1}{9}a^{3}-\frac{4}{9}a^{2}+\frac{1}{3}a+\frac{1}{9}$, $\frac{1}{123498}a^{6}-\frac{107}{2287}a^{5}+\frac{19}{20583}a^{4}+\frac{6839}{61749}a^{3}-\frac{856}{2287}a^{2}-\frac{304}{20583}a+\frac{47899}{123498}$, $\frac{1}{370494}a^{7}-\frac{1}{370494}a^{6}+\frac{3326}{61749}a^{5}-\frac{6922}{185247}a^{4}+\frac{14458}{185247}a^{3}-\frac{8153}{61749}a^{2}-\frac{5741}{370494}a-\frac{101383}{370494}$, $\frac{1}{25\cdots 22}a^{8}-\frac{339306459274595}{25\cdots 22}a^{7}+\frac{347197696429343}{12\cdots 61}a^{6}-\frac{58\cdots 15}{12\cdots 61}a^{5}-\frac{45\cdots 05}{12\cdots 61}a^{4}+\frac{14\cdots 60}{12\cdots 61}a^{3}+\frac{36\cdots 19}{25\cdots 22}a^{2}+\frac{87\cdots 87}{25\cdots 22}a+\frac{78\cdots 81}{12\cdots 61}$ 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:  $C_{3}\times C_{6}\times C_{6}\times C_{54}$, which has order $5832$ (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:  $C_{3}\times C_{6}\times C_{6}\times C_{54}$, which has order $5832$ (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:  $4$
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:   $\frac{1279258081123}{42\cdots 87}a^{8}+\frac{21531690966268}{42\cdots 87}a^{7}+\frac{297189793571680}{42\cdots 87}a^{6}-\frac{702219929598758}{42\cdots 87}a^{5}-\frac{50\cdots 18}{42\cdots 87}a^{4}+\frac{12\cdots 36}{42\cdots 87}a^{3}-\frac{51\cdots 93}{42\cdots 87}a^{2}+\frac{25\cdots 22}{42\cdots 87}a-\frac{99\cdots 97}{42\cdots 87}$, $\frac{91\cdots 23}{42\cdots 87}a^{8}-\frac{34\cdots 24}{42\cdots 87}a^{7}+\frac{64\cdots 42}{42\cdots 87}a^{6}-\frac{24\cdots 74}{42\cdots 87}a^{5}+\frac{62\cdots 48}{42\cdots 87}a^{4}-\frac{15\cdots 44}{42\cdots 87}a^{3}-\frac{10\cdots 13}{42\cdots 87}a^{2}-\frac{15\cdots 18}{42\cdots 87}a+\frac{15\cdots 73}{42\cdots 87}$, $\frac{51\cdots 62}{42\cdots 87}a^{8}-\frac{35\cdots 80}{14\cdots 29}a^{7}+\frac{25\cdots 87}{42\cdots 87}a^{6}-\frac{10\cdots 09}{42\cdots 87}a^{5}+\frac{64\cdots 12}{14\cdots 29}a^{4}+\frac{80\cdots 67}{42\cdots 87}a^{3}-\frac{32\cdots 23}{42\cdots 87}a^{2}-\frac{12\cdots 60}{15\cdots 81}a+\frac{25\cdots 97}{42\cdots 87}$, $\frac{18\cdots 89}{42\cdots 87}a^{8}-\frac{78\cdots 06}{14\cdots 29}a^{7}+\frac{84\cdots 87}{42\cdots 87}a^{6}-\frac{46\cdots 12}{42\cdots 87}a^{5}+\frac{23\cdots 95}{15\cdots 81}a^{4}-\frac{14\cdots 85}{42\cdots 87}a^{3}+\frac{23\cdots 37}{42\cdots 87}a^{2}-\frac{10\cdots 06}{14\cdots 29}a-\frac{29\cdots 04}{42\cdots 87}$ 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:  \( 91166245978.46117 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 91166245978.46117 \cdot 5832}{2\cdot\sqrt{194731833464812587366907938019256832}}\cr\approx \mathstrut & 1.87781405007302 \end{aligned}\] (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^9 + 342*x^7 - 20559*x^6 + 38988*x^5 + 2351934*x^4 + 143854347*x^3 - 267184764*x^2 - 32179357980*x - 332006485861) 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 + 342*x^7 - 20559*x^6 + 38988*x^5 + 2351934*x^4 + 143854347*x^3 - 267184764*x^2 - 32179357980*x - 332006485861, 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 + 342*x^7 - 20559*x^6 + 38988*x^5 + 2351934*x^4 + 143854347*x^3 - 267184764*x^2 - 32179357980*x - 332006485861); 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^9 + 342*x^7 - 20559*x^6 + 38988*x^5 + 2351934*x^4 + 143854347*x^3 - 267184764*x^2 - 32179357980*x - 332006485861); 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

$S_3^2$ (as 9T8):

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 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

3.1.1481976.3, 3.1.1270979667.1

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 6 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 siblings: 18.0.44402766688238530812346229902570152786470217145543488433324722627805452435456.2, 18.0.113761460893702510484735982000599907732061184340694743130306972736028672.2, some data not computed
Minimal sibling: 6.2.5674591274422324904667648.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R R ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ ${\href{/padicField/7.3.0.1}{3} }^{3}$ ${\href{/padicField/11.2.0.1}{2} }^{4}{,}\,{\href{/padicField/11.1.0.1}{1} }$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.3.0.1}{3} }^{3}$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.3.0.1}{3} }$ ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.3.0.1}{3} }^{3}$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.3.0.1}{3} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.2.0.1}{2} }^{3}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$

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 $\Q_{2}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
2.1.2.3a1.3$x^{2} + 4 x + 2$$2$$1$$3$$C_2$$$[3]$$
2.2.1.0a1.1$x^{2} + x + 1$$1$$2$$0$$C_2$$$[\ ]^{2}$$
2.2.2.6a1.5$x^{4} + 2 x^{3} + 7 x^{2} + 6 x + 7$$2$$2$$6$$C_2^2$$$[3]^{2}$$
\(3\) Copy content Toggle raw display 3.1.9.19c2.6$x^{9} + 6 x^{6} + 18 x^{2} + 12$$9$$1$$19$$S_3\times C_3$$$[2, \frac{5}{2}]_{2}$$
\(2287\) Copy content Toggle raw display Deg $3$$3$$1$$2$
Deg $6$$6$$1$$5$

Spectrum of ring of integers

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