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Properties

Label 9.3.210082347387.1
Degree $9$
Signature $[3, 3]$
Discriminant $-210082347387$
Root discriminant \(18.12\)
Ramified primes $3,11$
Class number $1$
Class group trivial
Galois group $((C_3^2:Q_8):C_3):C_2$ (as 9T26)

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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 - 3*x^8 + 9*x^6 + 3*x^5 - 6*x^3 - 12*x^2 - 9*x - 2)
 
Copy content gp:K = bnfinit(y^9 - 3*y^8 + 9*y^6 + 3*y^5 - 6*y^3 - 12*y^2 - 9*y - 2, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 3*x^8 + 9*x^6 + 3*x^5 - 6*x^3 - 12*x^2 - 9*x - 2);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - 3*x^8 + 9*x^6 + 3*x^5 - 6*x^3 - 12*x^2 - 9*x - 2)
 

\( x^{9} - 3x^{8} + 9x^{6} + 3x^{5} - 6x^{3} - 12x^{2} - 9x - 2 \) 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:  $[3, 3]$
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:   \(-210082347387\) \(\medspace = -\,3^{15}\cdot 11^{4}\) 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:  \(18.12\)
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:  $3^{97/54}11^{2/3}\approx 35.58876626592777$
Ramified primes:   \(3\), \(11\) 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:  \(\Q(\sqrt{-3}) \)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{271}a^{8}+\frac{112}{271}a^{7}-\frac{128}{271}a^{6}-\frac{77}{271}a^{5}+\frac{91}{271}a^{4}-\frac{104}{271}a^{3}-\frac{42}{271}a^{2}+\frac{36}{271}a+\frac{66}{271}$ 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:  Trivial group, which has order $1$
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:  Trivial group, which has order $1$
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:  $5$
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:   $\frac{260}{271}a^{8}-\frac{961}{271}a^{7}+\frac{595}{271}a^{6}+\frac{2202}{271}a^{5}-\frac{1001}{271}a^{4}+\frac{331}{271}a^{3}-\frac{1706}{271}a^{2}-\frac{2293}{271}a-\frac{455}{271}$, $\frac{296}{271}a^{8}-\frac{994}{271}a^{7}+\frac{323}{271}a^{6}+\frac{2682}{271}a^{5}-\frac{164}{271}a^{4}-\frac{161}{271}a^{3}-\frac{1592}{271}a^{2}-\frac{3165}{271}a-\frac{1331}{271}$, $\frac{672}{271}a^{8}-\frac{2242}{271}a^{7}+\frac{704}{271}a^{6}+\frac{5979}{271}a^{5}-\frac{94}{271}a^{4}-\frac{241}{271}a^{3}-\frac{4105}{271}a^{2}-\frac{6973}{271}a-\frac{3073}{271}$, $\frac{26}{271}a^{8}-\frac{69}{271}a^{7}-\frac{76}{271}a^{6}+\frac{437}{271}a^{5}-\frac{73}{271}a^{4}-\frac{265}{271}a^{3}+\frac{263}{271}a^{2}-\frac{419}{271}a-\frac{181}{271}$, $\frac{434}{271}a^{8}-\frac{1527}{271}a^{7}+\frac{816}{271}a^{6}+\frac{3438}{271}a^{5}-\frac{614}{271}a^{4}+\frac{663}{271}a^{3}-\frac{2510}{271}a^{2}-\frac{4430}{271}a-\frac{1979}{271}$ 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:  \( 406.606363501 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 406.606363501 \cdot 1}{2\cdot\sqrt{210082347387}}\cr\approx \mathstrut & 0.880195181742 \end{aligned}\]

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 - 3*x^8 + 9*x^6 + 3*x^5 - 6*x^3 - 12*x^2 - 9*x - 2) 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 - 3*x^8 + 9*x^6 + 3*x^5 - 6*x^3 - 12*x^2 - 9*x - 2, 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 - 3*x^8 + 9*x^6 + 3*x^5 - 6*x^3 - 12*x^2 - 9*x - 2); 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 - 3*x^8 + 9*x^6 + 3*x^5 - 6*x^3 - 12*x^2 - 9*x - 2); 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

$C_3^2:\GL(2,3)$ (as 9T26):

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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A solvable group of order 432
The 11 conjugacy class representatives for $((C_3^2:Q_8):C_3):C_2$
Character table for $((C_3^2:Q_8):C_3):C_2$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 sibling: data not computed
Degree 24 siblings: data not computed
Degree 27 sibling: 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 ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ R ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ R ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(3\) Copy content Toggle raw display 3.1.9.15b1.5$x^{9} + 3 x^{8} + 6 x^{7} + 6 x^{3} + 3$$9$$1$$15$$C_3^2 : D_{6} $$$[\frac{3}{2}, \frac{3}{2}, 2]_{2}^{2}$$
\(11\) Copy content Toggle raw display 11.3.1.0a1.1$x^{3} + 2 x + 9$$1$$3$$0$$C_3$$$[\ ]^{3}$$
11.2.3.4a1.1$x^{6} + 21 x^{5} + 153 x^{4} + 427 x^{3} + 306 x^{2} + 95 x + 8$$3$$2$$4$$S_3\times C_3$$$[\ ]_{3}^{6}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*432 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
2.3267.3t2.b.a$2$ $ 3^{3} \cdot 11^{2}$ 3.1.3267.1 $S_3$ (as 3T2) $1$ $0$
2.3267.24t22.b.a$2$ $ 3^{3} \cdot 11^{2}$ 8.2.32019867.2 $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
2.3267.24t22.b.b$2$ $ 3^{3} \cdot 11^{2}$ 8.2.32019867.2 $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
3.3267.4t5.c.a$3$ $ 3^{3} \cdot 11^{2}$ 4.2.3267.1 $S_4$ (as 4T5) $1$ $1$
3.9801.6t8.e.a$3$ $ 3^{4} \cdot 11^{2}$ 4.2.3267.1 $S_4$ (as 4T5) $1$ $-1$
4.9801.8t23.b.a$4$ $ 3^{4} \cdot 11^{2}$ 8.2.32019867.2 $\textrm{GL(2,3)}$ (as 8T23) $1$ $0$
*432 8.210082347387.9t26.a.a$8$ $ 3^{15} \cdot 11^{4}$ 9.3.210082347387.1 $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $2$
8.210082347387.18t157.a.a$8$ $ 3^{15} \cdot 11^{4}$ 9.3.210082347387.1 $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $-2$
16.646...929.24t1334.a.a$16$ $ 3^{30} \cdot 11^{12}$ 9.3.210082347387.1 $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $0$

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 *.

Spectrum of ring of integers

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