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Properties

Label 9.1.218889236736.1
Degree $9$
Signature $[1, 4]$
Discriminant $218889236736$
Root discriminant \(18.20\)
Ramified primes $2,3,19$
Class number $1$
Class group trivial
Galois group $(C_9:C_3):C_2$ (as 9T10)

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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 - 2*x^7 - 6*x^6 + 6*x^5 - 8*x^3 - 18*x^2 + 10*x - 6)
 
Copy content gp:K = bnfinit(y^9 - 2*y^7 - 6*y^6 + 6*y^5 - 8*y^3 - 18*y^2 + 10*y - 6, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 2*x^7 - 6*x^6 + 6*x^5 - 8*x^3 - 18*x^2 + 10*x - 6);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - 2*x^7 - 6*x^6 + 6*x^5 - 8*x^3 - 18*x^2 + 10*x - 6)
 

\( x^{9} - 2x^{7} - 6x^{6} + 6x^{5} - 8x^{3} - 18x^{2} + 10x - 6 \) 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:   \(218889236736\) \(\medspace = 2^{8}\cdot 3^{8}\cdot 19^{4}\) 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:  \(18.20\)
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:  $2^{8/9}3^{4/3}19^{1/2}\approx 34.92373561548968$
Ramified primes:   \(2\), \(3\), \(19\) 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:  \(\Q\)
$\Aut(K/\Q)$:   $C_1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a$, $\frac{1}{2049}a^{8}-\frac{139}{2049}a^{7}+\frac{65}{683}a^{6}-\frac{158}{683}a^{5}+\frac{108}{683}a^{4}+\frac{14}{683}a^{3}+\frac{301}{2049}a^{2}-\frac{877}{2049}a+\frac{113}{683}$ 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:  $4$
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:   $\frac{250}{2049}a^{8}+\frac{83}{2049}a^{7}-\frac{142}{683}a^{6}-\frac{569}{683}a^{5}+\frac{363}{683}a^{4}+\frac{85}{683}a^{3}-\frac{2612}{2049}a^{2}-\frac{6154}{2049}a+\frac{247}{683}$, $\frac{83}{2049}a^{8}+\frac{74}{2049}a^{7}-\frac{69}{683}a^{6}-\frac{137}{683}a^{5}+\frac{85}{683}a^{4}-\frac{204}{683}a^{3}-\frac{1654}{2049}a^{2}+\frac{290}{2049}a-\frac{183}{683}$, $\frac{14}{683}a^{8}+\frac{103}{683}a^{7}-\frac{2}{683}a^{6}-\frac{489}{683}a^{5}-\frac{245}{683}a^{4}+\frac{588}{683}a^{3}+\frac{116}{683}a^{2}-\frac{2033}{683}a-\frac{35}{683}$, $\frac{45}{683}a^{8}-\frac{108}{683}a^{7}-\frac{104}{683}a^{6}-\frac{157}{683}a^{5}+\frac{237}{683}a^{4}-\frac{159}{683}a^{3}-\frac{115}{683}a^{2}-\frac{534}{683}a+\frac{229}{683}$ 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:  \( 273.038407353 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 273.038407353 \cdot 1}{2\cdot\sqrt{218889236736}}\cr\approx \mathstrut & 0.909559285853 \end{aligned}\]

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 - 2*x^7 - 6*x^6 + 6*x^5 - 8*x^3 - 18*x^2 + 10*x - 6) 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 - 2*x^7 - 6*x^6 + 6*x^5 - 8*x^3 - 18*x^2 + 10*x - 6, 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 - 2*x^7 - 6*x^6 + 6*x^5 - 8*x^3 - 18*x^2 + 10*x - 6); 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 - 2*x^7 - 6*x^6 + 6*x^5 - 8*x^3 - 18*x^2 + 10*x - 6); 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

$C_9:C_6$ (as 9T10):

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 54
The 10 conjugacy class representatives for $(C_9:C_3):C_2$
Character table for $(C_9:C_3):C_2$

Intermediate fields

3.1.76.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 27 sibling: 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 ${\href{/padicField/5.9.0.1}{9} }$ ${\href{/padicField/7.9.0.1}{9} }$ ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.9.0.1}{9} }$ R ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.9.0.1}{9} }$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(2\) Copy content Toggle raw display 2.1.9.8a1.1$x^{9} + 2$$9$$1$$8$$(C_9:C_3):C_2$$$[\ ]_{9}^{6}$$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$$[\ ]$$
3.2.1.0a1.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$$[\ ]^{2}$$
3.2.3.8a2.1$x^{6} + 6 x^{5} + 24 x^{4} + 56 x^{3} + 84 x^{2} + 72 x + 35$$3$$2$$8$$C_6$$$[2]^{2}$$
\(19\) Copy content Toggle raw display $\Q_{19}$$x + 17$$1$$1$$0$Trivial$$[\ ]$$
19.1.2.1a1.1$x^{2} + 19$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.1.2.1a1.1$x^{2} + 19$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.1.2.1a1.1$x^{2} + 19$$2$$1$$1$$C_2$$$[\ ]_{2}$$
19.1.2.1a1.1$x^{2} + 19$$2$$1$$1$$C_2$$$[\ ]_{2}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*54 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.19.2t1.a.a$1$ $ 19 $ \(\Q(\sqrt{-19}) \) $C_2$ (as 2T1) $1$ $-1$
1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.171.6t1.f.a$1$ $ 3^{2} \cdot 19 $ 6.0.45001899.1 $C_6$ (as 6T1) $0$ $-1$
1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.171.6t1.f.b$1$ $ 3^{2} \cdot 19 $ 6.0.45001899.1 $C_6$ (as 6T1) $0$ $-1$
*54 2.76.3t2.a.a$2$ $ 2^{2} \cdot 19 $ 3.1.76.1 $S_3$ (as 3T2) $1$ $0$
2.6156.6t5.a.a$2$ $ 2^{2} \cdot 3^{4} \cdot 19 $ 6.0.720030384.6 $S_3\times C_3$ (as 6T5) $0$ $0$
2.6156.6t5.a.b$2$ $ 2^{2} \cdot 3^{4} \cdot 19 $ 6.0.720030384.6 $S_3\times C_3$ (as 6T5) $0$ $0$
*54 6.2880121536.9t10.a.a$6$ $ 2^{6} \cdot 3^{8} \cdot 19^{3}$ 9.1.218889236736.1 $(C_9:C_3):C_2$ (as 9T10) $1$ $0$

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)