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Properties

Label 9.3.866...504.1
Degree $9$
Signature $[3, 3]$
Discriminant $-8.666\times 10^{21}$
Root discriminant \(273.86\)
Ramified primes $2,3,7,31$
Class number $2457$ (GRH)
Class group [3, 819] (GRH)
Galois group $S_3\times C_3$ (as 9T4)

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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 - 117*x^7 - 1068*x^6 - 5292*x^5 - 15255*x^4 - 26055*x^3 - 25947*x^2 - 13932*x - 3123)
 
Copy content gp:K = bnfinit(y^9 - 3*y^8 - 117*y^7 - 1068*y^6 - 5292*y^5 - 15255*y^4 - 26055*y^3 - 25947*y^2 - 13932*y - 3123, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 3*x^8 - 117*x^7 - 1068*x^6 - 5292*x^5 - 15255*x^4 - 26055*x^3 - 25947*x^2 - 13932*x - 3123);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - 3*x^8 - 117*x^7 - 1068*x^6 - 5292*x^5 - 15255*x^4 - 26055*x^3 - 25947*x^2 - 13932*x - 3123)
 

\( x^{9} - 3x^{8} - 117x^{7} - 1068x^{6} - 5292x^{5} - 15255x^{4} - 26055x^{3} - 25947x^{2} - 13932x - 3123 \) 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:   \(-8666121461345916813504\) \(\medspace = -\,2^{6}\cdot 3^{15}\cdot 7^{3}\cdot 31^{7}\) 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:  \(273.86\)
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:  $2^{2/3}3^{31/18}7^{1/2}31^{5/6}\approx 487.24335268924307$
Ramified primes:   \(2\), \(3\), \(7\), \(31\) 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:  \(\Q(\sqrt{-651}) \)
$\Aut(K/\Q)$:   $C_3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{21}a^{6}-\frac{2}{21}a^{5}-\frac{3}{7}a^{4}+\frac{3}{7}a^{3}-\frac{3}{7}a^{2}-\frac{3}{7}a+\frac{3}{7}$, $\frac{1}{21}a^{7}+\frac{1}{21}a^{5}-\frac{3}{7}a^{4}+\frac{3}{7}a^{3}-\frac{2}{7}a^{2}-\frac{3}{7}a-\frac{1}{7}$, $\frac{1}{21}a^{8}-\frac{1}{7}a^{4}+\frac{2}{7}a^{3}+\frac{2}{7}a-\frac{3}{7}$ 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:  $C_{3}\times C_{819}$, which has order $2457$ (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:  $C_{3}\times C_{819}$, which has order $2457$ (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:  $5$
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:   $a^{8}-4a^{7}-113a^{6}-955a^{5}-4337a^{4}-10918a^{3}-15137a^{2}-10810a-3121$, $a^{8}-5a^{7}-107a^{6}-854a^{5}-3584a^{4}-8087a^{3}-9881a^{2}-6185a-1562$, $\frac{16}{21}a^{8}-\frac{62}{21}a^{7}-\frac{260}{3}a^{6}-\frac{5161}{7}a^{5}-\frac{23630}{7}a^{4}-8584a^{3}-\frac{84044}{7}a^{2}-\frac{60199}{7}a-2456$, $\frac{1}{3}a^{8}-\frac{11}{7}a^{7}-\frac{109}{3}a^{6}-\frac{2055}{7}a^{5}-\frac{8805}{7}a^{4}-\frac{20343}{7}a^{3}-\frac{25281}{7}a^{2}-\frac{15847}{7}a-\frac{3922}{7}$, $3a^{8}-\frac{99}{7}a^{7}-327a^{6}-\frac{18495}{7}a^{5}-\frac{79245}{7}a^{4}-\frac{183087}{7}a^{3}-\frac{227529}{7}a^{2}-\frac{142560}{7}a-\frac{35165}{7}$ 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:  \( 10197.25168202796 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 10197.25168202796 \cdot 2457}{2\cdot\sqrt{8666121461345916813504}}\cr\approx \mathstrut & 0.267039499565836 \end{aligned}\] (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 - 3*x^8 - 117*x^7 - 1068*x^6 - 5292*x^5 - 15255*x^4 - 26055*x^3 - 25947*x^2 - 13932*x - 3123) 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 - 117*x^7 - 1068*x^6 - 5292*x^5 - 15255*x^4 - 26055*x^3 - 25947*x^2 - 13932*x - 3123, 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 - 117*x^7 - 1068*x^6 - 5292*x^5 - 15255*x^4 - 26055*x^3 - 25947*x^2 - 13932*x - 3123); 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 - 3*x^8 - 117*x^7 - 1068*x^6 - 5292*x^5 - 15255*x^4 - 26055*x^3 - 25947*x^2 - 13932*x - 3123); 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

$C_3\times S_3$ (as 9T4):

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 18
The 9 conjugacy class representatives for $S_3\times C_3$
Character table for $S_3\times C_3$

Intermediate fields

3.3.77841.2, 3.1.23436.2

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
Minimal sibling: 6.0.3092529594281904.2

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 ${\href{/padicField/5.3.0.1}{3} }^{3}$ R ${\href{/padicField/11.3.0.1}{3} }^{3}$ ${\href{/padicField/13.3.0.1}{3} }^{3}$ ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.3.0.1}{3} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.3.0.1}{3} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{3}$ R ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.3.0.1}{3} }$ ${\href{/padicField/41.3.0.1}{3} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ ${\href{/padicField/47.3.0.1}{3} }^{3}$ ${\href{/padicField/53.3.0.1}{3} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{3}$

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 3.1.9.15b2.9$x^{9} + 6 x^{8} + 6 x^{7} + 3 x^{3} + 21$$9$$1$$15$$S_3\times C_3$$$[\frac{3}{2}, 2]_{2}$$
\(7\) Copy content Toggle raw display 7.3.1.0a1.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$$[\ ]^{3}$$
7.3.2.3a1.2$x^{6} + 12 x^{5} + 36 x^{4} + 8 x^{3} + 48 x^{2} + 23$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(31\) Copy content Toggle raw display 31.1.3.2a1.3$x^{3} + 279$$3$$1$$2$$C_3$$$[\ ]_{3}$$
31.1.6.5a1.5$x^{6} + 806$$6$$1$$5$$C_6$$$[\ ]_{6}$$

Spectrum of ring of integers

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