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Properties

Label 9.3.125...112.1
Degree $9$
Signature $[3, 3]$
Discriminant $-1.259\times 10^{33}$
Root discriminant \(4762.10\)
Ramified primes $2,37,73,127$
Class number $82323540$ (GRH)
Class group [3, 18, 1524510] (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 - x^8 - 2085*x^7 - 77286*x^6 - 1437264*x^5 - 16258695*x^4 - 116782119*x^3 - 521681457*x^2 - 1324203408*x - 1460616767)
 
Copy content gp:K = bnfinit(y^9 - y^8 - 2085*y^7 - 77286*y^6 - 1437264*y^5 - 16258695*y^4 - 116782119*y^3 - 521681457*y^2 - 1324203408*y - 1460616767, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - x^8 - 2085*x^7 - 77286*x^6 - 1437264*x^5 - 16258695*x^4 - 116782119*x^3 - 521681457*x^2 - 1324203408*x - 1460616767);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - x^8 - 2085*x^7 - 77286*x^6 - 1437264*x^5 - 16258695*x^4 - 116782119*x^3 - 521681457*x^2 - 1324203408*x - 1460616767)
 

\( x^{9} - x^{8} - 2085 x^{7} - 77286 x^{6} - 1437264 x^{5} - 16258695 x^{4} - 116782119 x^{3} + \cdots - 1460616767 \) 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:   \(-1259466642471540279228280776978112\) \(\medspace = -\,2^{6}\cdot 37^{7}\cdot 73^{3}\cdot 127^{7}\) 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:  \(4762.10\)
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:  $2^{2/3}37^{5/6}73^{1/2}127^{5/6}\approx 15572.295322709964$
Ramified primes:   \(2\), \(37\), \(73\), \(127\) 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:  \(\Q(\sqrt{-343027}) \)
$\Aut(K/\Q)$:   $C_3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{73}a^{6}-\frac{25}{73}a^{5}+\frac{5}{73}a^{4}-\frac{24}{73}a^{3}-\frac{10}{73}a^{2}+\frac{6}{73}a-\frac{27}{73}$, $\frac{1}{73}a^{7}-\frac{36}{73}a^{5}+\frac{28}{73}a^{4}-\frac{26}{73}a^{3}-\frac{25}{73}a^{2}-\frac{23}{73}a-\frac{18}{73}$, $\frac{1}{73}a^{8}+\frac{4}{73}a^{5}+\frac{8}{73}a^{4}-\frac{13}{73}a^{3}-\frac{18}{73}a^{2}-\frac{21}{73}a-\frac{23}{73}$ 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:  $C_{3}\times C_{18}\times C_{1524510}$, which has order $82323540$ (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:  $C_{3}\times C_{18}\times C_{1524510}$, which has order $82323540$ (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:   $a^{8}-8a^{7}-2029a^{6}-63083a^{5}-995683a^{4}-9288914a^{3}-51759721a^{2}-159363410a-208659537$, $a^{8}-9a^{7}-2013a^{6}-61182a^{5}-947808a^{4}-8676231a^{3}-47372271a^{2}-142703289a-182577096$, $\frac{5}{73}a^{8}-\frac{43}{73}a^{7}-\frac{10098}{73}a^{6}-\frac{309688}{73}a^{5}-\frac{4833081}{73}a^{4}-\frac{44572317}{73}a^{3}-\frac{245297244}{73}a^{2}-\frac{745162528}{73}a-\frac{961793038}{73}$, $\frac{68}{73}a^{8}-\frac{614}{73}a^{7}-\frac{136851}{73}a^{6}-\frac{4156598}{73}a^{5}-\frac{64356903}{73}a^{4}-\frac{588792546}{73}a^{3}-\frac{3212878539}{73}a^{2}-\frac{9672177569}{73}a-\frac{12366334897}{73}$, $\frac{245}{73}a^{8}-\frac{2135}{73}a^{7}-\frac{494375}{73}a^{6}-\frac{15121015}{73}a^{5}-\frac{235443495}{73}a^{4}-\frac{2166117975}{73}a^{3}-\frac{11888762235}{73}a^{2}-\frac{36005217780}{73}a-\frac{46312621947}{73}$ 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:  \( 77054.60654044396 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 77054.60654044396 \cdot 82323540}{2\cdot\sqrt{1259466642471540279228280776978112}}\cr\approx \mathstrut & 0.177349066195908 \end{aligned}\] (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 - x^8 - 2085*x^7 - 77286*x^6 - 1437264*x^5 - 16258695*x^4 - 116782119*x^3 - 521681457*x^2 - 1324203408*x - 1460616767) 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 - x^8 - 2085*x^7 - 77286*x^6 - 1437264*x^5 - 16258695*x^4 - 116782119*x^3 - 521681457*x^2 - 1324203408*x - 1460616767, 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 - x^8 - 2085*x^7 - 77286*x^6 - 1437264*x^5 - 16258695*x^4 - 116782119*x^3 - 521681457*x^2 - 1324203408*x - 1460616767); 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 - x^8 - 2085*x^7 - 77286*x^6 - 1437264*x^5 - 16258695*x^4 - 116782119*x^3 - 521681457*x^2 - 1324203408*x - 1460616767); 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); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
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

3.3.22080601.1, 3.1.1372108.2

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(L)]
 
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: 6.0.14259877284041547139367728.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.3.0.1}{3} }$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ ${\href{/padicField/7.3.0.1}{3} }^{3}$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.3.0.1}{3} }^{3}$ ${\href{/padicField/17.3.0.1}{3} }^{3}$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.3.0.1}{3} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.3.0.1}{3} }^{3}$ R ${\href{/padicField/41.3.0.1}{3} }^{3}$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.3.0.1}{3} }$ ${\href{/padicField/47.2.0.1}{2} }^{3}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ ${\href{/padicField/53.3.0.1}{3} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$

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$$$[\ ]_{3}^{6}$$
\(37\) Copy content Toggle raw display 37.1.3.2a1.2$x^{3} + 74$$3$$1$$2$$C_3$$$[\ ]_{3}$$
37.1.6.5a1.5$x^{6} + 592$$6$$1$$5$$C_6$$$[\ ]_{6}$$
\(73\) Copy content Toggle raw display $\Q_{73}$$x + 68$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{73}$$x + 68$$1$$1$$0$Trivial$$[\ ]$$
$\Q_{73}$$x + 68$$1$$1$$0$Trivial$$[\ ]$$
73.1.2.1a1.1$x^{2} + 73$$2$$1$$1$$C_2$$$[\ ]_{2}$$
73.1.2.1a1.1$x^{2} + 73$$2$$1$$1$$C_2$$$[\ ]_{2}$$
73.1.2.1a1.1$x^{2} + 73$$2$$1$$1$$C_2$$$[\ ]_{2}$$
\(127\) Copy content Toggle raw display 127.1.3.2a1.3$x^{3} + 1143$$3$$1$$2$$C_3$$$[\ ]_{3}$$
127.1.6.5a1.3$x^{6} + 1143$$6$$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)