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Properties

Label 9.9.471655843734321.1
Degree $9$
Signature $[9, 0]$
Discriminant $4.717\times 10^{14}$
Root discriminant \(42.70\)
Ramified primes $3,31$
Class number $1$
Class group trivial
Galois group $C_3^2$ (as 9T2)

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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 - 36*x^7 + 101*x^6 + 327*x^5 - 921*x^4 - 575*x^3 + 1632*x^2 + 480*x - 449)
 
Copy content gp:K = bnfinit(y^9 - 3*y^8 - 36*y^7 + 101*y^6 + 327*y^5 - 921*y^4 - 575*y^3 + 1632*y^2 + 480*y - 449, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 3*x^8 - 36*x^7 + 101*x^6 + 327*x^5 - 921*x^4 - 575*x^3 + 1632*x^2 + 480*x - 449);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - 3*x^8 - 36*x^7 + 101*x^6 + 327*x^5 - 921*x^4 - 575*x^3 + 1632*x^2 + 480*x - 449)
 

\( x^{9} - 3x^{8} - 36x^{7} + 101x^{6} + 327x^{5} - 921x^{4} - 575x^{3} + 1632x^{2} + 480x - 449 \) 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:  $[9, 0]$
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:   \(471655843734321\) \(\medspace = 3^{12}\cdot 31^{6}\) 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:  \(42.70\)
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:  $3^{4/3}31^{2/3}\approx 42.697534899657064$
Ramified primes:   \(3\), \(31\) 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:  \(\Q\)
$\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$:   $C_3^2$
Copy content comment:Autmorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphisms(K)
 
This field is Galois and abelian over $\Q$.
Conductor:  \(279=3^{2}\cdot 31\)
Dirichlet character group:    $\lbrace$$\chi_{279}(160,·)$, $\chi_{279}(1,·)$, $\chi_{279}(67,·)$, $\chi_{279}(211,·)$, $\chi_{279}(118,·)$, $\chi_{279}(25,·)$, $\chi_{279}(187,·)$, $\chi_{279}(253,·)$, $\chi_{279}(94,·)$$\rbrace$
This is not a CM field.
This field has no CM subfields.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{1306507354}a^{8}-\frac{37989939}{1306507354}a^{7}+\frac{42896737}{1306507354}a^{6}+\frac{218118735}{1306507354}a^{5}-\frac{318467565}{1306507354}a^{4}-\frac{934207}{11986306}a^{3}-\frac{550249579}{1306507354}a^{2}-\frac{110649025}{653253677}a-\frac{196756513}{653253677}$ 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:  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:  $8$
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:   $\frac{3950274}{653253677}a^{8}-\frac{7583430}{653253677}a^{7}-\frac{138957862}{653253677}a^{6}+\frac{232893930}{653253677}a^{5}+\frac{1136050113}{653253677}a^{4}-\frac{17745132}{5993153}a^{3}-\frac{973838523}{653253677}a^{2}+\frac{1363425116}{653253677}a+\frac{134281107}{653253677}$, $\frac{1357941}{653253677}a^{8}+\frac{370768}{653253677}a^{7}-\frac{45693250}{653253677}a^{6}-\frac{29817612}{653253677}a^{5}+\frac{303027105}{653253677}a^{4}+\frac{2673488}{5993153}a^{3}+\frac{117030332}{653253677}a^{2}-\frac{1245328864}{653253677}a-\frac{738224050}{653253677}$, $\frac{2592333}{1306507354}a^{8}-\frac{3977099}{653253677}a^{7}-\frac{46632306}{653253677}a^{6}+\frac{131355771}{653253677}a^{5}+\frac{416511504}{653253677}a^{4}-\frac{10209310}{5993153}a^{3}-\frac{872061266}{653253677}a^{2}+\frac{3262007657}{1306507354}a+\frac{1416133094}{653253677}$, $\frac{3408291}{1306507354}a^{8}-\frac{4559878}{653253677}a^{7}-\frac{142802903}{1306507354}a^{6}+\frac{291454407}{1306507354}a^{5}+\frac{1802540891}{1306507354}a^{4}-\frac{18970703}{11986306}a^{3}-\frac{6917185207}{1306507354}a^{2}-\frac{728334175}{653253677}a+\frac{1937677351}{1306507354}$, $\frac{11279890}{653253677}a^{8}-\frac{12215469}{1306507354}a^{7}-\frac{403118063}{653253677}a^{6}+\frac{134770957}{653253677}a^{5}+\frac{3369783124}{653253677}a^{4}-\frac{15249942}{5993153}a^{3}-\frac{5361191365}{653253677}a^{2}+\frac{515367316}{653253677}a+\frac{3194658395}{1306507354}$, $\frac{10772719}{1306507354}a^{8}-\frac{665721}{653253677}a^{7}-\frac{211830141}{653253677}a^{6}-\frac{39003597}{653253677}a^{5}+\frac{2282403583}{653253677}a^{4}+\frac{5362173}{5993153}a^{3}-\frac{7577416794}{653253677}a^{2}-\frac{3618337243}{1306507354}a+\frac{2394002606}{653253677}$, $\frac{12672643}{1306507354}a^{8}+\frac{287826}{653253677}a^{7}-\frac{465508491}{1306507354}a^{6}-\frac{166385283}{1306507354}a^{5}+\frac{4201343493}{1306507354}a^{4}+\frac{14623405}{11986306}a^{3}-\frac{8441202895}{1306507354}a^{2}-\frac{2654920513}{653253677}a+\frac{286623887}{1306507354}$, $\frac{3685345}{1306507354}a^{8}-\frac{25217819}{653253677}a^{7}-\frac{37187883}{653253677}a^{6}+\frac{864035106}{653253677}a^{5}-\frac{254417984}{653253677}a^{4}-\frac{63167512}{5993153}a^{3}+\frac{6495456325}{653253677}a^{2}+\frac{12528822679}{1306507354}a-\frac{3396678308}{653253677}$ 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:  \( 18035.7055684 \)
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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{9}\cdot(2\pi)^{0}\cdot 18035.7055684 \cdot 1}{2\cdot\sqrt{471655843734321}}\cr\approx \mathstrut & 0.212598645070 \end{aligned}\]

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 - 36*x^7 + 101*x^6 + 327*x^5 - 921*x^4 - 575*x^3 + 1632*x^2 + 480*x - 449) 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 - 36*x^7 + 101*x^6 + 327*x^5 - 921*x^4 - 575*x^3 + 1632*x^2 + 480*x - 449, 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 - 36*x^7 + 101*x^6 + 327*x^5 - 921*x^4 - 575*x^3 + 1632*x^2 + 480*x - 449); 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 - 36*x^7 + 101*x^6 + 327*x^5 - 921*x^4 - 575*x^3 + 1632*x^2 + 480*x - 449); 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

$C_3^2$ (as 9T2):

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)
 
An abelian group of order 9
The 9 conjugacy class representatives for $C_3^2$
Character table for $C_3^2$

Intermediate fields

\(\Q(\zeta_{9})^+\), 3.3.77841.2, 3.3.961.1, 3.3.77841.1

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]
 

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.3.0.1}{3} }^{3}$ R ${\href{/padicField/5.3.0.1}{3} }^{3}$ ${\href{/padicField/7.3.0.1}{3} }^{3}$ ${\href{/padicField/11.3.0.1}{3} }^{3}$ ${\href{/padicField/13.3.0.1}{3} }^{3}$ ${\href{/padicField/17.3.0.1}{3} }^{3}$ ${\href{/padicField/19.3.0.1}{3} }^{3}$ ${\href{/padicField/23.3.0.1}{3} }^{3}$ ${\href{/padicField/29.3.0.1}{3} }^{3}$ R ${\href{/padicField/37.3.0.1}{3} }^{3}$ ${\href{/padicField/41.3.0.1}{3} }^{3}$ ${\href{/padicField/43.3.0.1}{3} }^{3}$ ${\href{/padicField/47.3.0.1}{3} }^{3}$ ${\href{/padicField/53.3.0.1}{3} }^{3}$ ${\href{/padicField/59.3.0.1}{3} }^{3}$

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
\(3\) Copy content Toggle raw display 3.3.3.12a2.1$x^{9} + 6 x^{7} + 9 x^{6} + 12 x^{5} + 36 x^{4} + 23 x^{3} + 36 x^{2} + 30 x + 10$$3$$3$$12$$C_3^2$$$[2]^{3}$$
\(31\) Copy content Toggle raw display 31.3.3.6a1.3$x^{9} + 3 x^{7} + 84 x^{6} + 3 x^{5} + 168 x^{4} + 2353 x^{3} + 84 x^{2} + 2352 x + 21983$$3$$3$$6$$C_3^2$$$[\ ]_{3}^{3}$$

Spectrum of ring of integers

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