Show commands: Magma / Oscar / Pari/GP / SageMath

Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^9 + 9*x^7 - 63*x^6 - 81*x^5 - 1296*x^4 - 1359*x^3 - 6885*x^2 - 5265*x - 1125)
 
Copy content gp:K = bnfinit(y^9 + 9*y^7 - 63*y^6 - 81*y^5 - 1296*y^4 - 1359*y^3 - 6885*y^2 - 5265*y - 1125, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 + 9*x^7 - 63*x^6 - 81*x^5 - 1296*x^4 - 1359*x^3 - 6885*x^2 - 5265*x - 1125);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 + 9*x^7 - 63*x^6 - 81*x^5 - 1296*x^4 - 1359*x^3 - 6885*x^2 - 5265*x - 1125)
 

\( x^{9} + 9x^{7} - 63x^{6} - 81x^{5} - 1296x^{4} - 1359x^{3} - 6885x^{2} - 5265x - 1125 \) 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:   \(47091202575755625\) \(\medspace = 3^{22}\cdot 5^{4}\cdot 7^{4}\) 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:  \(71.21\)
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:  $3^{22/9}5^{1/2}7^{1/2}\approx 86.76217340159114$
Ramified primes:   \(3\), \(5\), \(7\) 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:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3}a^{5}$, $\frac{1}{3}a^{6}$, $\frac{1}{3}a^{7}$, $\frac{1}{28733890485}a^{8}-\frac{472207895}{5746778097}a^{7}+\frac{1067277793}{9577963495}a^{6}+\frac{832518307}{28733890485}a^{5}+\frac{2670865738}{9577963495}a^{4}-\frac{1679177157}{9577963495}a^{3}-\frac{2901029378}{9577963495}a^{2}-\frac{34988759}{1915592699}a-\frac{100071575}{1915592699}$ 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:  $C_{9}$, which has order $9$
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:  $C_{9}$, which has order $9$
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:   $\frac{157716848}{28733890485}a^{8}+\frac{76832660}{5746778097}a^{7}-\frac{119397156}{9577963495}a^{6}-\frac{3987574258}{9577963495}a^{5}-\frac{13759048286}{9577963495}a^{4}-\frac{35347383621}{9577963495}a^{3}-\frac{54328419504}{9577963495}a^{2}+\frac{19220361822}{1915592699}a+\frac{8631688384}{1915592699}$, $\frac{22806}{31403159}a^{8}-\frac{43274}{31403159}a^{7}+\frac{94590}{31403159}a^{6}-\frac{228276}{31403159}a^{5}-\frac{4190106}{31403159}a^{4}+\frac{2221154}{31403159}a^{3}-\frac{29231910}{31403159}a^{2}+\frac{9486540}{31403159}a+\frac{9225761}{31403159}$, $\frac{37427474}{9577963495}a^{8}+\frac{179478355}{5746778097}a^{7}-\frac{231413672}{28733890485}a^{6}-\frac{2286057967}{9577963495}a^{5}-\frac{21430951319}{9577963495}a^{4}-\frac{48737265659}{9577963495}a^{3}-\frac{249559609156}{9577963495}a^{2}-\frac{36978920944}{1915592699}a-\frac{7791350039}{1915592699}$, $\frac{99851844}{9577963495}a^{8}+\frac{634434895}{1915592699}a^{7}+\frac{3255622571}{9577963495}a^{6}-\frac{7548249687}{9577963495}a^{5}-\frac{237696792044}{9577963495}a^{4}-\frac{572069708694}{9577963495}a^{3}-\frac{2275119449031}{9577963495}a^{2}-\frac{326547301971}{1915592699}a-\frac{67637451004}{1915592699}$ 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:  \( 24173.5233815 \)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 1 \)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 24173.5233815 \cdot 9}{2\cdot\sqrt{47091202575755625}}\cr\approx \mathstrut & 1.5625432507 \end{aligned}\]

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 + 9*x^7 - 63*x^6 - 81*x^5 - 1296*x^4 - 1359*x^3 - 6885*x^2 - 5265*x - 1125) 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 + 9*x^7 - 63*x^6 - 81*x^5 - 1296*x^4 - 1359*x^3 - 6885*x^2 - 5265*x - 1125, 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 + 9*x^7 - 63*x^6 - 81*x^5 - 1296*x^4 - 1359*x^3 - 6885*x^2 - 5265*x - 1125); 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 + 9*x^7 - 63*x^6 - 81*x^5 - 1296*x^4 - 1359*x^3 - 6885*x^2 - 5265*x - 1125); 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()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma: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

3.1.2835.1

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 ${\href{/padicField/2.6.0.1}{6} }{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ R R R ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.9.0.1}{9} }$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{3}$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.6.0.1}{6} }{,}\,{\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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
\(3\) Copy content Toggle raw display 3.1.9.22a3.8$x^{9} + 18 x^{8} + 9 x^{7} + 6 x^{6} + 18 x^{5} + 57$$9$$1$$22$$C_9$$$[2, 3]$$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$$[\ ]$$
5.1.2.1a1.2$x^{2} + 10$$2$$1$$1$$C_2$$$[\ ]_{2}$$
5.3.2.3a1.1$x^{6} + 6 x^{4} + 6 x^{3} + 9 x^{2} + 23 x + 9$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$
\(7\) Copy content Toggle raw display $\Q_{7}$$x + 4$$1$$1$$0$Trivial$$[\ ]$$
7.1.2.1a1.2$x^{2} + 21$$2$$1$$1$$C_2$$$[\ ]_{2}$$
7.3.2.3a1.1$x^{6} + 12 x^{5} + 36 x^{4} + 8 x^{3} + 48 x^{2} + 7 x + 16$$2$$3$$3$$C_6$$$[\ ]_{2}^{3}$$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
*54 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.35.2t1.a.a$1$ $ 5 \cdot 7 $ \(\Q(\sqrt{-35}) \) $C_2$ (as 2T1) $1$ $-1$
1.315.6t1.h.a$1$ $ 3^{2} \cdot 5 \cdot 7 $ 6.0.281302875.3 $C_6$ (as 6T1) $0$ $-1$
1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.315.6t1.h.b$1$ $ 3^{2} \cdot 5 \cdot 7 $ 6.0.281302875.3 $C_6$ (as 6T1) $0$ $-1$
1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
*54 2.2835.3t2.a.a$2$ $ 3^{4} \cdot 5 \cdot 7 $ 3.1.2835.1 $S_3$ (as 3T2) $1$ $0$
2.315.6t5.a.a$2$ $ 3^{2} \cdot 5 \cdot 7 $ 6.0.3472875.1 $S_3\times C_3$ (as 6T5) $0$ $0$
2.315.6t5.a.b$2$ $ 3^{2} \cdot 5 \cdot 7 $ 6.0.3472875.1 $S_3\times C_3$ (as 6T5) $0$ $0$
*54 6.166...875.9t10.a.a$6$ $ 3^{18} \cdot 5^{3} \cdot 7^{3}$ 9.1.47091202575755625.1 $(C_9:C_3):C_2$ (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)