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Properties

Label 9.1.114897217668536241.1
Degree $9$
Signature $[1, 4]$
Discriminant $1.149\times 10^{17}$
Root discriminant \(78.63\)
Ramified primes $3,17,19$
Class number $9$ (GRH)
Class group [9] (GRH)
Galois group $D_{9}$ (as 9T3)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 - 13*x^7 - 39*x^6 + 127*x^5 + 185*x^4 + 2178*x^3 - 449*x^2 + 24992*x + 22592)
 
gp: K = bnfinit(y^9 - 2*y^8 - 13*y^7 - 39*y^6 + 127*y^5 + 185*y^4 + 2178*y^3 - 449*y^2 + 24992*y + 22592, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 2*x^8 - 13*x^7 - 39*x^6 + 127*x^5 + 185*x^4 + 2178*x^3 - 449*x^2 + 24992*x + 22592);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - 2*x^8 - 13*x^7 - 39*x^6 + 127*x^5 + 185*x^4 + 2178*x^3 - 449*x^2 + 24992*x + 22592)
 

\( x^{9} - 2x^{8} - 13x^{7} - 39x^{6} + 127x^{5} + 185x^{4} + 2178x^{3} - 449x^{2} + 24992x + 22592 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $9$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[1, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(114897217668536241\) \(\medspace = 3^{4}\cdot 17^{4}\cdot 19^{8}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(78.63\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $3^{1/2}17^{1/2}19^{8/9}\approx 97.82614855618401$
Ramified primes:   \(3\), \(17\), \(19\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $1$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6}a^{4}-\frac{1}{3}a^{2}-\frac{1}{2}a-\frac{1}{3}$, $\frac{1}{18}a^{5}-\frac{1}{18}a^{4}-\frac{4}{9}a^{3}-\frac{1}{18}a^{2}-\frac{5}{18}a-\frac{2}{9}$, $\frac{1}{144}a^{6}+\frac{1}{48}a^{5}-\frac{1}{16}a^{4}+\frac{1}{48}a^{3}+\frac{19}{48}a^{2}-\frac{17}{48}a-\frac{5}{18}$, $\frac{1}{432}a^{7}-\frac{1}{216}a^{5}-\frac{17}{216}a^{4}+\frac{13}{27}a^{3}+\frac{1}{216}a^{2}-\frac{37}{144}a+\frac{19}{54}$, $\frac{1}{273910896}a^{8}-\frac{43249}{273910896}a^{7}+\frac{567193}{273910896}a^{6}+\frac{2748913}{273910896}a^{5}-\frac{3009361}{273910896}a^{4}-\frac{134936717}{273910896}a^{3}+\frac{4571975}{19565064}a^{2}-\frac{32852453}{136955448}a-\frac{5714438}{17119431}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

Monogenic:  No
Index:  Not computed
Inessential primes:  $2$, $3$

Class group and class number

$C_{9}$, which has order $9$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $4$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{617663}{273910896}a^{8}-\frac{654197}{273910896}a^{7}-\frac{1806251}{136955448}a^{6}-\frac{17789983}{68477724}a^{5}+\frac{33826769}{136955448}a^{4}+\frac{213252139}{136955448}a^{3}+\frac{296601755}{39130128}a^{2}-\frac{3744187157}{273910896}a+\frac{4245984685}{34238862}$, $\frac{3824995}{273910896}a^{8}-\frac{3609713}{136955448}a^{7}-\frac{54123185}{273910896}a^{6}-\frac{184916507}{273910896}a^{5}+\frac{528936119}{273910896}a^{4}+\frac{1581651973}{273910896}a^{3}+\frac{366689575}{9782532}a^{2}-\frac{7151534473}{273910896}a+\frac{7173529955}{34238862}$, $\frac{391471}{136955448}a^{8}-\frac{6198623}{273910896}a^{7}-\frac{655663}{68477724}a^{6}+\frac{54425557}{136955448}a^{5}-\frac{71759437}{136955448}a^{4}-\frac{160167961}{68477724}a^{3}+\frac{53152525}{4891266}a^{2}-\frac{3982440839}{273910896}a-\frac{633939491}{34238862}$, $\frac{23623}{136955448}a^{8}+\frac{210565}{273910896}a^{7}-\frac{1917673}{273910896}a^{6}-\frac{3638869}{273910896}a^{5}+\frac{13090027}{273910896}a^{4}+\frac{75628661}{273910896}a^{3}-\frac{30554659}{39130128}a^{2}+\frac{178569091}{68477724}a+\frac{49324157}{17119431}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 291204.143728 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 291204.143728 \cdot 9}{2\cdot\sqrt{114897217668536241}}\cr\approx \mathstrut & 12.0504919629 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 - 13*x^7 - 39*x^6 + 127*x^5 + 185*x^4 + 2178*x^3 - 449*x^2 + 24992*x + 22592)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^9 - 2*x^8 - 13*x^7 - 39*x^6 + 127*x^5 + 185*x^4 + 2178*x^3 - 449*x^2 + 24992*x + 22592, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^9 - 2*x^8 - 13*x^7 - 39*x^6 + 127*x^5 + 185*x^4 + 2178*x^3 - 449*x^2 + 24992*x + 22592);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 2*x^8 - 13*x^7 - 39*x^6 + 127*x^5 + 185*x^4 + 2178*x^3 - 449*x^2 + 24992*x + 22592);
 
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

$D_9$ (as 9T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A solvable group of order 18
The 6 conjugacy class representatives for $D_{9}$
Character table for $D_{9}$

Intermediate fields

3.1.18411.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Galois closure: 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 ${\href{/padicField/2.2.0.1}{2} }^{4}{,}\,{\href{/padicField/2.1.0.1}{1} }$ R ${\href{/padicField/5.9.0.1}{9} }$ ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }$ ${\href{/padicField/11.9.0.1}{9} }$ ${\href{/padicField/13.3.0.1}{3} }^{3}$ R R ${\href{/padicField/23.9.0.1}{9} }$ ${\href{/padicField/29.3.0.1}{3} }^{3}$ ${\href{/padicField/31.2.0.1}{2} }^{4}{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.9.0.1}{9} }$ ${\href{/padicField/47.2.0.1}{2} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.2.0.1}{2} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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 $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
\(17\) Copy content Toggle raw display $\Q_{17}$$x + 14$$1$$1$$0$Trivial$[\ ]$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
17.2.1.2$x^{2} + 51$$2$$1$$1$$C_2$$[\ ]_{2}$
\(19\) Copy content Toggle raw display 19.9.8.2$x^{9} + 57$$9$$1$$8$$C_9$$[\ ]_{9}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.51.2t1.a.a$1$ $ 3 \cdot 17 $ \(\Q(\sqrt{-51}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.18411.3t2.a.a$2$ $ 3 \cdot 17 \cdot 19^{2}$ 3.1.18411.1 $S_3$ (as 3T2) $1$ $0$
* 2.18411.9t3.a.a$2$ $ 3 \cdot 17 \cdot 19^{2}$ 9.1.114897217668536241.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.18411.9t3.a.b$2$ $ 3 \cdot 17 \cdot 19^{2}$ 9.1.114897217668536241.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.18411.9t3.a.c$2$ $ 3 \cdot 17 \cdot 19^{2}$ 9.1.114897217668536241.1 $D_{9}$ (as 9T3) $1$ $0$

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 *.