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Properties

Label 9.1.495755401801.1
Degree $9$
Signature $[1, 4]$
Discriminant $495755401801$
Root discriminant \(19.93\)
Ramified primes $11,23$
Class number $1$
Class group trivial
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 - 4*x^8 + 4*x^7 - x^6 + 5*x^5 + 40*x^4 - 17*x^3 - 82*x^2 - 38*x - 23)
 
gp: K = bnfinit(y^9 - 4*y^8 + 4*y^7 - y^6 + 5*y^5 + 40*y^4 - 17*y^3 - 82*y^2 - 38*y - 23, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 4*x^8 + 4*x^7 - x^6 + 5*x^5 + 40*x^4 - 17*x^3 - 82*x^2 - 38*x - 23);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 4*x^8 + 4*x^7 - x^6 + 5*x^5 + 40*x^4 - 17*x^3 - 82*x^2 - 38*x - 23)
 

\( x^{9} - 4x^{8} + 4x^{7} - x^{6} + 5x^{5} + 40x^{4} - 17x^{3} - 82x^{2} - 38x - 23 \) 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:   \(495755401801\) \(\medspace = 11^{6}\cdot 23^{4}\) 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:  \(19.93\)
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:  $11^{2/3}23^{1/2}\approx 23.72060207739333$
Ramified primes:   \(11\), \(23\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{7}a^{7}+\frac{3}{7}a^{6}+\frac{2}{7}a^{5}+\frac{1}{7}a^{3}-\frac{2}{7}a^{2}+\frac{2}{7}a-\frac{1}{7}$, $\frac{1}{950635}a^{8}-\frac{8973}{190127}a^{7}-\frac{348746}{950635}a^{6}-\frac{36983}{190127}a^{5}+\frac{19001}{190127}a^{4}-\frac{9216}{27161}a^{3}+\frac{205588}{950635}a^{2}+\frac{62645}{190127}a-\frac{386633}{950635}$ Copy content Toggle raw display

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

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Trivial group, which has order $1$

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:   $\frac{5322}{950635}a^{8}-\frac{5268}{190127}a^{7}+\frac{20723}{950635}a^{6}+\frac{12241}{190127}a^{5}-\frac{24242}{190127}a^{4}+\frac{63659}{190127}a^{3}-\frac{44737}{135805}a^{2}-\frac{221873}{190127}a+\frac{328144}{950635}$, $\frac{11812}{950635}a^{8}-\frac{6854}{190127}a^{7}-\frac{14687}{950635}a^{6}+\frac{5927}{27161}a^{5}-\frac{100175}{190127}a^{4}+\frac{284882}{190127}a^{3}-\frac{1281799}{950635}a^{2}-\frac{38705}{190127}a-\frac{66553}{135805}$, $\frac{15597}{950635}a^{8}-\frac{18409}{190127}a^{7}+\frac{142108}{950635}a^{6}+\frac{21467}{190127}a^{5}-\frac{49396}{190127}a^{4}-\frac{5940}{27161}a^{3}+\frac{64181}{950635}a^{2}+\frac{11412}{190127}a+\frac{513539}{950635}$, $\frac{11346}{950635}a^{8}-\frac{8230}{190127}a^{7}-\frac{57636}{950635}a^{6}-\frac{25990}{190127}a^{5}-\frac{18672}{190127}a^{4}+\frac{117281}{190127}a^{3}+\frac{118514}{135805}a^{2}+\frac{48283}{190127}a+\frac{35092}{950635}$ Copy content Toggle raw display
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:  \( 182.493109815 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 182.493109815 \cdot 1}{2\cdot\sqrt{495755401801}}\cr\approx \mathstrut & 0.403954283634 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^8 + 4*x^7 - x^6 + 5*x^5 + 40*x^4 - 17*x^3 - 82*x^2 - 38*x - 23)
 
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 - 4*x^8 + 4*x^7 - x^6 + 5*x^5 + 40*x^4 - 17*x^3 - 82*x^2 - 38*x - 23, 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 - 4*x^8 + 4*x^7 - x^6 + 5*x^5 + 40*x^4 - 17*x^3 - 82*x^2 - 38*x - 23);
 
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 - 4*x^8 + 4*x^7 - x^6 + 5*x^5 + 40*x^4 - 17*x^3 - 82*x^2 - 38*x - 23);
 
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

3.1.23.1

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 ${\href{/padicField/2.9.0.1}{9} }$ ${\href{/padicField/3.9.0.1}{9} }$ ${\href{/padicField/5.2.0.1}{2} }^{4}{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }$ R ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.2.0.1}{2} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.2.0.1}{2} }^{4}{,}\,{\href{/padicField/19.1.0.1}{1} }$ R ${\href{/padicField/29.9.0.1}{9} }$ ${\href{/padicField/31.9.0.1}{9} }$ ${\href{/padicField/37.2.0.1}{2} }^{4}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.2.0.1}{2} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }$ ${\href{/padicField/59.3.0.1}{3} }^{3}$

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

# 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
\(11\) Copy content Toggle raw display 11.3.2.1$x^{3} + 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 21 x^{5} + 153 x^{4} + 449 x^{3} + 537 x^{2} + 1569 x + 3440$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
\(23\) Copy content Toggle raw display $\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$
23.2.1.2$x^{2} + 23$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.23.2t1.a.a$1$ $ 23 $ \(\Q(\sqrt{-23}) \) $C_2$ (as 2T1) $1$ $-1$
* 2.23.3t2.b.a$2$ $ 23 $ 3.1.23.1 $S_3$ (as 3T2) $1$ $0$
* 2.2783.9t3.a.a$2$ $ 11^{2} \cdot 23 $ 9.1.495755401801.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.2783.9t3.a.b$2$ $ 11^{2} \cdot 23 $ 9.1.495755401801.1 $D_{9}$ (as 9T3) $1$ $0$
* 2.2783.9t3.a.c$2$ $ 11^{2} \cdot 23 $ 9.1.495755401801.1 $D_{9}$ (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 *.