Properties

Label 9.3.1191988823.1
Degree $9$
Signature $[3, 3]$
Discriminant $-1191988823$
Root discriminant \(10.20\)
Ramified primes $23,313$
Class number $1$
Class group trivial
Galois group $C_3 \wr S_3 $ (as 9T20)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 6*x^7 - 3*x^6 + 12*x^5 + 14*x^4 - 7*x^3 - 22*x^2 - 9*x + 1)
 
gp: K = bnfinit(y^9 - 6*y^7 - 3*y^6 + 12*y^5 + 14*y^4 - 7*y^3 - 22*y^2 - 9*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 6*x^7 - 3*x^6 + 12*x^5 + 14*x^4 - 7*x^3 - 22*x^2 - 9*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 6*x^7 - 3*x^6 + 12*x^5 + 14*x^4 - 7*x^3 - 22*x^2 - 9*x + 1)
 

\( x^{9} - 6x^{7} - 3x^{6} + 12x^{5} + 14x^{4} - 7x^{3} - 22x^{2} - 9x + 1 \) 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:  $[3, 3]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(-1191988823\) \(\medspace = -\,23^{3}\cdot 313^{2}\) 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:  \(10.20\)
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:  $23^{1/2}313^{2/3}\approx 221.0854404093889$
Ramified primes:   \(23\), \(313\) 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:  \(\Q(\sqrt{-23}) \)
$\card{ \Aut(K/\Q) }$:  $3$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{391}a^{8}+\frac{192}{391}a^{7}+\frac{104}{391}a^{6}+\frac{24}{391}a^{5}-\frac{72}{391}a^{4}-\frac{125}{391}a^{3}-\frac{156}{391}a^{2}+\frac{133}{391}a+\frac{112}{391}$ 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:  $5$
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:   $\frac{75}{391}a^{8}-\frac{67}{391}a^{7}-\frac{411}{391}a^{6}+\frac{236}{391}a^{5}+\frac{856}{391}a^{4}+\frac{9}{391}a^{3}-\frac{1143}{391}a^{2}-\frac{582}{391}a+\frac{580}{391}$, $\frac{41}{391}a^{8}+\frac{52}{391}a^{7}-\frac{428}{391}a^{6}-\frac{189}{391}a^{5}+\frac{958}{391}a^{4}+\frac{740}{391}a^{3}-\frac{531}{391}a^{2}-\frac{1585}{391}a-\frac{100}{391}$, $\frac{128}{391}a^{8}-\frac{57}{391}a^{7}-\frac{764}{391}a^{6}-\frac{56}{391}a^{5}+\frac{1732}{391}a^{4}+\frac{1204}{391}a^{3}-\frac{1591}{391}a^{2}-\frac{2526}{391}a-\frac{522}{391}$, $\frac{93}{391}a^{8}-\frac{130}{391}a^{7}-\frac{494}{391}a^{6}+\frac{277}{391}a^{5}+\frac{1124}{391}a^{4}+\frac{496}{391}a^{3}-\frac{1214}{391}a^{2}-\frac{1316}{391}a-\frac{141}{391}$, $\frac{144}{391}a^{8}-\frac{113}{391}a^{7}-\frac{664}{391}a^{6}-\frac{63}{391}a^{5}+\frac{1362}{391}a^{4}+\frac{1159}{391}a^{3}-\frac{959}{391}a^{2}-\frac{1962}{391}a-\frac{685}{391}$ 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:  \( 7.82049862039 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 7.82049862039 \cdot 1}{2\cdot\sqrt{1191988823}}\cr\approx \mathstrut & 0.224749093420 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 6*x^7 - 3*x^6 + 12*x^5 + 14*x^4 - 7*x^3 - 22*x^2 - 9*x + 1)
 
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 - 6*x^7 - 3*x^6 + 12*x^5 + 14*x^4 - 7*x^3 - 22*x^2 - 9*x + 1, 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 - 6*x^7 - 3*x^6 + 12*x^5 + 14*x^4 - 7*x^3 - 22*x^2 - 9*x + 1);
 
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 - 6*x^7 - 3*x^6 + 12*x^5 + 14*x^4 - 7*x^3 - 22*x^2 - 9*x + 1);
 
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

$C_3\wr S_3$ (as 9T20):

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 162
The 22 conjugacy class representatives for $C_3 \wr S_3 $
Character table for $C_3 \wr S_3 $ is not computed

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

Degree 9 siblings: data not computed
Degree 18 siblings: data not computed
Degree 27 siblings: 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 ${\href{/padicField/2.9.0.1}{9} }$ ${\href{/padicField/3.9.0.1}{9} }$ ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.9.0.1}{9} }$ ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}$ ${\href{/padicField/19.2.0.1}{2} }^{3}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ R ${\href{/padicField/29.9.0.1}{9} }$ ${\href{/padicField/31.9.0.1}{9} }$ ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ ${\href{/padicField/41.9.0.1}{9} }$ ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}$ ${\href{/padicField/47.9.0.1}{9} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.3.0.1}{3} }^{3}$

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

# 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
\(23\) Copy content Toggle raw display $\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
$\Q_{23}$$x + 18$$1$$1$$0$Trivial$[\ ]$
$\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}$
\(313\) Copy content Toggle raw display Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $2$$1$$2$$0$$C_2$$[\ ]^{2}$
Deg $3$$3$$1$$2$