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Properties

Label 9.3.21793521976875.2
Degree $9$
Signature $[3, 3]$
Discriminant $-2.179\times 10^{13}$
Root discriminant \(30.34\)
Ramified primes $3,5,11$
Class number $1$
Class group trivial
Galois group $((C_3^2:Q_8):C_3):C_2$ (as 9T26)

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Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 16*x^6 - 33*x^5 - 36*x^4 + 120*x^3 + 99*x^2 - 297*x + 77)
 
gp: K = bnfinit(y^9 - 3*y^8 + 16*y^6 - 33*y^5 - 36*y^4 + 120*y^3 + 99*y^2 - 297*y + 77, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 3*x^8 + 16*x^6 - 33*x^5 - 36*x^4 + 120*x^3 + 99*x^2 - 297*x + 77);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 3*x^8 + 16*x^6 - 33*x^5 - 36*x^4 + 120*x^3 + 99*x^2 - 297*x + 77)
 

\( x^{9} - 3x^{8} + 16x^{6} - 33x^{5} - 36x^{4} + 120x^{3} + 99x^{2} - 297x + 77 \) 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:   \(-21793521976875\) \(\medspace = -\,3^{9}\cdot 5^{4}\cdot 11^{6}\) 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:  \(30.34\)
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:  $3^{7/6}5^{1/2}11^{2/3}\approx 39.84632346086567$
Ramified primes:   \(3\), \(5\), \(11\) 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:  \(\Q(\sqrt{-3}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{7326961}a^{8}+\frac{2887079}{7326961}a^{7}+\frac{2383307}{7326961}a^{6}+\frac{2376363}{7326961}a^{5}+\frac{1044202}{7326961}a^{4}-\frac{3285805}{7326961}a^{3}-\frac{3544087}{7326961}a^{2}+\frac{1944621}{7326961}a-\frac{3580625}{7326961}$ 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:   $\frac{257883}{7326961}a^{8}-\frac{548258}{7326961}a^{7}-\frac{437443}{7326961}a^{6}+\frac{3928450}{7326961}a^{5}-\frac{5853267}{7326961}a^{4}-\frac{12192048}{7326961}a^{3}+\frac{19981241}{7326961}a^{2}+\frac{34813464}{7326961}a-\frac{53345577}{7326961}$, $\frac{147464}{7326961}a^{8}-\frac{178210}{7326961}a^{7}-\frac{354839}{7326961}a^{6}+\frac{1429685}{7326961}a^{5}-\frac{1208648}{7326961}a^{4}-\frac{6017590}{7326961}a^{3}-\frac{444199}{7326961}a^{2}+\frac{13645448}{7326961}a-\frac{3167496}{7326961}$, $\frac{67296}{7326961}a^{8}-\frac{156453}{7326961}a^{7}-\frac{148418}{7326961}a^{6}+\frac{1473662}{7326961}a^{5}-\frac{2265159}{7326961}a^{4}-\frac{1177261}{7326961}a^{3}+\frac{4355720}{7326961}a^{2}-\frac{1635605}{7326961}a+\frac{26407}{7326961}$, $\frac{631771}{7326961}a^{8}-\frac{884431}{7326961}a^{7}-\frac{892725}{7326961}a^{6}+\frac{8266051}{7326961}a^{5}-\frac{8372776}{7326961}a^{4}-\frac{31027979}{7326961}a^{3}+\frac{23931757}{7326961}a^{2}+\frac{80237726}{7326961}a-\frac{67713423}{7326961}$, $\frac{726450}{7326961}a^{8}-\frac{2065817}{7326961}a^{7}-\frac{187189}{7326961}a^{6}+\frac{10947101}{7326961}a^{5}-\frac{21710313}{7326961}a^{4}-\frac{24995514}{7326961}a^{3}+\frac{70113367}{7326961}a^{2}+\frac{83133377}{7326961}a-\frac{169126743}{7326961}$ 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:  \( 4612.22721653 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 4612.22721653 \cdot 1}{2\cdot\sqrt{21793521976875}}\cr\approx \mathstrut & 0.980271960238 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 16*x^6 - 33*x^5 - 36*x^4 + 120*x^3 + 99*x^2 - 297*x + 77)
 
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 - 3*x^8 + 16*x^6 - 33*x^5 - 36*x^4 + 120*x^3 + 99*x^2 - 297*x + 77, 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 - 3*x^8 + 16*x^6 - 33*x^5 - 36*x^4 + 120*x^3 + 99*x^2 - 297*x + 77);
 
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 - 3*x^8 + 16*x^6 - 33*x^5 - 36*x^4 + 120*x^3 + 99*x^2 - 297*x + 77);
 
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

$C_3^2:\GL(2,3)$ (as 9T26):

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 432
The 11 conjugacy class representatives for $((C_3^2:Q_8):C_3):C_2$
Character table for $((C_3^2:Q_8):C_3):C_2$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.
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 12 sibling: data not computed
Degree 18 sibling: data not computed
Degree 24 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 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 ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.1.0.1}{1} }$ R R ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{3}$ R ${\href{/padicField/13.3.0.1}{3} }^{3}$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 3.9.9.6$x^{9} - 6 x^{8} + 45 x^{7} + 594 x^{6} + 99 x^{5} + 108 x^{4} - 54 x^{3} + 27 x^{2} + 81 x + 27$$3$$3$$9$$S_3\times C_3$$[3/2]_{2}^{3}$
\(5\) Copy content Toggle raw display $\Q_{5}$$x + 3$$1$$1$$0$Trivial$[\ ]$
5.8.4.2$x^{8} + 100 x^{4} - 500 x^{2} + 1250$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
\(11\) Copy content Toggle raw display 11.3.2.1$x^{3} + 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.3.2.1$x^{3} + 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
11.3.2.1$x^{3} + 11$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
2.3267.3t2.b.a$2$ $ 3^{3} \cdot 11^{2}$ 3.1.3267.1 $S_3$ (as 3T2) $1$ $0$
2.81675.24t22.a.a$2$ $ 3^{3} \cdot 5^{2} \cdot 11^{2}$ 8.2.20012416875.4 $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
2.81675.24t22.a.b$2$ $ 3^{3} \cdot 5^{2} \cdot 11^{2}$ 8.2.20012416875.4 $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
3.3267.4t5.c.a$3$ $ 3^{3} \cdot 11^{2}$ 4.2.3267.1 $S_4$ (as 4T5) $1$ $1$
3.9801.6t8.e.a$3$ $ 3^{4} \cdot 11^{2}$ 4.2.3267.1 $S_4$ (as 4T5) $1$ $-1$
4.6125625.8t23.a.a$4$ $ 3^{4} \cdot 5^{4} \cdot 11^{2}$ 8.2.20012416875.4 $\textrm{GL(2,3)}$ (as 8T23) $1$ $0$
* 8.217...875.9t26.a.a$8$ $ 3^{9} \cdot 5^{4} \cdot 11^{6}$ 9.3.21793521976875.2 $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $2$
8.196...875.18t157.a.a$8$ $ 3^{11} \cdot 5^{4} \cdot 11^{6}$ 9.3.21793521976875.2 $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $-2$
16.392...625.24t1334.a.a$16$ $ 3^{18} \cdot 5^{8} \cdot 11^{10}$ 9.3.21793521976875.2 $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $0$

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 *.