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Properties

Label 9.3.2824440448203.2
Degree $9$
Signature $[3, 3]$
Discriminant $-2.824\times 10^{12}$
Root discriminant \(24.18\)
Ramified primes $3,11$
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^7 - 9*x^6 + 3*x^5 + 18*x^4 + 15*x^3 - 9*x^2 - 27*x + 6)
 
gp: K = bnfinit(y^9 - 3*y^7 - 9*y^6 + 3*y^5 + 18*y^4 + 15*y^3 - 9*y^2 - 27*y + 6, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 3*x^7 - 9*x^6 + 3*x^5 + 18*x^4 + 15*x^3 - 9*x^2 - 27*x + 6);
 
oscar: Qx, x = polynomial_ring(QQ); K, a = number_field(x^9 - 3*x^7 - 9*x^6 + 3*x^5 + 18*x^4 + 15*x^3 - 9*x^2 - 27*x + 6)
 

\( x^{9} - 3x^{7} - 9x^{6} + 3x^{5} + 18x^{4} + 15x^{3} - 9x^{2} - 27x + 6 \) 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:   \(-2824440448203\) \(\medspace = -\,3^{13}\cdot 11^{6}\) 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:  \(24.18\)
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:  $3^{85/54}11^{2/3}\approx 27.879564620720714$
Ramified primes:   \(3\), \(11\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{193}a^{8}-\frac{31}{193}a^{7}-\frac{7}{193}a^{6}+\frac{15}{193}a^{5}-\frac{76}{193}a^{4}+\frac{58}{193}a^{3}-\frac{46}{193}a^{2}+\frac{66}{193}a+\frac{50}{193}$ 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:   $\frac{35}{193}a^{8}+\frac{73}{193}a^{7}-\frac{52}{193}a^{6}-\frac{440}{193}a^{5}-\frac{537}{193}a^{4}+\frac{100}{193}a^{3}+\frac{899}{193}a^{2}+\frac{766}{193}a+\frac{13}{193}$, $\frac{1}{193}a^{8}-\frac{31}{193}a^{7}-\frac{7}{193}a^{6}+\frac{15}{193}a^{5}+\frac{310}{193}a^{4}+\frac{58}{193}a^{3}-\frac{46}{193}a^{2}-\frac{513}{193}a-\frac{143}{193}$, $\frac{7}{193}a^{8}-\frac{24}{193}a^{7}-\frac{49}{193}a^{6}-\frac{88}{193}a^{5}+\frac{240}{193}a^{4}+\frac{406}{193}a^{3}+\frac{450}{193}a^{2}-\frac{117}{193}a-\frac{229}{193}$, $\frac{6}{193}a^{8}+\frac{7}{193}a^{7}-\frac{42}{193}a^{6}-\frac{103}{193}a^{5}+\frac{123}{193}a^{4}+\frac{155}{193}a^{3}+\frac{110}{193}a^{2}-\frac{376}{193}a+\frac{107}{193}$, $\frac{167}{193}a^{8}+\frac{227}{193}a^{7}-\frac{204}{193}a^{6}-\frac{1741}{193}a^{5}-\frac{1884}{193}a^{4}+\frac{422}{193}a^{3}+\frac{2933}{193}a^{2}+\frac{2723}{193}a-\frac{721}{193}$ 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:  \( 1522.69940225 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{3}\cdot 1522.69940225 \cdot 1}{2\cdot\sqrt{2824440448203}}\cr\approx \mathstrut & 0.898974892546 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^7 - 9*x^6 + 3*x^5 + 18*x^4 + 15*x^3 - 9*x^2 - 27*x + 6)
 
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^7 - 9*x^6 + 3*x^5 + 18*x^4 + 15*x^3 - 9*x^2 - 27*x + 6, 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^7 - 9*x^6 + 3*x^5 + 18*x^4 + 15*x^3 - 9*x^2 - 27*x + 6);
 
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^7 - 9*x^6 + 3*x^5 + 18*x^4 + 15*x^3 - 9*x^2 - 27*x + 6);
 
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 ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ R ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }$ ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.1.0.1}{1} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.6.0.1}{6} }{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }$ ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }$ ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.3.0.1}{3} }$ ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.1.0.1}{1} }$

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 3.9.13.7$x^{9} + 3 x^{5} + 6 x^{3} + 3$$9$$1$$13$$C_3^2 : D_{6} $$[3/2, 3/2, 5/3]_{2}^{2}$
\(11\) Copy content Toggle raw display 11.9.6.1$x^{9} + 6 x^{7} + 60 x^{6} + 12 x^{5} + 42 x^{4} - 1465 x^{3} + 240 x^{2} - 1560 x + 8088$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$

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) $1$ $0$
2.3267.24t22.b.a$2$ $ 3^{3} \cdot 11^{2}$ 8.2.32019867.2 $\textrm{GL(2,3)}$ (as 8T23) $0$ $0$
2.3267.24t22.b.b$2$ $ 3^{3} \cdot 11^{2}$ 8.2.32019867.2 $\textrm{GL(2,3)}$ (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) $1$ $-1$
4.9801.8t23.b.a$4$ $ 3^{4} \cdot 11^{2}$ 8.2.32019867.2 $\textrm{GL(2,3)}$ (as 8T23) $1$ $0$
* 8.282...203.9t26.a.a$8$ $ 3^{13} \cdot 11^{6}$ 9.3.2824440448203.2 $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $2$
8.282...203.18t157.a.a$8$ $ 3^{13} \cdot 11^{6}$ 9.3.2824440448203.2 $((C_3^2:Q_8):C_3):C_2$ (as 9T26) $1$ $-2$
16.659...729.24t1334.a.a$16$ $ 3^{26} \cdot 11^{10}$ 9.3.2824440448203.2 $((C_3^2:Q_8):C_3):C_2$ (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 *.