Properties

Label 9.1.39217774848.1
Degree $9$
Signature $[1, 4]$
Discriminant $39217774848$
Root discriminant \(15.03\)
Ramified primes $2,3,7,53$
Class number $1$
Class group trivial
Galois group $S_3\wr S_3$ (as 9T31)

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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 + 20*x^5 - 16*x^4 - 4*x^3 - 4*x^2 + 20*x - 12)
 
gp: K = bnfinit(y^9 - 6*y^7 + 20*y^5 - 16*y^4 - 4*y^3 - 4*y^2 + 20*y - 12, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - 6*x^7 + 20*x^5 - 16*x^4 - 4*x^3 - 4*x^2 + 20*x - 12);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - 6*x^7 + 20*x^5 - 16*x^4 - 4*x^3 - 4*x^2 + 20*x - 12)
 

\( x^{9} - 6x^{7} + 20x^{5} - 16x^{4} - 4x^{3} - 4x^{2} + 20x - 12 \) 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:   \(39217774848\) \(\medspace = 2^{8}\cdot 3\cdot 7^{3}\cdot 53^{3}\) 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:  \(15.03\)
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:  $2^{8/9}3^{1/2}7^{1/2}53^{1/2}\approx 61.777424773632966$
Ramified primes:   \(2\), \(3\), \(7\), \(53\) 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:  \(\Q(\sqrt{1113}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{78}a^{8}+\frac{11}{78}a^{7}-\frac{1}{39}a^{6}+\frac{17}{78}a^{5}+\frac{2}{13}a^{4}+\frac{19}{39}a^{3}+\frac{4}{13}a^{2}+\frac{1}{3}a-\frac{1}{13}$ 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:   $a^{8}+a^{7}-5a^{6}-5a^{5}+15a^{4}-a^{3}-5a^{2}-9a+11$, $\frac{1}{13}a^{8}-\frac{2}{13}a^{7}-\frac{17}{26}a^{6}+\frac{21}{26}a^{5}+\frac{38}{13}a^{4}-\frac{40}{13}a^{3}-\frac{28}{13}a^{2}-a+\frac{59}{13}$, $\frac{35}{78}a^{8}+\frac{17}{39}a^{7}-\frac{187}{78}a^{6}-\frac{185}{78}a^{5}+\frac{96}{13}a^{4}+\frac{2}{39}a^{3}-\frac{55}{13}a^{2}-\frac{13}{3}a+\frac{69}{13}$, $\frac{10}{13}a^{8}+\frac{51}{26}a^{7}-\frac{27}{26}a^{6}-\frac{167}{26}a^{5}+\frac{16}{13}a^{4}+\frac{42}{13}a^{3}+\frac{32}{13}a^{2}-4a+\frac{5}{13}$ 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:  \( 181.78033132 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 181.78033132 \cdot 1}{2\cdot\sqrt{39217774848}}\cr\approx \mathstrut & 1.4306219682 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - 6*x^7 + 20*x^5 - 16*x^4 - 4*x^3 - 4*x^2 + 20*x - 12)
 
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 + 20*x^5 - 16*x^4 - 4*x^3 - 4*x^2 + 20*x - 12, 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 + 20*x^5 - 16*x^4 - 4*x^3 - 4*x^2 + 20*x - 12);
 
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 + 20*x^5 - 16*x^4 - 4*x^3 - 4*x^2 + 20*x - 12);
 
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

$S_3\wr S_3$ (as 9T31):

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 1296
The 22 conjugacy class representatives for $S_3\wr S_3$
Character table for $S_3\wr S_3$ is not computed

Intermediate fields

3.1.1484.1

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 12 sibling: data not computed
Degree 18 siblings: data not computed
Degree 24 siblings: data not computed
Degree 27 siblings: 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 R R ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ R ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.3.0.1}{3} }$ ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }$ ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ ${\href{/padicField/19.9.0.1}{9} }$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.3.0.1}{3} }$ ${\href{/padicField/31.9.0.1}{9} }$ ${\href{/padicField/37.9.0.1}{9} }$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }$ R ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }$

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
\(2\) Copy content Toggle raw display 2.9.8.1$x^{9} + 2$$9$$1$$8$$(C_9:C_3):C_2$$[\ ]_{9}^{6}$
\(3\) Copy content Toggle raw display $\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.0.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.2.1.1$x^{2} + 6$$2$$1$$1$$C_2$$[\ ]_{2}$
3.3.0.1$x^{3} + 2 x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
\(7\) Copy content Toggle raw display 7.2.1.2$x^{2} + 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.3.0.1$x^{3} + 6 x^{2} + 4$$1$$3$$0$$C_3$$[\ ]^{3}$
7.4.2.1$x^{4} + 12 x^{3} + 56 x^{2} + 120 x + 268$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
\(53\) Copy content Toggle raw display $\Q_{53}$$x + 51$$1$$1$$0$Trivial$[\ ]$
53.2.0.1$x^{2} + 49 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
53.6.3.1$x^{6} + 7314 x^{5} + 17831697 x^{4} + 14491896314 x^{3} + 998947242 x^{2} + 44403234204 x + 739971484841$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.371.2t1.a.a$1$ $ 7 \cdot 53 $ \(\Q(\sqrt{-371}) \) $C_2$ (as 2T1) $1$ $-1$
1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
1.1113.2t1.a.a$1$ $ 3 \cdot 7 \cdot 53 $ \(\Q(\sqrt{1113}) \) $C_2$ (as 2T1) $1$ $1$
2.13356.6t3.a.a$2$ $ 2^{2} \cdot 3^{2} \cdot 7 \cdot 53 $ 6.2.22059998352.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.1484.3t2.a.a$2$ $ 2^{2} \cdot 7 \cdot 53 $ 3.1.1484.1 $S_3$ (as 3T2) $1$ $0$
3.13356.4t5.a.a$3$ $ 2^{2} \cdot 3^{2} \cdot 7 \cdot 53 $ 4.2.13356.1 $S_4$ (as 4T5) $1$ $1$
3.1651692.6t11.a.a$3$ $ 2^{2} \cdot 3 \cdot 7^{2} \cdot 53^{2}$ 6.0.6606768.1 $S_4\times C_2$ (as 6T11) $1$ $1$
3.4955076.6t8.a.a$3$ $ 2^{2} \cdot 3^{2} \cdot 7^{2} \cdot 53^{2}$ 4.2.13356.1 $S_4$ (as 4T5) $1$ $-1$
3.4452.6t11.a.a$3$ $ 2^{2} \cdot 3 \cdot 7 \cdot 53 $ 6.0.6606768.1 $S_4\times C_2$ (as 6T11) $1$ $-1$
* 6.26427072.9t31.a.a$6$ $ 2^{6} \cdot 3 \cdot 7^{2} \cdot 53^{2}$ 9.1.39217774848.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
6.294...312.18t300.a.a$6$ $ 2^{6} \cdot 3^{5} \cdot 7^{4} \cdot 53^{4}$ 9.1.39217774848.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
6.363...152.18t319.a.a$6$ $ 2^{6} \cdot 3 \cdot 7^{4} \cdot 53^{4}$ 9.1.39217774848.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2140592832.18t311.a.a$6$ $ 2^{6} \cdot 3^{5} \cdot 7^{2} \cdot 53^{2}$ 9.1.39217774848.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
8.540...656.24t2893.a.a$8$ $ 2^{8} \cdot 3^{4} \cdot 7^{6} \cdot 53^{6}$ 9.1.39217774848.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2854123776.12t213.a.a$8$ $ 2^{8} \cdot 3^{4} \cdot 7^{2} \cdot 53^{2}$ 9.1.39217774848.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
12.649...824.36t2217.a.a$12$ $ 2^{10} \cdot 3^{8} \cdot 7^{7} \cdot 53^{7}$ 9.1.39217774848.1 $S_3\wr S_3$ (as 9T31) $1$ $2$
12.472...864.36t2214.a.a$12$ $ 2^{10} \cdot 3^{8} \cdot 7^{5} \cdot 53^{5}$ 9.1.39217774848.1 $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.778...464.36t2210.a.a$12$ $ 2^{12} \cdot 3^{6} \cdot 7^{6} \cdot 53^{6}$ 9.1.39217774848.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
12.802...504.36t2216.a.a$12$ $ 2^{10} \cdot 3^{4} \cdot 7^{7} \cdot 53^{7}$ 9.1.39217774848.1 $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.582...344.18t315.a.a$12$ $ 2^{10} \cdot 3^{4} \cdot 7^{5} \cdot 53^{5}$ 9.1.39217774848.1 $S_3\wr S_3$ (as 9T31) $1$ $2$
16.385...264.24t2912.a.a$16$ $ 2^{14} \cdot 3^{8} \cdot 7^{8} \cdot 53^{8}$ 9.1.39217774848.1 $S_3\wr S_3$ (as 9T31) $1$ $0$

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 *.