Properties

Label 9.1.6915325248.1
Degree $9$
Signature $[1, 4]$
Discriminant $6915325248$
Root discriminant \(12.40\)
Ramified primes $2,3,13,31$
Class number $1$
Class group trivial
Galois group $S_3\wr S_3$ (as 9T31)

Related objects

Downloads

Learn more

Show commands: Magma / Oscar / PariGP / SageMath

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 + 3*x^7 + 2*x^6 + 8*x^5 - 3*x^4 + 15*x^3 + 13*x^2 - 1)
 
gp: K = bnfinit(y^9 - y^8 + 3*y^7 + 2*y^6 + 8*y^5 - 3*y^4 + 15*y^3 + 13*y^2 - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^9 - x^8 + 3*x^7 + 2*x^6 + 8*x^5 - 3*x^4 + 15*x^3 + 13*x^2 - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^9 - x^8 + 3*x^7 + 2*x^6 + 8*x^5 - 3*x^4 + 15*x^3 + 13*x^2 - 1)
 

\( x^{9} - x^{8} + 3x^{7} + 2x^{6} + 8x^{5} - 3x^{4} + 15x^{3} + 13x^{2} - 1 \) 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:  $[1, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(6915325248\) \(\medspace = 2^{6}\cdot 3^{2}\cdot 13\cdot 31^{4}\) 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:  \(12.40\)
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))
 
Ramified primes:   \(2\), \(3\), \(13\), \(31\) 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:  \(\Q(\sqrt{13}) \)
$\card{ \Aut(K/\Q) }$:  $1$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{8}-\frac{1}{3}a^{6}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}$ 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:   $\frac{19}{3}a^{8}-\frac{28}{3}a^{7}+\frac{70}{3}a^{6}+2a^{5}+\frac{149}{3}a^{4}-42a^{3}+\frac{344}{3}a^{2}+\frac{92}{3}a-\frac{44}{3}$, $\frac{31}{3}a^{8}-15a^{7}+\frac{113}{3}a^{6}+4a^{5}+\frac{242}{3}a^{4}-\frac{202}{3}a^{3}+185a^{2}+53a-\frac{73}{3}$, $2a^{8}-\frac{8}{3}a^{7}+\frac{22}{3}a^{6}+a^{5}+16a^{4}-\frac{34}{3}a^{3}+\frac{106}{3}a^{2}+\frac{28}{3}a-\frac{14}{3}$, $\frac{49}{3}a^{8}-\frac{71}{3}a^{7}+60a^{6}+6a^{5}+\frac{386}{3}a^{4}-\frac{317}{3}a^{3}+\frac{886}{3}a^{2}+\frac{244}{3}a-37$ 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:  \( 18.2781597266 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{4}\cdot 18.2781597266 \cdot 1}{2\cdot\sqrt{6915325248}}\cr\approx \mathstrut & 0.342567088077 \end{aligned}\]

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 + 3*x^7 + 2*x^6 + 8*x^5 - 3*x^4 + 15*x^3 + 13*x^2 - 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 - x^8 + 3*x^7 + 2*x^6 + 8*x^5 - 3*x^4 + 15*x^3 + 13*x^2 - 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 - x^8 + 3*x^7 + 2*x^6 + 8*x^5 - 3*x^4 + 15*x^3 + 13*x^2 - 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 - x^8 + 3*x^7 + 2*x^6 + 8*x^5 - 3*x^4 + 15*x^3 + 13*x^2 - 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

$S_3\wr S_3$ (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

3.1.31.1

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

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 ${\href{/padicField/5.6.0.1}{6} }{,}\,{\href{/padicField/5.3.0.1}{3} }$ ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.3.0.1}{3} }$ ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }$ R ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ ${\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.3.0.1}{3} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ R ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.3.0.1}{3} }$ ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.2.0.1}{2} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{3}$ ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.3.0.1}{3} }$

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
\(2\) Copy content Toggle raw display 2.9.6.1$x^{9} + 3 x^{7} + 9 x^{6} + 3 x^{5} - 26 x^{3} + 9 x^{2} - 27 x + 29$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
\(3\) Copy content Toggle raw display 3.2.0.1$x^{2} + 2 x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
3.3.0.1$x^{3} + 2 x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
3.4.2.2$x^{4} - 6 x^{3} + 12 x^{2} + 36 x + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
\(13\) Copy content Toggle raw display $\Q_{13}$$x + 11$$1$$1$$0$Trivial$[\ ]$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.6.0.1$x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
\(31\) Copy content Toggle raw display $\Q_{31}$$x + 28$$1$$1$$0$Trivial$[\ ]$
31.2.1.2$x^{2} + 31$$2$$1$$1$$C_2$$[\ ]_{2}$
31.2.1.1$x^{2} + 93$$2$$1$$1$$C_2$$[\ ]_{2}$
31.4.2.1$x^{4} + 58 x^{3} + 909 x^{2} + 1972 x + 26855$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.31.2t1.a.a$1$ $ 31 $ \(\Q(\sqrt{-31}) \) $C_2$ (as 2T1) $1$ $-1$
1.403.2t1.a.a$1$ $ 13 \cdot 31 $ \(\Q(\sqrt{-403}) \) $C_2$ (as 2T1) $1$ $-1$
1.13.2t1.a.a$1$ $ 13 $ \(\Q(\sqrt{13}) \) $C_2$ (as 2T1) $1$ $1$
2.5239.6t3.b.a$2$ $ 13^{2} \cdot 31 $ 6.2.2111317.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.31.3t2.b.a$2$ $ 31 $ 3.1.31.1 $S_3$ (as 3T2) $1$ $0$
3.47151.4t5.a.a$3$ $ 3^{2} \cdot 13^{2} \cdot 31 $ 4.2.47151.1 $S_4$ (as 4T5) $1$ $1$
3.112437.6t11.b.a$3$ $ 3^{2} \cdot 13 \cdot 31^{2}$ 6.0.3485547.1 $S_4\times C_2$ (as 6T11) $1$ $-1$
3.1461681.6t8.a.a$3$ $ 3^{2} \cdot 13^{2} \cdot 31^{2}$ 4.2.47151.1 $S_4$ (as 4T5) $1$ $-1$
3.3627.6t11.b.a$3$ $ 3^{2} \cdot 13 \cdot 31 $ 6.0.3485547.1 $S_4\times C_2$ (as 6T11) $1$ $1$
* 6.223075008.9t31.a.a$6$ $ 2^{6} \cdot 3^{2} \cdot 13 \cdot 31^{3}$ 9.1.6915325248.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
6.637...488.18t300.a.a$6$ $ 2^{6} \cdot 3^{2} \cdot 13^{5} \cdot 31^{3}$ 9.1.6915325248.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
6.223075008.18t319.a.a$6$ $ 2^{6} \cdot 3^{2} \cdot 13 \cdot 31^{3}$ 9.1.6915325248.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
6.637...488.18t311.a.a$6$ $ 2^{6} \cdot 3^{2} \cdot 13^{5} \cdot 31^{3}$ 9.1.6915325248.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
8.136...704.24t2893.a.a$8$ $ 2^{6} \cdot 3^{4} \cdot 13^{4} \cdot 31^{4}$ 9.1.6915325248.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
8.136...704.12t213.a.a$8$ $ 2^{6} \cdot 3^{4} \cdot 13^{4} \cdot 31^{4}$ 9.1.6915325248.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
12.108...376.36t2217.a.a$12$ $ 2^{6} \cdot 3^{6} \cdot 13^{8} \cdot 31^{5}$ 9.1.6915325248.1 $S_3\wr S_3$ (as 9T31) $1$ $2$
12.104...336.36t2214.a.a$12$ $ 2^{6} \cdot 3^{6} \cdot 13^{8} \cdot 31^{7}$ 9.1.6915325248.1 $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.115...224.36t2210.a.a$12$ $ 2^{12} \cdot 3^{8} \cdot 13^{6} \cdot 31^{6}$ 9.1.6915325248.1 $S_3\wr S_3$ (as 9T31) $1$ $0$
12.366...776.36t2216.a.a$12$ $ 2^{6} \cdot 3^{6} \cdot 13^{4} \cdot 31^{7}$ 9.1.6915325248.1 $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.381...416.18t315.a.a$12$ $ 2^{6} \cdot 3^{6} \cdot 13^{4} \cdot 31^{5}$ 9.1.6915325248.1 $S_3\wr S_3$ (as 9T31) $1$ $2$
16.186...616.24t2912.a.a$16$ $ 2^{12} \cdot 3^{8} \cdot 13^{8} \cdot 31^{8}$ 9.1.6915325248.1 $S_3\wr S_3$ (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 *.