Properties

Label 9.1.7148031401.1
Degree $9$
Signature $[1, 4]$
Discriminant $7\cdot 19^{3}\cdot 53^{3}$
Root discriminant $12.44$
Ramified primes $7, 19, 53$
Class number $1$
Class group Trivial
Galois group $S_3\wr S_3$ (as 9T31)

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Show commands for: SageMath / Pari/GP / Magma

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

Normalized defining polynomial

\( x^{9} - 3 x^{8} + 5 x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 2 x + 1 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $9$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[1, 4]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(7148031401=7\cdot 19^{3}\cdot 53^{3}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $12.44$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $7, 19, 53$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $1$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{149} a^{8} + \frac{54}{149} a^{7} - \frac{46}{149} a^{6} + \frac{58}{149} a^{5} + \frac{23}{149} a^{4} - \frac{22}{149} a^{3} - \frac{58}{149} a^{2} - \frac{32}{149} a - \frac{34}{149}$

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 

Class group and class number

Trivial group, which has order $1$

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $4$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 20.5268225838 \)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

3.1.1007.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

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 ${\href{/LocalNumberField/2.9.0.1}{9} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }$ ${\href{/LocalNumberField/5.6.0.1}{6} }{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ R ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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

sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 
magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.0.1$x^{2} - x + 3$$1$$2$$0$$C_2$$[\ ]^{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.3.0.1$x^{3} - x + 2$$1$$3$$0$$C_3$$[\ ]^{3}$
$19$$\Q_{19}$$x + 4$$1$$1$$0$Trivial$[\ ]$
19.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
19.6.3.1$x^{6} - 38 x^{4} + 361 x^{2} - 109744$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$53$53.2.1.2$x^{2} + 106$$2$$1$$1$$C_2$$[\ ]_{2}$
53.3.0.1$x^{3} - x + 8$$1$$3$$0$$C_3$$[\ ]^{3}$
53.4.2.1$x^{4} + 477 x^{2} + 70225$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ $x$ $C_1$ $1$ $1$
1.7049.2t1.a.a$1$ $ 7 \cdot 19 \cdot 53 $ $x^{2} - x - 1762$ $C_2$ (as 2T1) $1$ $1$
1.1007.2t1.a.a$1$ $ 19 \cdot 53 $ $x^{2} - x + 252$ $C_2$ (as 2T1) $1$ $-1$
1.7.2t1.a.a$1$ $ 7 $ $x^{2} - x + 2$ $C_2$ (as 2T1) $1$ $-1$
2.49343.6t3.a.a$2$ $ 7^{2} \cdot 19 \cdot 53 $ $x^{6} - 2 x^{5} + 25 x^{4} - 132 x^{3} + 252 x^{2} - 1296 x - 4133$ $D_{6}$ (as 6T3) $1$ $0$
* 2.1007.3t2.a.a$2$ $ 19 \cdot 53 $ $x^{3} - x^{2} - 2 x + 7$ $S_3$ (as 3T2) $1$ $0$
3.49688401.6t8.a.a$3$ $ 7^{2} \cdot 19^{2} \cdot 53^{2}$ $x^{4} - x^{3} + x - 6$ $S_4$ (as 4T5) $1$ $-1$
3.49343.4t5.a.a$3$ $ 7^{2} \cdot 19 \cdot 53 $ $x^{4} - x^{3} + x - 6$ $S_4$ (as 4T5) $1$ $1$
3.7098343.6t11.a.a$3$ $ 7 \cdot 19^{2} \cdot 53^{2}$ $x^{6} - x^{5} + x^{4} + 5 x^{3} + x^{2} - x + 1$ $S_4\times C_2$ (as 6T11) $1$ $1$
3.7049.6t11.a.a$3$ $ 7 \cdot 19 \cdot 53 $ $x^{6} - x^{5} + x^{4} + 5 x^{3} + x^{2} - x + 1$ $S_4\times C_2$ (as 6T11) $1$ $-1$
6.7198067620807.18t319.a.a$6$ $ 7 \cdot 19^{4} \cdot 53^{4}$ $x^{9} - 3 x^{8} + 5 x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 2 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.17043121543.18t311.a.a$6$ $ 7^{5} \cdot 19^{2} \cdot 53^{2}$ $x^{9} - 3 x^{8} + 5 x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 2 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
* 6.7098343.9t31.a.a$6$ $ 7 \cdot 19^{2} \cdot 53^{2}$ $x^{9} - 3 x^{8} + 5 x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 2 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.17282560357557607.18t300.a.a$6$ $ 7^{5} \cdot 19^{4} \cdot 53^{4}$ $x^{9} - 3 x^{8} + 5 x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 2 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2503623292574419117249.24t2893.a.a$8$ $ 7^{4} \cdot 19^{6} \cdot 53^{6}$ $x^{9} - 3 x^{8} + 5 x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 2 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2434731649.12t213.a.a$8$ $ 7^{4} \cdot 19^{2} \cdot 53^{2}$ $x^{9} - 3 x^{8} + 5 x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 2 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.6053277922149478562618452943.36t2217.a.a$12$ $ 7^{8} \cdot 19^{7} \cdot 53^{7}$ $x^{9} - 3 x^{8} + 5 x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 2 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
12.122677541336146536745201.36t2210.a.a$12$ $ 7^{6} \cdot 19^{6} \cdot 53^{6}$ $x^{9} - 3 x^{8} + 5 x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 2 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.5969413630060755015407.36t2214.a.a$12$ $ 7^{8} \cdot 19^{5} \cdot 53^{5}$ $x^{9} - 3 x^{8} + 5 x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 2 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.2521148655622440051069743.36t2216.a.a$12$ $ 7^{4} \cdot 19^{7} \cdot 53^{7}$ $x^{9} - 3 x^{8} + 5 x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 2 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.2486219754294358607.18t315.a.a$12$ $ 7^{4} \cdot 19^{5} \cdot 53^{5}$ $x^{9} - 3 x^{8} + 5 x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 2 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
16.6095650867604524912556782113601.24t2912.a.a$16$ $ 7^{8} \cdot 19^{8} \cdot 53^{8}$ $x^{9} - 3 x^{8} + 5 x^{7} - 2 x^{6} - 5 x^{5} + 8 x^{4} + 4 x^{3} - 4 x^{2} + 2 x + 1$ $S_3\wr S_3$ (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 *.