Properties

Label 9.1.7952095936.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{6}\cdot 499^{3}$
Root discriminant $12.59$
Ramified primes $2, 499$
Class number $1$
Class group Trivial
Galois Group $S_3\wr S_3$ (as 9T31)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 6, 14, 5, -10, -2, 3, 2, -2, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 + 2*x^7 + 3*x^6 - 2*x^5 - 10*x^4 + 5*x^3 + 14*x^2 + 6*x + 1)
gp: K = bnfinit(x^9 - 2*x^8 + 2*x^7 + 3*x^6 - 2*x^5 - 10*x^4 + 5*x^3 + 14*x^2 + 6*x + 1, 1)

Normalized defining polynomial

\(x^{9} \) \(\mathstrut -\mathstrut 2 x^{8} \) \(\mathstrut +\mathstrut 2 x^{7} \) \(\mathstrut +\mathstrut 3 x^{6} \) \(\mathstrut -\mathstrut 2 x^{5} \) \(\mathstrut -\mathstrut 10 x^{4} \) \(\mathstrut +\mathstrut 5 x^{3} \) \(\mathstrut +\mathstrut 14 x^{2} \) \(\mathstrut +\mathstrut 6 x \) \(\mathstrut +\mathstrut 1 \)

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

Invariants

Degree:  $9$
magma: Degree(K);
sage: K.degree()
gp: poldegree(K.pol)
Signature:  $[1, 4]$
magma: Signature(K);
sage: K.signature()
gp: K.sign
Discriminant:  \(7952095936=2^{6}\cdot 499^{3}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $12.59$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 499$
magma: PrimeDivisors(Discriminant(K));
sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{318} a^{8} - \frac{31}{318} a^{7} + \frac{1}{3} a^{6} - \frac{25}{159} a^{5} - \frac{71}{159} a^{4} - \frac{13}{159} a^{3} + \frac{41}{106} a^{2} - \frac{55}{318} a - \frac{74}{159}$

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

Class group and class number

Trivial group, which has order $1$

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

Unit group

magma: UK, f := UnitGroup(K);
sage: UK = K.unit_group()
Rank:  $4$
magma: UnitRank(K);
sage: UK.rank()
gp: K.fu
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
sage: UK.torsion_generator()
gp: K.tu[2]
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
magma: [K!f(g): g in Generators(UK)];
sage: UK.fundamental_units()
gp: K.fu
Regulator:  \( 55.5982508837 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

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

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

3.1.499.1

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 R ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.4.0.1}{4} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }$

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

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];
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]])

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
$2$2.2.0.1$x^{2} - x + 1$$1$$2$$0$$C_2$$[\ ]^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.4.4.4$x^{4} - 5$$2$$2$$4$$D_{4}$$[2, 2]^{2}$
499Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.499.2t1.1c1$1$ $ 499 $ $x^{2} - x + 125$ $C_2$ (as 2T1) $1$ $-1$
1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
1.2e2_499.2t1.1c1$1$ $ 2^{2} \cdot 499 $ $x^{2} - 499$ $C_2$ (as 2T1) $1$ $1$
2.2e4_499.6t3.1c1$2$ $ 2^{4} \cdot 499 $ $x^{6} - 24 x^{4} + 144 x^{2} - 499$ $D_{6}$ (as 6T3) $1$ $0$
* 2.499.3t2.1c1$2$ $ 499 $ $x^{3} + 4 x - 3$ $S_3$ (as 3T2) $1$ $0$
3.2e6_499.4t5.2c1$3$ $ 2^{6} \cdot 499 $ $x^{4} - 2 x^{3} - 4 x^{2} - 4 x + 6$ $S_4$ (as 4T5) $1$ $1$
3.2e4_499.6t11.1c1$3$ $ 2^{4} \cdot 499 $ $x^{6} + 4 x^{4} - 5 x^{2} + 4$ $S_4\times C_2$ (as 6T11) $1$ $-1$
3.2e6_499e2.6t8.2c1$3$ $ 2^{6} \cdot 499^{2}$ $x^{4} - 2 x^{3} - 4 x^{2} - 4 x + 6$ $S_4$ (as 4T5) $1$ $-1$
3.2e4_499e2.6t11.1c1$3$ $ 2^{4} \cdot 499^{2}$ $x^{6} + 4 x^{4} - 5 x^{2} + 4$ $S_4\times C_2$ (as 6T11) $1$ $1$
* 6.2e6_499e2.9t31.1c1$6$ $ 2^{6} \cdot 499^{2}$ $x^{9} - 2 x^{8} + 2 x^{7} + 3 x^{6} - 2 x^{5} - 10 x^{4} + 5 x^{3} + 14 x^{2} + 6 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e12_499e4.18t303.1c1$6$ $ 2^{12} \cdot 499^{4}$ $x^{9} - 2 x^{8} + 2 x^{7} + 3 x^{6} - 2 x^{5} - 10 x^{4} + 5 x^{3} + 14 x^{2} + 6 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e6_499e4.18t320.1c1$6$ $ 2^{6} \cdot 499^{4}$ $x^{9} - 2 x^{8} + 2 x^{7} + 3 x^{6} - 2 x^{5} - 10 x^{4} + 5 x^{3} + 14 x^{2} + 6 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
6.2e12_499e2.18t312.1c1$6$ $ 2^{12} \cdot 499^{2}$ $x^{9} - 2 x^{8} + 2 x^{7} + 3 x^{6} - 2 x^{5} - 10 x^{4} + 5 x^{3} + 14 x^{2} + 6 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2e14_499e6.24t2895.1c1$8$ $ 2^{14} \cdot 499^{6}$ $x^{9} - 2 x^{8} + 2 x^{7} + 3 x^{6} - 2 x^{5} - 10 x^{4} + 5 x^{3} + 14 x^{2} + 6 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
8.2e14_499e2.12t213.1c1$8$ $ 2^{14} \cdot 499^{2}$ $x^{9} - 2 x^{8} + 2 x^{7} + 3 x^{6} - 2 x^{5} - 10 x^{4} + 5 x^{3} + 14 x^{2} + 6 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2e22_499e7.36t2305.1c1$12$ $ 2^{22} \cdot 499^{7}$ $x^{9} - 2 x^{8} + 2 x^{7} + 3 x^{6} - 2 x^{5} - 10 x^{4} + 5 x^{3} + 14 x^{2} + 6 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
12.2e22_499e5.36t2215.1c1$12$ $ 2^{22} \cdot 499^{5}$ $x^{9} - 2 x^{8} + 2 x^{7} + 3 x^{6} - 2 x^{5} - 10 x^{4} + 5 x^{3} + 14 x^{2} + 6 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.2e20_499e6.36t2210.1c1$12$ $ 2^{20} \cdot 499^{6}$ $x^{9} - 2 x^{8} + 2 x^{7} + 3 x^{6} - 2 x^{5} - 10 x^{4} + 5 x^{3} + 14 x^{2} + 6 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $0$
12.2e18_499e7.36t2218.1c1$12$ $ 2^{18} \cdot 499^{7}$ $x^{9} - 2 x^{8} + 2 x^{7} + 3 x^{6} - 2 x^{5} - 10 x^{4} + 5 x^{3} + 14 x^{2} + 6 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $-2$
12.2e18_499e5.18t315.1c1$12$ $ 2^{18} \cdot 499^{5}$ $x^{9} - 2 x^{8} + 2 x^{7} + 3 x^{6} - 2 x^{5} - 10 x^{4} + 5 x^{3} + 14 x^{2} + 6 x + 1$ $S_3\wr S_3$ (as 9T31) $1$ $2$
16.2e28_499e8.24t2912.1c1$16$ $ 2^{28} \cdot 499^{8}$ $x^{9} - 2 x^{8} + 2 x^{7} + 3 x^{6} - 2 x^{5} - 10 x^{4} + 5 x^{3} + 14 x^{2} + 6 x + 1$ $S_3\wr S_3$ (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 *.