Properties

Label 9.1.43192866448384.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{10}\cdot 59^{6}$
Root discriminant $32.74$
Ramified primes $2, 59$
Class number $2$
Class group $[2]$
Galois Group $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![44, 48, -25, -77, -25, 39, 21, -7, -3, 1]);
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 - 7*x^7 + 21*x^6 + 39*x^5 - 25*x^4 - 77*x^3 - 25*x^2 + 48*x + 44)
gp: K = bnfinit(x^9 - 3*x^8 - 7*x^7 + 21*x^6 + 39*x^5 - 25*x^4 - 77*x^3 - 25*x^2 + 48*x + 44, 1)

Normalized defining polynomial

\(x^{9} \) \(\mathstrut -\mathstrut 3 x^{8} \) \(\mathstrut -\mathstrut 7 x^{7} \) \(\mathstrut +\mathstrut 21 x^{6} \) \(\mathstrut +\mathstrut 39 x^{5} \) \(\mathstrut -\mathstrut 25 x^{4} \) \(\mathstrut -\mathstrut 77 x^{3} \) \(\mathstrut -\mathstrut 25 x^{2} \) \(\mathstrut +\mathstrut 48 x \) \(\mathstrut +\mathstrut 44 \)

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:  \(43192866448384=2^{10}\cdot 59^{6}\)
magma: Discriminant(K);
sage: K.disc()
gp: K.disc
Root discriminant:  $32.74$
magma: Abs(Discriminant(K))^(1/Degree(K));
sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
Ramified primes:  $2, 59$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{5} - \frac{1}{2} a^{3} - \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{68} a^{8} - \frac{3}{34} a^{7} - \frac{3}{34} a^{6} + \frac{5}{68} a^{5} + \frac{7}{68} a^{4} + \frac{11}{34} a^{3} - \frac{6}{17} a^{2} - \frac{21}{68} a + \frac{13}{34}$

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

Class group and class number

$C_{2}$, which has order $2$

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:  \( 5397.11697558 \)
magma: Regulator(K);
sage: K.regulator()
gp: K.reg

Galois group

$(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30):

magma: GaloisGroup(K);
sage: K.galois_group(type='pari')
gp: polgalois(K.pol)
A solvable group of order 648
The 14 conjugacy class representatives for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$
Character table for $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$

Intermediate fields

3.1.13924.1

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

Sibling fields

Degree 12 siblings: 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 ${\href{/LocalNumberField/3.4.0.1}{4} }{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.4.0.1}{4} }{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{5}$ ${\href{/LocalNumberField/31.4.0.1}{4} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }$ ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }$ R

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.6.8.1$x^{6} + 2 x^{3} + 2$$6$$1$$8$$D_{6}$$[2]_{3}^{2}$
59Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2.2t1.1c1$1$ $ 2^{2}$ $x^{2} + 1$ $C_2$ (as 2T1) $1$ $-1$
* 2.2e2_59e2.3t2.1c1$2$ $ 2^{2} \cdot 59^{2}$ $x^{3} - x^{2} + 20 x + 24$ $S_3$ (as 3T2) $1$ $0$
3.2e4_59e2.6t8.1c1$3$ $ 2^{4} \cdot 59^{2}$ $x^{4} - x^{3} - 2 x^{2} - x - 5$ $S_4$ (as 4T5) $1$ $-1$
3.2e2_59e2.4t5.1c1$3$ $ 2^{2} \cdot 59^{2}$ $x^{4} - x^{3} - 2 x^{2} - x - 5$ $S_4$ (as 4T5) $1$ $1$
6.2e8_59e4.18t222.1c1$6$ $ 2^{8} \cdot 59^{4}$ $x^{9} - 3 x^{8} - 7 x^{7} + 21 x^{6} + 39 x^{5} - 25 x^{4} - 77 x^{3} - 25 x^{2} + 48 x + 44$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
* 6.2e8_59e4.9t30.1c1$6$ $ 2^{8} \cdot 59^{4}$ $x^{9} - 3 x^{8} - 7 x^{7} + 21 x^{6} + 39 x^{5} - 25 x^{4} - 77 x^{3} - 25 x^{2} + 48 x + 44$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
6.2e8_59e4.36t1130.1c1$6$ $ 2^{8} \cdot 59^{4}$ $x^{9} - 3 x^{8} - 7 x^{7} + 21 x^{6} + 39 x^{5} - 25 x^{4} - 77 x^{3} - 25 x^{2} + 48 x + 44$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
6.2e8_59e4.36t1130.1c2$6$ $ 2^{8} \cdot 59^{4}$ $x^{9} - 3 x^{8} - 7 x^{7} + 21 x^{6} + 39 x^{5} - 25 x^{4} - 77 x^{3} - 25 x^{2} + 48 x + 44$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.2e10_59e6.12t177.1c1$8$ $ 2^{10} \cdot 59^{6}$ $x^{9} - 3 x^{8} - 7 x^{7} + 21 x^{6} + 39 x^{5} - 25 x^{4} - 77 x^{3} - 25 x^{2} + 48 x + 44$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.2e10_59e6.12t177.2c1$8$ $ 2^{10} \cdot 59^{6}$ $x^{9} - 3 x^{8} - 7 x^{7} + 21 x^{6} + 39 x^{5} - 25 x^{4} - 77 x^{3} - 25 x^{2} + 48 x + 44$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
8.2e10_59e4.12t178.1c1$8$ $ 2^{10} \cdot 59^{4}$ $x^{9} - 3 x^{8} - 7 x^{7} + 21 x^{6} + 39 x^{5} - 25 x^{4} - 77 x^{3} - 25 x^{2} + 48 x + 44$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $0$
12.2e18_59e8.36t1124.1c1$12$ $ 2^{18} \cdot 59^{8}$ $x^{9} - 3 x^{8} - 7 x^{7} + 21 x^{6} + 39 x^{5} - 25 x^{4} - 77 x^{3} - 25 x^{2} + 48 x + 44$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $-2$
12.2e16_59e8.18t218.1c1$12$ $ 2^{16} \cdot 59^{8}$ $x^{9} - 3 x^{8} - 7 x^{7} + 21 x^{6} + 39 x^{5} - 25 x^{4} - 77 x^{3} - 25 x^{2} + 48 x + 44$ $(((C_3 \times (C_3^2 : C_2)) : C_2) : C_3) : C_2$ (as 9T30) $1$ $2$

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 *.