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Properties

Label 9.1.1355457106081.1
Degree $9$
Signature $[1, 4]$
Discriminant $13^{4}\cdot 83^{4}$
Root discriminant $22.28$
Ramified primes $13, 83$
Class number $1$
Class group Trivial
Galois group $(C_9:C_3):C_2$ (as 9T10)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, -137, 100, -30, 18, 6, -5, 4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 + 4*x^7 - 5*x^6 + 6*x^5 + 18*x^4 - 30*x^3 + 100*x^2 - 137*x + 64)
 
gp: K = bnfinit(x^9 - x^8 + 4*x^7 - 5*x^6 + 6*x^5 + 18*x^4 - 30*x^3 + 100*x^2 - 137*x + 64, 1)
 

Normalized defining polynomial

\( x^{9} - x^{8} + 4 x^{7} - 5 x^{6} + 6 x^{5} + 18 x^{4} - 30 x^{3} + 100 x^{2} - 137 x + 64 \)

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:  \(1355457106081=13^{4}\cdot 83^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $22.28$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $13, 83$
magma: PrimeDivisors(Discriminant(Integers(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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{1676002} a^{8} - \frac{161009}{838001} a^{7} - \frac{324716}{838001} a^{6} - \frac{33217}{1676002} a^{5} + \frac{193931}{1676002} a^{4} + \frac{431713}{1676002} a^{3} + \frac{412743}{1676002} a^{2} + \frac{48073}{1676002} a - \frac{384453}{838001}$

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:  \( 597.385656575 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9:C_3$ (as 9T10):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $(C_9:C_3):C_2$
Character table for $(C_9:C_3):C_2$

Intermediate fields

3.1.83.1

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

Sibling fields

Degree 18 sibling: data not computed
Degree 27 sibling: 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 ${\href{/LocalNumberField/2.6.0.1}{6} }{,}\,{\href{/LocalNumberField/2.2.0.1}{2} }{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.9.0.1}{9} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.9.0.1}{9} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }$ R ${\href{/LocalNumberField/17.9.0.1}{9} }$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
$83$$\Q_{83}$$x + 3$$1$$1$$0$Trivial$[\ ]$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$
83.2.1.2$x^{2} + 249$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.83.2t1.1c1$1$ $ 83 $ $x^{2} - x + 21$ $C_2$ (as 2T1) $1$ $-1$
1.13.3t1.1c1$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.13_83.6t1.1c1$1$ $ 13 \cdot 83 $ $x^{6} - x^{5} + 54 x^{4} - 31 x^{3} + 1317 x^{2} - 758 x + 13151$ $C_6$ (as 6T1) $0$ $-1$
1.13.3t1.1c2$1$ $ 13 $ $x^{3} - x^{2} - 4 x - 1$ $C_3$ (as 3T1) $0$ $1$
1.13_83.6t1.1c2$1$ $ 13 \cdot 83 $ $x^{6} - x^{5} + 54 x^{4} - 31 x^{3} + 1317 x^{2} - 758 x + 13151$ $C_6$ (as 6T1) $0$ $-1$
* 2.83.3t2.1c1$2$ $ 83 $ $x^{3} - x^{2} + x - 2$ $S_3$ (as 3T2) $1$ $0$
2.13e2_83.6t5.1c1$2$ $ 13^{2} \cdot 83 $ $x^{6} - x^{5} + 3 x^{4} + 68 x^{3} + 176 x^{2} - 199 x + 3879$ $S_3\times C_3$ (as 6T5) $0$ $0$
2.13e2_83.6t5.1c2$2$ $ 13^{2} \cdot 83 $ $x^{6} - x^{5} + 3 x^{4} + 68 x^{3} + 176 x^{2} - 199 x + 3879$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 6.13e4_83e3.9t10.1c1$6$ $ 13^{4} \cdot 83^{3}$ $x^{9} - x^{8} + 4 x^{7} - 5 x^{6} + 6 x^{5} + 18 x^{4} - 30 x^{3} + 100 x^{2} - 137 x + 64$ $(C_9:C_3):C_2$ (as 9T10) $1$ $0$

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 *.