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Properties

Label 9.1.61334280470528.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{31}\cdot 13^{4}$
Root discriminant $34.04$
Ramified primes $2, 13$
Class number $1$
Class group Trivial
Galois group $C_3^2:C_8$ (as 9T15)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![100, -98, -32, 8, 16, 40, 0, 0, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 + 40*x^5 + 16*x^4 + 8*x^3 - 32*x^2 - 98*x + 100)
 
gp: K = bnfinit(x^9 - 2*x^8 + 40*x^5 + 16*x^4 + 8*x^3 - 32*x^2 - 98*x + 100, 1)
 

Normalized defining polynomial

\( x^{9} - 2 x^{8} + 40 x^{5} + 16 x^{4} + 8 x^{3} - 32 x^{2} - 98 x + 100 \)

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:  \(61334280470528=2^{31}\cdot 13^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.04$
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:  $2, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{20701726} a^{8} - \frac{2295416}{10350863} a^{7} + \frac{2944322}{10350863} a^{6} - \frac{301450}{10350863} a^{5} - \frac{4679580}{10350863} a^{4} + \frac{2200086}{10350863} a^{3} - \frac{3958921}{10350863} a^{2} + \frac{4882056}{10350863} a + \frac{3454782}{10350863}$

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:  \( 9790.60405402 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$F_9$ (as 9T15):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 72
The 9 conjugacy class representatives for $C_3^2:C_8$
Character table for $C_3^2:C_8$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 12 sibling: data not computed
Degree 18 sibling: data not computed
Degree 24 sibling: data not computed
Degree 36 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 R ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.8.0.1}{8} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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

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$$0$Trivial$[\ ]$
2.8.31.2$x^{8} + 24 x^{6} + 4 x^{4} + 16 x^{2} + 34$$8$$1$$31$$C_8$$[3, 4, 5]$
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.8.4.2$x^{8} + 169 x^{4} - 2197 x^{2} + 57122$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
1.2e4.4t1.1c1$1$ $ 2^{4}$ $x^{4} - 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $1$
1.2e4.4t1.1c2$1$ $ 2^{4}$ $x^{4} - 4 x^{2} + 2$ $C_4$ (as 4T1) $0$ $1$
1.2e5_13.8t1.1c1$1$ $ 2^{5} \cdot 13 $ $x^{8} + 104 x^{6} + 3380 x^{4} + 35152 x^{2} + 57122$ $C_8$ (as 8T1) $0$ $-1$
1.2e5_13.8t1.1c2$1$ $ 2^{5} \cdot 13 $ $x^{8} + 104 x^{6} + 3380 x^{4} + 35152 x^{2} + 57122$ $C_8$ (as 8T1) $0$ $-1$
1.2e5_13.8t1.1c3$1$ $ 2^{5} \cdot 13 $ $x^{8} + 104 x^{6} + 3380 x^{4} + 35152 x^{2} + 57122$ $C_8$ (as 8T1) $0$ $-1$
1.2e5_13.8t1.1c4$1$ $ 2^{5} \cdot 13 $ $x^{8} + 104 x^{6} + 3380 x^{4} + 35152 x^{2} + 57122$ $C_8$ (as 8T1) $0$ $-1$
* 8.2e31_13e4.9t15.1c1$8$ $ 2^{31} \cdot 13^{4}$ $x^{9} - 2 x^{8} + 40 x^{5} + 16 x^{4} + 8 x^{3} - 32 x^{2} - 98 x + 100$ $C_3^2:C_8$ (as 9T15) $1$ $0$

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 *.