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Properties

Label 9.1.6904624074435625.1
Degree $9$
Signature $[1, 4]$
Discriminant $5^{4}\cdot 73^{7}$
Root discriminant $57.53$
Ramified primes $5, 73$
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![-608, 1920, -1654, 1121, -434, 108, -18, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 + 2*x^7 - 18*x^6 + 108*x^5 - 434*x^4 + 1121*x^3 - 1654*x^2 + 1920*x - 608)
 
gp: K = bnfinit(x^9 - 2*x^8 + 2*x^7 - 18*x^6 + 108*x^5 - 434*x^4 + 1121*x^3 - 1654*x^2 + 1920*x - 608, 1)
 

Normalized defining polynomial

\( x^{9} - 2 x^{8} + 2 x^{7} - 18 x^{6} + 108 x^{5} - 434 x^{4} + 1121 x^{3} - 1654 x^{2} + 1920 x - 608 \)

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:  \(6904624074435625=5^{4}\cdot 73^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $57.53$
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:  $5, 73$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{6} - \frac{1}{6} a^{5} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{6} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{5037024} a^{8} + \frac{35461}{2518512} a^{7} - \frac{118019}{2518512} a^{6} + \frac{314819}{2518512} a^{5} - \frac{80805}{419752} a^{4} + \frac{703}{2518512} a^{3} - \frac{1020215}{5037024} a^{2} - \frac{20525}{2518512} a + \frac{68775}{209876}$

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:  \( 170059.082386 \)
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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.8.0.1}{8} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.8.0.1}{8} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.8.0.1}{8} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.8.0.1}{8} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.8.4.2$x^{8} + 25 x^{4} - 250 x^{2} + 1250$$2$$4$$4$$C_8$$[\ ]_{2}^{4}$
$73$$\Q_{73}$$x + 5$$1$$1$$0$Trivial$[\ ]$
73.8.7.1$x^{8} - 73$$8$$1$$7$$C_8$$[\ ]_{8}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.73.2t1.1c1$1$ $ 73 $ $x^{2} - x - 18$ $C_2$ (as 2T1) $1$ $1$
1.73.4t1.1c1$1$ $ 73 $ $x^{4} - x^{3} - 27 x^{2} + 41 x + 2$ $C_4$ (as 4T1) $0$ $1$
1.73.4t1.1c2$1$ $ 73 $ $x^{4} - x^{3} - 27 x^{2} + 41 x + 2$ $C_4$ (as 4T1) $0$ $1$
1.5_73.8t1.1c1$1$ $ 5 \cdot 73 $ $x^{8} - x^{7} + 78 x^{6} + 17 x^{5} + 1706 x^{4} + 3421 x^{3} + 14117 x^{2} + 45478 x + 272444$ $C_8$ (as 8T1) $0$ $-1$
1.5_73.8t1.1c2$1$ $ 5 \cdot 73 $ $x^{8} - x^{7} + 78 x^{6} + 17 x^{5} + 1706 x^{4} + 3421 x^{3} + 14117 x^{2} + 45478 x + 272444$ $C_8$ (as 8T1) $0$ $-1$
1.5_73.8t1.1c3$1$ $ 5 \cdot 73 $ $x^{8} - x^{7} + 78 x^{6} + 17 x^{5} + 1706 x^{4} + 3421 x^{3} + 14117 x^{2} + 45478 x + 272444$ $C_8$ (as 8T1) $0$ $-1$
1.5_73.8t1.1c4$1$ $ 5 \cdot 73 $ $x^{8} - x^{7} + 78 x^{6} + 17 x^{5} + 1706 x^{4} + 3421 x^{3} + 14117 x^{2} + 45478 x + 272444$ $C_8$ (as 8T1) $0$ $-1$
* 8.5e4_73e7.9t15.1c1$8$ $ 5^{4} \cdot 73^{7}$ $x^{9} - 2 x^{8} + 2 x^{7} - 18 x^{6} + 108 x^{5} - 434 x^{4} + 1121 x^{3} - 1654 x^{2} + 1920 x - 608$ $C_3^2:C_8$ (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 *.