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Properties

Label 9.1.33554432000000.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{31}\cdot 5^{6}$
Root discriminant $31.83$
Ramified primes $2, 5$
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![-25, 121, -84, 132, -82, 50, -28, 12, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 + 12*x^7 - 28*x^6 + 50*x^5 - 82*x^4 + 132*x^3 - 84*x^2 + 121*x - 25)
 
gp: K = bnfinit(x^9 - x^8 + 12*x^7 - 28*x^6 + 50*x^5 - 82*x^4 + 132*x^3 - 84*x^2 + 121*x - 25, 1)
 

Normalized defining polynomial

\( x^{9} - x^{8} + 12 x^{7} - 28 x^{6} + 50 x^{5} - 82 x^{4} + 132 x^{3} - 84 x^{2} + 121 x - 25 \)

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:  \(33554432000000=2^{31}\cdot 5^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.83$
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:  $2, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{20} a^{5} - \frac{3}{20} a^{4} + \frac{1}{5} a^{3} + \frac{2}{5} a^{2} + \frac{1}{20} a + \frac{1}{4}$, $\frac{1}{20} a^{6} - \frac{1}{4} a^{4} + \frac{1}{4} a^{2} + \frac{2}{5} a - \frac{1}{4}$, $\frac{1}{20} a^{7} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{2}{5} a^{2} - \frac{1}{4}$, $\frac{1}{140} a^{8} + \frac{1}{140} a^{7} + \frac{3}{28} a^{4} + \frac{53}{140} a^{3} + \frac{1}{5} a^{2} + \frac{3}{14}$

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:  \( 7776.97162299 \)
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 ${\href{/LocalNumberField/3.8.0.1}{8} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ R ${\href{/LocalNumberField/7.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.8.0.1}{8} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.8.0.1}{8} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.8.0.1}{8} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.8.0.1}{8} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.8.0.1}{8} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.8.0.1}{8} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.8.0.1}{8} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.8.0.1}{8} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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$$0$Trivial$[\ ]$
2.8.31.7$x^{8} + 28 x^{4} + 16 x^{3} + 50$$8$$1$$31$$C_8$$[3, 4, 5]$
$5$$\Q_{5}$$x + 2$$1$$1$$0$Trivial$[\ ]$
5.8.6.4$x^{8} - 5 x^{4} + 50$$4$$2$$6$$C_8$$[\ ]_{4}^{2}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e3.2t1.1c1$1$ $ 2^{3}$ $x^{2} - 2$ $C_2$ (as 2T1) $1$ $1$
1.2e4_5.4t1.3c1$1$ $ 2^{4} \cdot 5 $ $x^{4} - 20 x^{2} + 50$ $C_4$ (as 4T1) $0$ $1$
1.2e4_5.4t1.3c2$1$ $ 2^{4} \cdot 5 $ $x^{4} - 20 x^{2} + 50$ $C_4$ (as 4T1) $0$ $1$
1.2e5_5.8t1.2c1$1$ $ 2^{5} \cdot 5 $ $x^{8} + 40 x^{6} + 500 x^{4} + 2000 x^{2} + 50$ $C_8$ (as 8T1) $0$ $-1$
1.2e5_5.8t1.2c2$1$ $ 2^{5} \cdot 5 $ $x^{8} + 40 x^{6} + 500 x^{4} + 2000 x^{2} + 50$ $C_8$ (as 8T1) $0$ $-1$
1.2e5_5.8t1.2c3$1$ $ 2^{5} \cdot 5 $ $x^{8} + 40 x^{6} + 500 x^{4} + 2000 x^{2} + 50$ $C_8$ (as 8T1) $0$ $-1$
1.2e5_5.8t1.2c4$1$ $ 2^{5} \cdot 5 $ $x^{8} + 40 x^{6} + 500 x^{4} + 2000 x^{2} + 50$ $C_8$ (as 8T1) $0$ $-1$
* 8.2e31_5e6.9t15.1c1$8$ $ 2^{31} \cdot 5^{6}$ $x^{9} - x^{8} + 12 x^{7} - 28 x^{6} + 50 x^{5} - 82 x^{4} + 132 x^{3} - 84 x^{2} + 121 x - 25$ $C_3^2:C_8$ (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 *.