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Properties

Label 9.9.1499682086581441.1
Degree $9$
Signature $[9, 0]$
Discriminant $7^{8}\cdot 127^{4}$
Root discriminant $48.55$
Ramified primes $7, 127$
Class number $1$
Class group Trivial
Galois group $C_9:C_3$ (as 9T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![601, -3757, 6873, -2877, -1386, 707, 91, -48, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 - 48*x^7 + 91*x^6 + 707*x^5 - 1386*x^4 - 2877*x^3 + 6873*x^2 - 3757*x + 601)
 
gp: K = bnfinit(x^9 - 2*x^8 - 48*x^7 + 91*x^6 + 707*x^5 - 1386*x^4 - 2877*x^3 + 6873*x^2 - 3757*x + 601, 1)
 

Normalized defining polynomial

\( x^{9} - 2 x^{8} - 48 x^{7} + 91 x^{6} + 707 x^{5} - 1386 x^{4} - 2877 x^{3} + 6873 x^{2} - 3757 x + 601 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[9, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(1499682086581441=7^{8}\cdot 127^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $48.55$
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:  $7, 127$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{12156062477} a^{8} + \frac{4375628963}{12156062477} a^{7} + \frac{3230463433}{12156062477} a^{6} - \frac{4652571613}{12156062477} a^{5} + \frac{3978384864}{12156062477} a^{4} + \frac{1815656959}{12156062477} a^{3} + \frac{2897640125}{12156062477} a^{2} - \frac{1726659479}{12156062477} a + \frac{709167361}{12156062477}$

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:  $8$
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:  \( 41351.8820738 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_9:C_3$ (as 9T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 27
The 11 conjugacy class representatives for $C_9:C_3$
Character table for $C_9:C_3$

Intermediate fields

\(\Q(\zeta_{7})^+\)

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

Sibling fields

Galois closure: 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 ${\href{/LocalNumberField/2.9.0.1}{9} }$ ${\href{/LocalNumberField/3.9.0.1}{9} }$ ${\href{/LocalNumberField/5.9.0.1}{9} }$ R ${\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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
$7$7.9.8.2$x^{9} - 7$$9$$1$$8$$C_9:C_3$$[\ ]_{9}^{3}$
$127$127.3.0.1$x^{3} - x + 9$$1$$3$$0$$C_3$$[\ ]^{3}$
127.3.2.3$x^{3} - 10287$$3$$1$$2$$C_3$$[\ ]_{3}$
127.3.2.2$x^{3} + 1143$$3$$1$$2$$C_3$$[\ ]_{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.127.3t1.1c1$1$ $ 127 $ $x^{3} - x^{2} - 42 x - 80$ $C_3$ (as 3T1) $0$ $1$
* 1.7.3t1.1c1$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
* 1.7.3t1.1c2$1$ $ 7 $ $x^{3} - x^{2} - 2 x + 1$ $C_3$ (as 3T1) $0$ $1$
1.7_127.3t1.2c1$1$ $ 7 \cdot 127 $ $x^{3} - x^{2} - 296 x + 1317$ $C_3$ (as 3T1) $0$ $1$
1.7_127.3t1.1c1$1$ $ 7 \cdot 127 $ $x^{3} - x^{2} - 296 x - 461$ $C_3$ (as 3T1) $0$ $1$
1.127.3t1.1c2$1$ $ 127 $ $x^{3} - x^{2} - 42 x - 80$ $C_3$ (as 3T1) $0$ $1$
1.7_127.3t1.1c2$1$ $ 7 \cdot 127 $ $x^{3} - x^{2} - 296 x - 461$ $C_3$ (as 3T1) $0$ $1$
1.7_127.3t1.2c2$1$ $ 7 \cdot 127 $ $x^{3} - x^{2} - 296 x + 1317$ $C_3$ (as 3T1) $0$ $1$
* 3.7e3_127e2.9t6.1c1$3$ $ 7^{3} \cdot 127^{2}$ $x^{9} - 2 x^{8} - 48 x^{7} + 91 x^{6} + 707 x^{5} - 1386 x^{4} - 2877 x^{3} + 6873 x^{2} - 3757 x + 601$ $C_9:C_3$ (as 9T6) $0$ $3$
* 3.7e3_127e2.9t6.1c2$3$ $ 7^{3} \cdot 127^{2}$ $x^{9} - 2 x^{8} - 48 x^{7} + 91 x^{6} + 707 x^{5} - 1386 x^{4} - 2877 x^{3} + 6873 x^{2} - 3757 x + 601$ $C_9:C_3$ (as 9T6) $0$ $3$

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 *.