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Properties

Label 9.1.38564894884096.3
Degree $9$
Signature $[1, 4]$
Discriminant $2^{8}\cdot 7^{4}\cdot 89^{4}$
Root discriminant $32.33$
Ramified primes $2, 7, 89$
Class number $3$
Class group $[3]$
Galois group $C_3^2 : C_6$ (as 9T11)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![500, -75, 280, 147, -74, 79, -32, 2, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 + 2*x^7 - 32*x^6 + 79*x^5 - 74*x^4 + 147*x^3 + 280*x^2 - 75*x + 500)
 
gp: K = bnfinit(x^9 - 2*x^8 + 2*x^7 - 32*x^6 + 79*x^5 - 74*x^4 + 147*x^3 + 280*x^2 - 75*x + 500, 1)
 

Normalized defining polynomial

\( x^{9} - 2 x^{8} + 2 x^{7} - 32 x^{6} + 79 x^{5} - 74 x^{4} + 147 x^{3} + 280 x^{2} - 75 x + 500 \)

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:  \(38564894884096=2^{8}\cdot 7^{4}\cdot 89^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.33$
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:  $2, 7, 89$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{130} a^{7} - \frac{16}{65} a^{6} + \frac{21}{65} a^{5} + \frac{3}{130} a^{4} + \frac{12}{65} a^{3} + \frac{21}{130} a^{2} + \frac{57}{130} a + \frac{5}{13}$, $\frac{1}{2371850} a^{8} - \frac{1811}{1185925} a^{7} + \frac{78846}{1185925} a^{6} - \frac{962497}{2371850} a^{5} + \frac{132622}{1185925} a^{4} + \frac{689571}{2371850} a^{3} + \frac{1128727}{2371850} a^{2} - \frac{93596}{237185} a - \frac{14848}{47437}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{3}$, which has order $3$

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:  \( 2680.36449494 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$He_3:C_2$ (as 9T11):

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 $C_3^2 : C_6$
Character table for $C_3^2 : C_6$

Intermediate fields

3.1.356.1

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

Sibling fields

Degree 9 sibling: data not computed
Degree 18 siblings: 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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$$0$Trivial$[\ ]$
2.2.2.1$x^{2} + 2 x + 2$$2$$1$$2$$C_2$$[2]$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
89Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.2e2_89.2t1.1c1$1$ $ 2^{2} \cdot 89 $ $x^{2} + 89$ $C_2$ (as 2T1) $1$ $-1$
1.2e2_7_89.6t1.1c1$1$ $ 2^{2} \cdot 7 \cdot 89 $ $x^{6} - 2 x^{5} + 264 x^{4} - 350 x^{3} + 23943 x^{2} - 16024 x + 745109$ $C_6$ (as 6T1) $0$ $-1$
1.2e2_7_89.6t1.1c2$1$ $ 2^{2} \cdot 7 \cdot 89 $ $x^{6} - 2 x^{5} + 264 x^{4} - 350 x^{3} + 23943 x^{2} - 16024 x + 745109$ $C_6$ (as 6T1) $0$ $-1$
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) $0$ $1$
* 2.2e2_89.3t2.1c1$2$ $ 2^{2} \cdot 89 $ $x^{3} - x^{2} + x + 7$ $S_3$ (as 3T2) $1$ $0$
2.2e2_7e2_89.6t5.1c1$2$ $ 2^{2} \cdot 7^{2} \cdot 89 $ $x^{6} - 3 x^{4} - 56 x^{3} + 514 x^{2} + 2576 x + 7993$ $S_3\times C_3$ (as 6T5) $0$ $0$
2.2e2_7e2_89.6t5.1c2$2$ $ 2^{2} \cdot 7^{2} \cdot 89 $ $x^{6} - 3 x^{4} - 56 x^{3} + 514 x^{2} + 2576 x + 7993$ $S_3\times C_3$ (as 6T5) $0$ $0$
* 6.2e6_7e4_89e3.9t13.1c1$6$ $ 2^{6} \cdot 7^{4} \cdot 89^{3}$ $x^{9} - 2 x^{8} + 2 x^{7} - 32 x^{6} + 79 x^{5} - 74 x^{4} + 147 x^{3} + 280 x^{2} - 75 x + 500$ $C_3^2 : C_6$ (as 9T11) $1$ $0$

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 *.