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Properties

Label 9.1.4545340842854449.1
Degree $9$
Signature $[1, 4]$
Discriminant $11^{6}\cdot 37^{6}$
Root discriminant $54.92$
Ramified primes $11, 37$
Class number $3$
Class group $[3]$
Galois group $(C_3^2:Q_8):C_3$ (as 9T23)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![199, -513, 608, 62, 139, -87, -3, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 5*x^7 - 3*x^6 - 87*x^5 + 139*x^4 + 62*x^3 + 608*x^2 - 513*x + 199)
 
gp: K = bnfinit(x^9 - 3*x^8 + 5*x^7 - 3*x^6 - 87*x^5 + 139*x^4 + 62*x^3 + 608*x^2 - 513*x + 199, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} + 5 x^{7} - 3 x^{6} - 87 x^{5} + 139 x^{4} + 62 x^{3} + 608 x^{2} - 513 x + 199 \)

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:  \(4545340842854449=11^{6}\cdot 37^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $54.92$
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:  $11, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{6} + \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a - \frac{1}{4}$, $\frac{1}{140710288} a^{8} + \frac{189386}{8794393} a^{7} - \frac{3806787}{140710288} a^{6} - \frac{6920017}{35177572} a^{5} + \frac{12670797}{140710288} a^{4} - \frac{29698587}{70355144} a^{3} + \frac{1969971}{35177572} a^{2} + \frac{12207137}{35177572} a + \frac{51197827}{140710288}$

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:  \( 9546.6899706 \)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$ASL(2,3)$ (as 9T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 216
The 10 conjugacy class representatives for $(C_3^2:Q_8):C_3$
Character table for $(C_3^2:Q_8):C_3$

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 24 siblings: data not computed
Degree 27 sibling: data not computed
Degree 36 siblings: 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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ R ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
$11$$\Q_{11}$$x + 3$$1$$1$$0$Trivial$[\ ]$
11.8.6.1$x^{8} + 143 x^{4} + 5929$$4$$2$$6$$Q_8$$[\ ]_{4}^{2}$
$37$37.9.6.1$x^{9} + 222 x^{6} + 15059 x^{3} + 405224$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.37.3t1.1c1$1$ $ 37 $ $x^{3} - x^{2} - 12 x - 11$ $C_3$ (as 3T1) $0$ $1$
1.37.3t1.1c2$1$ $ 37 $ $x^{3} - x^{2} - 12 x - 11$ $C_3$ (as 3T1) $0$ $1$
2.11e2_37e2.24t7.2c1$2$ $ 11^{2} \cdot 37^{2}$ $x^{8} - 2 x^{7} + 13 x^{6} + 18 x^{5} + 5 x^{4} + 126 x^{3} + 142 x^{2} - 70 x + 47$ $\SL(2,3)$ (as 8T12) $-1$ $-2$
2.11e2_37.8t12.2c1$2$ $ 11^{2} \cdot 37 $ $x^{8} - 2 x^{7} + 13 x^{6} + 18 x^{5} + 5 x^{4} + 126 x^{3} + 142 x^{2} - 70 x + 47$ $\SL(2,3)$ (as 8T12) $0$ $-2$
2.11e2_37.8t12.2c2$2$ $ 11^{2} \cdot 37 $ $x^{8} - 2 x^{7} + 13 x^{6} + 18 x^{5} + 5 x^{4} + 126 x^{3} + 142 x^{2} - 70 x + 47$ $\SL(2,3)$ (as 8T12) $0$ $-2$
3.11e2_37e2.4t4.1c1$3$ $ 11^{2} \cdot 37^{2}$ $x^{4} - 2 x^{3} - 9 x^{2} - x + 3$ $A_4$ (as 4T4) $1$ $3$
* 8.11e6_37e6.9t23.1c1$8$ $ 11^{6} \cdot 37^{6}$ $x^{9} - 3 x^{8} + 5 x^{7} - 3 x^{6} - 87 x^{5} + 139 x^{4} + 62 x^{3} + 608 x^{2} - 513 x + 199$ $(C_3^2:Q_8):C_3$ (as 9T23) $1$ $0$
8.11e6_37e5.24t569.1c1$8$ $ 11^{6} \cdot 37^{5}$ $x^{9} - 3 x^{8} + 5 x^{7} - 3 x^{6} - 87 x^{5} + 139 x^{4} + 62 x^{3} + 608 x^{2} - 513 x + 199$ $(C_3^2:Q_8):C_3$ (as 9T23) $0$ $0$
8.11e6_37e5.24t569.1c2$8$ $ 11^{6} \cdot 37^{5}$ $x^{9} - 3 x^{8} + 5 x^{7} - 3 x^{6} - 87 x^{5} + 139 x^{4} + 62 x^{3} + 608 x^{2} - 513 x + 199$ $(C_3^2:Q_8):C_3$ (as 9T23) $0$ $0$

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 *.