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Properties

Label 9.9.754169853931...8161.1
Degree $9$
Signature $[9, 0]$
Discriminant $71^{6}\cdot 277^{4}$
Root discriminant $208.79$
Ramified primes $71, 277$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
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![-5714, -4931, 14194, -2711, -3614, 1022, 196, -66, -2, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 2*x^8 - 66*x^7 + 196*x^6 + 1022*x^5 - 3614*x^4 - 2711*x^3 + 14194*x^2 - 4931*x - 5714)
 
gp: K = bnfinit(x^9 - 2*x^8 - 66*x^7 + 196*x^6 + 1022*x^5 - 3614*x^4 - 2711*x^3 + 14194*x^2 - 4931*x - 5714, 1)
 

Normalized defining polynomial

\( x^{9} - 2 x^{8} - 66 x^{7} + 196 x^{6} + 1022 x^{5} - 3614 x^{4} - 2711 x^{3} + 14194 x^{2} - 4931 x - 5714 \)

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:  \(754169853931401428161=71^{6}\cdot 277^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $208.79$
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:  $71, 277$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{7} - \frac{1}{6} a^{6} + \frac{1}{6} a^{5} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{6} a - \frac{1}{3}$, $\frac{1}{737031330} a^{8} - \frac{9352877}{245677110} a^{7} + \frac{8057593}{35096730} a^{6} - \frac{4279474}{52645095} a^{5} - \frac{23390671}{52645095} a^{4} - \frac{58393858}{122838555} a^{3} + \frac{171040943}{368515665} a^{2} + \frac{22252463}{49135422} a - \frac{137629333}{368515665}$

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

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 72796312.5644 \) (assuming GRH)
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 ${\href{/LocalNumberField/2.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ ${\href{/LocalNumberField/3.6.0.1}{6} }{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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
71Data not computed
277Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.277.3t1.1c1$1$ $ 277 $ $x^{3} - x^{2} - 92 x - 236$ $C_3$ (as 3T1) $0$ $1$
1.277.3t1.1c2$1$ $ 277 $ $x^{3} - x^{2} - 92 x - 236$ $C_3$ (as 3T1) $0$ $1$
2.277e2.24t7.1c1$2$ $ 277^{2}$ $x^{8} - x^{7} - 11 x^{6} + 13 x^{5} + 32 x^{4} - 41 x^{3} - 23 x^{2} + 32 x - 1$ $\SL(2,3)$ (as 8T12) $-1$ $2$
2.277.8t12.1c1$2$ $ 277 $ $x^{8} - x^{7} - 11 x^{6} + 13 x^{5} + 32 x^{4} - 41 x^{3} - 23 x^{2} + 32 x - 1$ $\SL(2,3)$ (as 8T12) $0$ $2$
2.277.8t12.1c2$2$ $ 277 $ $x^{8} - x^{7} - 11 x^{6} + 13 x^{5} + 32 x^{4} - 41 x^{3} - 23 x^{2} + 32 x - 1$ $\SL(2,3)$ (as 8T12) $0$ $2$
3.277e2.4t4.1c1$3$ $ 277^{2}$ $x^{4} - x^{3} - 11 x^{2} + 4 x + 12$ $A_4$ (as 4T4) $1$ $3$
* 8.71e6_277e4.9t23.1c1$8$ $ 71^{6} \cdot 277^{4}$ $x^{9} - 2 x^{8} - 66 x^{7} + 196 x^{6} + 1022 x^{5} - 3614 x^{4} - 2711 x^{3} + 14194 x^{2} - 4931 x - 5714$ $(C_3^2:Q_8):C_3$ (as 9T23) $1$ $8$
8.71e6_277e6.24t569.1c1$8$ $ 71^{6} \cdot 277^{6}$ $x^{9} - 2 x^{8} - 66 x^{7} + 196 x^{6} + 1022 x^{5} - 3614 x^{4} - 2711 x^{3} + 14194 x^{2} - 4931 x - 5714$ $(C_3^2:Q_8):C_3$ (as 9T23) $0$ $8$
8.71e6_277e6.24t569.1c2$8$ $ 71^{6} \cdot 277^{6}$ $x^{9} - 2 x^{8} - 66 x^{7} + 196 x^{6} + 1022 x^{5} - 3614 x^{4} - 2711 x^{3} + 14194 x^{2} - 4931 x - 5714$ $(C_3^2:Q_8):C_3$ (as 9T23) $0$ $8$

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 *.