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Properties

Label 9.1.115646511237465664.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{6}\cdot 349^{6}$
Root discriminant $78.69$
Ramified primes $2, 349$
Class number $9$
Class group $[9]$
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![3454, 4066, 3378, 1333, 510, 257, 22, -25, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 4*x^8 - 25*x^7 + 22*x^6 + 257*x^5 + 510*x^4 + 1333*x^3 + 3378*x^2 + 4066*x + 3454)
 
gp: K = bnfinit(x^9 - 4*x^8 - 25*x^7 + 22*x^6 + 257*x^5 + 510*x^4 + 1333*x^3 + 3378*x^2 + 4066*x + 3454, 1)
 

Normalized defining polynomial

\( x^{9} - 4 x^{8} - 25 x^{7} + 22 x^{6} + 257 x^{5} + 510 x^{4} + 1333 x^{3} + 3378 x^{2} + 4066 x + 3454 \)

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:  \(115646511237465664=2^{6}\cdot 349^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $78.69$
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:  $2, 349$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{6} a^{4} - \frac{1}{6} a^{3} - \frac{1}{3}$, $\frac{1}{12} a^{5} + \frac{5}{12} a^{3} - \frac{1}{2} a^{2} - \frac{1}{6} a - \frac{1}{6}$, $\frac{1}{24} a^{6} - \frac{1}{24} a^{5} + \frac{1}{24} a^{4} + \frac{5}{24} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a + \frac{5}{12}$, $\frac{1}{48} a^{7} - \frac{1}{24} a^{4} + \frac{17}{48} a^{3} - \frac{1}{6} a^{2} + \frac{11}{24} a + \frac{1}{24}$, $\frac{1}{2600352} a^{8} - \frac{26285}{2600352} a^{7} - \frac{7697}{650088} a^{6} - \frac{569}{433392} a^{5} + \frac{11447}{2600352} a^{4} + \frac{1236127}{2600352} a^{3} + \frac{140893}{433392} a^{2} - \frac{25793}{216696} a - \frac{186889}{1300176}$

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

Class group and class number

$C_{9}$, which has order $9$

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:  \( 26174.2758378 \)
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 R ${\href{/LocalNumberField/3.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/5.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{3}$

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$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
349Data not computed

Artin representations

Label Dimension Conductor Defining polynomial of Artin field $G$ Ind $\chi(c)$
* 1.1.1t1.1c1$1$ $1$ $x$ $C_1$ $1$ $1$
1.349.3t1.1c1$1$ $ 349 $ $x^{3} - x^{2} - 116 x + 517$ $C_3$ (as 3T1) $0$ $1$
1.349.3t1.1c2$1$ $ 349 $ $x^{3} - x^{2} - 116 x + 517$ $C_3$ (as 3T1) $0$ $1$
2.349e2.24t7.1c1$2$ $ 349^{2}$ $x^{8} - 4 x^{7} + 5 x^{6} + x^{5} - 6 x^{4} + 7 x^{3} + 31 x^{2} + 34 x + 12$ $\SL(2,3)$ (as 8T12) $-1$ $-2$
2.349.8t12.1c1$2$ $ 349 $ $x^{8} - 4 x^{7} + 5 x^{6} + x^{5} - 6 x^{4} + 7 x^{3} + 31 x^{2} + 34 x + 12$ $\SL(2,3)$ (as 8T12) $0$ $-2$
2.349.8t12.1c2$2$ $ 349 $ $x^{8} - 4 x^{7} + 5 x^{6} + x^{5} - 6 x^{4} + 7 x^{3} + 31 x^{2} + 34 x + 12$ $\SL(2,3)$ (as 8T12) $0$ $-2$
3.349e2.4t4.1c1$3$ $ 349^{2}$ $x^{4} - x^{3} - 10 x^{2} + 3 x + 20$ $A_4$ (as 4T4) $1$ $3$
* 8.2e6_349e6.9t23.2c1$8$ $ 2^{6} \cdot 349^{6}$ $x^{9} - 4 x^{8} - 25 x^{7} + 22 x^{6} + 257 x^{5} + 510 x^{4} + 1333 x^{3} + 3378 x^{2} + 4066 x + 3454$ $(C_3^2:Q_8):C_3$ (as 9T23) $1$ $0$
8.2e6_349e5.24t569.2c1$8$ $ 2^{6} \cdot 349^{5}$ $x^{9} - 4 x^{8} - 25 x^{7} + 22 x^{6} + 257 x^{5} + 510 x^{4} + 1333 x^{3} + 3378 x^{2} + 4066 x + 3454$ $(C_3^2:Q_8):C_3$ (as 9T23) $0$ $0$
8.2e6_349e5.24t569.2c2$8$ $ 2^{6} \cdot 349^{5}$ $x^{9} - 4 x^{8} - 25 x^{7} + 22 x^{6} + 257 x^{5} + 510 x^{4} + 1333 x^{3} + 3378 x^{2} + 4066 x + 3454$ $(C_3^2:Q_8):C_3$ (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 *.