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Properties

Label 9.9.163391343832...4809.1
Degree $9$
Signature $[9, 0]$
Discriminant $3^{4}\cdot 37^{6}\cdot 2977717^{4}$
Root discriminant $13{,}639.65$
Ramified primes $3, 37, 2977717$
Class number $21$ (GRH)
Class group $[21]$ (GRH)
Galois group $\PSL(2,8)$ (as 9T27)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1186350537195, -848281897593, -82305140040, 7683210630, 1157871573, 27805575, -295072, -12000, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 - 12000*x^7 - 295072*x^6 + 27805575*x^5 + 1157871573*x^4 + 7683210630*x^3 - 82305140040*x^2 - 848281897593*x - 1186350537195)
 
gp: K = bnfinit(x^9 - 3*x^8 - 12000*x^7 - 295072*x^6 + 27805575*x^5 + 1157871573*x^4 + 7683210630*x^3 - 82305140040*x^2 - 848281897593*x - 1186350537195, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} - 12000 x^{7} - 295072 x^{6} + 27805575 x^{5} + 1157871573 x^{4} + 7683210630 x^{3} - 82305140040 x^{2} - 848281897593 x - 1186350537195 \)

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:  \(16339134383250936197651486708561524809=3^{4}\cdot 37^{6}\cdot 2977717^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $13{,}639.65$
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:  $3, 37, 2977717$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{6} a^{3} + \frac{1}{3} a^{2} - \frac{1}{2}$, $\frac{1}{6} a^{4} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{36} a^{5} + \frac{1}{36} a^{4} + \frac{1}{36} a^{3} + \frac{1}{12} a^{2} + \frac{1}{12} a - \frac{1}{4}$, $\frac{1}{180} a^{6} + \frac{1}{90} a^{5} + \frac{7}{90} a^{4} + \frac{1}{45} a^{3} - \frac{13}{30} a^{2} + \frac{11}{30} a - \frac{1}{4}$, $\frac{1}{25920} a^{7} + \frac{1}{1080} a^{6} + \frac{61}{8640} a^{5} - \frac{43}{25920} a^{4} + \frac{89}{1728} a^{3} - \frac{685}{1728} a^{2} + \frac{719}{1440} a - \frac{167}{576}$, $\frac{1}{61662905710112566418911773120} a^{8} + \frac{880652372263325969197313}{61662905710112566418911773120} a^{7} - \frac{23488502144965886157324283}{20554301903370855472970591040} a^{6} + \frac{142126109980933322146206557}{15415726427528141604727943280} a^{5} - \frac{1171319478579222145135852751}{15415726427528141604727943280} a^{4} + \frac{26810947050652628330967943}{1712858491947571289414215920} a^{3} - \frac{1009591725114133611421545823}{20554301903370855472970591040} a^{2} + \frac{151923720800337086186577311}{1370286793558057031531372736} a + \frac{362730291667242617973677105}{1370286793558057031531372736}$

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

Class group and class number

$C_{21}$, which has order $21$ (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:  \( 13435625715600000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$\PSL(2,8)$ (as 9T27):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A non-solvable group of order 504
The 9 conjugacy class representatives for $\PSL(2,8)$
Character table for $\PSL(2,8)$

Intermediate fields

The extension is primitive: there are no intermediate fields between this field and $\Q$.

Sibling fields

Degree 28 sibling: data not computed
Degree 36 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/2.1.0.1}{1} }$ R ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ ${\href{/LocalNumberField/7.7.0.1}{7} }{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/17.9.0.1}{9} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.7.0.1}{7} }{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.9.0.1}{9} }$ R ${\href{/LocalNumberField/41.7.0.1}{7} }{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/53.9.0.1}{9} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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
$3$$\Q_{3}$$x + 1$$1$$1$$0$Trivial$[\ ]$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$37$37.9.6.3$x^{9} - 74 x^{6} + 1369 x^{3} - 202612$$3$$3$$6$$C_9$$[\ ]_{3}^{3}$
2977717Data not computed