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Properties

Label 9.1.180593175951...9376.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{14}\cdot 3^{22}\cdot 37^{8}$
Root discriminant $1067.88$
Ramified primes $2, 3, 37$
Class number $63$ (GRH)
Class group $[63]$ (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![-26643404, -1299699, -693972, -151848, -27972, -4662, -444, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 444*x^6 - 4662*x^5 - 27972*x^4 - 151848*x^3 - 693972*x^2 - 1299699*x - 26643404)
 
gp: K = bnfinit(x^9 - 444*x^6 - 4662*x^5 - 27972*x^4 - 151848*x^3 - 693972*x^2 - 1299699*x - 26643404, 1)
 

Normalized defining polynomial

\( x^{9} - 444 x^{6} - 4662 x^{5} - 27972 x^{4} - 151848 x^{3} - 693972 x^{2} - 1299699 x - 26643404 \)

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:  \(1805931759515773661624549376=2^{14}\cdot 3^{22}\cdot 37^{8}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1067.88$
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:  $2, 3, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{4} a^{2} - \frac{1}{4} a$, $\frac{1}{8} a^{5} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{16} a^{6} - \frac{1}{16} a^{5} - \frac{1}{8} a^{4} - \frac{1}{8} a^{3} - \frac{7}{16} a^{2} - \frac{1}{16} a - \frac{1}{4}$, $\frac{1}{352} a^{7} + \frac{1}{88} a^{6} + \frac{15}{352} a^{5} - \frac{9}{88} a^{4} - \frac{69}{352} a^{3} - \frac{13}{88} a^{2} + \frac{7}{32} a - \frac{43}{88}$, $\frac{1}{1446734975920960} a^{8} + \frac{550017674241}{1446734975920960} a^{7} - \frac{43225946705209}{1446734975920960} a^{6} - \frac{2753038725463}{131521361447360} a^{5} - \frac{25306404670685}{289346995184192} a^{4} - \frac{257606738130557}{1446734975920960} a^{3} - \frac{106977048971847}{289346995184192} a^{2} - \frac{699920365225407}{1446734975920960} a - \frac{149321238313139}{361683743980240}$

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

Class group and class number

$C_{63}$, which has order $63$ (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:  $4$
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 (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 893886967.791 \) (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 R R ${\href{/LocalNumberField/5.7.0.1}{7} }{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.9.0.1}{9} }$ ${\href{/LocalNumberField/17.9.0.1}{9} }$ ${\href{/LocalNumberField/19.7.0.1}{7} }{,}\,{\href{/LocalNumberField/19.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/23.9.0.1}{9} }$ ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/43.7.0.1}{7} }{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/47.7.0.1}{7} }{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.7.0.1}{7} }{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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
$2$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.8.14.6$x^{8} + 4 x^{7} + 4$$8$$1$$14$$C_2^3:C_7$$[2, 2, 2]^{7}$
$3$3.9.22.10$x^{9} + 6 x^{6} + 18 x^{5} + 6$$9$$1$$22$$D_{9}$$[2, 3]^{2}$
$37$37.9.8.2$x^{9} + 74$$9$$1$$8$$C_9$$[\ ]_{9}$