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Properties

Label 9.9.823197013411...9136.1
Degree $9$
Signature $[9, 0]$
Discriminant $2^{6}\cdot 7^{10}\cdot 37^{8}\cdot 263^{4}\cdot 1283^{4}$
Root discriminant $97{,}861.42$
Ramified primes $2, 7, 37, 263, 1283$
Class number $9$ (GRH)
Class group $[9]$ (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![75454628055435091, -8789861772156959, 258358461584640, -1082243626268, -54192964154, 604538774, 2290792, -46044, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - x^8 - 46044*x^7 + 2290792*x^6 + 604538774*x^5 - 54192964154*x^4 - 1082243626268*x^3 + 258358461584640*x^2 - 8789861772156959*x + 75454628055435091)
 
gp: K = bnfinit(x^9 - x^8 - 46044*x^7 + 2290792*x^6 + 604538774*x^5 - 54192964154*x^4 - 1082243626268*x^3 + 258358461584640*x^2 - 8789861772156959*x + 75454628055435091, 1)
 

Normalized defining polynomial

\( x^{9} - x^{8} - 46044 x^{7} + 2290792 x^{6} + 604538774 x^{5} - 54192964154 x^{4} - 1082243626268 x^{3} + 258358461584640 x^{2} - 8789861772156959 x + 75454628055435091 \)

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:  \(823197013411488700177278327969031167198339136=2^{6}\cdot 7^{10}\cdot 37^{8}\cdot 263^{4}\cdot 1283^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $97{,}861.42$
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:  $2, 7, 37, 263, 1283$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{4} a + \frac{1}{4}$, $\frac{1}{8} a^{4} - \frac{1}{4} a^{2} - \frac{1}{2} a - \frac{3}{8}$, $\frac{1}{3552} a^{5} + \frac{131}{3552} a^{4} - \frac{31}{1776} a^{3} - \frac{235}{1776} a^{2} + \frac{439}{1184} a - \frac{1381}{3552}$, $\frac{1}{184704} a^{6} - \frac{1}{11544} a^{5} + \frac{7321}{184704} a^{4} + \frac{2605}{46176} a^{3} + \frac{8669}{61568} a^{2} - \frac{7453}{46176} a - \frac{8107}{61568}$, $\frac{1}{738816} a^{7} + \frac{1}{738816} a^{6} - \frac{23}{738816} a^{5} - \frac{6563}{738816} a^{4} + \frac{87499}{738816} a^{3} - \frac{50285}{738816} a^{2} + \frac{129067}{738816} a + \frac{348655}{738816}$, $\frac{1}{31852662746895004330423231488} a^{8} + \frac{35068074485834352271}{612551206671057775585062144} a^{7} + \frac{911267489660129030573}{1137595098103393011800829696} a^{6} - \frac{14057440124392465305761}{142199387262924126475103712} a^{5} + \frac{2830302829728086712605981}{758396732068928674533886464} a^{4} + \frac{16910430083678960998583773}{1137595098103393011800829696} a^{3} + \frac{11158839230759996781024623}{126399455344821445755647744} a^{2} - \frac{112962729992611083319096339}{442398093706875060144767104} a - \frac{8058729287184286904161938419}{31852662746895004330423231488}$

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

Class group and class number

$C_{9}$, which has order $9$ (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:  \( 33057799402800000000 \) (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 ${\href{/LocalNumberField/3.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/3.1.0.1}{1} }$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.9.0.1}{9} }$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.7.0.1}{7} }{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/19.9.0.1}{9} }$ ${\href{/LocalNumberField/23.7.0.1}{7} }{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/29.9.0.1}{9} }$ ${\href{/LocalNumberField/31.7.0.1}{7} }{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{2}$ R ${\href{/LocalNumberField/41.9.0.1}{9} }$ ${\href{/LocalNumberField/43.9.0.1}{9} }$ ${\href{/LocalNumberField/47.9.0.1}{9} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }$ ${\href{/LocalNumberField/59.9.0.1}{9} }$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$7$7.2.1.2$x^{2} + 14$$2$$1$$1$$C_2$$[\ ]_{2}$
7.7.9.4$x^{7} + 35 x^{3} + 7$$7$$1$$9$$D_{7}$$[3/2]_{2}$
$37$37.9.8.4$x^{9} + 296$$9$$1$$8$$C_9$$[\ ]_{9}$
263Data not computed
1283Data not computed