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Properties

Label 9.1.101857357464...0000.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{6}\cdot 3^{10}\cdot 5^{6}\cdot 23^{7}\cdot 47^{7}$
Root discriminant $3601.17$
Ramified primes $2, 3, 5, 23, 47$
Class number $116640$ (GRH)
Class group $[3, 3, 3, 3, 1440]$ (GRH)
Galois group $S_3^2$ (as 9T8)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![3977418013950, 507469495050, 27619200930, 1882373241, 46441767, 2850606, 26953, 2196, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 + 2196*x^7 + 26953*x^6 + 2850606*x^5 + 46441767*x^4 + 1882373241*x^3 + 27619200930*x^2 + 507469495050*x + 3977418013950)
 
gp: K = bnfinit(x^9 - 3*x^8 + 2196*x^7 + 26953*x^6 + 2850606*x^5 + 46441767*x^4 + 1882373241*x^3 + 27619200930*x^2 + 507469495050*x + 3977418013950, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} + 2196 x^{7} + 26953 x^{6} + 2850606 x^{5} + 46441767 x^{4} + 1882373241 x^{3} + 27619200930 x^{2} + 507469495050 x + 3977418013950 \)

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:  \(101857357464204117967975089000000=2^{6}\cdot 3^{10}\cdot 5^{6}\cdot 23^{7}\cdot 47^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3601.17$
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:  $2, 3, 5, 23, 47$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{26} a^{7} - \frac{6}{13} a^{6} - \frac{1}{26} a^{5} - \frac{2}{13} a^{4} - \frac{7}{26} a^{3} - \frac{2}{13} a^{2} - \frac{3}{13} a - \frac{6}{13}$, $\frac{1}{99468186260120728014812805873239610} a^{8} + \frac{293676945175728125602595280646187}{99468186260120728014812805873239610} a^{7} + \frac{16962180122284153164051588769257391}{99468186260120728014812805873239610} a^{6} + \frac{1564905339378116459279532484355531}{33156062086706909338270935291079870} a^{5} - \frac{744450200411337011756660182474103}{33156062086706909338270935291079870} a^{4} - \frac{14646306555878773311275648586252771}{33156062086706909338270935291079870} a^{3} - \frac{4649575920907236235422761945125284}{16578031043353454669135467645539935} a^{2} - \frac{26218818412303953708417029220535}{255046631436206994909776425315999} a - \frac{556475799731732644661806379743851}{3315606208670690933827093529107987}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{1440}$, which has order $116640$ (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:  \( 78747971.58348021 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3^2$ (as 9T8):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 9 conjugacy class representatives for $S_3^2$
Character table for $S_3^2$

Intermediate fields

3.1.324300.2, 3.1.788778675.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 6 sibling: data not computed
Degree 12 sibling: data not computed
Degree 18 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 R R ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/11.1.0.1}{1} }$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ R ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }$

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}$
$3$3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.7.4$x^{6} + 3 x^{2} + 3$$6$$1$$7$$S_3$$[3/2]_{2}$
$5$5.3.2.1$x^{3} - 5$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
$23$23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.6.5.2$x^{6} + 46$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$
$47$47.3.2.1$x^{3} - 47$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
47.6.5.1$x^{6} - 47$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$