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Properties

Label 9.1.174565121124...5424.1
Degree $9$
Signature $[1, 4]$
Discriminant $2^{6}\cdot 3^{10}\cdot 7^{7}\cdot 13^{7}\cdot 19^{7}$
Root discriminant $1774.62$
Ramified primes $2, 3, 7, 13, 19$
Class number $236196$ (GRH)
Class group $[3, 3, 9, 54, 54]$ (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![-5231495807, -645345360, -6714576, 9087609, 187164, 3888, -5181, 108, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 + 108*x^7 - 5181*x^6 + 3888*x^5 + 187164*x^4 + 9087609*x^3 - 6714576*x^2 - 645345360*x - 5231495807)
 
gp: K = bnfinit(x^9 + 108*x^7 - 5181*x^6 + 3888*x^5 + 187164*x^4 + 9087609*x^3 - 6714576*x^2 - 645345360*x - 5231495807, 1)
 

Normalized defining polynomial

\( x^{9} + 108 x^{7} - 5181 x^{6} + 3888 x^{5} + 187164 x^{4} + 9087609 x^{3} - 6714576 x^{2} - 645345360 x - 5231495807 \)

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:  \(174565121124349094450651855424=2^{6}\cdot 3^{10}\cdot 7^{7}\cdot 13^{7}\cdot 19^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1774.62$
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:  $2, 3, 7, 13, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{4}{9} a + \frac{4}{9}$, $\frac{1}{9} a^{5} - \frac{1}{9} a^{3} + \frac{4}{9} a^{2} - \frac{4}{9}$, $\frac{1}{46683} a^{6} + \frac{16}{1729} a^{5} + \frac{4}{5187} a^{4} + \frac{1718}{46683} a^{3} + \frac{32}{1729} a^{2} - \frac{1745}{5187} a + \frac{10366}{46683}$, $\frac{1}{140049} a^{7} + \frac{1}{140049} a^{6} + \frac{64}{15561} a^{5} + \frac{1763}{140049} a^{4} + \frac{386}{7371} a^{3} - \frac{524}{1197} a^{2} - \frac{67619}{140049} a + \frac{13822}{140049}$, $\frac{1}{44588312935630191} a^{8} - \frac{137204422525}{44588312935630191} a^{7} + \frac{271947674023}{44588312935630191} a^{6} + \frac{1238542536772727}{44588312935630191} a^{5} + \frac{1912568532415939}{44588312935630191} a^{4} - \frac{1936296362418532}{44588312935630191} a^{3} + \frac{12748397698031101}{44588312935630191} a^{2} + \frac{154931676162671}{44588312935630191} a - \frac{6931574293447877}{44588312935630191}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{9}\times C_{54}\times C_{54}$, which has order $236196$ (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:  \( 3012686.824087748 \) (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

3.1.186732.1, 3.1.8968323.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/5.1.0.1}{1} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ R ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }$ ${\href{/LocalNumberField/29.6.0.1}{6} }{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$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}$
$7$7.3.2.1$x^{3} + 14$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.3$x^{6} - 112$$6$$1$$5$$C_6$$[\ ]_{6}$
$13$13.3.2.1$x^{3} + 26$$3$$1$$2$$C_3$$[\ ]_{3}$
13.6.5.3$x^{6} - 208$$6$$1$$5$$C_6$$[\ ]_{6}$
$19$19.3.2.1$x^{3} + 76$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.5.4$x^{6} + 76$$6$$1$$5$$C_6$$[\ ]_{6}$