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Properties

Label 9.3.173205145504...0736.1
Degree $9$
Signature $[3, 3]$
Discriminant $-\,2^{13}\cdot 3^{10}\cdot 7^{6}\cdot 19^{7}\cdot 23^{7}$
Root discriminant $3819.99$
Ramified primes $2, 3, 7, 19, 23$
Class number $104976$ (GRH)
Class group $[3, 3, 6, 6, 324]$ (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![-27749527672, 18278886120, -545206164, -769203028, -70352358, -2192826, -47993, -1779, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^9 - 3*x^8 - 1779*x^7 - 47993*x^6 - 2192826*x^5 - 70352358*x^4 - 769203028*x^3 - 545206164*x^2 + 18278886120*x - 27749527672)
 
gp: K = bnfinit(x^9 - 3*x^8 - 1779*x^7 - 47993*x^6 - 2192826*x^5 - 70352358*x^4 - 769203028*x^3 - 545206164*x^2 + 18278886120*x - 27749527672, 1)
 

Normalized defining polynomial

\( x^{9} - 3 x^{8} - 1779 x^{7} - 47993 x^{6} - 2192826 x^{5} - 70352358 x^{4} - 769203028 x^{3} - 545206164 x^{2} + 18278886120 x - 27749527672 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $9$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[3, 3]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-173205145504055940264382658420736=-\,2^{13}\cdot 3^{10}\cdot 7^{6}\cdot 19^{7}\cdot 23^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $3819.99$
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:  $2, 3, 7, 19, 23$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2}$, $\frac{1}{60} a^{7} + \frac{11}{60} a^{6} + \frac{1}{20} a^{5} + \frac{17}{60} a^{4} + \frac{4}{15} a^{3} - \frac{3}{10} a^{2} + \frac{7}{15} a - \frac{4}{15}$, $\frac{1}{10471081198630092907788205105320} a^{8} - \frac{11769344497077948769719468817}{2094216239726018581557641021064} a^{7} + \frac{78511978831432172676675696697}{10471081198630092907788205105320} a^{6} - \frac{1931351391828496129259726357161}{10471081198630092907788205105320} a^{5} - \frac{1538455963252025162669174153273}{5235540599315046453894102552660} a^{4} - \frac{34991704477729597140635942153}{1308885149828761613473525638165} a^{3} - \frac{162178510593735353340283587461}{2617770299657523226947051276330} a^{2} + \frac{294532894477486644277737248209}{2617770299657523226947051276330} a - \frac{399509772813516503789790434593}{1308885149828761613473525638165}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{6}\times C_{6}\times C_{324}$, which has order $104976$ (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:  $5$
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:  \( 329616308.2006819 \) (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

3.1.1010607948.5, 3.3.513912.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }$ R R ${\href{/LocalNumberField/29.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }$ ${\href{/LocalNumberField/41.6.0.1}{6} }{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{3}$

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$$[\ ]_{3}^{2}$
2.6.11.9$x^{6} + 6 x^{4} + 2$$6$$1$$11$$D_{6}$$[3]_{3}^{2}$
$3$3.3.3.2$x^{3} + 3 x + 3$$3$$1$$3$$S_3$$[3/2]_{2}$
3.6.7.2$x^{6} + 3 x^{2} + 6$$6$$1$$7$$D_{6}$$[3/2]_{2}^{2}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$19$19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.6.5.2$x^{6} - 19$$6$$1$$5$$C_6$$[\ ]_{6}$
$23$23.3.2.1$x^{3} - 23$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
23.6.5.2$x^{6} + 46$$6$$1$$5$$D_{6}$$[\ ]_{6}^{2}$