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Properties

Label 18.6.91338670682...8704.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{34}\cdot 3^{25}\cdot 13^{7}$
Root discriminant $46.18$
Ramified primes $2, 3, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $S_3\times S_3\wr C_2$ (as 18T150)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-956, 4392, -2208, -11364, 14976, -5520, -798, 8226, -3573, -4672, 2619, 1212, -834, -72, 204, -6, -27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 27*x^16 - 6*x^15 + 204*x^14 - 72*x^13 - 834*x^12 + 1212*x^11 + 2619*x^10 - 4672*x^9 - 3573*x^8 + 8226*x^7 - 798*x^6 - 5520*x^5 + 14976*x^4 - 11364*x^3 - 2208*x^2 + 4392*x - 956)
 
gp: K = bnfinit(x^18 - 27*x^16 - 6*x^15 + 204*x^14 - 72*x^13 - 834*x^12 + 1212*x^11 + 2619*x^10 - 4672*x^9 - 3573*x^8 + 8226*x^7 - 798*x^6 - 5520*x^5 + 14976*x^4 - 11364*x^3 - 2208*x^2 + 4392*x - 956, 1)
 

Normalized defining polynomial

\( x^{18} - 27 x^{16} - 6 x^{15} + 204 x^{14} - 72 x^{13} - 834 x^{12} + 1212 x^{11} + 2619 x^{10} - 4672 x^{9} - 3573 x^{8} + 8226 x^{7} - 798 x^{6} - 5520 x^{5} + 14976 x^{4} - 11364 x^{3} - 2208 x^{2} + 4392 x - 956 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(913386706821601472305592008704=2^{34}\cdot 3^{25}\cdot 13^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $46.18$
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:  $2, 3, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{2}$, $\frac{1}{6} a^{12} - \frac{1}{2} a^{4} - \frac{1}{3} a^{3}$, $\frac{1}{6} a^{13} - \frac{1}{2} a^{5} - \frac{1}{3} a^{4}$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{13} + \frac{1}{18} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} - \frac{1}{6} a^{6} + \frac{7}{18} a^{5} - \frac{7}{18} a^{4} - \frac{1}{9} a^{3} - \frac{1}{9} a^{2} + \frac{1}{9} a - \frac{1}{9}$, $\frac{1}{36} a^{15} - \frac{1}{36} a^{14} - \frac{1}{18} a^{13} - \frac{1}{18} a^{12} + \frac{1}{18} a^{11} - \frac{1}{18} a^{10} - \frac{1}{6} a^{9} - \frac{1}{6} a^{8} - \frac{1}{12} a^{7} + \frac{7}{36} a^{6} - \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{18} a^{3} + \frac{1}{18} a^{2} + \frac{4}{9} a - \frac{1}{3}$, $\frac{1}{148212} a^{16} + \frac{75}{16468} a^{15} - \frac{35}{37053} a^{14} + \frac{5813}{74106} a^{13} - \frac{2357}{37053} a^{12} + \frac{2621}{74106} a^{11} + \frac{1147}{8234} a^{10} + \frac{3281}{74106} a^{9} + \frac{2759}{49404} a^{8} - \frac{13709}{148212} a^{7} - \frac{11303}{24702} a^{6} - \frac{15482}{37053} a^{5} - \frac{16394}{37053} a^{4} - \frac{29425}{74106} a^{3} - \frac{1313}{37053} a^{2} + \frac{5647}{12351} a - \frac{12008}{37053}$, $\frac{1}{1109522130946888768179698520} a^{17} - \frac{875760094041984187853}{369840710315629589393232840} a^{16} - \frac{1265996524746914893111543}{554761065473444384089849260} a^{15} - \frac{1837678357672822637398573}{277380532736722192044924630} a^{14} + \frac{361416190657344796304596}{15410029596484566224718035} a^{13} - \frac{2260507306671640328707024}{27738053273672219204492463} a^{12} - \frac{2921454157015970554257629}{24120046324932364525645620} a^{11} + \frac{8051327740076937282414571}{61640118385938264898872140} a^{10} - \frac{92918462529428749527705743}{1109522130946888768179698520} a^{9} - \frac{35880263963155428662593459}{221904426189377753635939704} a^{8} - \frac{119929240061881509798202}{757870308023831125805805} a^{7} - \frac{193104108206439771523852891}{554761065473444384089849260} a^{6} - \frac{1213428740202375507630191}{55476106547344438408984926} a^{5} - \frac{1842751673728250464950251}{6164011838593826489887214} a^{4} - \frac{68834427646359712200387998}{138690266368361096022462315} a^{3} - \frac{117102863738301088072402817}{277380532736722192044924630} a^{2} + \frac{7273076950543078413333189}{30820059192969132449436070} a + \frac{101670105763836698189949499}{277380532736722192044924630}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $11$
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:  \( 2310293755.43 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$S_3\times S_3\wr C_2$ (as 18T150):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 432
The 27 conjugacy class representatives for $S_3\times S_3\wr C_2$
Character table for $S_3\times S_3\wr C_2$ is not computed

Intermediate fields

\(\Q(\sqrt{3}) \), 3.3.2808.1, 6.6.378473472.1, 6.2.7278336.2

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

Sibling fields

Degree 12 siblings: data not computed
Degree 18 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }$ ${\href{/LocalNumberField/11.6.0.1}{6} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{4}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{4}$

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$2.2.2.2$x^{2} + 2 x - 2$$2$$1$$2$$C_2$$[2]$
2.4.8.4$x^{4} + 6 x^{2} + 4 x + 6$$4$$1$$8$$C_2^2$$[2, 3]$
2.4.6.9$x^{4} + 2 x^{3} + 6$$4$$1$$6$$D_{4}$$[2, 2]^{2}$
2.8.18.54$x^{8} + 6 x^{6} + 4 x^{3} + 6$$8$$1$$18$$D_4\times C_2$$[2, 2, 3]^{2}$
3Data not computed
$13$$\Q_{13}$$x + 2$$1$$1$$0$Trivial$[\ ]$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.1.2$x^{2} + 26$$2$$1$$1$$C_2$$[\ ]_{2}$
13.3.0.1$x^{3} - 2 x + 6$$1$$3$$0$$C_3$$[\ ]^{3}$
13.4.2.1$x^{4} + 39 x^{2} + 676$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
13.6.3.2$x^{6} - 338 x^{2} + 13182$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$