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Properties

Label 18.0.21137726688...3008.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{6}\cdot 7^{15}\cdot 13^{12}$
Root discriminant $80.71$
Ramified primes $2, 3, 7, 13$
Class number $19656$ (GRH)
Class group $[3, 3, 2184]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![24890048, 42657746, 32238933, 18327094, 10123484, 4190046, 1272160, 435918, 94236, -7126, -766, -2142, -1868, 42, -56, -38, 20, -4, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 4*x^17 + 20*x^16 - 38*x^15 - 56*x^14 + 42*x^13 - 1868*x^12 - 2142*x^11 - 766*x^10 - 7126*x^9 + 94236*x^8 + 435918*x^7 + 1272160*x^6 + 4190046*x^5 + 10123484*x^4 + 18327094*x^3 + 32238933*x^2 + 42657746*x + 24890048)
 
gp: K = bnfinit(x^18 - 4*x^17 + 20*x^16 - 38*x^15 - 56*x^14 + 42*x^13 - 1868*x^12 - 2142*x^11 - 766*x^10 - 7126*x^9 + 94236*x^8 + 435918*x^7 + 1272160*x^6 + 4190046*x^5 + 10123484*x^4 + 18327094*x^3 + 32238933*x^2 + 42657746*x + 24890048, 1)
 

Normalized defining polynomial

\( x^{18} - 4 x^{17} + 20 x^{16} - 38 x^{15} - 56 x^{14} + 42 x^{13} - 1868 x^{12} - 2142 x^{11} - 766 x^{10} - 7126 x^{9} + 94236 x^{8} + 435918 x^{7} + 1272160 x^{6} + 4190046 x^{5} + 10123484 x^{4} + 18327094 x^{3} + 32238933 x^{2} + 42657746 x + 24890048 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-21137726688804112981348984293163008=-\,2^{18}\cdot 3^{6}\cdot 7^{15}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.71$
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:  $2, 3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{8} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{3} + \frac{1}{8} a^{2} + \frac{1}{4} a$, $\frac{1}{16} a^{7} - \frac{1}{16} a^{6} - \frac{1}{8} a^{5} + \frac{3}{16} a^{3} + \frac{1}{16} a^{2} - \frac{1}{8} a$, $\frac{1}{64} a^{8} - \frac{3}{64} a^{6} - \frac{1}{32} a^{5} - \frac{5}{64} a^{4} + \frac{3}{16} a^{3} + \frac{7}{64} a^{2} - \frac{5}{32} a$, $\frac{1}{128} a^{9} - \frac{1}{128} a^{8} - \frac{3}{128} a^{7} + \frac{1}{128} a^{6} - \frac{3}{128} a^{5} + \frac{17}{128} a^{4} - \frac{5}{128} a^{3} - \frac{17}{128} a^{2} - \frac{27}{64} a - \frac{1}{2}$, $\frac{1}{512} a^{10} - \frac{1}{128} a^{8} - \frac{1}{256} a^{7} - \frac{1}{256} a^{6} + \frac{7}{256} a^{5} - \frac{29}{128} a^{4} - \frac{11}{256} a^{3} - \frac{71}{512} a^{2} - \frac{123}{256} a - \frac{1}{8}$, $\frac{1}{2048} a^{11} - \frac{1}{2048} a^{10} - \frac{1}{512} a^{9} + \frac{1}{1024} a^{8} + \frac{1}{128} a^{6} + \frac{63}{1024} a^{5} - \frac{81}{1024} a^{4} - \frac{305}{2048} a^{3} - \frac{431}{2048} a^{2} + \frac{347}{1024} a + \frac{1}{32}$, $\frac{1}{8192} a^{12} - \frac{1}{4096} a^{11} - \frac{3}{8192} a^{10} - \frac{13}{4096} a^{9} + \frac{15}{4096} a^{8} - \frac{9}{512} a^{7} + \frac{167}{4096} a^{6} + \frac{5}{128} a^{5} + \frac{1361}{8192} a^{4} + \frac{657}{4096} a^{3} + \frac{3461}{8192} a^{2} + \frac{1829}{4096} a - \frac{33}{128}$, $\frac{1}{65536} a^{13} - \frac{3}{65536} a^{12} - \frac{1}{65536} a^{11} - \frac{23}{65536} a^{10} + \frac{23}{8192} a^{9} + \frac{105}{32768} a^{8} - \frac{977}{32768} a^{7} - \frac{711}{32768} a^{6} - \frac{367}{65536} a^{5} - \frac{12719}{65536} a^{4} - \frac{10781}{65536} a^{3} + \frac{5701}{65536} a^{2} + \frac{5435}{32768} a + \frac{161}{1024}$, $\frac{1}{131072} a^{14} + \frac{3}{65536} a^{12} + \frac{3}{65536} a^{11} + \frac{3}{131072} a^{10} + \frac{45}{65536} a^{9} - \frac{179}{32768} a^{8} - \frac{349}{32768} a^{7} - \frac{6457}{131072} a^{6} + \frac{3937}{32768} a^{5} - \frac{2381}{65536} a^{4} + \frac{16103}{65536} a^{3} - \frac{34347}{131072} a^{2} - \frac{799}{65536} a + \frac{19}{2048}$, $\frac{1}{524288} a^{15} - \frac{1}{524288} a^{14} - \frac{1}{262144} a^{13} + \frac{3}{65536} a^{12} + \frac{5}{524288} a^{11} + \frac{271}{524288} a^{10} - \frac{115}{262144} a^{9} + \frac{473}{65536} a^{8} - \frac{11957}{524288} a^{7} + \frac{27437}{524288} a^{6} + \frac{29101}{262144} a^{5} - \frac{2001}{16384} a^{4} + \frac{124143}{524288} a^{3} - \frac{72251}{524288} a^{2} - \frac{121945}{262144} a + \frac{2789}{8192}$, $\frac{1}{1432354816} a^{16} + \frac{497}{716177408} a^{15} + \frac{2043}{1432354816} a^{14} + \frac{873}{716177408} a^{13} + \frac{22541}{1432354816} a^{12} - \frac{82017}{716177408} a^{11} - \frac{940665}{1432354816} a^{10} + \frac{688587}{716177408} a^{9} - \frac{10406621}{1432354816} a^{8} - \frac{1187433}{716177408} a^{7} + \frac{693601}{1432354816} a^{6} + \frac{23232855}{716177408} a^{5} + \frac{89250639}{1432354816} a^{4} + \frac{131724601}{716177408} a^{3} - \frac{72726307}{1432354816} a^{2} + \frac{310001941}{716177408} a + \frac{8038703}{22380544}$, $\frac{1}{22917677056} a^{17} + \frac{5}{22917677056} a^{16} + \frac{2497}{22917677056} a^{15} - \frac{8029}{22917677056} a^{14} + \frac{515}{22917677056} a^{13} - \frac{644795}{22917677056} a^{12} + \frac{3554209}{22917677056} a^{11} + \frac{585547}{22917677056} a^{10} - \frac{73748187}{22917677056} a^{9} - \frac{30892169}{22917677056} a^{8} - \frac{147152949}{22917677056} a^{7} + \frac{1331905521}{22917677056} a^{6} + \frac{1442194009}{22917677056} a^{5} + \frac{5165077887}{22917677056} a^{4} - \frac{764343085}{22917677056} a^{3} + \frac{6844519073}{22917677056} a^{2} - \frac{1201578945}{11458838528} a + \frac{100511469}{358088704}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{2184}$, which has order $19656$ (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:  \( 1090042674.5636587 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-7}) \), 3.3.28392.1, 3.3.8281.1, 6.0.5642739648.7, 6.0.480024727.1, 9.9.54951571781503488.2

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 12 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
$\Q_{2}$$x + 1$$1$$1$$0$Trivial$[\ ]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
2.2.3.2$x^{2} + 6$$2$$1$$3$$C_2$$[3]$
$3$3.6.0.1$x^{6} - x + 2$$1$$6$$0$$C_6$$[\ ]^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.4$x^{6} + 14$$6$$1$$5$$C_6$$[\ ]_{6}$
13Data not computed