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Properties

Label 18.0.63472850187...9123.5
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{9}\cdot 7^{12}\cdot 13^{12}$
Root discriminant $35.04$
Ramified primes $3, 7, 13$
Class number $9$ (GRH)
Class group $[3, 3]$ (GRH)
Galois group $S_3 \times C_3$ (as 18T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1728, -4752, 9468, -14058, 19809, -18939, 13099, -7926, 5814, -4338, 2544, -852, 220, -210, 198, -108, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 198*x^14 - 210*x^13 + 220*x^12 - 852*x^11 + 2544*x^10 - 4338*x^9 + 5814*x^8 - 7926*x^7 + 13099*x^6 - 18939*x^5 + 19809*x^4 - 14058*x^3 + 9468*x^2 - 4752*x + 1728)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 108*x^15 + 198*x^14 - 210*x^13 + 220*x^12 - 852*x^11 + 2544*x^10 - 4338*x^9 + 5814*x^8 - 7926*x^7 + 13099*x^6 - 18939*x^5 + 19809*x^4 - 14058*x^3 + 9468*x^2 - 4752*x + 1728, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 108 x^{15} + 198 x^{14} - 210 x^{13} + 220 x^{12} - 852 x^{11} + 2544 x^{10} - 4338 x^{9} + 5814 x^{8} - 7926 x^{7} + 13099 x^{6} - 18939 x^{5} + 19809 x^{4} - 14058 x^{3} + 9468 x^{2} - 4752 x + 1728 \)

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:  \(-6347285018761982937208599123=-\,3^{9}\cdot 7^{12}\cdot 13^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.04$
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:  $3, 7, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{6} + \frac{1}{6} a^{4} - \frac{1}{2} a^{3} - \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{7} + \frac{1}{6} a^{5} - \frac{1}{3} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{12} a^{8} - \frac{1}{12} a^{2}$, $\frac{1}{12} a^{9} - \frac{1}{12} a^{3}$, $\frac{1}{24} a^{10} - \frac{1}{24} a^{9} - \frac{1}{12} a^{7} - \frac{1}{12} a^{5} + \frac{5}{24} a^{4} - \frac{7}{24} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{24} a^{11} - \frac{1}{24} a^{9} - \frac{1}{12} a^{7} - \frac{1}{12} a^{6} + \frac{1}{8} a^{5} - \frac{1}{12} a^{4} - \frac{1}{24} a^{3} + \frac{1}{6} a^{2}$, $\frac{1}{144} a^{12} + \frac{1}{72} a^{10} + \frac{1}{48} a^{9} - \frac{1}{24} a^{8} - \frac{1}{12} a^{7} + \frac{11}{144} a^{6} + \frac{1}{6} a^{5} - \frac{13}{72} a^{4} - \frac{5}{48} a^{3} + \frac{3}{8} a^{2} - \frac{1}{4} a$, $\frac{1}{144} a^{13} + \frac{1}{72} a^{11} - \frac{1}{48} a^{10} - \frac{1}{144} a^{7} + \frac{17}{72} a^{5} + \frac{1}{48} a^{4} - \frac{1}{2} a^{3} - \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{1152} a^{14} + \frac{1}{1152} a^{13} + \frac{1}{576} a^{12} - \frac{1}{1152} a^{11} - \frac{7}{384} a^{10} - \frac{7}{192} a^{9} - \frac{37}{1152} a^{8} + \frac{11}{1152} a^{7} + \frac{17}{576} a^{6} + \frac{121}{1152} a^{5} - \frac{5}{384} a^{4} - \frac{9}{64} a^{3} + \frac{7}{32} a^{2} - \frac{1}{8} a - \frac{1}{2}$, $\frac{1}{1152} a^{15} + \frac{1}{1152} a^{13} - \frac{1}{384} a^{12} - \frac{5}{288} a^{11} - \frac{7}{384} a^{10} + \frac{5}{1152} a^{9} - \frac{1}{24} a^{8} + \frac{23}{1152} a^{7} + \frac{29}{384} a^{6} - \frac{17}{144} a^{5} - \frac{49}{384} a^{4} + \frac{23}{64} a^{3} - \frac{25}{96} a^{2} - \frac{3}{8} a - \frac{1}{2}$, $\frac{1}{20809728} a^{16} - \frac{1}{2601216} a^{15} - \frac{2219}{5202432} a^{14} + \frac{973}{325152} a^{13} - \frac{11929}{3468288} a^{12} - \frac{47557}{2601216} a^{11} + \frac{20081}{3468288} a^{10} - \frac{416123}{10404864} a^{9} - \frac{70421}{2601216} a^{8} + \frac{64495}{2601216} a^{7} + \frac{163985}{3468288} a^{6} + \frac{606025}{2601216} a^{5} + \frac{914321}{20809728} a^{4} + \frac{1620757}{3468288} a^{3} - \frac{225965}{1734144} a^{2} + \frac{57001}{144512} a + \frac{9321}{36128}$, $\frac{1}{56789747712} a^{17} + \frac{113}{4732478976} a^{16} - \frac{4945451}{14197436928} a^{15} - \frac{81385}{394373248} a^{14} - \frac{85351019}{28394873856} a^{13} - \frac{435209}{147889968} a^{12} + \frac{227793395}{28394873856} a^{11} + \frac{196776395}{9464957952} a^{10} - \frac{9023215}{221834952} a^{9} - \frac{16312033}{788746496} a^{8} + \frac{1074756035}{28394873856} a^{7} - \frac{6718915}{591559872} a^{6} + \frac{9463671889}{56789747712} a^{5} + \frac{431414561}{3154985984} a^{4} - \frac{1242322619}{4732478976} a^{3} + \frac{37858687}{197186624} a^{2} + \frac{10694271}{49296656} a + \frac{10115037}{24648328}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( -\frac{144099}{197186624} a^{17} + \frac{2449683}{394373248} a^{16} - \frac{11149223}{443669904} a^{15} + \frac{9500435}{147889968} a^{14} - \frac{46247657}{443669904} a^{13} + \frac{47374457}{591559872} a^{12} - \frac{38118467}{443669904} a^{11} + \frac{325798451}{591559872} a^{10} - \frac{2738540605}{1774679616} a^{9} + \frac{652623833}{295779936} a^{8} - \frac{293295781}{110917476} a^{7} + \frac{2196812551}{591559872} a^{6} - \frac{11961780337}{1774679616} a^{5} + \frac{10970630825}{1183119744} a^{4} - \frac{4395510373}{591559872} a^{3} + \frac{324815003}{98593312} a^{2} - \frac{56597075}{24648328} a + \frac{8262617}{6162082} \) (order $6$)
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:  \( 5831495.31751 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3$ (as 18T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 9 conjugacy class representatives for $S_3 \times C_3$
Character table for $S_3 \times C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.24843.1 x3, 3.3.8281.1, 6.0.1851523947.3, 6.0.223587.2 x2, 6.0.1851523947.2, 9.3.15332469805107.4 x3

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

Sibling fields

Degree 6 sibling: 6.0.223587.2
Degree 9 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 ${\href{/LocalNumberField/2.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
$3$3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
3.6.3.2$x^{6} - 9 x^{2} + 27$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$13$13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
13.9.6.1$x^{9} + 234 x^{6} + 16900 x^{3} + 474552$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$