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Properties

Label 18.0.71860663336...6879.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 239^{9}$
Root discriminant $66.89$
Ramified primes $3, 239$
Class number $1215$ (GRH)
Class group $[3, 3, 3, 45]$ (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![512000, 0, 56880, 185679, -107280, -44469, 71814, 48843, 29970, -10260, -14661, -2511, 2301, 648, 117, -54, -18, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 18*x^16 - 54*x^15 + 117*x^14 + 648*x^13 + 2301*x^12 - 2511*x^11 - 14661*x^10 - 10260*x^9 + 29970*x^8 + 48843*x^7 + 71814*x^6 - 44469*x^5 - 107280*x^4 + 185679*x^3 + 56880*x^2 + 512000)
 
gp: K = bnfinit(x^18 - 18*x^16 - 54*x^15 + 117*x^14 + 648*x^13 + 2301*x^12 - 2511*x^11 - 14661*x^10 - 10260*x^9 + 29970*x^8 + 48843*x^7 + 71814*x^6 - 44469*x^5 - 107280*x^4 + 185679*x^3 + 56880*x^2 + 512000, 1)
 

Normalized defining polynomial

\( x^{18} - 18 x^{16} - 54 x^{15} + 117 x^{14} + 648 x^{13} + 2301 x^{12} - 2511 x^{11} - 14661 x^{10} - 10260 x^{9} + 29970 x^{8} + 48843 x^{7} + 71814 x^{6} - 44469 x^{5} - 107280 x^{4} + 185679 x^{3} + 56880 x^{2} + 512000 \)

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:  \(-718606633367888171454789870996879=-\,3^{24}\cdot 239^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $66.89$
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:  $3, 239$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} - \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a$, $\frac{1}{6} a^{8} + \frac{1}{3} a^{2} - \frac{1}{2} a$, $\frac{1}{6} a^{9} + \frac{1}{3} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{6} a^{10} + \frac{1}{3} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{12} a^{11} - \frac{1}{12} a^{10} - \frac{1}{12} a^{8} - \frac{1}{6} a^{7} - \frac{1}{3} a^{5} - \frac{5}{12} a^{4} + \frac{1}{4} a^{3} - \frac{1}{6} a^{2} + \frac{5}{12} a$, $\frac{1}{4356} a^{12} - \frac{3}{242} a^{11} - \frac{1}{484} a^{10} + \frac{83}{1452} a^{9} - \frac{119}{1452} a^{8} - \frac{5}{66} a^{7} - \frac{10}{99} a^{6} - \frac{41}{484} a^{5} - \frac{2}{11} a^{4} - \frac{677}{1452} a^{3} - \frac{301}{1452} a^{2} + \frac{5}{12} a + \frac{67}{1089}$, $\frac{1}{4356} a^{13} - \frac{7}{1452} a^{11} - \frac{79}{1452} a^{10} + \frac{7}{1452} a^{9} - \frac{1}{726} a^{8} + \frac{14}{99} a^{7} + \frac{185}{1452} a^{6} - \frac{307}{726} a^{5} - \frac{413}{1452} a^{4} - \frac{559}{1452} a^{3} + \frac{323}{1452} a^{2} - \frac{296}{1089} a - \frac{125}{363}$, $\frac{1}{261360} a^{14} - \frac{1}{14520} a^{13} + \frac{1}{52272} a^{12} + \frac{313}{14520} a^{11} - \frac{241}{4356} a^{10} - \frac{307}{14520} a^{9} - \frac{481}{23760} a^{8} - \frac{4463}{29040} a^{7} + \frac{1447}{32670} a^{6} - \frac{13169}{29040} a^{5} - \frac{239}{1980} a^{4} - \frac{67}{7260} a^{3} - \frac{43921}{130680} a^{2} + \frac{13051}{29040} a - \frac{1}{297}$, $\frac{1}{31624560} a^{15} - \frac{1}{1171280} a^{14} + \frac{227}{31624560} a^{13} + \frac{161}{3513840} a^{12} - \frac{10111}{5270760} a^{11} + \frac{42083}{1756920} a^{10} + \frac{2499643}{31624560} a^{9} - \frac{14492}{219615} a^{8} - \frac{2231281}{31624560} a^{7} - \frac{21467}{234256} a^{6} - \frac{3858013}{10541520} a^{5} + \frac{86401}{175692} a^{4} - \frac{3240847}{15812280} a^{3} + \frac{1093853}{3513840} a^{2} - \frac{107431}{261360} a - \frac{19684}{43923}$, $\frac{1}{7589894400} a^{16} - \frac{1}{843321600} a^{15} - \frac{17}{7589894400} a^{14} - \frac{4709}{843321600} a^{13} - \frac{8777}{344995200} a^{12} + \frac{1171781}{140553600} a^{11} + \frac{93428051}{1517978880} a^{10} + \frac{17032693}{421660800} a^{9} + \frac{248119873}{7589894400} a^{8} - \frac{35785493}{843321600} a^{7} - \frac{1058446937}{7589894400} a^{6} + \frac{1323847}{6388800} a^{5} - \frac{29656123}{151797888} a^{4} - \frac{50935111}{843321600} a^{3} + \frac{1309658311}{7589894400} a^{2} + \frac{26023}{527076} a - \frac{105551}{1185921}$, $\frac{1}{724242990697815916800} a^{17} + \frac{42706400069}{724242990697815916800} a^{16} - \frac{2561031581759}{724242990697815916800} a^{15} - \frac{827047330637707}{724242990697815916800} a^{14} + \frac{18816951616564237}{181060747674453979200} a^{13} - \frac{1379947496838059}{362121495348907958400} a^{12} + \frac{9004699936012961347}{724242990697815916800} a^{11} + \frac{1958986825149886451}{181060747674453979200} a^{10} - \frac{9149786295935972359}{144848598139563183360} a^{9} - \frac{57594913355359880143}{724242990697815916800} a^{8} - \frac{2196461490294911293}{65840271881619628800} a^{7} - \frac{751203041687237413}{72424299069781591680} a^{6} - \frac{68758499727400311751}{362121495348907958400} a^{5} - \frac{60870993722542703059}{724242990697815916800} a^{4} + \frac{286806844835844913909}{724242990697815916800} a^{3} + \frac{50009032696752114169}{362121495348907958400} a^{2} - \frac{4259333983509641341}{9053037383722698960} a - \frac{2162635710973609}{113162967296533737}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{45}$, which has order $1215$ (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:  \( 10885172.9386 \) (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

\(\Q(\sqrt{-239}) \), 3.1.239.1 x3, \(\Q(\zeta_{9})^+\), 6.0.13651919.1, 6.0.89570240559.2 x2, 6.0.89570240559.4, 9.3.7255189485279.4 x3

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

Sibling fields

Degree 6 sibling: data not computed
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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$3$3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
3.3.4.2$x^{3} - 3 x^{2} + 3$$3$$1$$4$$C_3$$[2]$
239Data not computed