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Properties

Label 18.0.45028390589...0000.5
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{37}\cdot 5^{12}$
Root discriminant $44.40$
Ramified primes $2, 3, 5$
Class number $27$ (GRH)
Class group $[3, 3, 3]$ (GRH)
Galois group $C_3^2 : C_2$ (as 18T4)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4356, 8316, 13500, 45324, 17064, -113832, 123129, -82809, 43227, -14664, 1557, 1071, -516, -27, 189, -126, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 516*x^12 + 1071*x^11 + 1557*x^10 - 14664*x^9 + 43227*x^8 - 82809*x^7 + 123129*x^6 - 113832*x^5 + 17064*x^4 + 45324*x^3 + 13500*x^2 + 8316*x + 4356)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 126*x^15 + 189*x^14 - 27*x^13 - 516*x^12 + 1071*x^11 + 1557*x^10 - 14664*x^9 + 43227*x^8 - 82809*x^7 + 123129*x^6 - 113832*x^5 + 17064*x^4 + 45324*x^3 + 13500*x^2 + 8316*x + 4356, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 126 x^{15} + 189 x^{14} - 27 x^{13} - 516 x^{12} + 1071 x^{11} + 1557 x^{10} - 14664 x^{9} + 43227 x^{8} - 82809 x^{7} + 123129 x^{6} - 113832 x^{5} + 17064 x^{4} + 45324 x^{3} + 13500 x^{2} + 8316 x + 4356 \)

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:  \(-450283905890997363000000000000=-\,2^{12}\cdot 3^{37}\cdot 5^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $44.40$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{5} a^{6} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} - \frac{2}{5} a^{3} + \frac{1}{5} a^{2} + \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{5} a^{7} + \frac{2}{5} a^{5} + \frac{1}{5} a^{4} + \frac{2}{5} a - \frac{2}{5}$, $\frac{1}{5} a^{8} + \frac{2}{5} a^{5} - \frac{2}{5} a^{4} - \frac{1}{5} a^{3} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{45} a^{9} + \frac{1}{5} a^{5} - \frac{2}{5} a^{4} - \frac{2}{15} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a + \frac{1}{15}$, $\frac{1}{45} a^{10} + \frac{1}{5} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a - \frac{1}{5}$, $\frac{1}{45} a^{11} + \frac{4}{15} a^{5} - \frac{1}{5} a^{4} + \frac{2}{5} a^{3} + \frac{7}{15} a^{2} + \frac{2}{5} a - \frac{1}{5}$, $\frac{1}{450} a^{12} + \frac{2}{225} a^{11} - \frac{2}{225} a^{10} - \frac{1}{90} a^{9} - \frac{2}{25} a^{7} + \frac{7}{150} a^{6} + \frac{14}{75} a^{5} - \frac{7}{15} a^{4} + \frac{1}{30} a^{3} - \frac{4}{75} a^{2} - \frac{11}{75} a - \frac{4}{75}$, $\frac{1}{2700} a^{13} - \frac{1}{900} a^{12} + \frac{1}{150} a^{11} - \frac{1}{100} a^{10} + \frac{1}{180} a^{9} - \frac{2}{25} a^{8} - \frac{59}{900} a^{7} + \frac{1}{100} a^{6} - \frac{31}{150} a^{5} + \frac{53}{180} a^{4} + \frac{49}{300} a^{3} - \frac{6}{25} a^{2} + \frac{17}{50} a + \frac{31}{150}$, $\frac{1}{2700} a^{14} - \frac{1}{900} a^{12} - \frac{7}{900} a^{11} - \frac{1}{150} a^{10} + \frac{1}{300} a^{9} + \frac{17}{180} a^{8} - \frac{2}{75} a^{7} - \frac{7}{100} a^{6} - \frac{89}{900} a^{5} - \frac{21}{50} a^{4} + \frac{7}{60} a^{3} + \frac{19}{150} a^{2} - \frac{2}{25} a + \frac{13}{50}$, $\frac{1}{491400} a^{15} + \frac{43}{491400} a^{14} + \frac{1}{10920} a^{13} + \frac{1}{20475} a^{12} + \frac{193}{32760} a^{11} + \frac{1669}{163800} a^{10} - \frac{59}{6825} a^{9} - \frac{1513}{32760} a^{8} - \frac{4621}{54600} a^{7} - \frac{493}{8190} a^{6} + \frac{81691}{163800} a^{5} + \frac{1129}{4200} a^{4} - \frac{125}{312} a^{3} - \frac{5387}{27300} a^{2} + \frac{577}{5460} a - \frac{5489}{27300}$, $\frac{1}{982800} a^{16} - \frac{1}{982800} a^{15} - \frac{1}{36400} a^{14} - \frac{17}{122850} a^{13} - \frac{23}{65520} a^{12} + \frac{599}{109200} a^{11} + \frac{67}{8190} a^{10} - \frac{389}{36400} a^{9} - \frac{281}{5200} a^{8} - \frac{269}{5460} a^{7} + \frac{19763}{327600} a^{6} + \frac{10513}{21840} a^{5} + \frac{19031}{327600} a^{4} + \frac{281}{18200} a^{3} - \frac{1769}{3640} a^{2} + \frac{4633}{18200} a + \frac{5597}{13650}$, $\frac{1}{5332528797195600} a^{17} + \frac{1810860823}{5332528797195600} a^{16} + \frac{1893923791}{5332528797195600} a^{15} + \frac{21553673837}{2666264398597800} a^{14} - \frac{24307891321}{152357965634160} a^{13} + \frac{343092561479}{592503199688400} a^{12} - \frac{4719475837261}{888754799532600} a^{11} - \frac{2705797562153}{253929942723600} a^{10} + \frac{4500814941499}{592503199688400} a^{9} + \frac{1134149756321}{25392994272360} a^{8} - \frac{7888085742499}{136731507620400} a^{7} + \frac{113580175230727}{1777509599065200} a^{6} - \frac{169809727697399}{1777509599065200} a^{5} + \frac{85301523097897}{444377399766300} a^{4} + \frac{14558309510611}{29625159984420} a^{3} - \frac{2330361425611}{14107219040200} a^{2} - \frac{15192830396421}{49375266640700} a + \frac{903761133139}{1923711687300}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}$, which has order $27$ (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:  \( -\frac{2330596735}{30471593126832} a^{17} + \frac{8337083525}{11850063993768} a^{16} - \frac{42205415257}{11850063993768} a^{15} + \frac{242116368005}{23700127987536} a^{14} - \frac{1137037775285}{71100383962608} a^{13} + \frac{2885146570}{634824856809} a^{12} + \frac{2705471032685}{71100383962608} a^{11} - \frac{6092379038987}{71100383962608} a^{10} - \frac{42410881345}{390661450344} a^{9} + \frac{81252868483685}{71100383962608} a^{8} - \frac{11756117569715}{3385732569648} a^{7} + \frac{40567998752245}{5925031996884} a^{6} - \frac{5198272583684}{493752666407} a^{5} + \frac{251078784934655}{23700127987536} a^{4} - \frac{10210731386615}{2962515998442} a^{3} - \frac{3430466263275}{1975010665628} a^{2} - \frac{2955645331495}{1692866284824} a - \frac{816500707}{134659818111} \) (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:  \( 8347574.07937 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3:S_3$ (as 18T4):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 18
The 6 conjugacy class representatives for $C_3^2 : C_2$
Character table for $C_3^2 : C_2$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.6075.1 x3, 3.1.108.1 x3, 3.1.24300.4 x3, 3.1.24300.3 x3, 6.0.110716875.1, 6.0.34992.1, 6.0.1771470000.2, 6.0.1771470000.1, 9.1.387420489000000.13 x9

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

Sibling fields

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 R R R ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
3Data not computed
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$