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Properties

Label 18.0.61773685738...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 11^{12}$
Root discriminant $39.76$
Ramified primes $2, 3, 5, 11$
Class number $3$ (GRH)
Class group $[3]$ (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![8100, -5400, 2700, 16140, -16910, 9090, 6239, -11791, 10815, -4988, 1341, -27, -258, 169, -81, 14, 5, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 5*x^16 + 14*x^15 - 81*x^14 + 169*x^13 - 258*x^12 - 27*x^11 + 1341*x^10 - 4988*x^9 + 10815*x^8 - 11791*x^7 + 6239*x^6 + 9090*x^5 - 16910*x^4 + 16140*x^3 + 2700*x^2 - 5400*x + 8100)
 
gp: K = bnfinit(x^18 - 3*x^17 + 5*x^16 + 14*x^15 - 81*x^14 + 169*x^13 - 258*x^12 - 27*x^11 + 1341*x^10 - 4988*x^9 + 10815*x^8 - 11791*x^7 + 6239*x^6 + 9090*x^5 - 16910*x^4 + 16140*x^3 + 2700*x^2 - 5400*x + 8100, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 5 x^{16} + 14 x^{15} - 81 x^{14} + 169 x^{13} - 258 x^{12} - 27 x^{11} + 1341 x^{10} - 4988 x^{9} + 10815 x^{8} - 11791 x^{7} + 6239 x^{6} + 9090 x^{5} - 16910 x^{4} + 16140 x^{3} + 2700 x^{2} - 5400 x + 8100 \)

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:  \(-61773685738999443000000000000=-\,2^{12}\cdot 3^{9}\cdot 5^{12}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $39.76$
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:  $2, 3, 5, 11$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{3} + \frac{1}{3} a$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{7} + \frac{1}{9} a^{6} + \frac{1}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2}$, $\frac{1}{9} a^{8} - \frac{1}{9} a^{2}$, $\frac{1}{9} a^{9} - \frac{1}{9} a^{3}$, $\frac{1}{9} a^{10} - \frac{1}{9} a^{4}$, $\frac{1}{27} a^{11} + \frac{1}{27} a^{10} - \frac{1}{27} a^{9} - \frac{1}{27} a^{8} - \frac{1}{9} a^{6} - \frac{4}{27} a^{5} + \frac{5}{27} a^{4} - \frac{2}{27} a^{3} + \frac{7}{27} a^{2} - \frac{1}{9} a + \frac{1}{3}$, $\frac{1}{162} a^{12} + \frac{1}{81} a^{11} + \frac{1}{162} a^{9} + \frac{4}{81} a^{8} + \frac{23}{162} a^{6} + \frac{2}{81} a^{5} + \frac{10}{27} a^{4} + \frac{5}{162} a^{3} + \frac{26}{81} a^{2} + \frac{1}{27} a + \frac{2}{9}$, $\frac{1}{162} a^{13} + \frac{1}{81} a^{11} + \frac{7}{162} a^{10} - \frac{2}{81} a^{8} + \frac{5}{162} a^{7} - \frac{4}{27} a^{6} + \frac{5}{81} a^{5} - \frac{49}{162} a^{4} + \frac{2}{27} a^{3} - \frac{19}{81} a^{2} + \frac{1}{27} a - \frac{1}{9}$, $\frac{1}{486} a^{14} - \frac{1}{486} a^{12} + \frac{7}{486} a^{11} + \frac{1}{81} a^{10} - \frac{13}{486} a^{9} + \frac{11}{486} a^{8} - \frac{1}{81} a^{7} - \frac{59}{486} a^{6} + \frac{41}{486} a^{5} + \frac{7}{81} a^{4} - \frac{101}{486} a^{3} + \frac{2}{27} a^{2} - \frac{2}{9} a - \frac{4}{9}$, $\frac{1}{53460} a^{15} - \frac{13}{53460} a^{14} - \frac{5}{2673} a^{13} - \frac{91}{53460} a^{12} - \frac{901}{53460} a^{11} - \frac{293}{26730} a^{10} - \frac{871}{17820} a^{9} + \frac{2023}{53460} a^{8} - \frac{686}{13365} a^{7} - \frac{4823}{53460} a^{6} - \frac{1105}{10692} a^{5} + \frac{1697}{26730} a^{4} + \frac{2246}{13365} a^{3} - \frac{427}{1782} a^{2} + \frac{17}{297} a + \frac{35}{99}$, $\frac{1}{80991900} a^{16} - \frac{122}{20247975} a^{15} - \frac{10313}{16198380} a^{14} - \frac{63397}{26997300} a^{13} - \frac{93403}{40495950} a^{12} - \frac{1169371}{80991900} a^{11} - \frac{348913}{7362900} a^{10} + \frac{69457}{1840725} a^{9} - \frac{3076109}{80991900} a^{8} + \frac{57313}{8999100} a^{7} + \frac{899243}{8099190} a^{6} + \frac{3812029}{80991900} a^{5} + \frac{4657607}{40495950} a^{4} + \frac{4082}{24543} a^{3} - \frac{143599}{299970} a^{2} + \frac{1208}{149985} a - \frac{2399}{9999}$, $\frac{1}{11685430340100} a^{17} - \frac{41077}{11685430340100} a^{16} - \frac{3967993}{1947571723350} a^{15} + \frac{11081269019}{11685430340100} a^{14} + \frac{414728423}{285010496100} a^{13} - \frac{980813401}{973785861675} a^{12} + \frac{20489097467}{3895143446700} a^{11} + \frac{14720688989}{779028689340} a^{10} + \frac{62864764459}{1947571723350} a^{9} + \frac{232830473743}{11685430340100} a^{8} + \frac{165940398467}{11685430340100} a^{7} - \frac{32198495743}{973785861675} a^{6} - \frac{570763617221}{5842715170050} a^{5} - \frac{2740140194323}{5842715170050} a^{4} - \frac{23010226843}{194757172335} a^{3} - \frac{1978642118}{7213228605} a^{2} - \frac{505118462}{1967244165} a - \frac{306965642}{1442645721}$

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

Class group and class number

$C_{3}$, which has order $3$ (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:  \( \frac{5567302}{28924332525} a^{17} - \frac{17535853}{28924332525} a^{16} + \frac{35365543}{38565776700} a^{15} + \frac{320357357}{115697330100} a^{14} - \frac{2272106}{141094305} a^{13} + \frac{247798129}{7713155340} a^{12} - \frac{1787670457}{38565776700} a^{11} - \frac{105082927}{19282888350} a^{10} + \frac{10115550743}{38565776700} a^{9} - \frac{109961557007}{115697330100} a^{8} + \frac{59367751406}{28924332525} a^{7} - \frac{83483745331}{38565776700} a^{6} + \frac{2116186211}{2103587820} a^{5} + \frac{68880808559}{57848665050} a^{4} - \frac{8609495837}{3856577670} a^{3} + \frac{38745623}{28567242} a^{2} + \frac{125997296}{214254315} a - \frac{1722530}{14283621} \) (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:  \( 430259533.331 \) (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

\(\Q(\sqrt{-3}) \), 3.1.300.1 x3, 3.1.36300.1 x3, 3.1.1452.1 x3, 3.1.9075.1 x3, 6.0.270000.1, 6.0.3953070000.1, 6.0.6324912.1, 6.0.247066875.1, 9.1.143496441000000.1 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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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$$[\ ]_{3}^{2}$
$3$3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
3.2.1.2$x^{2} + 3$$2$$1$$1$$C_2$$[\ ]_{2}$
$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$$[\ ]_{3}^{2}$
$11$11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
11.6.4.1$x^{6} + 220 x^{3} + 41503$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$