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Properties

Label 18.0.52111185803...0000.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{9}\cdot 5^{6}\cdot 7^{6}\cdot 181^{6}$
Root discriminant $50.87$
Ramified primes $2, 3, 5, 7, 181$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_3\times S_3^2$ (as 18T46)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![9261, 17199, 8820, -4725, 9324, -25590, 28780, -23991, 16119, -7263, 1758, 597, -1142, 717, -189, -27, 33, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 33*x^16 - 27*x^15 - 189*x^14 + 717*x^13 - 1142*x^12 + 597*x^11 + 1758*x^10 - 7263*x^9 + 16119*x^8 - 23991*x^7 + 28780*x^6 - 25590*x^5 + 9324*x^4 - 4725*x^3 + 8820*x^2 + 17199*x + 9261)
 
gp: K = bnfinit(x^18 - 9*x^17 + 33*x^16 - 27*x^15 - 189*x^14 + 717*x^13 - 1142*x^12 + 597*x^11 + 1758*x^10 - 7263*x^9 + 16119*x^8 - 23991*x^7 + 28780*x^6 - 25590*x^5 + 9324*x^4 - 4725*x^3 + 8820*x^2 + 17199*x + 9261, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 33 x^{16} - 27 x^{15} - 189 x^{14} + 717 x^{13} - 1142 x^{12} + 597 x^{11} + 1758 x^{10} - 7263 x^{9} + 16119 x^{8} - 23991 x^{7} + 28780 x^{6} - 25590 x^{5} + 9324 x^{4} - 4725 x^{3} + 8820 x^{2} + 17199 x + 9261 \)

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:  \(-5211118580339502644200128000000=-\,2^{12}\cdot 3^{9}\cdot 5^{6}\cdot 7^{6}\cdot 181^{6}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $50.87$
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:  $2, 3, 5, 7, 181$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{4} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{7} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{8} - \frac{1}{3} a^{2}$, $\frac{1}{21} a^{9} + \frac{1}{21} a^{8} + \frac{2}{21} a^{7} - \frac{1}{21} a^{6} - \frac{10}{21} a^{5} + \frac{5}{21} a^{4} + \frac{1}{3} a^{3} - \frac{5}{21} a^{2}$, $\frac{1}{21} a^{10} + \frac{1}{21} a^{8} - \frac{1}{7} a^{7} - \frac{2}{21} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} + \frac{3}{7} a^{3} - \frac{3}{7} a^{2}$, $\frac{1}{21} a^{11} + \frac{1}{7} a^{8} + \frac{1}{7} a^{7} + \frac{2}{21} a^{6} + \frac{5}{21} a^{5} - \frac{10}{21} a^{4} - \frac{3}{7} a^{3} + \frac{5}{21} a^{2}$, $\frac{1}{63} a^{12} - \frac{1}{63} a^{10} + \frac{2}{21} a^{8} + \frac{2}{21} a^{7} - \frac{4}{63} a^{6} - \frac{10}{21} a^{5} + \frac{16}{63} a^{4} + \frac{8}{21} a^{3} - \frac{2}{7} a^{2}$, $\frac{1}{63} a^{13} - \frac{1}{63} a^{11} + \frac{5}{63} a^{7} - \frac{1}{21} a^{6} - \frac{29}{63} a^{5} + \frac{5}{21} a^{4} + \frac{8}{21} a^{3} - \frac{4}{21} a^{2}$, $\frac{1}{315} a^{14} + \frac{1}{315} a^{13} + \frac{1}{315} a^{12} - \frac{4}{315} a^{11} - \frac{2}{315} a^{10} - \frac{2}{105} a^{9} - \frac{19}{315} a^{8} - \frac{1}{45} a^{7} + \frac{23}{315} a^{6} + \frac{97}{315} a^{5} - \frac{118}{315} a^{4} - \frac{2}{35} a^{3} - \frac{4}{105} a^{2} - \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{315} a^{15} + \frac{2}{315} a^{11} + \frac{2}{105} a^{10} + \frac{2}{315} a^{9} - \frac{11}{105} a^{8} + \frac{1}{7} a^{7} + \frac{1}{35} a^{6} + \frac{5}{63} a^{5} + \frac{5}{21} a^{4} + \frac{17}{105} a^{3} + \frac{2}{105} a^{2} - \frac{1}{5} a - \frac{2}{5}$, $\frac{1}{2205} a^{16} + \frac{2}{2205} a^{15} - \frac{1}{2205} a^{14} + \frac{4}{2205} a^{13} + \frac{16}{2205} a^{12} + \frac{13}{735} a^{11} + \frac{1}{2205} a^{10} - \frac{8}{2205} a^{9} + \frac{193}{2205} a^{8} + \frac{341}{2205} a^{7} + \frac{40}{441} a^{6} + \frac{1}{35} a^{5} + \frac{619}{2205} a^{4} + \frac{12}{35} a^{3} + \frac{11}{105} a^{2} - \frac{1}{5} a + \frac{2}{5}$, $\frac{1}{6286531916987390460735} a^{17} + \frac{530755030440389036}{6286531916987390460735} a^{16} - \frac{6460239112568641471}{6286531916987390460735} a^{15} + \frac{1242110867906840377}{1257306383397478092147} a^{14} - \frac{37728037121879172353}{6286531916987390460735} a^{13} - \frac{684578681808571159}{2095510638995796820245} a^{12} - \frac{126219752075488712444}{6286531916987390460735} a^{11} + \frac{135131246308892473}{13347201522266221785} a^{10} - \frac{4614341862453198388}{1257306383397478092147} a^{9} + \frac{189659600355681489367}{6286531916987390460735} a^{8} - \frac{171220562026823930281}{6286531916987390460735} a^{7} + \frac{9939826237830185817}{99786220904561753345} a^{6} + \frac{1872967412082972565426}{6286531916987390460735} a^{5} - \frac{327002774540462847104}{898075988141055780105} a^{4} - \frac{29985860486858307721}{99786220904561753345} a^{3} + \frac{27306775188274640309}{59871732542737052007} a^{2} + \frac{2624076115425993487}{14255174414937393335} a - \frac{3366197543054570484}{14255174414937393335}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( \frac{6441944}{836771511429} a^{17} - \frac{76578442}{597693936735} a^{16} + \frac{1086255938}{1394619185715} a^{15} - \frac{9224244817}{4183857557145} a^{14} + \frac{948840241}{1394619185715} a^{13} + \frac{1776170164}{119538787347} a^{12} - \frac{30675492329}{597693936735} a^{11} + \frac{72209529734}{836771511429} a^{10} - \frac{25731915022}{464873061905} a^{9} - \frac{565023254654}{4183857557145} a^{8} + \frac{830379202316}{1394619185715} a^{7} - \frac{1075822199002}{836771511429} a^{6} + \frac{8023366991071}{4183857557145} a^{5} - \frac{3161566698781}{1394619185715} a^{4} + \frac{384875280871}{199231312245} a^{3} - \frac{12969009892}{13282087483} a^{2} + \frac{2069078899}{9487205345} a + \frac{5709138424}{9487205345} \) (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:  \( 1263791154.7914963 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times S_3^2$ (as 18T46):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 108
The 27 conjugacy class representatives for $C_3\times S_3^2$
Character table for $C_3\times S_3^2$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.140.1, 6.0.884547.1, 6.0.529200.1

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

Sibling fields

Degree 12 sibling: data not computed
Degree 18 siblings: 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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }{,}\,{\href{/LocalNumberField/37.3.0.1}{3} }{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/37.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{9}$

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
2Data not computed
$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$$[\ ]_{2}^{3}$
$5$5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
5.4.2.1$x^{4} + 15 x^{2} + 100$$2$$2$$2$$C_2^2$$[\ ]_{2}^{2}$
$7$$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{7}$$x + 2$$1$$1$$0$Trivial$[\ ]$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
7.2.1.1$x^{2} - 7$$2$$1$$1$$C_2$$[\ ]_{2}$
$181$$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
$\Q_{181}$$x + 2$$1$$1$$0$Trivial$[\ ]$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.2.0.1$x^{2} - x + 18$$1$$2$$0$$C_2$$[\ ]^{2}$
181.3.2.1$x^{3} - 181$$3$$1$$2$$C_3$$[\ ]_{3}$
181.6.4.1$x^{6} + 9593 x^{3} + 191062152$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$