↑

Properties

Label 18.0.32388323041...0000.3
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{33}\cdot 5^{12}\cdot 17^{12}$
Root discriminant $229.98$
Ramified primes $2, 3, 5, 17$
Class number $243$ (GRH)
Class group $[3, 3, 3, 3, 3]$ (GRH)
Galois group $C_3\times C_3:S_3$ (as 18T23)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![385910388, 162804708, -287391240, 75997794, 91169694, -59070312, 13853359, 3557637, -912039, -66486, 62061, -207, -7292, 2889, -579, -18, 39, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 39*x^16 - 18*x^15 - 579*x^14 + 2889*x^13 - 7292*x^12 - 207*x^11 + 62061*x^10 - 66486*x^9 - 912039*x^8 + 3557637*x^7 + 13853359*x^6 - 59070312*x^5 + 91169694*x^4 + 75997794*x^3 - 287391240*x^2 + 162804708*x + 385910388)
 
gp: K = bnfinit(x^18 - 9*x^17 + 39*x^16 - 18*x^15 - 579*x^14 + 2889*x^13 - 7292*x^12 - 207*x^11 + 62061*x^10 - 66486*x^9 - 912039*x^8 + 3557637*x^7 + 13853359*x^6 - 59070312*x^5 + 91169694*x^4 + 75997794*x^3 - 287391240*x^2 + 162804708*x + 385910388, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 39 x^{16} - 18 x^{15} - 579 x^{14} + 2889 x^{13} - 7292 x^{12} - 207 x^{11} + 62061 x^{10} - 66486 x^{9} - 912039 x^{8} + 3557637 x^{7} + 13853359 x^{6} - 59070312 x^{5} + 91169694 x^{4} + 75997794 x^{3} - 287391240 x^{2} + 162804708 x + 385910388 \)

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:  \(-3238832304182321509797314520003000000000000=-\,2^{12}\cdot 3^{33}\cdot 5^{12}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $229.98$
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:  $2, 3, 5, 17$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} - \frac{1}{3} a^{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{4}$, $\frac{1}{9} a^{11} - \frac{1}{9} a^{10} + \frac{1}{9} a^{9} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{2}{9} a^{5} + \frac{1}{9} a^{4} - \frac{4}{9} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{18} a^{12} + \frac{1}{18} a^{9} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} - \frac{1}{18} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{7}{18} a^{3} - \frac{1}{3}$, $\frac{1}{18} a^{13} + \frac{1}{18} a^{10} + \frac{1}{3} a^{8} - \frac{1}{18} a^{7} - \frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{7}{18} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3} a$, $\frac{1}{18} a^{14} - \frac{1}{18} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{7}{18} a^{8} - \frac{1}{3} a^{7} - \frac{1}{3} a^{6} + \frac{7}{18} a^{5} - \frac{4}{9} a^{4} - \frac{2}{9} a^{3} + \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{2626110} a^{15} + \frac{1199}{262611} a^{14} - \frac{12799}{1313055} a^{13} - \frac{32479}{1313055} a^{12} + \frac{10016}{262611} a^{11} - \frac{90478}{1313055} a^{10} - \frac{87806}{1313055} a^{9} + \frac{251621}{1313055} a^{8} + \frac{349108}{1313055} a^{7} + \frac{76829}{262611} a^{6} + \frac{456479}{1313055} a^{5} - \frac{76847}{1313055} a^{4} + \frac{179629}{875370} a^{3} - \frac{32404}{87537} a^{2} - \frac{122548}{437685} a - \frac{43163}{145895}$, $\frac{1}{970568264751864617790} a^{16} + \frac{34964761935076}{485284132375932308895} a^{15} - \frac{9757977779917609294}{485284132375932308895} a^{14} - \frac{13109686766904401482}{485284132375932308895} a^{13} + \frac{13044598252865012012}{485284132375932308895} a^{12} - \frac{5953822258843016816}{161761377458644102965} a^{11} + \frac{26066255280770857606}{161761377458644102965} a^{10} - \frac{34864934330440553561}{485284132375932308895} a^{9} + \frac{11987208717785328173}{97056826475186461779} a^{8} - \frac{14543583015986170637}{69326304625133186985} a^{7} + \frac{51315761432350742269}{485284132375932308895} a^{6} + \frac{6125983282977870602}{161761377458644102965} a^{5} + \frac{368220975769856716709}{970568264751864617790} a^{4} + \frac{102032055010147137817}{485284132375932308895} a^{3} - \frac{26741146125308382276}{53920459152881367655} a^{2} - \frac{11307477579760725119}{32352275491728820593} a + \frac{78718240012346450447}{161761377458644102965}$, $\frac{1}{316046746792766072925755280130594912830} a^{17} - \frac{1863679204188679}{7524922542684906498232268574537974115} a^{16} + \frac{14228870412193055395482634801067}{105348915597588690975251760043531637610} a^{15} - \frac{3849221454979318026173659045736346812}{158023373396383036462877640065297456415} a^{14} - \frac{1824789461348174356781206823267926694}{158023373396383036462877640065297456415} a^{13} - \frac{2857242881694582909755931552631689343}{316046746792766072925755280130594912830} a^{12} + \frac{489363382192607935478409523099523896}{52674457798794345487625880021765818805} a^{11} - \frac{1695129295812662271664234511518441868}{158023373396383036462877640065297456415} a^{10} - \frac{37487019049921343471748673108828779493}{316046746792766072925755280130594912830} a^{9} + \frac{13339373508419758102101020752422001552}{31604674679276607292575528013059491283} a^{8} + \frac{26054974257508129612387257389812155686}{158023373396383036462877640065297456415} a^{7} - \frac{90673366107476670499324935722501951603}{316046746792766072925755280130594912830} a^{6} + \frac{14994284616859196508967143920802754421}{316046746792766072925755280130594912830} a^{5} - \frac{3175897049453221984852219190407690021}{158023373396383036462877640065297456415} a^{4} - \frac{14112790711706123593912870992776000135}{31604674679276607292575528013059491283} a^{3} + \frac{4072253785911604102386179727981871747}{10534891559758869097525176004353163761} a^{2} - \frac{1862346924131065868285599019999716741}{10534891559758869097525176004353163761} a + \frac{14287441687471492405453669848970582804}{52674457798794345487625880021765818805}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}$, which has order $243$ (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:  \( -\frac{322613093892650873}{13540489471792812627627885} a^{17} + \frac{6416222392224513221}{27080978943585625255255770} a^{16} - \frac{3179461939910923999}{2708097894358562525525577} a^{15} + \frac{9296390486531632823}{5416195788717125051051154} a^{14} + \frac{34356121266500029119}{3008997660398402806139530} a^{13} - \frac{428983748103555977723}{5416195788717125051051154} a^{12} + \frac{776314451010393725379}{3008997660398402806139530} a^{11} - \frac{7687795363768885552441}{27080978943585625255255770} a^{10} - \frac{29635967101534573574633}{27080978943585625255255770} a^{9} + \frac{14186262304587396430969}{5416195788717125051051154} a^{8} + \frac{165733773107215295620363}{9026992981195208418418590} a^{7} - \frac{2773865953876668672112463}{27080978943585625255255770} a^{6} - \frac{1154013138366596768222449}{5416195788717125051051154} a^{5} + \frac{20921298042257590218660022}{13540489471792812627627885} a^{4} - \frac{35409505473404777045170151}{9026992981195208418418590} a^{3} + \frac{12092639429623220885692801}{4513496490597604209209295} a^{2} + \frac{3048315327674360689540792}{902699298119520841841859} a - \frac{11305076408578940112936246}{1504498830199201403069765} \) (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:  \( 3978163078155.4883 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_3:S_3$ (as 18T23):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 18 conjugacy class representatives for $C_3\times C_3:S_3$
Character table for $C_3\times C_3:S_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.867.1 x3, 6.0.1771470000.6, Deg 6, Deg 6, 6.0.2255067.2

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

Sibling fields

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 ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
2.6.4.2$x^{6} - 2 x^{3} + 4$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$3$3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
3.6.11.15$x^{6} + 3 x^{3} + 3$$6$$1$$11$$S_3\times C_3$$[2, 5/2]_{2}$
$5$5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
5.6.4.2$x^{6} - 5 x^{3} + 50$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
$17$17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
17.6.4.1$x^{6} + 136 x^{3} + 7803$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$