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Properties

Label 18.0.16413889305...6807.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,3^{24}\cdot 13^{9}\cdot 19^{17}$
Root discriminant $251.68$
Ramified primes $3, 13, 19$
Class number $12842226$ (GRH)
Class group $[3, 4280742]$ (GRH)
Galois group $C_{18}$ (as 18T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![176288881193, -143569732788, 121351374567, -51037054889, 16443414363, -171617115, 233470852, -291557100, 207680922, -5227070, -2483370, 1401495, 277369, 51456, 7548, -8, 66, -3, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 3*x^17 + 66*x^16 - 8*x^15 + 7548*x^14 + 51456*x^13 + 277369*x^12 + 1401495*x^11 - 2483370*x^10 - 5227070*x^9 + 207680922*x^8 - 291557100*x^7 + 233470852*x^6 - 171617115*x^5 + 16443414363*x^4 - 51037054889*x^3 + 121351374567*x^2 - 143569732788*x + 176288881193)
 
gp: K = bnfinit(x^18 - 3*x^17 + 66*x^16 - 8*x^15 + 7548*x^14 + 51456*x^13 + 277369*x^12 + 1401495*x^11 - 2483370*x^10 - 5227070*x^9 + 207680922*x^8 - 291557100*x^7 + 233470852*x^6 - 171617115*x^5 + 16443414363*x^4 - 51037054889*x^3 + 121351374567*x^2 - 143569732788*x + 176288881193, 1)
 

Normalized defining polynomial

\( x^{18} - 3 x^{17} + 66 x^{16} - 8 x^{15} + 7548 x^{14} + 51456 x^{13} + 277369 x^{12} + 1401495 x^{11} - 2483370 x^{10} - 5227070 x^{9} + 207680922 x^{8} - 291557100 x^{7} + 233470852 x^{6} - 171617115 x^{5} + 16443414363 x^{4} - 51037054889 x^{3} + 121351374567 x^{2} - 143569732788 x + 176288881193 \)

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:  \(-16413889305350509964853040604122672586106807=-\,3^{24}\cdot 13^{9}\cdot 19^{17}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $251.68$
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:  $3, 13, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2223=3^{2}\cdot 13\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{2223}(1600,·)$, $\chi_{2223}(1,·)$, $\chi_{2223}(196,·)$, $\chi_{2223}(337,·)$, $\chi_{2223}(1234,·)$, $\chi_{2223}(157,·)$, $\chi_{2223}(625,·)$, $\chi_{2223}(1000,·)$, $\chi_{2223}(235,·)$, $\chi_{2223}(2092,·)$, $\chi_{2223}(1390,·)$, $\chi_{2223}(1327,·)$, $\chi_{2223}(1585,·)$, $\chi_{2223}(1780,·)$, $\chi_{2223}(376,·)$, $\chi_{2223}(313,·)$, $\chi_{2223}(1873,·)$, $\chi_{2223}(1663,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{19} a^{9} + \frac{8}{19} a^{8} + \frac{1}{19} a^{7} + \frac{7}{19} a^{6} + \frac{3}{19} a^{5} + \frac{9}{19} a^{4} - \frac{1}{19} a^{3} - \frac{4}{19} a^{2} - \frac{3}{19} a - \frac{1}{19}$, $\frac{1}{19} a^{10} - \frac{6}{19} a^{8} - \frac{1}{19} a^{7} + \frac{4}{19} a^{6} + \frac{4}{19} a^{5} + \frac{3}{19} a^{4} + \frac{4}{19} a^{3} - \frac{9}{19} a^{2} + \frac{4}{19} a + \frac{8}{19}$, $\frac{1}{19} a^{11} + \frac{9}{19} a^{8} - \frac{9}{19} a^{7} + \frac{8}{19} a^{6} + \frac{2}{19} a^{5} + \frac{1}{19} a^{4} + \frac{4}{19} a^{3} - \frac{1}{19} a^{2} + \frac{9}{19} a - \frac{6}{19}$, $\frac{1}{19} a^{12} - \frac{5}{19} a^{8} - \frac{1}{19} a^{7} - \frac{4}{19} a^{6} - \frac{7}{19} a^{5} - \frac{1}{19} a^{4} + \frac{8}{19} a^{3} + \frac{7}{19} a^{2} + \frac{2}{19} a + \frac{9}{19}$, $\frac{1}{589} a^{13} + \frac{15}{589} a^{12} - \frac{10}{589} a^{11} + \frac{1}{589} a^{10} + \frac{4}{589} a^{9} + \frac{90}{589} a^{8} + \frac{193}{589} a^{7} - \frac{156}{589} a^{6} - \frac{11}{31} a^{5} + \frac{238}{589} a^{4} + \frac{272}{589} a^{3} + \frac{91}{589} a^{2} + \frac{59}{589} a - \frac{6}{19}$, $\frac{1}{589} a^{14} + \frac{13}{589} a^{12} - \frac{4}{589} a^{11} - \frac{11}{589} a^{10} - \frac{1}{589} a^{9} + \frac{83}{589} a^{8} - \frac{168}{589} a^{7} + \frac{271}{589} a^{6} + \frac{56}{589} a^{5} + \frac{143}{589} a^{4} - \frac{238}{589} a^{3} + \frac{120}{589} a^{2} - \frac{110}{589} a + \frac{3}{19}$, $\frac{1}{21793} a^{15} - \frac{7}{21793} a^{14} - \frac{4}{21793} a^{13} - \frac{257}{21793} a^{12} - \frac{433}{21793} a^{11} - \frac{406}{21793} a^{10} + \frac{363}{21793} a^{9} + \frac{8385}{21793} a^{8} - \frac{842}{21793} a^{7} - \frac{9884}{21793} a^{6} + \frac{8822}{21793} a^{5} - \frac{201}{21793} a^{4} - \frac{6775}{21793} a^{3} - \frac{6062}{21793} a^{2} - \frac{474}{1147} a + \frac{2}{703}$, $\frac{1}{239723} a^{16} - \frac{1}{239723} a^{15} + \frac{65}{239723} a^{14} - \frac{2}{21793} a^{13} + \frac{4500}{239723} a^{12} + \frac{181}{21793} a^{11} + \frac{2700}{239723} a^{10} - \frac{4570}{239723} a^{9} - \frac{36150}{239723} a^{8} + \frac{51960}{239723} a^{7} - \frac{25248}{239723} a^{6} - \frac{77768}{239723} a^{5} - \frac{35}{1147} a^{4} + \frac{77608}{239723} a^{3} - \frac{61251}{239723} a^{2} + \frac{94766}{239723} a - \frac{1135}{7733}$, $\frac{1}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a^{17} - \frac{17518916718225978370844398351306926147377264979084701851219632721697598272}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a^{16} + \frac{2379661397320037502556542915957844864681841811388349887591802400838381251323}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a^{15} + \frac{8940941756694433973343969912873324194843507394901861991723921463954762479164}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a^{14} + \frac{91436484505654709686982539457970139696516417923497852395468040093609628897268}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a^{13} + \frac{3039831644814481749885421355007709761476104792497551581064504926616665266276205}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a^{12} + \frac{145629252928705908896573548963349710841717047092355414174828004525516936614800}{6826453034437236194491702726370936079035098911236170797182777906216881749747977} a^{11} - \frac{2045799456605498774273568512770381212641126952972847313845275003428889652223735}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a^{10} - \frac{919372991733017758618296262612845575953974251298039873248168414350739949793662}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a^{9} - \frac{9161051456540618111805826971235538061985211743120589786625669367242001929437890}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a^{8} + \frac{5251792046737075962517396856832182288250642181183875295938548637401823618454579}{11791146150391589790485668345549798681969716301226113195133889110738250295019233} a^{7} + \frac{55643512176626091564943501489014914100516597688629394415683264444366695122592882}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a^{6} + \frac{50690719187059399929755927282226419448009528217437956250794714756886092459254128}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a^{5} + \frac{2774246244686329182492889468631389823174693162556425211728108963973373953495646}{6826453034437236194491702726370936079035098911236170797182777906216881749747977} a^{4} - \frac{6410817129842301246175726513536498145438246844091103133585372111044500786660846}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a^{3} + \frac{19777498149129503634333144949926115696724162316438813035998414967819613632034255}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a^{2} - \frac{51695112171742153384083494237729644445585566205921497844800457127235303525235380}{129702607654307487695342351801047785501666879313487245146472780218120753245211563} a + \frac{1640876075703404466241537665382021250004650828893661365411649890636207495481205}{4183955085622822183720721025840251145215060623015717585370089684455508169200373}$

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

Class group and class number

$C_{3}\times C_{4280742}$, which has order $12842226$ (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:  \( 15010229.973756868 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{18}$ (as 18T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 18
The 18 conjugacy class representatives for $C_{18}$
Character table for $C_{18}$

Intermediate fields

\(\Q(\sqrt{-247}) \), 3.3.361.1, 6.0.5439989503.1, 9.9.9025761726072081.1

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

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.9.0.1}{9} }^{2}$ R $18$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{9}$ R ${\href{/LocalNumberField/17.9.0.1}{9} }^{2}$ R ${\href{/LocalNumberField/23.9.0.1}{9} }^{2}$ $18$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ $18$ $18$ ${\href{/LocalNumberField/59.9.0.1}{9} }^{2}$

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
3Data not computed
13Data not computed
19Data not computed