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Properties

Label 18.0.12639964587...1824.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{27}\cdot 3^{24}\cdot 37^{15}$
Root discriminant $248.05$
Ramified primes $2, 3, 37$
Class number $15331680$ (GRH)
Class group $[2, 2, 2, 234, 8190]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1084437811, 752535174, 1056948351, 248399886, 191128917, -22106286, 50853848, 12264582, 18104562, -952408, 863727, -51156, 387161, -354, 14787, -6, 204, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 204*x^16 - 6*x^15 + 14787*x^14 - 354*x^13 + 387161*x^12 - 51156*x^11 + 863727*x^10 - 952408*x^9 + 18104562*x^8 + 12264582*x^7 + 50853848*x^6 - 22106286*x^5 + 191128917*x^4 + 248399886*x^3 + 1056948351*x^2 + 752535174*x + 1084437811)
 
gp: K = bnfinit(x^18 + 204*x^16 - 6*x^15 + 14787*x^14 - 354*x^13 + 387161*x^12 - 51156*x^11 + 863727*x^10 - 952408*x^9 + 18104562*x^8 + 12264582*x^7 + 50853848*x^6 - 22106286*x^5 + 191128917*x^4 + 248399886*x^3 + 1056948351*x^2 + 752535174*x + 1084437811, 1)
 

Normalized defining polynomial

\( x^{18} + 204 x^{16} - 6 x^{15} + 14787 x^{14} - 354 x^{13} + 387161 x^{12} - 51156 x^{11} + 863727 x^{10} - 952408 x^{9} + 18104562 x^{8} + 12264582 x^{7} + 50853848 x^{6} - 22106286 x^{5} + 191128917 x^{4} + 248399886 x^{3} + 1056948351 x^{2} + 752535174 x + 1084437811 \)

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:  \(-12639964587166968704098411698997775327821824=-\,2^{27}\cdot 3^{24}\cdot 37^{15}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $248.05$
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:  $2, 3, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2664=2^{3}\cdot 3^{2}\cdot 37\)
Dirichlet character group:    $\lbrace$$\chi_{2664}(1,·)$, $\chi_{2664}(619,·)$, $\chi_{2664}(1897,·)$, $\chi_{2664}(1507,·)$, $\chi_{2664}(433,·)$, $\chi_{2664}(2515,·)$, $\chi_{2664}(889,·)$, $\chi_{2664}(2395,·)$, $\chi_{2664}(2209,·)$, $\chi_{2664}(1627,·)$, $\chi_{2664}(2083,·)$, $\chi_{2664}(1777,·)$, $\chi_{2664}(1321,·)$, $\chi_{2664}(1195,·)$, $\chi_{2664}(1009,·)$, $\chi_{2664}(307,·)$, $\chi_{2664}(739,·)$, $\chi_{2664}(121,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{44} a^{12} - \frac{3}{22} a^{10} - \frac{1}{11} a^{9} + \frac{1}{22} a^{8} - \frac{2}{11} a^{7} - \frac{5}{22} a^{6} + \frac{19}{44} a^{4} - \frac{5}{11} a^{3} - \frac{3}{11} a^{2} - \frac{1}{11} a - \frac{9}{44}$, $\frac{1}{44} a^{13} - \frac{3}{22} a^{11} - \frac{1}{11} a^{10} + \frac{1}{22} a^{9} - \frac{2}{11} a^{8} - \frac{5}{22} a^{7} + \frac{19}{44} a^{5} - \frac{5}{11} a^{4} - \frac{3}{11} a^{3} - \frac{1}{11} a^{2} - \frac{9}{44} a$, $\frac{1}{44} a^{14} - \frac{1}{11} a^{11} + \frac{5}{22} a^{10} - \frac{5}{22} a^{9} + \frac{1}{22} a^{8} - \frac{1}{11} a^{7} + \frac{3}{44} a^{6} + \frac{1}{22} a^{5} + \frac{7}{22} a^{4} - \frac{7}{22} a^{3} + \frac{7}{44} a^{2} + \frac{5}{11} a - \frac{5}{22}$, $\frac{1}{44} a^{15} + \frac{5}{22} a^{11} + \frac{5}{22} a^{10} + \frac{2}{11} a^{9} + \frac{1}{11} a^{8} - \frac{7}{44} a^{7} + \frac{3}{22} a^{6} - \frac{2}{11} a^{5} + \frac{9}{22} a^{4} + \frac{15}{44} a^{3} + \frac{4}{11} a^{2} - \frac{1}{11} a + \frac{2}{11}$, $\frac{1}{258332184255918873676820999084} a^{16} - \frac{2511731054565820909416397953}{258332184255918873676820999084} a^{15} + \frac{2077573508248552029893209985}{258332184255918873676820999084} a^{14} + \frac{2525877283324885584573779359}{258332184255918873676820999084} a^{13} + \frac{2174716061844842923800432975}{258332184255918873676820999084} a^{12} + \frac{30056020431958439502035335465}{129166092127959436838410499542} a^{11} + \frac{2720323386011592350144006017}{129166092127959436838410499542} a^{10} - \frac{13391970709614298051672638483}{64583046063979718419205249771} a^{9} + \frac{30572000925656287893480468361}{258332184255918873676820999084} a^{8} - \frac{39345260928186318887212283497}{258332184255918873676820999084} a^{7} - \frac{18146572361608683819457413381}{258332184255918873676820999084} a^{6} - \frac{89501943995618815152416844151}{258332184255918873676820999084} a^{5} - \frac{4313174887553968417668181677}{11742372011632676076219136322} a^{4} - \frac{4778548921519401557558305955}{23484744023265352152438272644} a^{3} + \frac{107834100759411554923594987263}{258332184255918873676820999084} a^{2} - \frac{1199123893132204806114529197}{23484744023265352152438272644} a + \frac{85780573036326797938567383509}{258332184255918873676820999084}$, $\frac{1}{38994625800773568384235229582947967104521676} a^{17} + \frac{57519419695313}{38994625800773568384235229582947967104521676} a^{16} + \frac{319322593587002806340226742497243025852571}{38994625800773568384235229582947967104521676} a^{15} - \frac{53637239249695147265845746637474056470909}{38994625800773568384235229582947967104521676} a^{14} + \frac{36809831068600593392702983577575739172274}{9748656450193392096058807395736991776130419} a^{13} - \frac{36900884458637758097236028195399414643469}{19497312900386784192117614791473983552260838} a^{12} + \frac{3941073986391495611244820602837857699232473}{19497312900386784192117614791473983552260838} a^{11} - \frac{572590908572157331114621857482032742650459}{19497312900386784192117614791473983552260838} a^{10} + \frac{913593930012456160218316761724642070076791}{38994625800773568384235229582947967104521676} a^{9} + \frac{4023935267788377479851351932144396454174873}{38994625800773568384235229582947967104521676} a^{8} - \frac{4841075089526701034547618887010543542820841}{38994625800773568384235229582947967104521676} a^{7} + \frac{75357596156903653300120674945391134120409}{3544965981888506216748657234813451554956516} a^{6} - \frac{877769698808275513310470799340005259079549}{38994625800773568384235229582947967104521676} a^{5} - \frac{10580440210133920912109217467180768025565001}{38994625800773568384235229582947967104521676} a^{4} + \frac{14861972771514181859026073200050676232767159}{38994625800773568384235229582947967104521676} a^{3} + \frac{8895926069215996740539199753535107148553909}{38994625800773568384235229582947967104521676} a^{2} - \frac{2674228432146209705364517187340626108843557}{19497312900386784192117614791473983552260838} a - \frac{688992706257715977707400169407309481062970}{9748656450193392096058807395736991776130419}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{234}\times C_{8190}$, which has order $15331680$ (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:  \( 8790607.764011372 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_3\times C_6$ (as 18T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 18
The 18 conjugacy class representatives for $C_6 \times C_3$
Character table for $C_6 \times C_3$

Intermediate fields

\(\Q(\sqrt{-74}) \), 3.3.1369.1, \(\Q(\zeta_{9})^+\), 3.3.110889.1, 3.3.110889.2, 6.0.35504105984.1, 6.0.170155178496.8, 6.0.232942439361024.3, 6.0.232942439361024.2, 9.9.1363532208525369.2

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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.7$x^{6} + 4 x^{4} + 4 x^{2} - 24$$2$$3$$9$$C_6$$[3]^{3}$
$3$3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
3.9.12.1$x^{9} + 18 x^{5} + 18 x^{3} + 27 x^{2} + 216$$3$$3$$12$$C_3^2$$[2]^{3}$
$37$37.6.5.5$x^{6} + 296$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.5$x^{6} + 296$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.5$x^{6} + 296$$6$$1$$5$$C_6$$[\ ]_{6}$