↑

Properties

Label 18.0.17854157885...3056.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 7^{9}\cdot 19^{12}$
Root discriminant $195.78$
Ramified primes $2, 3, 7, 19$
Class number $16595712$ (GRH)
Class group $[2, 2, 2, 42, 84, 588]$ (GRH)
Galois group $C_6 \times C_3$ (as 18T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![47051159113, -36740356506, 23166778845, -6699277158, 2508975903, -646277772, 340487281, -94591272, 37870875, -7706476, 2729481, -481404, 150340, -19992, 5088, -470, 105, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 105*x^16 - 470*x^15 + 5088*x^14 - 19992*x^13 + 150340*x^12 - 481404*x^11 + 2729481*x^10 - 7706476*x^9 + 37870875*x^8 - 94591272*x^7 + 340487281*x^6 - 646277772*x^5 + 2508975903*x^4 - 6699277158*x^3 + 23166778845*x^2 - 36740356506*x + 47051159113)
 
gp: K = bnfinit(x^18 - 6*x^17 + 105*x^16 - 470*x^15 + 5088*x^14 - 19992*x^13 + 150340*x^12 - 481404*x^11 + 2729481*x^10 - 7706476*x^9 + 37870875*x^8 - 94591272*x^7 + 340487281*x^6 - 646277772*x^5 + 2508975903*x^4 - 6699277158*x^3 + 23166778845*x^2 - 36740356506*x + 47051159113, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 105 x^{16} - 470 x^{15} + 5088 x^{14} - 19992 x^{13} + 150340 x^{12} - 481404 x^{11} + 2729481 x^{10} - 7706476 x^{9} + 37870875 x^{8} - 94591272 x^{7} + 340487281 x^{6} - 646277772 x^{5} + 2508975903 x^{4} - 6699277158 x^{3} + 23166778845 x^{2} - 36740356506 x + 47051159113 \)

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:  \(-178541578852998866709992697969426969133056=-\,2^{18}\cdot 3^{27}\cdot 7^{9}\cdot 19^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $195.78$
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:  $2, 3, 7, 19$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(4788=2^{2}\cdot 3^{2}\cdot 7\cdot 19\)
Dirichlet character group:    $\lbrace$$\chi_{4788}(1,·)$, $\chi_{4788}(505,·)$, $\chi_{4788}(3275,·)$, $\chi_{4788}(1679,·)$, $\chi_{4788}(83,·)$, $\chi_{4788}(2519,·)$, $\chi_{4788}(923,·)$, $\chi_{4788}(2015,·)$, $\chi_{4788}(3611,·)$, $\chi_{4788}(419,·)$, $\chi_{4788}(4453,·)$, $\chi_{4788}(2857,·)$, $\chi_{4788}(1261,·)$, $\chi_{4788}(3697,·)$, $\chi_{4788}(4115,·)$, $\chi_{4788}(2101,·)$, $\chi_{4788}(3193,·)$, $\chi_{4788}(1597,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{7} a^{6} - \frac{2}{7} a^{5} + \frac{3}{7} a^{4} - \frac{2}{7} a^{3} + \frac{1}{7} a^{2}$, $\frac{1}{7} a^{7} - \frac{1}{7} a^{5} - \frac{3}{7} a^{4} - \frac{3}{7} a^{3} + \frac{2}{7} a^{2}$, $\frac{1}{14} a^{8} + \frac{1}{7} a^{5} - \frac{1}{2} a^{4} + \frac{1}{14} a^{2} - \frac{1}{2}$, $\frac{1}{14} a^{9} - \frac{3}{14} a^{5} - \frac{3}{7} a^{4} + \frac{5}{14} a^{3} - \frac{1}{7} a^{2} - \frac{1}{2} a$, $\frac{1}{14} a^{10} - \frac{1}{14} a^{6} + \frac{2}{7} a^{5} - \frac{3}{14} a^{4} - \frac{3}{7} a^{3} - \frac{5}{14} a^{2}$, $\frac{1}{14} a^{11} - \frac{1}{14} a^{7} + \frac{5}{14} a^{5} - \frac{2}{7} a^{4} + \frac{3}{14} a^{3} - \frac{2}{7} a^{2}$, $\frac{1}{490} a^{12} - \frac{2}{245} a^{11} + \frac{17}{490} a^{10} + \frac{6}{245} a^{9} - \frac{9}{490} a^{8} + \frac{13}{245} a^{7} - \frac{2}{245} a^{6} - \frac{121}{245} a^{5} + \frac{4}{245} a^{4} + \frac{13}{35} a^{3} + \frac{3}{14} a^{2} + \frac{2}{5} a + \frac{2}{5}$, $\frac{1}{490} a^{13} + \frac{1}{490} a^{11} + \frac{1}{49} a^{10} + \frac{2}{245} a^{9} - \frac{1}{49} a^{8} + \frac{3}{49} a^{7} + \frac{11}{245} a^{6} - \frac{3}{98} a^{5} + \frac{2}{245} a^{4} - \frac{8}{35} a^{3} + \frac{9}{35} a^{2} - \frac{1}{2} a - \frac{2}{5}$, $\frac{1}{490} a^{14} + \frac{1}{35} a^{11} - \frac{13}{490} a^{10} + \frac{13}{490} a^{9} + \frac{2}{245} a^{8} - \frac{2}{245} a^{7} - \frac{11}{490} a^{6} + \frac{71}{490} a^{5} - \frac{17}{98} a^{4} + \frac{17}{70} a^{3} + \frac{1}{14} a^{2} - \frac{3}{10} a + \frac{1}{10}$, $\frac{1}{490} a^{15} + \frac{4}{245} a^{11} - \frac{3}{98} a^{10} + \frac{11}{490} a^{9} + \frac{17}{490} a^{8} + \frac{1}{49} a^{7} - \frac{13}{490} a^{6} - \frac{197}{490} a^{5} - \frac{12}{35} a^{4} + \frac{31}{70} a^{3} + \frac{17}{35} a^{2} - \frac{1}{10}$, $\frac{1}{36043748510501558912638420} a^{16} + \frac{83030121332500495606}{183896676073987545472645} a^{15} - \frac{7537117920193128622472}{9010937127625389728159605} a^{14} - \frac{7956061229309155665291}{18021874255250779456319210} a^{13} + \frac{2031385066682574082793}{9010937127625389728159605} a^{12} + \frac{531356361075048372637969}{18021874255250779456319210} a^{11} + \frac{61228187167027409805687}{2574553465035825636617030} a^{10} + \frac{38025266444284324552406}{9010937127625389728159605} a^{9} + \frac{1246595542293973258321189}{36043748510501558912638420} a^{8} + \frac{134734534059900639251157}{9010937127625389728159605} a^{7} - \frac{410661532524715721736443}{9010937127625389728159605} a^{6} - \frac{3391687771318877776830243}{18021874255250779456319210} a^{5} + \frac{3822231336632303248505113}{36043748510501558912638420} a^{4} - \frac{22527094879436693282997}{257455346503582563661703} a^{3} - \frac{1217848998106803183199101}{2574553465035825636617030} a^{2} + \frac{50733858808854106364047}{367793352147975090945290} a - \frac{529794649869335215523}{3976144347545676658868}$, $\frac{1}{174654025586240421485673548004079755048340} a^{17} - \frac{1800850433855093}{174654025586240421485673548004079755048340} a^{16} - \frac{1478224897221670770695285693490156552}{8732701279312021074283677400203987752417} a^{15} - \frac{11843546599900633325758722681680526724}{43663506396560105371418387001019938762085} a^{14} - \frac{83079493178912717195935881215298777673}{87327012793120210742836774002039877524170} a^{13} - \frac{36977069009585852654251081351470736256}{43663506396560105371418387001019938762085} a^{12} + \frac{1565546296308291390001739455367206384439}{87327012793120210742836774002039877524170} a^{11} - \frac{3084794279166140849646274452485070915519}{87327012793120210742836774002039877524170} a^{10} + \frac{399445683089420023993950776587478614013}{34930805117248084297134709600815951009668} a^{9} + \frac{3395622990188187243782009914782746825003}{174654025586240421485673548004079755048340} a^{8} + \frac{2814771393072072917225519789039588604713}{87327012793120210742836774002039877524170} a^{7} - \frac{237114127791774802832953882486088230693}{6237643770937157910202626714431419823155} a^{6} - \frac{4395443586874064047995847563571119349227}{34930805117248084297134709600815951009668} a^{5} - \frac{7559479176739367117742839128088558135787}{24950575083748631640810506857725679292620} a^{4} + \frac{534574032536881699312921978678353531888}{1247528754187431582040525342886283964631} a^{3} - \frac{115189254591745581258572322919742227751}{1782183934553473688629321918408977092330} a^{2} - \frac{1730093648827697292974719113141490688669}{3564367869106947377258643836817954184660} a - \frac{6211902270151792297801792887587696273}{19266853346524039877073750469286238836}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{42}\times C_{84}\times C_{588}$, which has order $16595712$ (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:  \( 1472619.082400847 \) (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

\(\Q(\sqrt{-21}) \), \(\Q(\zeta_{9})^+\), 3.3.29241.1, 3.3.361.1, 3.3.29241.2, 6.0.432081216.1, 6.0.56309256150336.3, 6.0.77241777984.6, 6.0.56309256150336.2, 9.9.25002110044521.1

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 R R ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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$2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.5$x^{6} - 2 x^{4} + x^{2} - 3$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$7$7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.1$x^{6} - 14 x^{4} + 49 x^{2} - 1372$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
$19$19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$
19.3.2.2$x^{3} - 19$$3$$1$$2$$C_3$$[\ ]_{3}$