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Properties

Label 18.0.65241576375...2048.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{18}\cdot 3^{27}\cdot 7^{12}\cdot 11^{9}$
Root discriminant $126.13$
Ramified primes $2, 3, 7, 11$
Class number $5660928$ (GRH)
Class group $[2, 2, 2, 2, 36, 9828]$ (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![203458741597, -72603966546, 108873336285, -31704369010, 26626577955, -6889678704, 4043926949, -852307704, 378327639, -62431200, 21979977, -2760612, 791808, -71736, 17052, -1010, 201, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 + 201*x^16 - 1010*x^15 + 17052*x^14 - 71736*x^13 + 791808*x^12 - 2760612*x^11 + 21979977*x^10 - 62431200*x^9 + 378327639*x^8 - 852307704*x^7 + 4043926949*x^6 - 6889678704*x^5 + 26626577955*x^4 - 31704369010*x^3 + 108873336285*x^2 - 72603966546*x + 203458741597)
 
gp: K = bnfinit(x^18 - 6*x^17 + 201*x^16 - 1010*x^15 + 17052*x^14 - 71736*x^13 + 791808*x^12 - 2760612*x^11 + 21979977*x^10 - 62431200*x^9 + 378327639*x^8 - 852307704*x^7 + 4043926949*x^6 - 6889678704*x^5 + 26626577955*x^4 - 31704369010*x^3 + 108873336285*x^2 - 72603966546*x + 203458741597, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} + 201 x^{16} - 1010 x^{15} + 17052 x^{14} - 71736 x^{13} + 791808 x^{12} - 2760612 x^{11} + 21979977 x^{10} - 62431200 x^{9} + 378327639 x^{8} - 852307704 x^{7} + 4043926949 x^{6} - 6889678704 x^{5} + 26626577955 x^{4} - 31704369010 x^{3} + 108873336285 x^{2} - 72603966546 x + 203458741597 \)

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:  \(-65241576375887238794986513680935682048=-\,2^{18}\cdot 3^{27}\cdot 7^{12}\cdot 11^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $126.13$
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:  $2, 3, 7, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(2772=2^{2}\cdot 3^{2}\cdot 7\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{2772}(1,·)$, $\chi_{2772}(1451,·)$, $\chi_{2772}(2375,·)$, $\chi_{2772}(2377,·)$, $\chi_{2772}(2507,·)$, $\chi_{2772}(527,·)$, $\chi_{2772}(529,·)$, $\chi_{2772}(659,·)$, $\chi_{2772}(793,·)$, $\chi_{2772}(925,·)$, $\chi_{2772}(1187,·)$, $\chi_{2772}(2641,·)$, $\chi_{2772}(263,·)$, $\chi_{2772}(1453,·)$, $\chi_{2772}(1583,·)$, $\chi_{2772}(1717,·)$, $\chi_{2772}(1849,·)$, $\chi_{2772}(2111,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{11} a^{6} - \frac{2}{11} a^{5} - \frac{3}{11} a^{4} - \frac{5}{11} a^{3} + \frac{2}{11} a^{2} - \frac{4}{11} a + \frac{1}{11}$, $\frac{1}{11} a^{7} + \frac{4}{11} a^{5} + \frac{3}{11} a^{3} + \frac{4}{11} a + \frac{2}{11}$, $\frac{1}{22} a^{8} + \frac{4}{11} a^{5} - \frac{7}{22} a^{4} - \frac{1}{11} a^{3} + \frac{7}{22} a^{2} - \frac{2}{11} a + \frac{7}{22}$, $\frac{1}{22} a^{9} + \frac{9}{22} a^{5} + \frac{3}{22} a^{3} + \frac{1}{11} a^{2} - \frac{5}{22} a - \frac{4}{11}$, $\frac{1}{22} a^{10} - \frac{1}{22} a^{6} - \frac{1}{11} a^{5} - \frac{1}{2} a^{4} + \frac{4}{11} a^{3} - \frac{3}{22} a^{2} + \frac{5}{11} a - \frac{5}{11}$, $\frac{1}{22} a^{11} - \frac{1}{22} a^{7} + \frac{7}{22} a^{5} + \frac{1}{11} a^{4} + \frac{9}{22} a^{3} - \frac{4}{11} a^{2} + \frac{2}{11} a + \frac{1}{11}$, $\frac{1}{242} a^{12} - \frac{2}{121} a^{11} - \frac{1}{121} a^{10} + \frac{1}{121} a^{9} - \frac{4}{121} a^{7} + \frac{9}{242} a^{6} - \frac{3}{11} a^{5} - \frac{14}{121} a^{4} + \frac{42}{121} a^{3} - \frac{13}{242} a^{2} - \frac{59}{121} a + \frac{67}{242}$, $\frac{1}{242} a^{13} + \frac{2}{121} a^{11} + \frac{5}{242} a^{10} - \frac{3}{242} a^{9} + \frac{3}{242} a^{8} - \frac{1}{242} a^{7} + \frac{3}{242} a^{6} - \frac{83}{242} a^{5} - \frac{36}{121} a^{4} - \frac{9}{121} a^{3} + \frac{3}{121} a^{2} + \frac{23}{121} a + \frac{15}{242}$, $\frac{1}{242} a^{14} - \frac{1}{242} a^{11} + \frac{5}{242} a^{10} - \frac{5}{242} a^{9} - \frac{1}{242} a^{8} - \frac{9}{242} a^{7} - \frac{9}{242} a^{6} + \frac{19}{121} a^{5} - \frac{19}{121} a^{4} - \frac{3}{11} a^{3} + \frac{5}{121} a^{2} - \frac{63}{242} a - \frac{46}{121}$, $\frac{1}{242} a^{15} + \frac{1}{242} a^{11} + \frac{2}{121} a^{10} + \frac{1}{242} a^{9} + \frac{1}{121} a^{8} + \frac{5}{242} a^{7} - \frac{4}{121} a^{6} - \frac{52}{121} a^{5} + \frac{41}{121} a^{4} - \frac{19}{121} a^{3} - \frac{60}{121} a^{2} + \frac{60}{121} a + \frac{17}{121}$, $\frac{1}{3877503100580917049606634292} a^{16} - \frac{707794974695496817676506}{969375775145229262401658573} a^{15} - \frac{3260866152388363886014405}{1938751550290458524803317146} a^{14} - \frac{1203278292574662498502653}{1938751550290458524803317146} a^{13} + \frac{556491582923868489140030}{969375775145229262401658573} a^{12} + \frac{2276706379579146924753785}{969375775145229262401658573} a^{11} + \frac{1074932156214661467104638}{969375775145229262401658573} a^{10} - \frac{42306178538583355866690609}{1938751550290458524803317146} a^{9} + \frac{7415820244515568475460111}{3877503100580917049606634292} a^{8} - \frac{899926040618547883165097}{1938751550290458524803317146} a^{7} + \frac{39077567222946244837717205}{1938751550290458524803317146} a^{6} + \frac{110641892644611768047798631}{1938751550290458524803317146} a^{5} - \frac{661396988252272006462751135}{3877503100580917049606634292} a^{4} - \frac{74680072019176753205750350}{969375775145229262401658573} a^{3} + \frac{541779004172429977837557841}{1938751550290458524803317146} a^{2} + \frac{397747514052119079532801061}{969375775145229262401658573} a + \frac{1824463171172358720761562373}{3877503100580917049606634292}$, $\frac{1}{151825341642790884082419722028390635259684436} a^{17} - \frac{18906540592748529}{151825341642790884082419722028390635259684436} a^{16} - \frac{128610671921172538192738220053417294328215}{75912670821395442041209861014195317629842218} a^{15} + \frac{152488785804790046242097301744662281339411}{75912670821395442041209861014195317629842218} a^{14} + \frac{42009540789073129407875069958942074647978}{37956335410697721020604930507097658814921109} a^{13} - \frac{11306555242666059217612421915913531593905}{75912670821395442041209861014195317629842218} a^{12} - \frac{294113648267455571460024349431839541353603}{37956335410697721020604930507097658814921109} a^{11} - \frac{402008611661415730706469069082855009621325}{75912670821395442041209861014195317629842218} a^{10} + \frac{1376625135704736781392214184883817866002495}{151825341642790884082419722028390635259684436} a^{9} + \frac{252133860064040088136391680470157692544353}{13802303785708262189310883820762785023607676} a^{8} - \frac{91940610543504674324825138080485930693399}{6901151892854131094655441910381392511803838} a^{7} - \frac{133972612700550674632899908641852946885123}{6901151892854131094655441910381392511803838} a^{6} + \frac{61006529807138758276267738643636765366920249}{151825341642790884082419722028390635259684436} a^{5} + \frac{17711205451141578842700645166135674934675891}{151825341642790884082419722028390635259684436} a^{4} - \frac{21477232699032708239098009142445973118210853}{75912670821395442041209861014195317629842218} a^{3} - \frac{16822679140792404618254141702288024739159115}{37956335410697721020604930507097658814921109} a^{2} + \frac{11540157505156990209361669330513705109213881}{151825341642790884082419722028390635259684436} a - \frac{75360465949291043494627604441790330403838365}{151825341642790884082419722028390635259684436}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{2}\times C_{36}\times C_{9828}$, which has order $5660928$ (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:  \( 54408.48888868202 \) (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

\(\Q(\sqrt{-33}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{7})^+\), 3.3.3969.2, 3.3.3969.1, 6.0.1676676672.2, 6.0.5522223168.10, 6.0.4025700689472.9, 6.0.4025700689472.8, 9.9.62523502209.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ R R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
3Data not computed
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
$11$11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$