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Properties

Label 18.0.27186423427...9008.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{27}\cdot 11^{9}\cdot 37^{14}$
Root discriminant $720.22$
Ramified primes $2, 3, 11, 37$
Class number $12478006800$ (GRH)
Class group $[2, 6, 210, 4951590]$ (GRH)
Galois group $S_3 \times C_6$ (as 18T6)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![4052214442111177, 1475998825390308, 542518369755228, 88916584556742, 17528600923959, 1580920554276, 365903950891, 33622267188, 8555002947, 422945816, 97733346, 695970, 1453312, 38628, 14895, -518, 75, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 + 75*x^16 - 518*x^15 + 14895*x^14 + 38628*x^13 + 1453312*x^12 + 695970*x^11 + 97733346*x^10 + 422945816*x^9 + 8555002947*x^8 + 33622267188*x^7 + 365903950891*x^6 + 1580920554276*x^5 + 17528600923959*x^4 + 88916584556742*x^3 + 542518369755228*x^2 + 1475998825390308*x + 4052214442111177)
 
gp: K = bnfinit(x^18 + 75*x^16 - 518*x^15 + 14895*x^14 + 38628*x^13 + 1453312*x^12 + 695970*x^11 + 97733346*x^10 + 422945816*x^9 + 8555002947*x^8 + 33622267188*x^7 + 365903950891*x^6 + 1580920554276*x^5 + 17528600923959*x^4 + 88916584556742*x^3 + 542518369755228*x^2 + 1475998825390308*x + 4052214442111177, 1)
 

Normalized defining polynomial

\( x^{18} + 75 x^{16} - 518 x^{15} + 14895 x^{14} + 38628 x^{13} + 1453312 x^{12} + 695970 x^{11} + 97733346 x^{10} + 422945816 x^{9} + 8555002947 x^{8} + 33622267188 x^{7} + 365903950891 x^{6} + 1580920554276 x^{5} + 17528600923959 x^{4} + 88916584556742 x^{3} + 542518369755228 x^{2} + 1475998825390308 x + 4052214442111177 \)

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:  \(-2718642342706308433638679144477426237858460895019008=-\,2^{24}\cdot 3^{27}\cdot 11^{9}\cdot 37^{14}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $720.22$
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:  $2, 3, 11, 37$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{6} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} + \frac{1}{3} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} + \frac{1}{6}$, $\frac{1}{6} a^{10} - \frac{1}{6} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2} + \frac{1}{6} a - \frac{1}{2}$, $\frac{1}{6} a^{11} - \frac{1}{6} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} + \frac{1}{6} a^{2} - \frac{1}{2} a$, $\frac{1}{222} a^{12} + \frac{13}{222} a^{10} - \frac{31}{74} a^{8} - \frac{1}{6} a^{7} - \frac{11}{111} a^{6} - \frac{1}{2} a^{5} + \frac{11}{37} a^{4} - \frac{1}{3} a^{3} - \frac{13}{74} a^{2} + \frac{1}{6} a + \frac{50}{111}$, $\frac{1}{222} a^{13} + \frac{13}{222} a^{11} + \frac{3}{37} a^{9} + \frac{1}{3} a^{8} + \frac{89}{222} a^{7} - \frac{1}{2} a^{6} + \frac{11}{37} a^{5} + \frac{1}{6} a^{4} + \frac{12}{37} a^{3} - \frac{1}{3} a^{2} + \frac{50}{111} a - \frac{1}{2}$, $\frac{1}{222} a^{14} - \frac{1}{74} a^{10} - \frac{17}{111} a^{8} - \frac{3}{37} a^{6} - \frac{1}{3} a^{5} + \frac{17}{37} a^{4} - \frac{59}{222} a^{2} - \frac{7}{37}$, $\frac{1}{666} a^{15} + \frac{1}{666} a^{14} + \frac{1}{666} a^{13} + \frac{1}{666} a^{12} - \frac{3}{74} a^{11} + \frac{47}{666} a^{10} - \frac{8}{333} a^{9} - \frac{8}{333} a^{8} - \frac{25}{74} a^{7} + \frac{55}{111} a^{6} + \frac{47}{333} a^{5} - \frac{17}{666} a^{4} + \frac{25}{333} a^{3} - \frac{49}{333} a^{2} + \frac{7}{222} a + \frac{140}{333}$, $\frac{1}{666} a^{16} - \frac{1}{666} a^{12} - \frac{1}{18} a^{11} - \frac{5}{74} a^{10} + \frac{55}{666} a^{8} - \frac{1}{6} a^{7} + \frac{169}{666} a^{6} - \frac{1}{6} a^{5} - \frac{149}{666} a^{4} + \frac{5}{18} a^{3} - \frac{23}{333} a^{2} - \frac{1}{9} a + \frac{89}{666}$, $\frac{1}{658564178766586645356074681247530142134729286761522910454673904960549446247318} a^{17} - \frac{8969474820669608949605668368870218104477725965685602286017511501302944105}{73173797640731849484008297916392238014969920751280323383852656106727716249702} a^{16} - \frac{181977343805437631379844867825418418836431049233961593152035105472302950627}{658564178766586645356074681247530142134729286761522910454673904960549446247318} a^{15} - \frac{780134661356912955024683352448898713316227322089703822142841635055333813309}{658564178766586645356074681247530142134729286761522910454673904960549446247318} a^{14} - \frac{349630163022579813593129186230565834237548793295999094473645622169355949264}{329282089383293322678037340623765071067364643380761455227336952480274723123659} a^{13} + \frac{170078641209069365009428250025275459941811296893174006960056986125411303739}{658564178766586645356074681247530142134729286761522910454673904960549446247318} a^{12} - \frac{11358224729474400882168864233038222321441385202046871734928528324041124632003}{219521392922195548452024893749176714044909762253840970151557968320183148749106} a^{11} - \frac{799332236281902555918367644455628047115966109023258675441594581082609030977}{17799031858556395820434450844527841679317007750311430012288483917852687736414} a^{10} + \frac{2687515843427783551087776402042661897666374993126688155193406541622429946215}{329282089383293322678037340623765071067364643380761455227336952480274723123659} a^{9} + \frac{270633698481926586530395536818529640411593526294224460713952704087622448390629}{658564178766586645356074681247530142134729286761522910454673904960549446247318} a^{8} - \frac{296826680014453218895408707416399207054240537830849609299479460202512118808027}{658564178766586645356074681247530142134729286761522910454673904960549446247318} a^{7} - \frac{14636458198669123410949112370866999523854149222883386148129686263704509549063}{109760696461097774226012446874588357022454881126920485075778984160091574374553} a^{6} + \frac{32248232635352059627760880751171158896056725328546925006623899266599919940503}{219521392922195548452024893749176714044909762253840970151557968320183148749106} a^{5} - \frac{92177310410237474872239570363756974621885493178267384790216326070991374690026}{329282089383293322678037340623765071067364643380761455227336952480274723123659} a^{4} + \frac{15527925232383931359263832185802842065750019332991612266929108241088783553076}{36586898820365924742004148958196119007484960375640161691926328053363858124851} a^{3} - \frac{13204749579259362828836384136685345068481863208978195741163493039730748661555}{36586898820365924742004148958196119007484960375640161691926328053363858124851} a^{2} - \frac{8047645669366944037820255677652851313343433992025960218740788641720879846055}{658564178766586645356074681247530142134729286761522910454673904960549446247318} a + \frac{285085682471738338045151111114602964220020410627725918082632396233931797576993}{658564178766586645356074681247530142134729286761522910454673904960549446247318}$

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

Class group and class number

$C_{2}\times C_{6}\times C_{210}\times C_{4951590}$, which has order $12478006800$ (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:  \( 118546543.87559307 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_6\times S_3$ (as 18T6):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 36
The 18 conjugacy class representatives for $S_3 \times C_6$
Character table for $S_3 \times C_6$

Intermediate fields

\(\Q(\sqrt{-33}) \), 3.3.110889.1, 3.3.148.1, 6.0.3142362028272192.1, 6.0.12594624768.2, 9.9.3228844269788073792.1

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

Sibling fields

Galois closure: data not computed
Degree 12 sibling: data not computed
Degree 18 sibling: 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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/59.1.0.1}{1} }^{6}$

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
2Data not computed
3Data not computed
$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$$[\ ]_{2}^{3}$
$37$37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.3.2.1$x^{3} - 37$$3$$1$$2$$C_3$$[\ ]_{3}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$
37.6.5.1$x^{6} - 37$$6$$1$$5$$C_6$$[\ ]_{6}$