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Properties

Label 18.0.37765424774...1024.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{12}\cdot 3^{21}\cdot 7^{14}\cdot 37^{9}$
Root discriminant $158.03$
Ramified primes $2, 3, 7, 37$
Class number $11967264$ (GRH)
Class group $[18, 54, 12312]$ (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![486213973000, -303589255200, 201734400000, -26333009892, 16394855484, -3333464652, 2020760692, -771159882, 336386712, -81366637, 28639551, -4356336, 1221682, -123246, 25692, -1706, 258, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 258*x^16 - 1706*x^15 + 25692*x^14 - 123246*x^13 + 1221682*x^12 - 4356336*x^11 + 28639551*x^10 - 81366637*x^9 + 336386712*x^8 - 771159882*x^7 + 2020760692*x^6 - 3333464652*x^5 + 16394855484*x^4 - 26333009892*x^3 + 201734400000*x^2 - 303589255200*x + 486213973000)
 
gp: K = bnfinit(x^18 - 9*x^17 + 258*x^16 - 1706*x^15 + 25692*x^14 - 123246*x^13 + 1221682*x^12 - 4356336*x^11 + 28639551*x^10 - 81366637*x^9 + 336386712*x^8 - 771159882*x^7 + 2020760692*x^6 - 3333464652*x^5 + 16394855484*x^4 - 26333009892*x^3 + 201734400000*x^2 - 303589255200*x + 486213973000, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 258 x^{16} - 1706 x^{15} + 25692 x^{14} - 123246 x^{13} + 1221682 x^{12} - 4356336 x^{11} + 28639551 x^{10} - 81366637 x^{9} + 336386712 x^{8} - 771159882 x^{7} + 2020760692 x^{6} - 3333464652 x^{5} + 16394855484 x^{4} - 26333009892 x^{3} + 201734400000 x^{2} - 303589255200 x + 486213973000 \)

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:  \(-3776542477491033702973795200304073601024=-\,2^{12}\cdot 3^{21}\cdot 7^{14}\cdot 37^{9}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $158.03$
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:  $2, 3, 7, 37$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{12} - \frac{1}{4} a^{10} - \frac{1}{4} a^{8} - \frac{1}{4} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{11} - \frac{1}{4} a^{9} - \frac{1}{4} a^{7} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{14} - \frac{1}{4} a^{6}$, $\frac{1}{28} a^{15} - \frac{1}{28} a^{14} - \frac{1}{28} a^{12} + \frac{1}{14} a^{11} + \frac{3}{28} a^{10} - \frac{3}{14} a^{9} - \frac{1}{4} a^{8} - \frac{5}{28} a^{7} + \frac{3}{14} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3}$, $\frac{1}{140} a^{16} + \frac{1}{140} a^{15} - \frac{1}{70} a^{14} - \frac{1}{140} a^{13} + \frac{1}{20} a^{12} - \frac{3}{20} a^{11} + \frac{1}{20} a^{10} + \frac{9}{140} a^{9} + \frac{4}{35} a^{8} - \frac{8}{35} a^{7} - \frac{23}{140} a^{6} - \frac{3}{10} a^{5} + \frac{3}{10} a^{4} - \frac{3}{10} a^{3} - \frac{2}{5} a^{2} + \frac{1}{5} a$, $\frac{1}{14647924251638750433063601054508043215128591743612485076542759408755407014438496921762610100} a^{17} + \frac{2461293912001624945249730727282926047629858145564811817982676926022426728863788298027141}{14647924251638750433063601054508043215128591743612485076542759408755407014438496921762610100} a^{16} - \frac{3811699839749811916443072388964987712570612106993529820972967878368158845448148779226948}{3661981062909687608265900263627010803782147935903121269135689852188851753609624230440652525} a^{15} - \frac{183232763411856829077254571254061646352590177682611027288936883246750410061885164486821203}{7323962125819375216531800527254021607564295871806242538271379704377703507219248460881305050} a^{14} - \frac{23576313233633224616988846922904715775830752728278713126584661706521784817525977572559979}{7323962125819375216531800527254021607564295871806242538271379704377703507219248460881305050} a^{13} + \frac{1762312065534135403829129305151425749733649476931811440244700785884759334880092660971915729}{14647924251638750433063601054508043215128591743612485076542759408755407014438496921762610100} a^{12} + \frac{154915524178575474647835488011314863679773121547488738154002294302185605084855709333853691}{7323962125819375216531800527254021607564295871806242538271379704377703507219248460881305050} a^{11} + \frac{2132885970456765010297139212584511382261345978815401713375307479781063955945185965502085689}{14647924251638750433063601054508043215128591743612485076542759408755407014438496921762610100} a^{10} - \frac{2414160529917426332766575627375454373878152098358639397823861966188455233536898060185494449}{14647924251638750433063601054508043215128591743612485076542759408755407014438496921762610100} a^{9} - \frac{876259099788993181245005103246609571936553012918368062030982374902425486590115045516961653}{3661981062909687608265900263627010803782147935903121269135689852188851753609624230440652525} a^{8} - \frac{708932782346968216373791638445252434827901346772996621312516156626460545534733745342314547}{3661981062909687608265900263627010803782147935903121269135689852188851753609624230440652525} a^{7} - \frac{109143176024161371432751672305586399831940988369658771461014339500599333830701182584105301}{2092560607376964347580514436358291887875513106230355010934679915536486716348356703108944300} a^{6} + \frac{231340700754878007421923522728610313517201266697475784322650604887041414143787665926122153}{1046280303688482173790257218179145943937756553115177505467339957768243358174178351554472150} a^{5} - \frac{139396588713127242054039241591905400393958589886456549294913108884660638083293319941854043}{1046280303688482173790257218179145943937756553115177505467339957768243358174178351554472150} a^{4} + \frac{419167522609274273627930626953107084788754081717351484918825206919702028864029309662079381}{1046280303688482173790257218179145943937756553115177505467339957768243358174178351554472150} a^{3} + \frac{222803419693396025819022597516999924981036343644183628416476106024052399542699643430271586}{523140151844241086895128609089572971968878276557588752733669978884121679087089175777236075} a^{2} + \frac{8697134253425348983437522601081897070480991250163645581842951969849166805941984308198185}{20925606073769643475805144363582918878755131062303550109346799155364867163483567031089443} a - \frac{9514280821931761515866791252096076208767889878672764481162024395686096410202677598139329}{20925606073769643475805144363582918878755131062303550109346799155364867163483567031089443}$

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

Class group and class number

$C_{18}\times C_{54}\times C_{12312}$, which has order $11967264$ (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:  \( 296124.35954857944 \) (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

\(\Q(\sqrt{-111}) \), 3.3.756.1, \(\Q(\zeta_{7})^+\), 6.0.86850039024.2, 6.0.3283682031.3, 9.9.1037426999616.1

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 ${\href{/LocalNumberField/5.3.0.1}{3} }^{6}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{6}$

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$2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
2.9.6.1$x^{9} - 4 x^{3} + 8$$3$$3$$6$$S_3\times C_3$$[\ ]_{3}^{6}$
3Data not computed
$7$7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.3.2.2$x^{3} - 7$$3$$1$$2$$C_3$$[\ ]_{3}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
7.6.5.5$x^{6} + 56$$6$$1$$5$$C_6$$[\ ]_{6}$
$37$37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
37.6.3.1$x^{6} - 74 x^{4} + 1369 x^{2} - 202612$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$