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Properties

Label 18.0.15267461241...0000.2
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{16}\cdot 3^{53}\cdot 5^{16}\cdot 31^{12}$
Root discriminant $1940.83$
Ramified primes $2, 3, 5, 31$
Class number $6561$ (GRH)
Class group $[3, 3, 3, 3, 3, 3, 9]$ (GRH)
Galois group $D_9:C_3$ (as 18T18)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![37266634952753761, -1737411129, -6949644435, 32431674084, -24323755266, -24323756562, 32431675794, -6949646784, -1737408213, 386088221, 2907, -2304, 1554, -882, 414, -156, 45, -9, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 414*x^14 - 882*x^13 + 1554*x^12 - 2304*x^11 + 2907*x^10 + 386088221*x^9 - 1737408213*x^8 - 6949646784*x^7 + 32431675794*x^6 - 24323756562*x^5 - 24323755266*x^4 + 32431674084*x^3 - 6949644435*x^2 - 1737411129*x + 37266634952753761)
 
gp: K = bnfinit(x^18 - 9*x^17 + 45*x^16 - 156*x^15 + 414*x^14 - 882*x^13 + 1554*x^12 - 2304*x^11 + 2907*x^10 + 386088221*x^9 - 1737408213*x^8 - 6949646784*x^7 + 32431675794*x^6 - 24323756562*x^5 - 24323755266*x^4 + 32431674084*x^3 - 6949644435*x^2 - 1737411129*x + 37266634952753761, 1)
 

Normalized defining polynomial

\( x^{18} - 9 x^{17} + 45 x^{16} - 156 x^{15} + 414 x^{14} - 882 x^{13} + 1554 x^{12} - 2304 x^{11} + 2907 x^{10} + 386088221 x^{9} - 1737408213 x^{8} - 6949646784 x^{7} + 32431675794 x^{6} - 24323756562 x^{5} - 24323755266 x^{4} + 32431674084 x^{3} - 6949644435 x^{2} - 1737411129 x + 37266634952753761 \)

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:  \(-152674612414621913640596447375544095872332030000000000000000=-\,2^{16}\cdot 3^{53}\cdot 5^{16}\cdot 31^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1940.83$
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:  $2, 3, 5, 31$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{9} a^{5} - \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{558} a^{6} - \frac{1}{186} a^{5} + \frac{1}{93} a^{4} - \frac{7}{558} a^{3} + \frac{1}{93} a^{2} - \frac{1}{186} a + \frac{1}{558}$, $\frac{1}{1674} a^{7} + \frac{1}{1674} a^{6} - \frac{1}{279} a^{5} + \frac{17}{1674} a^{4} - \frac{11}{837} a^{3} + \frac{7}{558} a^{2} - \frac{11}{1674} a + \frac{2}{837}$, $\frac{1}{1674} a^{8} - \frac{1}{1674} a^{6} + \frac{5}{1674} a^{5} - \frac{1}{558} a^{4} + \frac{1}{1674} a^{3} + \frac{2}{837} a^{2} - \frac{1}{558} a + \frac{1}{837}$, $\frac{1}{5022} a^{9} + \frac{1}{1674} a^{6} + \frac{1}{1674} a^{3} + \frac{1}{5022}$, $\frac{1}{10044} a^{10} - \frac{1}{10044} a^{9} - \frac{1}{3348} a^{8} - \frac{1}{3348} a^{6} + \frac{1}{3348} a^{5} - \frac{13}{3348} a^{4} + \frac{5}{837} a^{3} - \frac{25}{3348} a^{2} + \frac{43}{10044} a - \frac{19}{10044}$, $\frac{1}{30132} a^{11} - \frac{1}{30132} a^{10} + \frac{1}{30132} a^{9} - \frac{1}{5022} a^{8} - \frac{1}{10044} a^{7} - \frac{5}{10044} a^{6} + \frac{13}{10044} a^{5} - \frac{23}{5022} a^{4} + \frac{61}{10044} a^{3} - \frac{10241}{30132} a^{2} + \frac{10151}{30132} a - \frac{2522}{7533}$, $\frac{1}{934092} a^{12} - \frac{1}{155682} a^{11} + \frac{7}{311364} a^{10} - \frac{25}{467046} a^{9} + \frac{5}{51894} a^{8} - \frac{7}{51894} a^{7} + \frac{47}{311364} a^{6} - \frac{7}{51894} a^{5} + \frac{5}{51894} a^{4} - \frac{25}{467046} a^{3} + \frac{7}{311364} a^{2} - \frac{1}{155682} a + \frac{1}{934092}$, $\frac{1}{2802276} a^{13} + \frac{1}{2802276} a^{12} - \frac{7}{934092} a^{11} + \frac{97}{2802276} a^{10} - \frac{65}{700569} a^{9} + \frac{14}{77841} a^{8} - \frac{247}{934092} a^{7} + \frac{287}{934092} a^{6} - \frac{22}{77841} a^{5} - \frac{77696}{700569} a^{4} - \frac{311693}{2802276} a^{3} + \frac{311411}{934092} a^{2} - \frac{1245497}{2802276} a + \frac{622735}{2802276}$, $\frac{1}{106486488} a^{14} + \frac{11}{106486488} a^{13} + \frac{5}{13310811} a^{12} + \frac{139}{106486488} a^{11} + \frac{5129}{106486488} a^{10} + \frac{3257}{53243244} a^{9} + \frac{6869}{35495496} a^{8} - \frac{9347}{35495496} a^{7} - \frac{27361}{35495496} a^{6} + \frac{860249}{53243244} a^{5} - \frac{386677}{106486488} a^{4} + \frac{9976363}{106486488} a^{3} + \frac{12657445}{26621622} a^{2} + \frac{23575907}{106486488} a + \frac{1925077}{5604552}$, $\frac{1}{319459464} a^{15} + \frac{11}{106486488} a^{13} + \frac{5}{10305144} a^{12} - \frac{47}{17747748} a^{11} + \frac{47}{106486488} a^{10} - \frac{2369}{319459464} a^{9} - \frac{329}{1478979} a^{8} - \frac{5113}{17747748} a^{7} + \frac{2665}{319459464} a^{6} - \frac{342067}{11831832} a^{5} - \frac{563389}{26621622} a^{4} - \frac{33783631}{319459464} a^{3} - \frac{2506207}{35495496} a^{2} + \frac{17933947}{53243244} a + \frac{1056607}{16813656}$, $\frac{1}{319459464} a^{16} + \frac{5}{79864866} a^{13} + \frac{1}{2802276} a^{12} - \frac{1}{934092} a^{11} - \frac{199}{159729732} a^{10} + \frac{163}{2802276} a^{9} - \frac{53}{622728} a^{8} + \frac{27617}{159729732} a^{7} - \frac{181}{934092} a^{6} - \frac{3517}{311364} a^{5} - \frac{17827595}{159729732} a^{4} + \frac{460453}{2802276} a^{3} + \frac{365651}{934092} a^{2} + \frac{77557193}{159729732} a - \frac{2425495}{5604552}$, $\frac{1}{36611769829399785144565546224666888427711982569279762592944152} a^{17} - \frac{49533751833785750409476468874372065466705131001349051}{36611769829399785144565546224666888427711982569279762592944152} a^{16} - \frac{1296188484800282455248047623857059141831124323944423}{18305884914699892572282773112333444213855991284639881296472076} a^{15} - \frac{28787466689601814812135512490501117106925715440625869}{18305884914699892572282773112333444213855991284639881296472076} a^{14} + \frac{1569842259589575777304091374840394692458270001030368251}{18305884914699892572282773112333444213855991284639881296472076} a^{13} - \frac{64247030613672543855410278410532126184857874435041832}{4576471228674973143070693278083361053463997821159970324118019} a^{12} - \frac{58294787010773270844938708984652214020556015116518748406}{4576471228674973143070693278083361053463997821159970324118019} a^{11} + \frac{44633711288005906951014597977464081135222354396956464693}{9152942457349946286141386556166722106927995642319940648236038} a^{10} + \frac{1096662666376876262149922441993249787620875346102790148345}{36611769829399785144565546224666888427711982569279762592944152} a^{9} - \frac{7749143371614054791780392871247911688798222032727342470717}{36611769829399785144565546224666888427711982569279762592944152} a^{8} + \frac{1983169138124157595811494327926976197869997178928427417923}{9152942457349946286141386556166722106927995642319940648236038} a^{7} - \frac{6439788903132658271469906572091955147163789849178007608401}{9152942457349946286141386556166722106927995642319940648236038} a^{6} - \frac{371882061298776810487731321191986991249726188681432344502783}{9152942457349946286141386556166722106927995642319940648236038} a^{5} - \frac{1153910854185344261965030531156422942749148186810068799514351}{18305884914699892572282773112333444213855991284639881296472076} a^{4} - \frac{267521959363548324283270428636818899388314132593678172382735}{18305884914699892572282773112333444213855991284639881296472076} a^{3} - \frac{1189855276519950721697638593701603792844287027570245371863647}{18305884914699892572282773112333444213855991284639881296472076} a^{2} - \frac{11804954176693358900457180982137867780642907924257164388219229}{36611769829399785144565546224666888427711982569279762592944152} a - \frac{71044729679365685231402760102796593871166399959924697}{189653400375218884822217525937122045365583613182621592}$

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

Class group and class number

$C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{3}\times C_{9}$, which has order $6561$ (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:  \( \frac{31552381975474622888611082941072198789630}{529132989787833638925967542846960464037923207441319265131} a^{17} - \frac{268194180377245855078107842866033391568895}{529132989787833638925967542846960464037923207441319265131} a^{16} + \frac{469075901603287556093569707483721484182168}{176377663262611212975322514282320154679307735813773088377} a^{15} - \frac{5190321604880708286564023157502804491678880}{529132989787833638925967542846960464037923207441319265131} a^{14} + \frac{15586709383051931112419406377643337479354220}{529132989787833638925967542846960464037923207441319265131} a^{13} - \frac{12954709160117849513657975005872846589187140}{176377663262611212975322514282320154679307735813773088377} a^{12} + \frac{87534446100740752571400248821481732356680400}{529132989787833638925967542846960464037923207441319265131} a^{11} + \frac{304376569759031084959263094883955213005951925224}{529132989787833638925967542846960464037923207441319265131} a^{10} - \frac{507471375498599344684017359330332080763685027090}{176377663262611212975322514282320154679307735813773088377} a^{9} + \frac{19795281676946100511187045945751777177501656221630}{529132989787833638925967542846960464037923207441319265131} a^{8} - \frac{70045427141213022981153252604257300402253428624880}{529132989787833638925967542846960464037923207441319265131} a^{7} - \frac{43450691809744272211612237177868092491806264459480}{176377663262611212975322514282320154679307735813773088377} a^{6} + \frac{629812596699414784214567875162154753247658246935452}{529132989787833638925967542846960464037923207441319265131} a^{5} - \frac{800960023349626914653743057752437707376780203629080}{529132989787833638925967542846960464037923207441319265131} a^{4} + \frac{157851708552897123478635608513269528961664638048840}{176377663262611212975322514282320154679307735813773088377} a^{3} - \frac{1424979959356582335534484407629193559884029462640440}{529132989787833638925967542846960464037923207441319265131} a^{2} + \frac{58793858745041571218512616225297505994725704332635651950}{529132989787833638925967542846960464037923207441319265131} a + \frac{406057412157315266901782737653817401249376951111}{913657650090660215160796652489314975553935007817} \) (order $6$)
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:  \( 28127663525441974000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$D_9:C_3$ (as 18T18):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 54
The 10 conjugacy class representatives for $D_9:C_3$
Character table for $D_9:C_3$

Intermediate fields

\(\Q(\sqrt{-3}) \), 3.1.24300.2 x3, 6.0.1771470000.4, 9.1.225591527925010157904900000000.1 x3

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

Sibling fields

Degree 9 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 R ${\href{/LocalNumberField/7.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/13.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{18}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ R ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/43.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{9}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{3}$

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
2Data not computed
3Data not computed
5Data not computed
$31$31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.3$x^{3} - 1519$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.1$x^{3} - 31$$3$$1$$2$$C_3$$[\ ]_{3}$
31.3.2.2$x^{3} + 217$$3$$1$$2$$C_3$$[\ ]_{3}$