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Properties

Label 18.18.2069651203...9536.1
Degree $18$
Signature $[18, 0]$
Discriminant $2^{16}\cdot 3^{24}\cdot 37^{9}\cdot 179^{4}\cdot 1433\cdot 15551^{4}$
Root discriminant $1973.91$
Ramified primes $2, 3, 37, 179, 1433, 15551$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T874

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-203877935399756689856, 29997087058481564736, 21225806692214506992, -1982681218316985312, -809958483143306820, 39378422425826340, 13983683787376737, -348984105127020, -126008992005570, 1566545909384, 640037186847, -3677605848, -1889868432, 4255380, 3191535, -1884, -2826, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 2826*x^16 - 1884*x^15 + 3191535*x^14 + 4255380*x^13 - 1889868432*x^12 - 3677605848*x^11 + 640037186847*x^10 + 1566545909384*x^9 - 126008992005570*x^8 - 348984105127020*x^7 + 13983683787376737*x^6 + 39378422425826340*x^5 - 809958483143306820*x^4 - 1982681218316985312*x^3 + 21225806692214506992*x^2 + 29997087058481564736*x - 203877935399756689856)
 
gp: K = bnfinit(x^18 - 2826*x^16 - 1884*x^15 + 3191535*x^14 + 4255380*x^13 - 1889868432*x^12 - 3677605848*x^11 + 640037186847*x^10 + 1566545909384*x^9 - 126008992005570*x^8 - 348984105127020*x^7 + 13983683787376737*x^6 + 39378422425826340*x^5 - 809958483143306820*x^4 - 1982681218316985312*x^3 + 21225806692214506992*x^2 + 29997087058481564736*x - 203877935399756689856, 1)
 

Normalized defining polynomial

\( x^{18} - 2826 x^{16} - 1884 x^{15} + 3191535 x^{14} + 4255380 x^{13} - 1889868432 x^{12} - 3677605848 x^{11} + 640037186847 x^{10} + 1566545909384 x^{9} - 126008992005570 x^{8} - 348984105127020 x^{7} + 13983683787376737 x^{6} + 39378422425826340 x^{5} - 809958483143306820 x^{4} - 1982681218316985312 x^{3} + 21225806692214506992 x^{2} + 29997087058481564736 x - 203877935399756689856 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[18, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(206965120322023780883776584258126018307696157475694936129536=2^{16}\cdot 3^{24}\cdot 37^{9}\cdot 179^{4}\cdot 1433\cdot 15551^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $1973.91$
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:  $2, 3, 37, 179, 1433, 15551$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{6} a^{3} - \frac{1}{2} a - \frac{1}{3}$, $\frac{1}{6} a^{4} - \frac{1}{2} a^{2} - \frac{1}{3} a$, $\frac{1}{12} a^{5} - \frac{1}{12} a^{4} - \frac{1}{12} a^{3} + \frac{1}{12} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{72} a^{6} - \frac{1}{12} a^{4} - \frac{1}{18} a^{3} - \frac{3}{8} a^{2} - \frac{1}{3} a + \frac{1}{18}$, $\frac{1}{72} a^{7} + \frac{1}{36} a^{4} + \frac{1}{24} a^{3} + \frac{1}{4} a^{2} - \frac{1}{9} a - \frac{1}{3}$, $\frac{1}{144} a^{8} - \frac{1}{144} a^{7} + \frac{1}{72} a^{5} - \frac{11}{144} a^{4} + \frac{1}{48} a^{3} - \frac{31}{72} a^{2} + \frac{11}{36} a - \frac{1}{6}$, $\frac{1}{864} a^{9} + \frac{1}{288} a^{7} - \frac{1}{144} a^{6} + \frac{1}{32} a^{5} + \frac{5}{72} a^{4} + \frac{7}{288} a^{3} - \frac{5}{16} a^{2} - \frac{11}{72} a - \frac{37}{108}$, $\frac{1}{1728} a^{10} - \frac{1}{1728} a^{9} + \frac{1}{576} a^{8} - \frac{1}{192} a^{7} + \frac{1}{192} a^{6} + \frac{11}{576} a^{5} - \frac{13}{576} a^{4} - \frac{17}{576} a^{3} - \frac{85}{288} a^{2} + \frac{67}{432} a - \frac{11}{216}$, $\frac{1}{1728} a^{11} - \frac{1}{288} a^{8} - \frac{1}{288} a^{7} + \frac{1}{288} a^{6} - \frac{5}{144} a^{5} + \frac{13}{288} a^{4} - \frac{41}{576} a^{3} - \frac{67}{864} a^{2} + \frac{61}{144} a - \frac{11}{72}$, $\frac{1}{10368} a^{12} + \frac{1}{2592} a^{9} + \frac{1}{576} a^{8} - \frac{1}{288} a^{7} + \frac{1}{432} a^{6} + \frac{7}{288} a^{5} - \frac{35}{1152} a^{4} - \frac{179}{2592} a^{3} - \frac{1}{18} a^{2} - \frac{7}{72} a - \frac{41}{648}$, $\frac{1}{10368} a^{13} - \frac{1}{5184} a^{10} + \frac{1}{576} a^{8} - \frac{11}{1728} a^{7} + \frac{1}{192} a^{6} - \frac{17}{1152} a^{5} - \frac{61}{5184} a^{4} - \frac{5}{192} a^{3} - \frac{79}{288} a^{2} + \frac{509}{1296} a + \frac{1}{8}$, $\frac{1}{20736} a^{14} - \frac{1}{20736} a^{13} + \frac{1}{5184} a^{11} + \frac{1}{10368} a^{10} - \frac{1}{3456} a^{9} + \frac{1}{864} a^{8} + \frac{1}{1728} a^{7} - \frac{1}{768} a^{6} - \frac{689}{20736} a^{5} - \frac{43}{1296} a^{4} + \frac{7}{288} a^{3} + \frac{67}{1296} a^{2} + \frac{407}{1296} a - \frac{25}{54}$, $\frac{1}{346327985664} a^{15} + \frac{2782687}{115442661888} a^{13} + \frac{2782687}{173163992832} a^{12} + \frac{161213}{712609024} a^{11} - \frac{5212643}{28860665472} a^{10} + \frac{54388595}{173163992832} a^{9} + \frac{1747693}{534456768} a^{8} - \frac{140092159}{38480887296} a^{7} + \frac{144111157}{86581996416} a^{6} - \frac{10448659}{4275654144} a^{5} + \frac{3331513871}{57721330944} a^{4} + \frac{1671449419}{43290998208} a^{3} + \frac{9}{32} a^{2} + \frac{367}{2592} a - \frac{1135}{3888}$, $\frac{1}{346327985664} a^{16} + \frac{2782687}{115442661888} a^{14} + \frac{2782687}{173163992832} a^{13} + \frac{1923737}{57721330944} a^{12} - \frac{5212643}{28860665472} a^{11} - \frac{45822049}{173163992832} a^{10} - \frac{712655}{1803791592} a^{9} - \frac{73285063}{38480887296} a^{8} - \frac{306836741}{86581996416} a^{7} - \frac{205383091}{38480887296} a^{6} - \frac{376279957}{57721330944} a^{5} + \frac{168289759}{43290998208} a^{4} + \frac{185}{5184} a^{3} + \frac{17}{1296} a^{2} + \frac{173}{972} a + \frac{205}{648}$, $\frac{1}{4818371409166875333704009042820255582321459979120798812889121511104832183879720784874873931128369664} a^{17} - \frac{187254451313685076859532967069632588762311088764104989890458167387035546540852861065453}{2409185704583437666852004521410127791160729989560399406444560755552416091939860392437436965564184832} a^{16} + \frac{3836628386657889794192222764359893264080720942335922503762762267366195007360173714443521}{4818371409166875333704009042820255582321459979120798812889121511104832183879720784874873931128369664} a^{15} + \frac{11054612595229926162724711972435930447017242308359906623986591647067491876751474999749680558339}{1204592852291718833426002260705063895580364994780199703222280377776208045969930196218718482782092416} a^{14} + \frac{63970116000169547074455009582194200054960899265434355822803648779732558287067806317284556397987}{2409185704583437666852004521410127791160729989560399406444560755552416091939860392437436965564184832} a^{13} - \frac{10836755256794565341325955480302437143522791228690389235736524309993414852629964934931080767511}{1204592852291718833426002260705063895580364994780199703222280377776208045969930196218718482782092416} a^{12} + \frac{546873699337837565670139377781346045507764115145193694254032581271654666909102375634913450812259}{2409185704583437666852004521410127791160729989560399406444560755552416091939860392437436965564184832} a^{11} + \frac{75974602711318929828744870288775782173230744198295082565786647784447870421424364161847393027129}{1204592852291718833426002260705063895580364994780199703222280377776208045969930196218718482782092416} a^{10} + \frac{1551462866127404172053029286627647004996038624621656857003982282667435805158229761526451070098793}{4818371409166875333704009042820255582321459979120798812889121511104832183879720784874873931128369664} a^{9} + \frac{2240990463953792060902448153344578416177827978768351727655581737670934874219848732140834488727117}{2409185704583437666852004521410127791160729989560399406444560755552416091939860392437436965564184832} a^{8} + \frac{15840735373526538088315860083804481524250429095701745508919825528116998752171308728111235584187409}{4818371409166875333704009042820255582321459979120798812889121511104832183879720784874873931128369664} a^{7} + \frac{578551826917832710089662714755593395982960086382771044090570565430411318697576836186375565466503}{150574106536464854178250282588132986947545624347524962902785047222026005746241274527339810347761552} a^{6} + \frac{165121264821652213528561158450185952680087437892019410696146998184284790156101936808597623513757}{9410881658529053386140642661758311684221601521720310181424065451376625359140079657958738146735097} a^{5} + \frac{51412595634755024206793285414654302545724787571477767854609335168608050873999194570271399280837975}{1204592852291718833426002260705063895580364994780199703222280377776208045969930196218718482782092416} a^{4} + \frac{48018784625242606053539273596503127429048144820216605762640144446786704572632429166351569707758875}{602296426145859416713001130352531947790182497390099851611140188888104022984965098109359241391046208} a^{3} + \frac{6056537175978375645984070324267185311023361758880208423054979049825079474438711708208394815}{108185470503766740595280680426977148856794942391766261166832970357778285645279076002829263776} a^{2} + \frac{22979750011697208466537294797620518133296645242806910678782697473704840202051629851518322527}{54092735251883370297640340213488574428397471195883130583416485178889142822639538001414631888} a + \frac{462154666210223385164581350118072771063884529688965701499876854146366524624393430496613061}{13523183812970842574410085053372143607099367798970782645854121294722285705659884500353657972}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $17$
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:  \( 1044636921100000000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T874:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 279936
The 174 conjugacy class representatives for t18n874 are not computed
Character table for t18n874 is not computed

Intermediate fields

\(\Q(\sqrt{37}) \), 3.3.148.1 x3, 6.6.810448.1

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

Sibling fields

Degree 18 siblings: 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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/5.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/7.9.0.1}{9} }{,}\,{\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }$ ${\href{/LocalNumberField/11.9.0.1}{9} }{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{3}$ ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }{,}\,{\href{/LocalNumberField/23.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }$ ${\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/31.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/47.9.0.1}{9} }{,}\,{\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }$ ${\href{/LocalNumberField/53.9.0.1}{9} }{,}\,{\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }$ ${\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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$2.6.4.1$x^{6} + 3 x^{5} + 6 x^{4} + 3 x^{3} + 9 x + 9$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
2.12.12.28$x^{12} - x^{10} + 2 x^{8} - x^{6} - 2 x^{4} + 3 x^{2} + 1$$6$$2$$12$$S_4$$[4/3, 4/3]_{3}^{2}$
$3$3.9.12.20$x^{9} + 6 x^{6} + 54 x^{2} + 27$$3$$3$$12$$(C_3^2:C_3):C_2$$[2, 2, 2]^{6}$
3.9.12.8$x^{9} + 12 x^{6} + 54 x^{2} + 54$$3$$3$$12$$C_3 \wr C_3 $$[2, 2, 2]^{3}$
37Data not computed
$179$179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.2.0.1$x^{2} - x + 7$$1$$2$$0$$C_2$$[\ ]^{2}$
179.6.4.2$x^{6} - 179 x^{3} + 224287$$3$$2$$4$$S_3\times C_3$$[\ ]_{3}^{6}$
179.6.0.1$x^{6} - x + 50$$1$$6$$0$$C_6$$[\ ]^{6}$
1433Data not computed
15551Data not computed