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Properties

Label 18.10.1241034522...7344.1
Degree $18$
Signature $[10, 4]$
Discriminant $2^{18}\cdot 7^{12}\cdot 41^{4}\cdot 73^{12}\cdot 4794733^{4}$
Root discriminant $8905.43$
Ramified primes $2, 7, 41, 73, 4794733$
Class number Not computed
Class group Not computed
Galois group 18T463

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5894493332285389666473057161, 408195892123775028048902443, -96033970663082701749723147, -3345773125455451750808938, 1840910048517668601690747, 6462730573786176922302, -3447917848391399824309, 3439078251571123287, -1073268117936735198, -3910634890096303, 1298486657446163, -367485308357, 62447385363, 304399221, -70915081, 2535, -1186, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 1186*x^16 + 2535*x^15 - 70915081*x^14 + 304399221*x^13 + 62447385363*x^12 - 367485308357*x^11 + 1298486657446163*x^10 - 3910634890096303*x^9 - 1073268117936735198*x^8 + 3439078251571123287*x^7 - 3447917848391399824309*x^6 + 6462730573786176922302*x^5 + 1840910048517668601690747*x^4 - 3345773125455451750808938*x^3 - 96033970663082701749723147*x^2 + 408195892123775028048902443*x - 5894493332285389666473057161)
 
gp: K = bnfinit(x^18 - 6*x^17 - 1186*x^16 + 2535*x^15 - 70915081*x^14 + 304399221*x^13 + 62447385363*x^12 - 367485308357*x^11 + 1298486657446163*x^10 - 3910634890096303*x^9 - 1073268117936735198*x^8 + 3439078251571123287*x^7 - 3447917848391399824309*x^6 + 6462730573786176922302*x^5 + 1840910048517668601690747*x^4 - 3345773125455451750808938*x^3 - 96033970663082701749723147*x^2 + 408195892123775028048902443*x - 5894493332285389666473057161, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 1186 x^{16} + 2535 x^{15} - 70915081 x^{14} + 304399221 x^{13} + 62447385363 x^{12} - 367485308357 x^{11} + 1298486657446163 x^{10} - 3910634890096303 x^{9} - 1073268117936735198 x^{8} + 3439078251571123287 x^{7} - 3447917848391399824309 x^{6} + 6462730573786176922302 x^{5} + 1840910048517668601690747 x^{4} - 3345773125455451750808938 x^{3} - 96033970663082701749723147 x^{2} + 408195892123775028048902443 x - 5894493332285389666473057161 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[10, 4]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(124103452202961465725793344698621400682587896362984419066163822801977344=2^{18}\cdot 7^{12}\cdot 41^{4}\cdot 73^{12}\cdot 4794733^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $8905.43$
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:  $2, 7, 41, 73, 4794733$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a^{2} - \frac{1}{3} a$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{8} - \frac{1}{3} a^{6} - \frac{1}{3} a^{5} - \frac{1}{3} a^{2} + \frac{1}{3} a$, $\frac{1}{123} a^{13} + \frac{5}{41} a^{12} - \frac{16}{123} a^{11} - \frac{20}{123} a^{10} + \frac{16}{123} a^{9} + \frac{19}{123} a^{8} + \frac{25}{123} a^{7} + \frac{52}{123} a^{6} + \frac{37}{123} a^{5} - \frac{52}{123} a^{4} + \frac{43}{123} a^{3} - \frac{19}{123} a^{2} + \frac{6}{41} a + \frac{47}{123}$, $\frac{1}{589752159} a^{14} - \frac{1341448}{589752159} a^{13} + \frac{8173985}{196584053} a^{12} - \frac{92447035}{589752159} a^{11} + \frac{16637660}{589752159} a^{10} + \frac{16447925}{196584053} a^{9} + \frac{45424816}{196584053} a^{8} + \frac{39930406}{196584053} a^{7} + \frac{58014866}{196584053} a^{6} - \frac{83020023}{196584053} a^{5} + \frac{69981250}{589752159} a^{4} - \frac{270410239}{589752159} a^{3} + \frac{3950105}{589752159} a^{2} + \frac{107422795}{589752159} a + \frac{131213995}{589752159}$, $\frac{1}{589752159} a^{15} + \frac{285418}{589752159} a^{13} - \frac{63607451}{589752159} a^{12} - \frac{14926951}{589752159} a^{11} + \frac{28900596}{196584053} a^{10} + \frac{63675310}{589752159} a^{9} + \frac{43848088}{196584053} a^{8} - \frac{260532032}{589752159} a^{7} + \frac{46170076}{589752159} a^{6} + \frac{114988514}{589752159} a^{5} - \frac{6699638}{196584053} a^{4} + \frac{616954}{196584053} a^{3} - \frac{3937739}{14384199} a^{2} - \frac{95307080}{589752159} a + \frac{88784007}{196584053}$, $\frac{1}{231871739363440854} a^{16} - \frac{93832657}{115935869681720427} a^{15} + \frac{17532519}{77290579787813618} a^{14} - \frac{488068461953929}{231871739363440854} a^{13} - \frac{2596139745360413}{38645289893906809} a^{12} + \frac{9948580963761730}{115935869681720427} a^{11} - \frac{14622132960624191}{231871739363440854} a^{10} + \frac{4081294650275457}{77290579787813618} a^{9} + \frac{10028426783606234}{38645289893906809} a^{8} + \frac{11288204708375968}{115935869681720427} a^{7} - \frac{32105020120987325}{115935869681720427} a^{6} - \frac{12808726258727243}{77290579787813618} a^{5} + \frac{96551718909402083}{231871739363440854} a^{4} + \frac{112259653952470763}{231871739363440854} a^{3} - \frac{47911142059513946}{115935869681720427} a^{2} + \frac{38778180217522991}{231871739363440854} a - \frac{44778919994585603}{231871739363440854}$, $\frac{1}{709131107631636947851341843799073479202773415188062462311433278483800363003748308124819488704243884736752881396249858306088665984979470230701452934681122377758313195347474865501759158233347553947636} a^{17} + \frac{134794690481888074901042451532836402307415133136498849791375917391654616996428842846926573499840592270205365039728011753687606416386964838085720630252486825544852761016984716069915}{236377035877212315950447281266357826400924471729354154103811092827933454334582769374939829568081294912250960465416619435362888661659823410233817644893707459252771065115824955167253052744449184649212} a^{16} + \frac{552846232078327631207361121782101374085197824325881189702401519346890676208103489999951010409738236488635374099613860978601598656616058401960296152243969077927436546267001353418288736637929}{709131107631636947851341843799073479202773415188062462311433278483800363003748308124819488704243884736752881396249858306088665984979470230701452934681122377758313195347474865501759158233347553947636} a^{15} + \frac{13957261532704654223437613543979592099264655073247611489371745941910922248462987083023011779635673235301641976589286208254961154138584870705396185455618726418840634815107930616615130737919}{118188517938606157975223640633178913200462235864677077051905546413966727167291384687469914784040647456125480232708309717681444330829911705116908822446853729626385532557912477583626526372224592324606} a^{14} - \frac{1552206285395064559186047451029924054927554644251838141545234442036584808202708303225370371815521368997646251315308004755830894599252508618835599519287015001506726731375913660920016922271107418627}{709131107631636947851341843799073479202773415188062462311433278483800363003748308124819488704243884736752881396249858306088665984979470230701452934681122377758313195347474865501759158233347553947636} a^{13} - \frac{3452257072496653850524926931373776424968578031999137268944311313130948334531871939137936558277033985073597200319916594565041087408309458796908758226538606764838622502499634361263673085602234024671}{177282776907909236962835460949768369800693353797015615577858319620950090750937077031204872176060971184188220349062464576522166496244867557675363233670280594439578298836868716375439789558336888486909} a^{12} - \frac{101824064931393799751397231105905066744572336237475710828118222694516589185499955275996750186518017273309173773511693025165369400027131735608650627642707598226256708702773984563580167394330822161353}{709131107631636947851341843799073479202773415188062462311433278483800363003748308124819488704243884736752881396249858306088665984979470230701452934681122377758313195347474865501759158233347553947636} a^{11} + \frac{15414756236613262468556831422282344787276078566501731440616986432154022183090841220597301258574945681408626501511901725325843390720863350959306603597976340488666372586809849147746834469501593478809}{177282776907909236962835460949768369800693353797015615577858319620950090750937077031204872176060971184188220349062464576522166496244867557675363233670280594439578298836868716375439789558336888486909} a^{10} + \frac{190241878913887095339967053922520409398753975790844335305046245699548517922725498233167337330547293035004388777409100987495853477737959025411802834776500417972779378092127252570090932523296723423}{17295880673942364581740044970709109248848132077757621031986177523995130804969470929873646065957167920408606863323167275758260145975109030017108608162954204335568614520670118670774613615447501315796} a^{9} + \frac{44449485053369971531268601767230617196211053923030664520509164086224136642517672893521599108285842308825764701469389493011132762832634398663469406833377889817788837903239108836806103126559452011307}{118188517938606157975223640633178913200462235864677077051905546413966727167291384687469914784040647456125480232708309717681444330829911705116908822446853729626385532557912477583626526372224592324606} a^{8} - \frac{12420192782412916774052451677031847124563623787681856360562875746917607191050728801836980843387517353659558993267173530039535619929106819056443512039535928414677891834175386430232370346953338218769}{59094258969303078987611820316589456600231117932338538525952773206983363583645692343734957392020323728062740116354154858840722165414955852558454411223426864813192766278956238791813263186112296162303} a^{7} - \frac{88214591637253134914770247361334110231198186374166699163369530807797642842840539546443465080575420979480992240825080344696219222660552053767086689369913571952381938070103141287863086263912250921165}{709131107631636947851341843799073479202773415188062462311433278483800363003748308124819488704243884736752881396249858306088665984979470230701452934681122377758313195347474865501759158233347553947636} a^{6} - \frac{2903255913220395206038071245088298409696053145069309074507844847903575051089571894578613505045512298280342646420000669149997835548046176818713384641200880662177149767622748660432042318927705571209}{6113199203721008171132257274129943786230805303345366054408907573136210025894381966593271454346930040834076563760774640569729879180857501988805628747251054980675113752995472978463441019252996154721} a^{5} - \frac{40643551823010665251115652503959896203655554370558630121262090557616576764581716737423587474231775270675043813186142201584967860747051495438329359308373131819364118603755928764006287089533508122953}{354565553815818473925670921899536739601386707594031231155716639241900181501874154062409744352121942368376440698124929153044332992489735115350726467340561188879156597673737432750879579116673776973818} a^{4} + \frac{165913949248977638065484230507061131509716475057035896774523341661950993838992937256094863065217685474785526865865913694667650955105509603161982173066174806829706748961329330750422371103767423063825}{709131107631636947851341843799073479202773415188062462311433278483800363003748308124819488704243884736752881396249858306088665984979470230701452934681122377758313195347474865501759158233347553947636} a^{3} + \frac{91251711664256154011840495185398391040547097959124702379901590004910240915918926956800137562622991882998795788371108008917311874792167182748996398652295632091375732533944224726078631391962924597051}{236377035877212315950447281266357826400924471729354154103811092827933454334582769374939829568081294912250960465416619435362888661659823410233817644893707459252771065115824955167253052744449184649212} a^{2} - \frac{18532772553618044414644920645249190965512414517227488775945790387638981763059294144919854318058364906160027195452288325893398374831008776827773249673674161614460070481686547687362130104581352721726}{59094258969303078987611820316589456600231117932338538525952773206983363583645692343734957392020323728062740116354154858840722165414955852558454411223426864813192766278956238791813263186112296162303} a - \frac{130699608338191261154933808731195167955090754529386571578036957418692412534790966477493015550062249343449462868315969372426840541613603913724659591275392941160382518643046683850104455070257777480413}{709131107631636947851341843799073479202773415188062462311433278483800363003748308124819488704243884736752881396249858306088665984979470230701452934681122377758313195347474865501759158233347553947636}$

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

Class group and class number

Not computed

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $13$
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:  Not computed
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  Not computed
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T463:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 48 conjugacy class representatives for t18n463
Character table for t18n463 is not computed

Intermediate fields

\(\Q(\zeta_{7})^+\), 3.3.2088968.1, 9.9.9115812039001375232.3

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ R ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.3.0.1$x^{3} - x + 1$$1$$3$$0$$C_3$$[\ ]^{3}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$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$$[\ ]_{3}^{3}$
41Data not computed
$73$73.9.6.1$x^{9} + 3066 x^{6} + 3128123 x^{3} + 1067462648$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
73.9.6.1$x^{9} + 3066 x^{6} + 3128123 x^{3} + 1067462648$$3$$3$$6$$C_3^2$$[\ ]_{3}^{3}$
4794733Data not computed