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Properties

Label 18.6.51624177137...0000.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{12}\cdot 3^{24}\cdot 5^{14}\cdot 13^{6}\cdot 1061117^{7}$
Root discriminant $12{,}449.69$
Ramified primes $2, 3, 5, 13, 1061117$
Class number Not computed
Class group Not computed
Galois group 18T461

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-371023633673372769266300083152445623019, 1957760134295180518228084208239496514, -887010722785755909642099142950505815, 244705662148773213221767445905616, -95464018377851901223533667416996, -66983200453112231194458550608, 5555830520763339794029338844, -10094756489315953842318336, 1064246929788629129834742, -315574620417688738476, 36115874586042869742, 5287125117175824, -222640923218036, 262385243232, -21346492116, 726016, -79635, -6, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 6*x^17 - 79635*x^16 + 726016*x^15 - 21346492116*x^14 + 262385243232*x^13 - 222640923218036*x^12 + 5287125117175824*x^11 + 36115874586042869742*x^10 - 315574620417688738476*x^9 + 1064246929788629129834742*x^8 - 10094756489315953842318336*x^7 + 5555830520763339794029338844*x^6 - 66983200453112231194458550608*x^5 - 95464018377851901223533667416996*x^4 + 244705662148773213221767445905616*x^3 - 887010722785755909642099142950505815*x^2 + 1957760134295180518228084208239496514*x - 371023633673372769266300083152445623019)
 
gp: K = bnfinit(x^18 - 6*x^17 - 79635*x^16 + 726016*x^15 - 21346492116*x^14 + 262385243232*x^13 - 222640923218036*x^12 + 5287125117175824*x^11 + 36115874586042869742*x^10 - 315574620417688738476*x^9 + 1064246929788629129834742*x^8 - 10094756489315953842318336*x^7 + 5555830520763339794029338844*x^6 - 66983200453112231194458550608*x^5 - 95464018377851901223533667416996*x^4 + 244705662148773213221767445905616*x^3 - 887010722785755909642099142950505815*x^2 + 1957760134295180518228084208239496514*x - 371023633673372769266300083152445623019, 1)
 

Normalized defining polynomial

\( x^{18} - 6 x^{17} - 79635 x^{16} + 726016 x^{15} - 21346492116 x^{14} + 262385243232 x^{13} - 222640923218036 x^{12} + 5287125117175824 x^{11} + 36115874586042869742 x^{10} - 315574620417688738476 x^{9} + 1064246929788629129834742 x^{8} - 10094756489315953842318336 x^{7} + 5555830520763339794029338844 x^{6} - 66983200453112231194458550608 x^{5} - 95464018377851901223533667416996 x^{4} + 244705662148773213221767445905616 x^{3} - 887010722785755909642099142950505815 x^{2} + 1957760134295180518228084208239496514 x - 371023633673372769266300083152445623019 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[6, 6]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(51624177137754227336536682168566437749832086648167693412837925000000000000=2^{12}\cdot 3^{24}\cdot 5^{14}\cdot 13^{6}\cdot 1061117^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $12{,}449.69$
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:  $2, 3, 5, 13, 1061117$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{4} a^{6} - \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{5} - \frac{1}{4} a^{3} + \frac{1}{4} a$, $\frac{1}{4} a^{8} - \frac{1}{4}$, $\frac{1}{8} a^{9} - \frac{1}{8} a^{8} - \frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{8} a + \frac{3}{8}$, $\frac{1}{8488936} a^{10} - \frac{54265}{4244468} a^{9} + \frac{641009}{8488936} a^{8} - \frac{430495}{4244468} a^{7} - \frac{236849}{4244468} a^{6} - \frac{640645}{4244468} a^{5} - \frac{279739}{4244468} a^{4} - \frac{1009003}{4244468} a^{3} - \frac{2108003}{8488936} a^{2} + \frac{98748}{1061117} a - \frac{2473071}{8488936}$, $\frac{1}{8488936} a^{11} + \frac{278809}{8488936} a^{9} + \frac{253815}{2122234} a^{8} + \frac{45857}{1061117} a^{7} - \frac{151645}{2122234} a^{6} + \frac{52459}{1061117} a^{5} + \frac{164512}{1061117} a^{4} + \frac{371851}{8488936} a^{3} + \frac{302037}{1061117} a^{2} - \frac{1640917}{8488936} a - \frac{167427}{1061117}$, $\frac{1}{84889360} a^{12} - \frac{1}{21222340} a^{11} - \frac{1}{21222340} a^{10} + \frac{490701}{8488936} a^{9} - \frac{704485}{16977872} a^{8} - \frac{842011}{21222340} a^{7} - \frac{306631}{21222340} a^{6} - \frac{129359}{5305585} a^{5} + \frac{554063}{16977872} a^{4} - \frac{251337}{2122234} a^{3} - \frac{3967333}{10611170} a^{2} - \frac{15242397}{42444680} a + \frac{17998021}{84889360}$, $\frac{1}{84889360} a^{13} + \frac{1}{21222340} a^{10} + \frac{536839}{16977872} a^{9} + \frac{2386483}{42444680} a^{8} + \frac{85017}{2122234} a^{7} - \frac{87395}{1061117} a^{6} - \frac{7901841}{84889360} a^{5} - \frac{263425}{4244468} a^{4} + \frac{1473967}{10611170} a^{3} - \frac{1705013}{4244468} a^{2} - \frac{322311}{16977872} a - \frac{10516713}{42444680}$, $\frac{1}{90077543015120} a^{14} - \frac{13567}{22519385753780} a^{13} - \frac{33089}{22519385753780} a^{12} - \frac{895583}{22519385753780} a^{11} + \frac{445197}{90077543015120} a^{10} - \frac{840730165407}{45038771507560} a^{9} + \frac{1160084682601}{45038771507560} a^{8} + \frac{2100385915011}{22519385753780} a^{7} + \frac{9631148836423}{90077543015120} a^{6} - \frac{4083918220667}{22519385753780} a^{5} - \frac{1695180179391}{22519385753780} a^{4} + \frac{75160527507}{5629846438445} a^{3} - \frac{28385339588981}{90077543015120} a^{2} + \frac{3478808288419}{45038771507560} a - \frac{2002688656289}{45038771507560}$, $\frac{1}{180155086030240} a^{15} - \frac{1}{180155086030240} a^{14} - \frac{494237}{180155086030240} a^{13} + \frac{800057}{180155086030240} a^{12} + \frac{3157889}{180155086030240} a^{11} - \frac{7972169}{180155086030240} a^{10} + \frac{912497044479}{180155086030240} a^{9} + \frac{11310651741573}{180155086030240} a^{8} + \frac{21035280932827}{180155086030240} a^{7} + \frac{12200642390469}{180155086030240} a^{6} - \frac{40049986490599}{180155086030240} a^{5} - \frac{44945747115381}{180155086030240} a^{4} - \frac{23859370940477}{180155086030240} a^{3} - \frac{56926728023323}{180155086030240} a^{2} + \frac{53861059639269}{180155086030240} a + \frac{8565510342795}{36031017206048}$, $\frac{1}{360310172060480} a^{16} - \frac{6521}{11259692876890} a^{13} - \frac{508167}{90077543015120} a^{12} - \frac{65589}{45038771507560} a^{11} + \frac{1143871}{45038771507560} a^{10} + \frac{132412357121}{2251938575378} a^{9} - \frac{11719942201261}{180155086030240} a^{8} + \frac{1556520431987}{22519385753780} a^{7} - \frac{282909752881}{11259692876890} a^{6} - \frac{466978814808}{5629846438445} a^{5} + \frac{2111418602225}{18015508603024} a^{4} - \frac{7859140093269}{45038771507560} a^{3} - \frac{12765538717571}{45038771507560} a^{2} - \frac{645689400227}{5629846438445} a + \frac{176621430270593}{360310172060480}$, $\frac{1}{3512487512057872439382603802471979169893418230724059217604835204425653227196589622641275473368525421430655434562243035046316644206762096678133774760263988152532150152849133227816722822526772675739648486873015816088102092719711431640685621335680} a^{17} - \frac{257447732099326500023908255476144676469164335068738229628086267692263567202988864709591764053048534897720866496053095524846471660115857576743846414684088856626446539718194256740087993873093129939391935041815556996119222690578929}{501783930293981777054657686067425595699059747246294173943547886346521889599512803234467924766932203061522204937463290720902377743823156668304824965751998307504592878978447603973817546075253239391378355267573688012586013245673061662955088762240} a^{16} - \frac{102662585208885684787300055166997714957910943622214343905448101498400910230277614674288974038955081593385912245337566221201400452382991344452113905015533273482376518963218614829213429867837894144301858445145658574261194143519021}{175624375602893621969130190123598958494670911536202960880241760221282661359829481132063773668426271071532771728112151752315832210338104833906688738013199407626607507642456661390836141126338633786982424343650790804405104635985571582034281066784} a^{15} + \frac{63174678895531607188482572932084985503281806865967216232741331467100952580403411212857039062079805666014013303844817048797795687103923642247945140720678318532159296399042082260570962953856327232301905181227007651346155862083445}{25089196514699088852732884303371279784952987362314708697177394317326094479975640161723396238346610153076110246873164536045118887191157833415241248287599915375229643948922380198690877303762661969568917763378684400629300662283653083147754438112} a^{14} + \frac{2541171828514564112469137639266418197932706749423884528111715660628239951912564281750427521818460311138521649576921212817016000725774712922427346142938244040233887294435547915363583823708605749399320823803029071229510449025451940785507}{439060939007234054922825475308997396236677278840507402200604400553206653399573702830159434171065677678831929320280379380789580525845262084766721845032998519066518769106141653477090352815846584467456060859126977011012761589963928955085702666960} a^{13} + \frac{143426808278472901024421319509558714190045730370145649836144946052418154756753135988371649005826120510600199810258364198919726652923930162835113604110490041544328337011267727978094670257434879568546189135588457516011884934335371372291}{87812187801446810984565095061799479247335455768101480440120880110641330679914740566031886834213135535766385864056075876157916105169052416953344369006599703813303753821228330695418070563169316893491212171825395402202552317992785791017140533392} a^{12} - \frac{88689523846907479673459555249532744654429714212583569422463121039150229901802973682351961818214748588901258694623336089588678864468710509086669791767127522602748476278048371888560725565717027870322438896716089129350829291386589585439}{175624375602893621969130190123598958494670911536202960880241760221282661359829481132063773668426271071532771728112151752315832210338104833906688738013199407626607507642456661390836141126338633786982424343650790804405104635985571582034281066784} a^{11} + \frac{36244683953066920708434833020170795580615900435336694761815689817994575482675835867386193328480154083289807096371345347028953664246102403688597658252159469904186996794527540113331680525950550675540528187905701697753386092880798582038929}{878121878014468109845650950617994792473354557681014804401208801106413306799147405660318868342131355357663858640560758761579161051690524169533443690065997038133037538212283306954180705631693168934912121718253954022025523179927857910171405333920} a^{10} + \frac{2860431938445737278030197958037472944882248771162731545765079886312599325907293037085165991606375506652109857611983105116644833102074723435725874739207403020317623637366561694059218536746051207577919275865052314785629478984877759963448323971}{50178393029398177705465768606742559569905974724629417394354788634652188959951280323446792476693220306152220493746329072090237774382315666830482496575199830750459287897844760397381754607525323939137835526757368801258601324567306166295508876224} a^{9} - \frac{157981884863398839189705556159209847921316930013464561600559192310188626574177646940660255355298474787044970647921702089298099161356848290112829132503350727313240946446219238437356123055943858793664005187179113294607579253094585253757949451727}{1756243756028936219691301901235989584946709115362029608802417602212826613598294811320637736684262710715327717281121517523158322103381048339066887380131994076266075076424566613908361411263386337869824243436507908044051046359855715820342810667840} a^{8} - \frac{5516090896861171067635382980197407295705041774547232676161925437803572311557257150435427774629702737625769585353996584318077651011451813483905621741840106296455342255225326443678459713702527278702526887832383454954424705870152273062771214443}{175624375602893621969130190123598958494670911536202960880241760221282661359829481132063773668426271071532771728112151752315832210338104833906688738013199407626607507642456661390836141126338633786982424343650790804405104635985571582034281066784} a^{7} + \frac{2479337367930800159220692171517892553207593591447882552781567666894600207873738924299260821138191260097232636682689978306428837395601491661733580793222974816646599313011153731157316261038149444037213791238376141743115339469058905757147394589}{175624375602893621969130190123598958494670911536202960880241760221282661359829481132063773668426271071532771728112151752315832210338104833906688738013199407626607507642456661390836141126338633786982424343650790804405104635985571582034281066784} a^{6} + \frac{20716889356597839294583408804868467312746977479085263031372096142883052565102063429750250608266709995309486753055547151427907053374605182409403414955639098244336106855653339002758686826522527859873931549939357488972802272822155672472961977831}{219530469503617027461412737654498698118338639420253701100302200276603326699786851415079717085532838839415964660140189690394790262922631042383360922516499259533259384553070826738545176407923292233728030429563488505506380794981964477542851333480} a^{5} + \frac{436707778747614781822006675505034896006085793258982138515369301077857596945839994650915181397449640404297397302119915152151425029292937456149715003446762389422385040000773367257839389564195807641503691099928691336215273435734650879210930007}{6272299128674772213183221075842819946238246840578677174294348579331523619993910040430849059586652538269027561718291134011279721797789458353810312071899978843807410987230595049672719325940665492392229440844671100157325165570913270786938609528} a^{4} - \frac{10783105808927307657936737282734059847641132112138414484093478973809236460653639072224277703717316962281376773921756991309891205026445181059139800141475716813523152305467293019718706808207627976320771497383555376592291597289149822034061273781}{878121878014468109845650950617994792473354557681014804401208801106413306799147405660318868342131355357663858640560758761579161051690524169533443690065997038133037538212283306954180705631693168934912121718253954022025523179927857910171405333920} a^{3} + \frac{64933018967619394865467162567122924751701809200904867091064133627501543272550563778625785889215823062742743853132658253267729750000152623428410008818637256957188120088691178796393620095133527586116725436543858750570033060751582819202602855711}{878121878014468109845650950617994792473354557681014804401208801106413306799147405660318868342131355357663858640560758761579161051690524169533443690065997038133037538212283306954180705631693168934912121718253954022025523179927857910171405333920} a^{2} - \frac{391443047914868373778251936115309994133190017649281540444698874260275231195208119071151843192890285795529442565937471994667481673912960744407224703467225727257655163659598849028518474402270318376005622419108764586577678196917402375958242658107}{3512487512057872439382603802471979169893418230724059217604835204425653227196589622641275473368525421430655434562243035046316644206762096678133774760263988152532150152849133227816722822526772675739648486873015816088102092719711431640685621335680} a + \frac{1281254977736332436023063587083899698500440696636275542360714339368177774912978579556937100231458396817812719569474188876084697122776594620083578266050018674815085260964235208913243942753941617950732935510908544101838784519093331697592949283}{74733776852295158284736251116425088721136558100511898246911387328205387812693396226410116454649477051716073075792405000985460515037491418683697335324765705373024471337215600591845166436739844164673372061127996086980895589781094290227353645440}$

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:  $11$
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

18T461:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 4608
The 60 conjugacy class representatives for t18n461 are not computed
Character table for t18n461 is not computed

Intermediate fields

3.3.1300.1, 3.3.1620.1, 9.9.5837879385000000.1

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 R ${\href{/LocalNumberField/7.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{3}$ R ${\href{/LocalNumberField/17.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/19.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.3.0.1}{3} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/37.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }{,}\,{\href{/LocalNumberField/43.6.0.1}{6} }$ ${\href{/LocalNumberField/47.4.0.1}{4} }{,}\,{\href{/LocalNumberField/47.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/53.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{3}$

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
$2$2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.3.2.1$x^{3} - 2$$3$$1$$2$$S_3$$[\ ]_{3}^{2}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
3Data not computed
$5$5.6.4.1$x^{6} + 25 x^{3} + 200$$3$$2$$4$$S_3$$[\ ]_{3}^{2}$
5.12.10.1$x^{12} + 6 x^{11} + 27 x^{10} + 80 x^{9} + 195 x^{8} + 366 x^{7} + 571 x^{6} + 702 x^{5} + 1005 x^{4} + 1140 x^{3} + 357 x^{2} - 138 x + 44$$6$$2$$10$$D_6$$[\ ]_{6}^{2}$
$13$13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.0.1$x^{2} - x + 2$$1$$2$$0$$C_2$$[\ ]^{2}$
13.2.1.1$x^{2} - 13$$2$$1$$1$$C_2$$[\ ]_{2}$
13.8.4.1$x^{8} + 26 x^{6} + 845 x^{4} + 6591 x^{2} + 114244$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
1061117Data not computed