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Properties

Label 18.6.48272376227...0256.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{27}\cdot 7^{12}\cdot 41^{7}\cdot 73^{12}\cdot 4794733^{7}$
Root discriminant $303{,}688.05$
Ramified primes $2, 7, 41, 73, 4794733$
Class number Not computed
Class group Not computed
Galois group 18T367

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-5809087981107339675113004179528265318506584446591366395820544, 0, -118466545789750935481669175656525993295564503629319250432, 0, -194342072854326719718697440564251332822089823755648, 0, -39740590777288682761741193269763171314300992, 0, 2101331127688893246617404284940269600, 0, 571202758172120436138505554688, 0, -6491171775323056058112, 0, -1680868814732124, 0, -880078, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 880078*x^16 - 1680868814732124*x^14 - 6491171775323056058112*x^12 + 571202758172120436138505554688*x^10 + 2101331127688893246617404284940269600*x^8 - 39740590777288682761741193269763171314300992*x^6 - 194342072854326719718697440564251332822089823755648*x^4 - 118466545789750935481669175656525993295564503629319250432*x^2 - 5809087981107339675113004179528265318506584446591366395820544)
 
gp: K = bnfinit(x^18 - 880078*x^16 - 1680868814732124*x^14 - 6491171775323056058112*x^12 + 571202758172120436138505554688*x^10 + 2101331127688893246617404284940269600*x^8 - 39740590777288682761741193269763171314300992*x^6 - 194342072854326719718697440564251332822089823755648*x^4 - 118466545789750935481669175656525993295564503629319250432*x^2 - 5809087981107339675113004179528265318506584446591366395820544, 1)
 

Normalized defining polynomial

\( x^{18} - 880078 x^{16} - 1680868814732124 x^{14} - 6491171775323056058112 x^{12} + 571202758172120436138505554688 x^{10} + 2101331127688893246617404284940269600 x^{8} - 39740590777288682761741193269763171314300992 x^{6} - 194342072854326719718697440564251332822089823755648 x^{4} - 118466545789750935481669175656525993295564503629319250432 x^{2} - 5809087981107339675113004179528265318506584446591366395820544 \)

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:  \(482723762275128238432243152436331745226533165017777364876172787462243456249389351866810619716960256=2^{27}\cdot 7^{12}\cdot 41^{7}\cdot 73^{12}\cdot 4794733^{7}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $303{,}688.05$
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 \(a\))

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{6290689696} a^{6} + \frac{392728067}{3145344848} a^{4} + \frac{139128805}{786336212} a^{2} - \frac{1}{4}$, $\frac{1}{6290689696} a^{7} + \frac{392728067}{3145344848} a^{5} + \frac{139128805}{786336212} a^{3} - \frac{1}{4} a$, $\frac{1}{2473298553210035776} a^{8} - \frac{440039}{1236649276605017888} a^{6} - \frac{2441594193599365}{38645289893906809} a^{4} - \frac{196407553}{1572672424} a^{2} - \frac{1}{2}$, $\frac{1}{2473298553210035776} a^{9} - \frac{440039}{1236649276605017888} a^{7} - \frac{2441594193599365}{38645289893906809} a^{5} - \frac{196407553}{1572672424} a^{3} - \frac{1}{2} a$, $\frac{1}{972422107738129986242160256} a^{10} - \frac{440039}{486211053869064993121080128} a^{8} - \frac{420217203683031}{243105526934532496560540064} a^{6} + \frac{38643226154606665}{309162319151254472} a^{4} - \frac{38503287}{393168106} a^{2} - \frac{1}{4}$, $\frac{1}{972422107738129986242160256} a^{11} - \frac{440039}{486211053869064993121080128} a^{9} - \frac{420217203683031}{243105526934532496560540064} a^{7} + \frac{38643226154606665}{309162319151254472} a^{5} - \frac{38503287}{393168106} a^{3} - \frac{1}{4} a$, $\frac{1}{9175808599966284256143268924799883264} a^{12} - \frac{1}{1944844215476259972484320512} a^{11} + \frac{884408219}{2293952149991571064035817231199970816} a^{10} + \frac{440039}{972422107738129986242160256} a^{9} + \frac{269318270202782029}{2293952149991571064035817231199970816} a^{8} + \frac{420217203683031}{486211053869064993121080128} a^{7} + \frac{36449482462689791}{2917266323214389958726480768} a^{6} + \frac{38647353633206953}{618324638302508944} a^{5} - \frac{303246563042660669}{2473298553210035776} a^{4} + \frac{38503287}{786336212} a^{3} + \frac{634895655}{3145344848} a^{2} - \frac{3}{8} a - \frac{43}{96}$, $\frac{1}{18351617199932568512286537849599766528} a^{13} + \frac{884408219}{4587904299983142128071634462399941632} a^{11} - \frac{1}{1944844215476259972484320512} a^{10} - \frac{658168687250981387}{4587904299983142128071634462399941632} a^{9} - \frac{98072007}{486211053869064993121080128} a^{8} - \frac{426255940463015113}{5834532646428779917452961536} a^{7} + \frac{253360926890549}{243105526934532496560540064} a^{6} + \frac{162523803099345431}{4946597106420071552} a^{5} - \frac{19110472605811745}{618324638302508944} a^{4} + \frac{471195541}{6290689696} a^{3} - \frac{435915511}{3145344848} a^{2} - \frac{67}{192} a + \frac{3}{8}$, $\frac{1}{39683988170939812093300146985934491851207757824} a^{14} - \frac{786776251}{19841994085469906046650073492967245925603878912} a^{12} + \frac{1313865657789540743}{9920997042734953023325036746483622962801939456} a^{10} - \frac{1}{4946597106420071552} a^{9} - \frac{837736086623715009}{8411157883302427234797996514399892992} a^{8} + \frac{440039}{2473298553210035776} a^{7} + \frac{451920369012552107}{16044964777679144772995644224} a^{6} - \frac{28878913119509349}{309162319151254472} a^{5} + \frac{1105241491791666355}{27206284085310393536} a^{4} - \frac{589928659}{3145344848} a^{3} - \frac{10647327223}{415185519936} a^{2} - \frac{1}{4} a + \frac{151}{1056}$, $\frac{1}{79367976341879624186600293971868983702415515648} a^{15} - \frac{786776251}{39683988170939812093300146985934491851207757824} a^{13} - \frac{8888490874201856833}{19841994085469906046650073492967245925603878912} a^{11} - \frac{1}{1944844215476259972484320512} a^{10} + \frac{2570661833248714079}{16822315766604854469595993028799785984} a^{9} + \frac{440039}{972422107738129986242160256} a^{8} - \frac{2076643735448689663}{32089929555358289545991288448} a^{7} - \frac{19112536345111889}{243105526934532496560540064} a^{6} - \frac{609653006839558985}{54412568170620787072} a^{5} + \frac{90632128698355}{1236649276605017888} a^{4} - \frac{95299459183}{830371039872} a^{3} - \frac{62122231}{1572672424} a^{2} + \frac{151}{2112} a - \frac{1}{4}$, $\frac{1}{44874294896587314157047170932473050698202636135283135830856218852289821068193035969884050265478588063766596902912} a^{16} + \frac{13060942309570996532626155108400000217870256421164988930125263867}{1869762287357804756543632122186377112425109838970130659619009118845409211174709832078502094394941169323608204288} a^{14} + \frac{26286963014877314333448506919125347453182462164031853829423971424220029177}{1019870338558802594478344793919842061322787184892798541610368610279314115186205362951910233306331546903786293248} a^{12} + \frac{167140710616314235017276235524186979746743066789137320646378963217278111927}{1783361485987820140486832993060458004595663183355465250676077015176374591764358900031016975212386475968} a^{10} - \frac{1}{4946597106420071552} a^{9} + \frac{171613769704716906844606194110348916922130323497068127461607741627815453591}{4535875262445169294802445122698884442563769868340910772340412106510935024226911477990071186528} a^{8} - \frac{98072007}{1236649276605017888} a^{7} - \frac{3664549735692758321187136330505997004958565307898577486571817701539643918505}{46146929959218709310082696511490527083229582903562485532236046700196327972004031049152} a^{6} + \frac{116442591535501525}{1236649276605017888} a^{5} + \frac{65933029198177032604180479323482265292579488193730349624576993072731886777}{3880066369776433939786366729399768648758094927538481258245871899885421595008} a^{4} + \frac{704486155}{3145344848} a^{3} - \frac{16773185495007523440400672407817882191774318157461933818243641126433}{597057624433743497386014284507314546134280645633239935601173229303264} a^{2} - \frac{1}{8} a + \frac{383475756175831904249367037761639945528908294863041411009529}{1518581022525116768718809262995799934327024597547696138916144}$, $\frac{1}{89748589793174628314094341864946101396405272270566271661712437704579642136386071939768100530957176127533193805824} a^{17} + \frac{13060942309570996532626155108400000217870256421164988930125263867}{3739524574715609513087264244372754224850219677940261319238018237690818422349419664157004188789882338647216408576} a^{15} + \frac{26286963014877314333448506919125347453182462164031853829423971424220029177}{2039740677117605188956689587839684122645574369785597083220737220558628230372410725903820466612663093807572586496} a^{13} - \frac{3333593818946656738669747327837344841285650477610445923142792853925881377677}{7133445943951280561947331972241832018382652733421861002704308060705498367057435600124067900849545903872} a^{11} - \frac{1}{1944844215476259972484320512} a^{10} - \frac{829109357692476910825285568762034037158981626040647598721661912818003400249}{4535875262445169294802445122698884442563769868340910772340412106510935024226911477990071186528} a^{9} - \frac{98072007}{486211053869064993121080128} a^{8} + \frac{3767388018522152409673824657704380665555578108122478405110264391432475565321}{92293859918437418620165393022981054166459165807124971064472093400392655944008062098304} a^{7} - \frac{38138568040125711}{486211053869064993121080128} a^{6} + \frac{310557166819641337070301631531013650134452726197671588371731641110624874929}{7760132739552867879572733458799537297516189855076962516491743799770843190016} a^{5} - \frac{115425020349339041}{1236649276605017888} a^{4} - \frac{76627259622443430824428154313636579557258454563646813980074560639869}{1194115248867486994772028569014629092268561291266479871202346458606528} a^{3} - \frac{714173121}{3145344848} a^{2} - \frac{375814755086726480110037593736260021634604003910806658448543}{3037162045050233537437618525991599868654049195095392277832288} a - \frac{1}{2}$

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

18T367:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 2304
The 48 conjugacy class representatives for t18n367
Character table for t18n367 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 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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/3.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }{,}\,{\href{/LocalNumberField/5.6.0.1}{6} }$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/11.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }{,}\,{\href{/LocalNumberField/19.6.0.1}{6} }$ ${\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/29.4.0.1}{4} }{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{2}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.6.0.1}{6} }$ R ${\href{/LocalNumberField/43.4.0.1}{4} }{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/43.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }{,}\,{\href{/LocalNumberField/47.3.0.1}{3} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }^{2}$

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$2.6.9.3$x^{6} - 4 x^{4} + 4 x^{2} + 24$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
2.6.9.1$x^{6} + 4 x^{4} + 4 x^{2} - 8$$2$$3$$9$$C_6$$[3]^{3}$
$7$7.9.6.1$x^{9} + 42 x^{6} + 539 x^{3} + 2744$$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
73Data not computed
4794733Data not computed