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Properties

Label 18.6.25935212617...7328.1
Degree $18$
Signature $[6, 6]$
Discriminant $2^{43}\cdot 3^{24}\cdot 11^{10}\cdot 47^{4}\cdot 953^{4}$
Root discriminant $927.77$
Ramified primes $2, 3, 11, 47, 953$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group 18T903

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1858061286256064, 583573253129280, 378744819233904, -88999185241248, -30871641520764, 7061043691332, 1895443582761, -226272859884, -65079800730, 3633226952, 1293156639, -32209920, -15726660, 153828, 115371, -324, -486, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 486*x^16 - 324*x^15 + 115371*x^14 + 153828*x^13 - 15726660*x^12 - 32209920*x^11 + 1293156639*x^10 + 3633226952*x^9 - 65079800730*x^8 - 226272859884*x^7 + 1895443582761*x^6 + 7061043691332*x^5 - 30871641520764*x^4 - 88999185241248*x^3 + 378744819233904*x^2 + 583573253129280*x - 1858061286256064)
 
gp: K = bnfinit(x^18 - 486*x^16 - 324*x^15 + 115371*x^14 + 153828*x^13 - 15726660*x^12 - 32209920*x^11 + 1293156639*x^10 + 3633226952*x^9 - 65079800730*x^8 - 226272859884*x^7 + 1895443582761*x^6 + 7061043691332*x^5 - 30871641520764*x^4 - 88999185241248*x^3 + 378744819233904*x^2 + 583573253129280*x - 1858061286256064, 1)
 

Normalized defining polynomial

\( x^{18} - 486 x^{16} - 324 x^{15} + 115371 x^{14} + 153828 x^{13} - 15726660 x^{12} - 32209920 x^{11} + 1293156639 x^{10} + 3633226952 x^{9} - 65079800730 x^{8} - 226272859884 x^{7} + 1895443582761 x^{6} + 7061043691332 x^{5} - 30871641520764 x^{4} - 88999185241248 x^{3} + 378744819233904 x^{2} + 583573253129280 x - 1858061286256064 \)

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:  \(259352126175392118892826453584331145475962636576227328=2^{43}\cdot 3^{24}\cdot 11^{10}\cdot 47^{4}\cdot 953^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $927.77$
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:  $2, 3, 11, 47, 953$
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 \(a\))

$1$, $a$, $a^{2}$, $\frac{1}{3} a^{3} + \frac{1}{3}$, $\frac{1}{3} a^{4} + \frac{1}{3} a$, $\frac{1}{3} a^{5} + \frac{1}{3} a^{2}$, $\frac{1}{9} a^{6} - \frac{1}{9} a^{3} - \frac{2}{9}$, $\frac{1}{18} a^{7} - \frac{1}{6} a^{5} + \frac{1}{9} a^{4} - \frac{1}{6} a^{3} + \frac{1}{3} a^{2} + \frac{1}{18} a + \frac{1}{3}$, $\frac{1}{36} a^{8} - \frac{1}{36} a^{6} - \frac{1}{9} a^{5} - \frac{1}{12} a^{4} + \frac{1}{9} a^{3} + \frac{13}{36} a^{2} - \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{108} a^{9} - \frac{1}{36} a^{7} - \frac{1}{12} a^{5} + \frac{1}{9} a^{4} + \frac{5}{36} a^{3} - \frac{1}{3} a^{2} + \frac{7}{18} a + \frac{4}{27}$, $\frac{1}{108} a^{10} + \frac{1}{18} a^{4} - \frac{1}{4} a^{2} - \frac{5}{27} a$, $\frac{1}{108} a^{11} + \frac{1}{18} a^{5} + \frac{1}{12} a^{3} - \frac{5}{27} a^{2} + \frac{1}{3}$, $\frac{1}{324} a^{12} + \frac{1}{324} a^{9} - \frac{1}{36} a^{7} + \frac{1}{54} a^{6} - \frac{1}{12} a^{5} + \frac{1}{36} a^{4} + \frac{13}{324} a^{3} - \frac{1}{3} a^{2} - \frac{4}{9} a + \frac{22}{81}$, $\frac{1}{324} a^{13} + \frac{1}{324} a^{10} + \frac{1}{54} a^{7} - \frac{1}{12} a^{5} - \frac{7}{162} a^{4} - \frac{1}{12} a^{2} - \frac{5}{81} a$, $\frac{1}{648} a^{14} - \frac{1}{648} a^{13} - \frac{1}{648} a^{12} + \frac{1}{648} a^{11} + \frac{1}{324} a^{10} + \frac{1}{324} a^{9} + \frac{1}{108} a^{8} + \frac{1}{54} a^{7} - \frac{11}{216} a^{6} - \frac{41}{648} a^{5} + \frac{95}{648} a^{4} + \frac{59}{648} a^{3} + \frac{17}{324} a^{2} - \frac{19}{162} a + \frac{22}{81}$, $\frac{1}{87073704} a^{15} - \frac{1}{179164} a^{13} + \frac{44629}{43536852} a^{12} + \frac{4273}{3224952} a^{11} + \frac{4273}{2418714} a^{10} + \frac{9943}{21768426} a^{9} - \frac{14197}{1612476} a^{8} - \frac{1819}{3224952} a^{7} - \frac{1408423}{43536852} a^{6} + \frac{54733}{1612476} a^{5} - \frac{21613}{268746} a^{4} - \frac{9772087}{87073704} a^{3} - \frac{17}{36} a^{2} - \frac{23}{54} a - \frac{40}{243}$, $\frac{1}{13409350416} a^{16} - \frac{17}{6704675208} a^{15} + \frac{166355}{319270248} a^{14} - \frac{658489}{478905372} a^{13} + \frac{7292947}{13409350416} a^{12} + \frac{537341}{319270248} a^{11} - \frac{4522547}{3352337604} a^{10} - \frac{701573}{1676168802} a^{9} - \frac{1317107}{135447984} a^{8} - \frac{7198337}{609515928} a^{7} + \frac{17114653}{6704675208} a^{6} + \frac{25831681}{159635124} a^{5} + \frac{299265665}{13409350416} a^{4} + \frac{113723729}{957810744} a^{3} + \frac{4423}{24948} a^{2} + \frac{1762}{18711} a - \frac{2962}{18711}$, $\frac{1}{19850683840377229324982226756212072698393239758317249127054188174160443759754117664} a^{17} + \frac{10690625085690060576264167973552657166265932242611843802176747088592343}{1240667740023576832811389172263254543649577484894828070440886760885027734984632354} a^{16} - \frac{17840946567191123884572544324138113439918049571845858233151150963873172787}{3308447306729538220830371126035345449732206626386208187842364695693407293292352944} a^{15} - \frac{523221119726149731240369399969141539656297356601249487629692801763323032051443}{708952994299186761606508098436145453514044277082758897394792434791444419991218488} a^{14} - \frac{25105579947787428750346700658044143591776048408970960213701225888777718438370261}{19850683840377229324982226756212072698393239758317249127054188174160443759754117664} a^{13} + \frac{114055641818294959065209534963539395289303472065828575203836996149897805747147}{150383968487706282765016869365242974987827573926645826720107486167882149695106952} a^{12} + \frac{6476511590108680168539227182501504569441021253108553982217513029391645284625235}{2481335480047153665622778344526509087299154969789656140881773521770055469969264708} a^{11} + \frac{742689115016326648205784851454955097894034850517362059951479178355530563117753}{225575952731559424147525304047864462481741360889968740080161229251823224542660428} a^{10} + \frac{18343626979002971413868785803693225216637183859086531658972706409624090376020261}{6616894613459076441660742252070690899464413252772416375684729391386814586584705888} a^{9} - \frac{31351361263128051480576022273606526153994661820824632277725914743144759427923}{16112568052254244581966093146276033034410097206426338577154373517987373181618602} a^{8} + \frac{22480884434656416699278951225620244982891028113203261351897695964787386884134319}{1417905988598373523213016196872290907028088554165517794789584869582888839982436976} a^{7} + \frac{91312771569869682359724394821412129665565323923978491886342863573134590731143253}{1654223653364769110415185563017672724866103313193104093921182347846703646646176472} a^{6} - \frac{2889652745502186902035616474245497641124954014123509949193436322788717739235123703}{19850683840377229324982226756212072698393239758317249127054188174160443759754117664} a^{5} + \frac{304741444953085047393406047953752879456145902066446043976602441273014583974418069}{4962670960094307331245556689053018174598309939579312281763547043540110939938529416} a^{4} - \frac{137036256210455039005884917776094813376058675227126178541192018182535948743383005}{827111826682384555207592781508836362433051656596552046960591173923351823323088236} a^{3} - \frac{5223959335965995994613314935481114415376643908063607729688466644107406838885}{27699040879274337094759866318306234369618393983050793026297398157777851242094} a^{2} + \frac{7274615970542403407526740823640157803194653292107240793507660877636861100261}{27699040879274337094759866318306234369618393983050793026297398157777851242094} a + \frac{1470862839801566035253971949870808247376304241180285730307680532814013836118}{4616506813212389515793311053051039061603065663841798837716233026296308540349}$

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:  $11$
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:  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:  \( 335661338521000000000 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

18T903:

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 559872
The 174 conjugacy class representatives for t18n903 are not computed
Character table for t18n903 is not computed

Intermediate fields

\(\Q(\sqrt{22}) \), 3.1.44.1, 6.2.2725888.3

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 R $18$ ${\href{/LocalNumberField/7.4.0.1}{4} }{,}\,{\href{/LocalNumberField/7.3.0.1}{3} }{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{5}{,}\,{\href{/LocalNumberField/7.1.0.1}{1} }$ R ${\href{/LocalNumberField/13.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/13.3.0.1}{3} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/13.1.0.1}{1} }$ ${\href{/LocalNumberField/17.6.0.1}{6} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{3}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/29.1.0.1}{1} }^{4}$ ${\href{/LocalNumberField/31.12.0.1}{12} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }$ $18$ ${\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{7}$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/53.6.0.1}{6} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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]])