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Properties

Label 18.0.68031822397...9056.1
Degree $18$
Signature $[0, 9]$
Discriminant $-\,2^{24}\cdot 3^{24}\cdot 11^{10}\cdot 862559^{4}$
Root discriminant $861.29$
Ramified primes $2, 3, 11, 862559$
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![8332747069445731648, -3861937353596896320, 1276967763570965616, 79282594732523072, -38890024289693124, -1193247825135372, 426471361916601, 10179941498940, -1243428132186, -26465161176, -11196401241, -120805560, 78880376, 693540, -5121, -836, -666, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^18 - 666*x^16 - 836*x^15 - 5121*x^14 + 693540*x^13 + 78880376*x^12 - 120805560*x^11 - 11196401241*x^10 - 26465161176*x^9 - 1243428132186*x^8 + 10179941498940*x^7 + 426471361916601*x^6 - 1193247825135372*x^5 - 38890024289693124*x^4 + 79282594732523072*x^3 + 1276967763570965616*x^2 - 3861937353596896320*x + 8332747069445731648)
 
gp: K = bnfinit(x^18 - 666*x^16 - 836*x^15 - 5121*x^14 + 693540*x^13 + 78880376*x^12 - 120805560*x^11 - 11196401241*x^10 - 26465161176*x^9 - 1243428132186*x^8 + 10179941498940*x^7 + 426471361916601*x^6 - 1193247825135372*x^5 - 38890024289693124*x^4 + 79282594732523072*x^3 + 1276967763570965616*x^2 - 3861937353596896320*x + 8332747069445731648, 1)
 

Normalized defining polynomial

\( x^{18} - 666 x^{16} - 836 x^{15} - 5121 x^{14} + 693540 x^{13} + 78880376 x^{12} - 120805560 x^{11} - 11196401241 x^{10} - 26465161176 x^{9} - 1243428132186 x^{8} + 10179941498940 x^{7} + 426471361916601 x^{6} - 1193247825135372 x^{5} - 38890024289693124 x^{4} + 79282594732523072 x^{3} + 1276967763570965616 x^{2} - 3861937353596896320 x + 8332747069445731648 \)

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

Invariants

Degree:  $18$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 9]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(-68031822397385704323327440417682068561570555605549056=-\,2^{24}\cdot 3^{24}\cdot 11^{10}\cdot 862559^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $861.29$
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:  $2, 3, 11, 862559$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{5} - \frac{1}{4} a^{4} + \frac{1}{4} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{6} - \frac{1}{2} a^{3} + \frac{1}{4} a^{2}$, $\frac{1}{4} a^{7} - \frac{1}{4} a^{3} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{4} a^{8} - \frac{1}{4} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{48} a^{9} + \frac{1}{16} a^{7} + \frac{1}{24} a^{6} + \frac{1}{16} a^{5} + \frac{17}{48} a^{3} + \frac{1}{8} a^{2} - \frac{1}{4} a + \frac{1}{6}$, $\frac{1}{48} a^{10} + \frac{1}{16} a^{8} + \frac{1}{24} a^{7} + \frac{1}{16} a^{6} - \frac{7}{48} a^{4} - \frac{3}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{3} a$, $\frac{1}{48} a^{11} + \frac{1}{24} a^{8} - \frac{1}{8} a^{7} - \frac{1}{8} a^{6} - \frac{1}{12} a^{5} - \frac{1}{8} a^{4} - \frac{1}{16} a^{3} + \frac{1}{24} a^{2} + \frac{1}{4} a - \frac{1}{2}$, $\frac{1}{96} a^{12} - \frac{1}{96} a^{11} + \frac{1}{24} a^{8} - \frac{1}{16} a^{7} - \frac{1}{48} a^{6} - \frac{1}{12} a^{5} + \frac{5}{32} a^{4} - \frac{29}{96} a^{3} - \frac{1}{48} a^{2} - \frac{3}{8} a - \frac{5}{12}$, $\frac{1}{192} a^{13} - \frac{1}{192} a^{12} + \frac{3}{32} a^{8} - \frac{7}{96} a^{7} - \frac{1}{12} a^{6} + \frac{1}{64} a^{5} + \frac{43}{192} a^{4} + \frac{37}{96} a^{3} - \frac{1}{16} a^{2} - \frac{11}{24} a - \frac{1}{6}$, $\frac{1}{192} a^{14} - \frac{1}{192} a^{12} - \frac{1}{96} a^{9} + \frac{1}{48} a^{8} + \frac{1}{32} a^{7} - \frac{5}{192} a^{6} - \frac{7}{96} a^{5} + \frac{7}{64} a^{4} + \frac{5}{96} a^{3} - \frac{19}{48} a^{2} + \frac{1}{8} a$, $\frac{1}{165611328} a^{15} - \frac{111}{27601888} a^{13} + \frac{287241}{55203776} a^{12} + \frac{428719}{82805664} a^{11} - \frac{515789}{82805664} a^{10} - \frac{118763}{41402832} a^{9} + \frac{5163529}{41402832} a^{8} + \frac{4789933}{165611328} a^{7} - \frac{3513205}{82805664} a^{6} + \frac{9835517}{82805664} a^{5} + \frac{19318381}{165611328} a^{4} + \frac{3960341}{10350708} a^{3} - \frac{1}{12} a^{2} - \frac{1}{12} a + \frac{1}{4}$, $\frac{1}{4140283200} a^{16} + \frac{11}{4140283200} a^{15} + \frac{3449903}{2070141600} a^{14} - \frac{1151439}{690047200} a^{13} + \frac{2441531}{690047200} a^{12} - \frac{1200103}{258767700} a^{11} + \frac{989267}{2070141600} a^{10} + \frac{2791127}{517535400} a^{9} - \frac{121663981}{4140283200} a^{8} - \frac{45768401}{4140283200} a^{7} + \frac{373731}{345023600} a^{6} - \frac{145661897}{2070141600} a^{5} + \frac{445436107}{2070141600} a^{4} - \frac{618418631}{2070141600} a^{3} + \frac{1}{400} a^{2} - \frac{47}{200} a + \frac{83}{300}$, $\frac{1}{8842892085430636898399530006123986454977250405754205682425666142947447265908583127078164216505244423107763521969230976160000} a^{17} + \frac{455613704123837543728015534794370635687318911282056275046135608753664032282295120750636020296461457887048033875259}{4421446042715318449199765003061993227488625202877102841212833071473723632954291563539082108252622211553881760984615488080000} a^{16} - \frac{300353893506267541533670995299711995367314400510400996279291330294382444576786584115505421166538465863830618925667}{340111234055024496092289615620153325191432707913623295477910236267209510227253197195314008327124785504144750844970422160000} a^{15} + \frac{235751999485953986254645764908922334728948736797844032914102738939647453686756340100399035123071291328874606436070148297}{138170188834853701537492656345687288359019537589909463787901033483553863529821611360596315882894444111058805030769234002500} a^{14} + \frac{2075231285637410801969718691562001447850910680544453602020294805838865878219084451531278985233923034842757991099691294841}{2947630695143545632799843335374662151659083468584735227475222047649149088636194375692721405501748141035921173989743658720000} a^{13} + \frac{1913669190326462994373212366583752837311928512923639848432638245940351480369089286003124288509443834319869195667995765659}{1473815347571772816399921667687331075829541734292367613737611023824574544318097187846360702750874070517960586994871829360000} a^{12} - \frac{7191754899468482159993971097069014099043328120950104067783529154826567309432129289682801794834042842422607458582411194113}{2210723021357659224599882501530996613744312601438551420606416535736861816477145781769541054126311105776940880492307744040000} a^{11} - \frac{65979563353577290226263513896519102592645016035822185473481971375827416467384669653600266862452139166267218928626416866}{34542547208713425384373164086421822089754884397477365946975258370888465882455402840149078970723611027764701257692308500625} a^{10} + \frac{16709887259469175822238755962358076009508739305341150309587624907673159141952997134167501162085726939964728523362566032231}{8842892085430636898399530006123986454977250405754205682425666142947447265908583127078164216505244423107763521969230976160000} a^{9} - \frac{85483378788537087879671412903014022396111851031954026210026977894776381100781817058254223388646085039788470531562733361753}{1473815347571772816399921667687331075829541734292367613737611023824574544318097187846360702750874070517960586994871829360000} a^{8} - \frac{20013964822149241068967258030594108201190449146458055651684514641282386704261020050775585432922611735422627765805019091017}{294763069514354563279984333537466215165908346858473522747522204764914908863619437569272140550174814103592117398974365872000} a^{7} - \frac{1144267378219090152896443866109879337269686518325947985447620575198506195324893429898752643167081440125262527925746405407}{27634037766970740307498531269137457671803907517981892757580206696710772705964322272119263176578888822211761006153846800500} a^{6} - \frac{358239017798561502102130124716598571783533325560846734237532601096982818421953722784176760155728153795372847511765638585219}{8842892085430636898399530006123986454977250405754205682425666142947447265908583127078164216505244423107763521969230976160000} a^{5} + \frac{831374035652126093389981602836159403037911114612372635724425223597197175734515101081371279260256406763481475849594382404343}{4421446042715318449199765003061993227488625202877102841212833071473723632954291563539082108252622211553881760984615488080000} a^{4} - \frac{6816919755795679880694450315767683200409552798814752710108135440122761155307701507125581648759762949932643410040832117111}{552680755339414806149970625382749153436078150359637855151604133934215454119286445442385263531577776444235220123076936010000} a^{3} - \frac{236702528970543346125720275824220917043569763040886550184108989827057133562052553518326442788991761393013070835675109}{640745450849640205655463133980109364618626842175013947047800943395426230691797831154025711321286748436031877382390000} a^{2} - \frac{4702754626336061172738758220126007876810045634160806734178436811661628985321347299829643759654251925149114208798723}{640745450849640205655463133980109364618626842175013947047800943395426230691797831154025711321286748436031877382390000} a - \frac{19288404269927116072742855388032831394816891241994035721080262626510829055759030198088744272986935186635997461834049}{106790908474940034275910522330018227436437807029168991174633490565904371781966305192337618553547791406005312897065000}$

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:  $8$
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:  \( 15164328511700000000 \) (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

\(\Q(\sqrt{-11}) \), 3.1.44.1 x3, 6.0.21296.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/5.3.0.1}{3} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/7.2.0.1}{2} }^{2}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }{,}\,{\href{/LocalNumberField/13.2.0.1}{2} }^{4}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }{,}\,{\href{/LocalNumberField/17.2.0.1}{2} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/19.2.0.1}{2} }^{5}$ ${\href{/LocalNumberField/23.9.0.1}{9} }{,}\,{\href{/LocalNumberField/23.6.0.1}{6} }{,}\,{\href{/LocalNumberField/23.3.0.1}{3} }$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{3}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{3}$ ${\href{/LocalNumberField/31.9.0.1}{9} }{,}\,{\href{/LocalNumberField/31.6.0.1}{6} }{,}\,{\href{/LocalNumberField/31.3.0.1}{3} }$ ${\href{/LocalNumberField/37.9.0.1}{9} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }$ ${\href{/LocalNumberField/43.6.0.1}{6} }{,}\,{\href{/LocalNumberField/43.4.0.1}{4} }^{2}{,}\,{\href{/LocalNumberField/43.2.0.1}{2} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{3}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/53.3.0.1}{3} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}{,}\,{\href{/LocalNumberField/53.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/59.9.0.1}{9} }{,}\,{\href{/LocalNumberField/59.6.0.1}{6} }{,}\,{\href{/LocalNumberField/59.3.0.1}{3} }$

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]])
 

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$$[\ ]_{3}^{2}$
2.12.20.21$x^{12} - 4 x^{10} + x^{8} + 7 x^{4} - 4 x^{2} - 5$$6$$2$$20$12T135$[4/3, 4/3, 2, 2, 8/3, 8/3]_{3}^{2}$
$3$3.9.12.4$x^{9} + 6 x^{6} + 18 x^{5} + 36 x^{3} + 27$$3$$3$$12$$C_3^2:C_3$$[2, 2]^{3}$
3.9.12.17$x^{9} + 6 x^{8} + 6 x^{6} + 27$$3$$3$$12$$C_3^2 : S_3 $$[2, 2]^{6}$
$11$11.2.1.2$x^{2} + 33$$2$$1$$1$$C_2$$[\ ]_{2}$
11.4.3.1$x^{4} + 33$$4$$1$$3$$D_{4}$$[\ ]_{4}^{2}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
11.6.3.2$x^{6} - 121 x^{2} + 3993$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
862559Data not computed