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Properties

Label 24.0.22191215949...2073.1
Degree $24$
Signature $[0, 12]$
Discriminant $13^{22}\cdot 17^{21}$
Root discriminant $125.24$
Ramified primes $13, 17$
Class number $198288$ (GRH)
Class group $[3, 6, 18, 612]$ (GRH)
Galois group $C_{24}$ (as 24T1)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![117887368, 66055116, 365222708, -6358985, 306255413, -15508935, 197154619, 144085698, -45630734, -12452743, 13524047, 552604, -1878824, -33644, 458052, -6400, -25004, 4681, -105, 247, 81, -22, 18, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 18*x^22 - 22*x^21 + 81*x^20 + 247*x^19 - 105*x^18 + 4681*x^17 - 25004*x^16 - 6400*x^15 + 458052*x^14 - 33644*x^13 - 1878824*x^12 + 552604*x^11 + 13524047*x^10 - 12452743*x^9 - 45630734*x^8 + 144085698*x^7 + 197154619*x^6 - 15508935*x^5 + 306255413*x^4 - 6358985*x^3 + 365222708*x^2 + 66055116*x + 117887368)
 
gp: K = bnfinit(x^24 - x^23 + 18*x^22 - 22*x^21 + 81*x^20 + 247*x^19 - 105*x^18 + 4681*x^17 - 25004*x^16 - 6400*x^15 + 458052*x^14 - 33644*x^13 - 1878824*x^12 + 552604*x^11 + 13524047*x^10 - 12452743*x^9 - 45630734*x^8 + 144085698*x^7 + 197154619*x^6 - 15508935*x^5 + 306255413*x^4 - 6358985*x^3 + 365222708*x^2 + 66055116*x + 117887368, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} + 18 x^{22} - 22 x^{21} + 81 x^{20} + 247 x^{19} - 105 x^{18} + 4681 x^{17} - 25004 x^{16} - 6400 x^{15} + 458052 x^{14} - 33644 x^{13} - 1878824 x^{12} + 552604 x^{11} + 13524047 x^{10} - 12452743 x^{9} - 45630734 x^{8} + 144085698 x^{7} + 197154619 x^{6} - 15508935 x^{5} + 306255413 x^{4} - 6358985 x^{3} + 365222708 x^{2} + 66055116 x + 117887368 \)

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

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 12]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(221912159494888970129181006529110805060115683312073=13^{22}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $125.24$
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:  $13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(221=13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{221}(128,·)$, $\chi_{221}(1,·)$, $\chi_{221}(2,·)$, $\chi_{221}(4,·)$, $\chi_{221}(70,·)$, $\chi_{221}(8,·)$, $\chi_{221}(76,·)$, $\chi_{221}(140,·)$, $\chi_{221}(15,·)$, $\chi_{221}(16,·)$, $\chi_{221}(19,·)$, $\chi_{221}(152,·)$, $\chi_{221}(30,·)$, $\chi_{221}(32,·)$, $\chi_{221}(35,·)$, $\chi_{221}(38,·)$, $\chi_{221}(166,·)$, $\chi_{221}(64,·)$, $\chi_{221}(111,·)$, $\chi_{221}(83,·)$, $\chi_{221}(118,·)$, $\chi_{221}(120,·)$, $\chi_{221}(59,·)$, $\chi_{221}(60,·)$$\rbrace$
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{5}$, $\frac{1}{332} a^{20} + \frac{23}{332} a^{19} - \frac{13}{166} a^{18} + \frac{15}{332} a^{17} + \frac{1}{166} a^{16} - \frac{31}{166} a^{15} + \frac{41}{166} a^{14} + \frac{7}{166} a^{13} + \frac{41}{166} a^{12} - \frac{41}{166} a^{11} - \frac{8}{83} a^{10} - \frac{11}{83} a^{9} + \frac{21}{166} a^{8} + \frac{15}{166} a^{7} + \frac{33}{332} a^{6} + \frac{127}{332} a^{5} - \frac{8}{83} a^{4} + \frac{105}{332} a^{3} - \frac{31}{166} a^{2} + \frac{25}{83} a - \frac{35}{83}$, $\frac{1}{664} a^{21} - \frac{1}{664} a^{20} + \frac{3}{664} a^{19} + \frac{29}{332} a^{18} + \frac{57}{664} a^{17} + \frac{7}{83} a^{16} + \frac{19}{166} a^{15} - \frac{16}{83} a^{14} - \frac{11}{83} a^{13} + \frac{27}{166} a^{12} - \frac{7}{83} a^{11} + \frac{15}{166} a^{10} - \frac{8}{83} a^{9} - \frac{37}{166} a^{8} + \frac{143}{664} a^{7} - \frac{167}{664} a^{6} - \frac{175}{664} a^{5} + \frac{63}{332} a^{4} + \frac{323}{664} a^{3} - \frac{9}{83} a^{2} - \frac{27}{83} a + \frac{5}{83}$, $\frac{1}{294152} a^{22} + \frac{139}{294152} a^{21} + \frac{207}{294152} a^{20} - \frac{2279}{147076} a^{19} + \frac{11517}{294152} a^{18} + \frac{14483}{147076} a^{17} + \frac{2061}{147076} a^{16} + \frac{2134}{36769} a^{15} - \frac{5697}{36769} a^{14} + \frac{611}{36769} a^{13} + \frac{6986}{36769} a^{12} + \frac{6275}{73538} a^{11} - \frac{7306}{36769} a^{10} + \frac{3651}{36769} a^{9} + \frac{55291}{294152} a^{8} + \frac{119813}{294152} a^{7} - \frac{88895}{294152} a^{6} - \frac{13085}{147076} a^{5} - \frac{134809}{294152} a^{4} + \frac{30425}{147076} a^{3} + \frac{17139}{147076} a^{2} + \frac{1299}{73538} a - \frac{1834}{36769}$, $\frac{1}{159039177195634551908618100307857095720319645285293260102652078731036182337434884277018363589889816} a^{23} + \frac{233661761889330926356000140392885361696525162561810963852387239165570758576490580485022836275}{159039177195634551908618100307857095720319645285293260102652078731036182337434884277018363589889816} a^{22} + \frac{12474131110918934369263135246762604347957542758719733040072961827086848816084744844345333041909}{39759794298908637977154525076964273930079911321323315025663019682759045584358721069254590897472454} a^{21} - \frac{95425572411668749125132871862932002483243764671423614502710191137653451520512889329107456073509}{159039177195634551908618100307857095720319645285293260102652078731036182337434884277018363589889816} a^{20} + \frac{644860830271114679186466070509982276647079747112760086562268367299683446242970383626816939964449}{19879897149454318988577262538482136965039955660661657512831509841379522792179360534627295448736227} a^{19} + \frac{766163893736834165010940793327639957995343593854588383377758462702732049000561251745095059195685}{39759794298908637977154525076964273930079911321323315025663019682759045584358721069254590897472454} a^{18} - \frac{7105202920425889913218116028659771334826999187868949721166782960479646724721884654663360313025489}{159039177195634551908618100307857095720319645285293260102652078731036182337434884277018363589889816} a^{17} + \frac{463741523515883126139365444682308613475100551219380125410704490619174651281425207333144212310560}{19879897149454318988577262538482136965039955660661657512831509841379522792179360534627295448736227} a^{16} - \frac{2365456947696379675495892128992754473817674743639092358610283061464258484321663318041630309591979}{19879897149454318988577262538482136965039955660661657512831509841379522792179360534627295448736227} a^{15} - \frac{9383597925424547328197297971229420941619252512113417369058497457713692540345307775575488629511013}{39759794298908637977154525076964273930079911321323315025663019682759045584358721069254590897472454} a^{14} - \frac{1658612128834792860454747266892803340117491205429015415922055678263548505082158700864073671659085}{19879897149454318988577262538482136965039955660661657512831509841379522792179360534627295448736227} a^{13} + \frac{7255322220879563754709233727602531402009026723892756804857830159986519734555562770705192239473017}{39759794298908637977154525076964273930079911321323315025663019682759045584358721069254590897472454} a^{12} - \frac{2158578865514453869422234019773810362369719595625517422335588691883426846640105635397376980382573}{19879897149454318988577262538482136965039955660661657512831509841379522792179360534627295448736227} a^{11} - \frac{3315192010672186719576354184784354403850671559277443303712286633673213538714483201553801122468325}{19879897149454318988577262538482136965039955660661657512831509841379522792179360534627295448736227} a^{10} - \frac{3929642659109611104178302123917493280769157499902007625769657063118623983974379918956211283397545}{159039177195634551908618100307857095720319645285293260102652078731036182337434884277018363589889816} a^{9} - \frac{20995527420123883053968473080598459001400194629688308208026063645721155885552255053537865586080591}{159039177195634551908618100307857095720319645285293260102652078731036182337434884277018363589889816} a^{8} - \frac{8344435201433647723298686413595884641037109561538020069975593098839917874426086407795704047399725}{19879897149454318988577262538482136965039955660661657512831509841379522792179360534627295448736227} a^{7} - \frac{10710581842373567764381682390891754463930452257874931207453896473321686210450309356461656381818659}{159039177195634551908618100307857095720319645285293260102652078731036182337434884277018363589889816} a^{6} + \frac{6550623378168585978158533972008667232955771901622169250661227428685722035584466798521313929751635}{39759794298908637977154525076964273930079911321323315025663019682759045584358721069254590897472454} a^{5} - \frac{1220061551374822800597234085007396855321672997335082716768352843569664181053583547108481229809971}{39759794298908637977154525076964273930079911321323315025663019682759045584358721069254590897472454} a^{4} + \frac{9252542526285565826385356403550579268887857640410789242966387486232491480667115079224906275840561}{159039177195634551908618100307857095720319645285293260102652078731036182337434884277018363589889816} a^{3} - \frac{6081163055546867666708271089699175973091457606599686118397252346970529393239318676896622907270394}{19879897149454318988577262538482136965039955660661657512831509841379522792179360534627295448736227} a^{2} + \frac{18115039933649833370961017216850979149666627438219518124010002926750691054287166449431807473404931}{39759794298908637977154525076964273930079911321323315025663019682759045584358721069254590897472454} a + \frac{9188932686405926692690424032310568488651917637156429048046727169007844795062159785120673413224952}{19879897149454318988577262538482136965039955660661657512831509841379522792179360534627295448736227}$

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

Class group and class number

$C_{3}\times C_{6}\times C_{18}\times C_{612}$, which has order $198288$ (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:  \( 20400053551.79016 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_{24}$ (as 24T1):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A cyclic group of order 24
The 24 conjugacy class representatives for $C_{24}$
Character table for $C_{24}$ is not computed

Intermediate fields

\(\Q(\sqrt{17}) \), 3.3.169.1, 4.4.830297.1, 6.6.140320193.1, 8.0.1980626399884457.1, 12.12.16348345805451893172953.1

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type ${\href{/LocalNumberField/2.3.0.1}{3} }^{8}$ $24$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{3}$ $24$ $24$ R R ${\href{/LocalNumberField/19.3.0.1}{3} }^{8}$ $24$ $24$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.3.0.1}{3} }^{8}$

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
$13$13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.1$x^{12} - 13$$12$$1$$11$$C_{12}$$[\ ]_{12}$
17Data not computed