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Properties

Label 24.0.21479659665...2352.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 3^{32}\cdot 17^{21}$
Root discriminant $103.24$
Ramified primes $2, 3, 17$
Class number $281488$ (GRH)
Class group $[2, 2, 2, 35186]$ (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![49248593, 132575034, 238003584, 165688332, 80638593, -5011404, -22218139, -8457624, 2598465, 2964574, 1143855, -434232, -34809, 30078, 125271, 4036, 13275, -1512, 2970, -36, 405, -8, 27, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 27*x^22 - 8*x^21 + 405*x^20 - 36*x^19 + 2970*x^18 - 1512*x^17 + 13275*x^16 + 4036*x^15 + 125271*x^14 + 30078*x^13 - 34809*x^12 - 434232*x^11 + 1143855*x^10 + 2964574*x^9 + 2598465*x^8 - 8457624*x^7 - 22218139*x^6 - 5011404*x^5 + 80638593*x^4 + 165688332*x^3 + 238003584*x^2 + 132575034*x + 49248593)
 
gp: K = bnfinit(x^24 + 27*x^22 - 8*x^21 + 405*x^20 - 36*x^19 + 2970*x^18 - 1512*x^17 + 13275*x^16 + 4036*x^15 + 125271*x^14 + 30078*x^13 - 34809*x^12 - 434232*x^11 + 1143855*x^10 + 2964574*x^9 + 2598465*x^8 - 8457624*x^7 - 22218139*x^6 - 5011404*x^5 + 80638593*x^4 + 165688332*x^3 + 238003584*x^2 + 132575034*x + 49248593, 1)
 

Normalized defining polynomial

\( x^{24} + 27 x^{22} - 8 x^{21} + 405 x^{20} - 36 x^{19} + 2970 x^{18} - 1512 x^{17} + 13275 x^{16} + 4036 x^{15} + 125271 x^{14} + 30078 x^{13} - 34809 x^{12} - 434232 x^{11} + 1143855 x^{10} + 2964574 x^{9} + 2598465 x^{8} - 8457624 x^{7} - 22218139 x^{6} - 5011404 x^{5} + 80638593 x^{4} + 165688332 x^{3} + 238003584 x^{2} + 132575034 x + 49248593 \)

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:  \(2147965966579880698889178764823700736525155172352=2^{24}\cdot 3^{32}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $103.24$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(612=2^{2}\cdot 3^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{612}(1,·)$, $\chi_{612}(427,·)$, $\chi_{612}(577,·)$, $\chi_{612}(535,·)$, $\chi_{612}(13,·)$, $\chi_{612}(205,·)$, $\chi_{612}(19,·)$, $\chi_{612}(151,·)$, $\chi_{612}(409,·)$, $\chi_{612}(331,·)$, $\chi_{612}(157,·)$, $\chi_{612}(223,·)$, $\chi_{612}(355,·)$, $\chi_{612}(421,·)$, $\chi_{612}(169,·)$, $\chi_{612}(43,·)$, $\chi_{612}(451,·)$, $\chi_{612}(559,·)$, $\chi_{612}(565,·)$, $\chi_{612}(361,·)$, $\chi_{612}(217,·)$, $\chi_{612}(127,·)$, $\chi_{612}(247,·)$, $\chi_{612}(373,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{52} a^{18} - \frac{7}{26} a^{16} - \frac{3}{26} a^{15} - \frac{9}{52} a^{14} - \frac{11}{26} a^{13} - \frac{1}{2} a^{12} - \frac{2}{13} a^{11} + \frac{11}{52} a^{10} + \frac{3}{26} a^{9} - \frac{15}{52} a^{8} - \frac{4}{13} a^{7} - \frac{6}{13} a^{6} + \frac{3}{26} a^{5} - \frac{4}{13} a^{4} + \frac{9}{26} a^{3} + \frac{4}{13} a^{2} + \frac{11}{26} a + \frac{9}{52}$, $\frac{1}{52} a^{19} - \frac{7}{26} a^{17} - \frac{3}{26} a^{16} - \frac{9}{52} a^{15} - \frac{11}{26} a^{14} - \frac{1}{2} a^{13} - \frac{2}{13} a^{12} + \frac{11}{52} a^{11} + \frac{3}{26} a^{10} - \frac{15}{52} a^{9} - \frac{4}{13} a^{8} - \frac{6}{13} a^{7} + \frac{3}{26} a^{6} - \frac{4}{13} a^{5} + \frac{9}{26} a^{4} + \frac{4}{13} a^{3} + \frac{11}{26} a^{2} + \frac{9}{52} a$, $\frac{1}{52} a^{20} - \frac{3}{26} a^{17} + \frac{3}{52} a^{16} - \frac{1}{26} a^{15} + \frac{1}{13} a^{14} - \frac{1}{13} a^{13} + \frac{11}{52} a^{12} - \frac{1}{26} a^{11} - \frac{17}{52} a^{10} + \frac{4}{13} a^{9} - \frac{1}{2} a^{8} - \frac{5}{26} a^{7} + \frac{3}{13} a^{6} - \frac{1}{26} a^{5} + \frac{7}{26} a^{3} + \frac{25}{52} a^{2} - \frac{1}{13} a + \frac{11}{26}$, $\frac{1}{52} a^{21} + \frac{3}{52} a^{17} + \frac{9}{26} a^{16} + \frac{5}{13} a^{15} - \frac{3}{26} a^{14} - \frac{17}{52} a^{13} - \frac{1}{26} a^{12} - \frac{1}{4} a^{11} - \frac{11}{26} a^{10} + \frac{5}{26} a^{9} + \frac{1}{13} a^{8} + \frac{5}{13} a^{7} + \frac{5}{26} a^{6} - \frac{4}{13} a^{5} + \frac{11}{26} a^{4} - \frac{23}{52} a^{3} - \frac{3}{13} a^{2} - \frac{1}{26} a + \frac{1}{26}$, $\frac{1}{3732291653890342221254089038508397128676} a^{22} - \frac{128236120930635763310641146989004313}{1866145826945171110627044519254198564338} a^{21} - \frac{11498303231027580545685057456113331201}{1866145826945171110627044519254198564338} a^{20} - \frac{12291639759031576131851425157335868175}{1866145826945171110627044519254198564338} a^{19} + \frac{5459210566182071056212153268956098603}{933072913472585555313522259627099282169} a^{18} - \frac{33141763198983223927817275379263813973}{933072913472585555313522259627099282169} a^{17} + \frac{354428661753597234409034117802206817752}{933072913472585555313522259627099282169} a^{16} - \frac{756435705082246532292150384257265828901}{1866145826945171110627044519254198564338} a^{15} - \frac{43396587791214507969610304482932114717}{1866145826945171110627044519254198564338} a^{14} + \frac{63853459027308542241030628863360058715}{1866145826945171110627044519254198564338} a^{13} + \frac{195071387462704900375101679878135658939}{3732291653890342221254089038508397128676} a^{12} - \frac{586246737731586920481460144986421549541}{1866145826945171110627044519254198564338} a^{11} - \frac{61569710532177227704557336421131357097}{287099357991564786250314541423722856052} a^{10} + \frac{403793248381577779403480326540651150607}{933072913472585555313522259627099282169} a^{9} + \frac{74647259293293967252055776220554612121}{287099357991564786250314541423722856052} a^{8} - \frac{764782859492472678244154089485095321215}{1866145826945171110627044519254198564338} a^{7} + \frac{342038380323150441591251105052006424918}{933072913472585555313522259627099282169} a^{6} - \frac{155453368724462884087177710948561580937}{933072913472585555313522259627099282169} a^{5} - \frac{1632170513037572629172051717462602682459}{3732291653890342221254089038508397128676} a^{4} - \frac{125683415394128026637330219329744830223}{933072913472585555313522259627099282169} a^{3} - \frac{309539866565109942181155694137449663762}{933072913472585555313522259627099282169} a^{2} + \frac{111099647780405137940815118350582782223}{1866145826945171110627044519254198564338} a - \frac{1190393715054231734357154186465614792331}{3732291653890342221254089038508397128676}$, $\frac{1}{1219215205279691676510909674327540773527184062146297332} a^{23} + \frac{21140994872187}{609607602639845838255454837163770386763592031073148666} a^{22} - \frac{2527009141538791186021612349051168944364972899032705}{1219215205279691676510909674327540773527184062146297332} a^{21} - \frac{1197443942577026385761020962676592656133204728577736}{304803801319922919127727418581885193381796015536574333} a^{20} + \frac{188091942837927573542323922417547898772058229622953}{46892892510757372173496525935674645135660925467165282} a^{19} + \frac{5333439478353570874406194799337384825414532990436085}{1219215205279691676510909674327540773527184062146297332} a^{18} - \frac{460795157269366116833147856978690329842404553579652161}{1219215205279691676510909674327540773527184062146297332} a^{17} + \frac{5731004601847387363297934729437192187490490756204387}{304803801319922919127727418581885193381796015536574333} a^{16} + \frac{95373166937308686612132872055435007524489602775628137}{609607602639845838255454837163770386763592031073148666} a^{15} - \frac{129436271535253676073462037944947163344602139825613825}{1219215205279691676510909674327540773527184062146297332} a^{14} + \frac{97822836712113064919916720121163742044760936337255439}{304803801319922919127727418581885193381796015536574333} a^{13} + \frac{72479598742093027011956599401876513157255820713142666}{304803801319922919127727418581885193381796015536574333} a^{12} - \frac{25574238578179967179809360226919392434191814787667423}{304803801319922919127727418581885193381796015536574333} a^{11} + \frac{481057654622924922312358724025341360884488297085669607}{1219215205279691676510909674327540773527184062146297332} a^{10} - \frac{562951868755465660065459344833220812855873208583023573}{1219215205279691676510909674327540773527184062146297332} a^{9} + \frac{353221619200223991051909461655428283118065650689977379}{1219215205279691676510909674327540773527184062146297332} a^{8} + \frac{47492770296189952600234931035832896065849550127423132}{304803801319922919127727418581885193381796015536574333} a^{7} + \frac{80786583587856023645847577305445564133935038158104991}{609607602639845838255454837163770386763592031073148666} a^{6} + \frac{3490031769032477897336713299590330967046994033476523}{93785785021514744346993051871349290271321850934330564} a^{5} - \frac{235114690598439356185378953839841144572832985732742939}{609607602639845838255454837163770386763592031073148666} a^{4} - \frac{325518293735583988735044555084588048098774094266973405}{1219215205279691676510909674327540773527184062146297332} a^{3} - \frac{66884770685767291112061813256822672461950721430304675}{304803801319922919127727418581885193381796015536574333} a^{2} + \frac{174640551265069728549586391895149461570616245724154603}{1219215205279691676510909674327540773527184062146297332} a + \frac{119489447071344840435770745480957738031978213219201327}{1219215205279691676510909674327540773527184062146297332}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{2}\times C_{35186}$, which has order $281488$ (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:  \( 363126034.10126877 \) (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

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{9})^+\), 4.4.4913.1, 6.6.32234193.1, 8.0.105046700288.1, 12.12.5104819233548816337.1

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 R R $24$ $24$ $24$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{6}$ $24$ $24$ $24$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{3}$ $24$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
3Data not computed
$17$17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.3$x^{8} - 17$$8$$1$$7$$C_8$$[\ ]_{8}$