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Properties

Label 24.0.24432957665...8177.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{12}\cdot 13^{16}\cdot 17^{21}$
Root discriminant $114.24$
Ramified primes $3, 13, 17$
Class number $251618$ (GRH)
Class group $[251618]$ (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![273883429, -2095819, 117512811, -66533111, -41535792, -2941847, 9837385, 18607066, 12801350, 8571783, 5137939, 2210291, 177954, -581885, 28113, 31732, 8894, 4630, -2197, 129, 81, 1, 6, -5, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 5*x^23 + 6*x^22 + x^21 + 81*x^20 + 129*x^19 - 2197*x^18 + 4630*x^17 + 8894*x^16 + 31732*x^15 + 28113*x^14 - 581885*x^13 + 177954*x^12 + 2210291*x^11 + 5137939*x^10 + 8571783*x^9 + 12801350*x^8 + 18607066*x^7 + 9837385*x^6 - 2941847*x^5 - 41535792*x^4 - 66533111*x^3 + 117512811*x^2 - 2095819*x + 273883429)
 
gp: K = bnfinit(x^24 - 5*x^23 + 6*x^22 + x^21 + 81*x^20 + 129*x^19 - 2197*x^18 + 4630*x^17 + 8894*x^16 + 31732*x^15 + 28113*x^14 - 581885*x^13 + 177954*x^12 + 2210291*x^11 + 5137939*x^10 + 8571783*x^9 + 12801350*x^8 + 18607066*x^7 + 9837385*x^6 - 2941847*x^5 - 41535792*x^4 - 66533111*x^3 + 117512811*x^2 - 2095819*x + 273883429, 1)
 

Normalized defining polynomial

\( x^{24} - 5 x^{23} + 6 x^{22} + x^{21} + 81 x^{20} + 129 x^{19} - 2197 x^{18} + 4630 x^{17} + 8894 x^{16} + 31732 x^{15} + 28113 x^{14} - 581885 x^{13} + 177954 x^{12} + 2210291 x^{11} + 5137939 x^{10} + 8571783 x^{9} + 12801350 x^{8} + 18607066 x^{7} + 9837385 x^{6} - 2941847 x^{5} - 41535792 x^{4} - 66533111 x^{3} + 117512811 x^{2} - 2095819 x + 273883429 \)

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:  \(24432957665017051467008966646668052403141068738177=3^{12}\cdot 13^{16}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $114.24$
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:  $3, 13, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(663=3\cdot 13\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{663}(256,·)$, $\chi_{663}(1,·)$, $\chi_{663}(263,·)$, $\chi_{663}(523,·)$, $\chi_{663}(16,·)$, $\chi_{663}(536,·)$, $\chi_{663}(217,·)$, $\chi_{663}(412,·)$, $\chi_{663}(157,·)$, $\chi_{663}(542,·)$, $\chi_{663}(287,·)$, $\chi_{663}(230,·)$, $\chi_{663}(614,·)$, $\chi_{663}(620,·)$, $\chi_{663}(365,·)$, $\chi_{663}(625,·)$, $\chi_{663}(562,·)$, $\chi_{663}(308,·)$, $\chi_{663}(53,·)$, $\chi_{663}(118,·)$, $\chi_{663}(55,·)$, $\chi_{663}(185,·)$, $\chi_{663}(638,·)$, $\chi_{663}(373,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{47} a^{18} - \frac{18}{47} a^{17} + \frac{8}{47} a^{16} + \frac{8}{47} a^{15} + \frac{15}{47} a^{14} + \frac{5}{47} a^{13} + \frac{10}{47} a^{12} + \frac{7}{47} a^{11} - \frac{7}{47} a^{10} + \frac{8}{47} a^{9} - \frac{17}{47} a^{8} - \frac{10}{47} a^{7} - \frac{14}{47} a^{6} - \frac{14}{47} a^{5} + \frac{17}{47} a^{4} - \frac{16}{47} a^{3} - \frac{21}{47} a^{2} + \frac{22}{47} a$, $\frac{1}{47} a^{19} + \frac{13}{47} a^{17} + \frac{11}{47} a^{16} + \frac{18}{47} a^{15} - \frac{7}{47} a^{14} + \frac{6}{47} a^{13} - \frac{1}{47} a^{12} - \frac{22}{47} a^{11} + \frac{23}{47} a^{10} - \frac{14}{47} a^{9} + \frac{13}{47} a^{8} - \frac{6}{47} a^{7} + \frac{16}{47} a^{6} + \frac{8}{47} a^{4} + \frac{20}{47} a^{3} + \frac{20}{47} a^{2} + \frac{20}{47} a$, $\frac{1}{47} a^{20} + \frac{10}{47} a^{17} + \frac{8}{47} a^{16} - \frac{17}{47} a^{15} - \frac{1}{47} a^{14} - \frac{19}{47} a^{13} - \frac{11}{47} a^{12} - \frac{21}{47} a^{11} - \frac{17}{47} a^{10} + \frac{3}{47} a^{9} - \frac{20}{47} a^{8} + \frac{5}{47} a^{7} - \frac{6}{47} a^{6} + \frac{2}{47} a^{5} - \frac{13}{47} a^{4} - \frac{7}{47} a^{3} + \frac{11}{47} a^{2} - \frac{4}{47} a$, $\frac{1}{82292159} a^{21} - \frac{705620}{82292159} a^{20} - \frac{305860}{82292159} a^{19} - \frac{3210}{82292159} a^{18} - \frac{5110269}{82292159} a^{17} - \frac{1033870}{82292159} a^{16} + \frac{20645656}{82292159} a^{15} + \frac{2648203}{82292159} a^{14} - \frac{13753051}{82292159} a^{13} - \frac{36991763}{82292159} a^{12} - \frac{257301}{798953} a^{11} + \frac{13735690}{82292159} a^{10} - \frac{40407397}{82292159} a^{9} - \frac{3956505}{82292159} a^{8} + \frac{18240276}{82292159} a^{7} - \frac{20347767}{82292159} a^{6} + \frac{12835958}{82292159} a^{5} - \frac{24992843}{82292159} a^{4} - \frac{11130373}{82292159} a^{3} - \frac{27032974}{82292159} a^{2} + \frac{4848019}{82292159} a - \frac{636268}{1750897}$, $\frac{1}{82292159} a^{22} - \frac{812164}{82292159} a^{20} - \frac{119499}{82292159} a^{19} + \frac{762940}{82292159} a^{18} - \frac{20717206}{82292159} a^{17} - \frac{10230355}{82292159} a^{16} + \frac{3876720}{82292159} a^{15} + \frac{477729}{1750897} a^{14} + \frac{35863053}{82292159} a^{13} + \frac{23816370}{82292159} a^{12} + \frac{32989789}{82292159} a^{11} - \frac{26710771}{82292159} a^{10} + \frac{9415700}{82292159} a^{9} + \frac{9936118}{82292159} a^{8} - \frac{36329960}{82292159} a^{7} + \frac{602492}{82292159} a^{6} - \frac{3186960}{82292159} a^{5} + \frac{2545732}{82292159} a^{4} - \frac{31253210}{82292159} a^{3} - \frac{5476503}{82292159} a^{2} + \frac{19501125}{82292159} a - \frac{168317}{1750897}$, $\frac{1}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{23} - \frac{139256218117366257383257325502647215226661614715340638633449842971643495014894910}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{22} - \frac{118686914002472808031612192465736081408491230361060839375333658742617027771327337}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{21} - \frac{245265515785107572398232930717127156174216203771631052204941922671508614514849789451559}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{20} - \frac{62059962824543545505609198462772351951893754234689598861498669546599530798535346549961}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{19} - \frac{258976324226568322908621139130889590182614980726290485573408334309288917305772702172464}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{18} + \frac{5859826776129025220661920007960630382934200760186703158412473580720660652480713560506202}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{17} - \frac{11510750525249563691550451796174510479286647564940248258107119600507070271740871143361230}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{16} - \frac{3246947191833445246233737451819250358749771590110599349402683697895793348149405508704230}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{15} + \frac{1326175972763759298685652370320082008645839420785981561422091411445103909881014599590283}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{14} + \frac{10248591371799600346467774683925598712735281109157029230890172371051145658375838671447359}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{13} - \frac{14189686805286537961390714154759668363639745807547450246610623691355214776803643603005784}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{12} - \frac{752889530883414910473053341646325567564946873309373323005014775518187331237683115516918}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{11} + \frac{10083553654672551948051140459144304687622352578745532598268189796748360017589099547194225}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{10} + \frac{7666660134808117802770641452411386236488194616758833276112088911976767179421166746437296}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{9} - \frac{606472652751164533368661495975725928195831396961971059078127980864388400734286588909988}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{8} - \frac{15721026474761991161648901496483268229049130895859204486415260356715421226795410907795994}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{7} - \frac{15989526345195006195437737652952539434812740140332834120739567959227720341402585640084268}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{6} - \frac{274099077854930388548383016247574358946599347108624435030702839185508106502431428069748}{701455123536730492987596882736433603230291255955882589245624039215400220224742132741623} a^{5} - \frac{14256405446958957631571964384477968896106944315918373603564388973718211554529861786249151}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{4} + \frac{15207353861593280632741141972185691840570488510568826387707886140886391558031098260036566}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{3} - \frac{7050854001482076411195356001839751455362529732565278682047344111337696400772928274453301}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a^{2} + \frac{6963025190916688154053800058137590731630100466256798648125291150024486813921227741777256}{32968390806226333170417053488612379351823689029926481694544329843123810350562880238856281} a + \frac{1386387750820766348980250519173431568159865779220265310019070393408729789776381440769}{6810243917832334883374727016858578672138750057824102808209942128304856507036331385841}$

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

Class group and class number

$C_{251618}$, which has order $251618$ (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:  \( 2336441224.9148784 \) (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, 4.4.4913.1, 6.6.140320193.1, 8.0.33237432513.1, 12.12.96735773996756764337.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/5.8.0.1}{8} }^{3}$ $24$ $24$ R R ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ $24$ $24$ ${\href{/LocalNumberField/31.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
3Data not computed
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
17Data not computed