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Properties

Label 24.24.1577886763...8912.1
Degree $24$
Signature $[24, 0]$
Discriminant $2^{36}\cdot 7^{16}\cdot 17^{21}$
Root discriminant $123.48$
Ramified primes $2, 7, 17$
Class number $2$ (GRH)
Class group $[2]$ (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![4464473, 38091612, 59502960, -197885496, -379624562, 505329876, 753516480, -773906888, -603277524, 595179728, 235875554, -245393056, -46687896, 58030784, 4000252, -8131192, 54533, 677528, -37878, -32480, 2884, 812, -90, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^23 - 90*x^22 + 812*x^21 + 2884*x^20 - 32480*x^19 - 37878*x^18 + 677528*x^17 + 54533*x^16 - 8131192*x^15 + 4000252*x^14 + 58030784*x^13 - 46687896*x^12 - 245393056*x^11 + 235875554*x^10 + 595179728*x^9 - 603277524*x^8 - 773906888*x^7 + 753516480*x^6 + 505329876*x^5 - 379624562*x^4 - 197885496*x^3 + 59502960*x^2 + 38091612*x + 4464473)
 
gp: K = bnfinit(x^24 - 8*x^23 - 90*x^22 + 812*x^21 + 2884*x^20 - 32480*x^19 - 37878*x^18 + 677528*x^17 + 54533*x^16 - 8131192*x^15 + 4000252*x^14 + 58030784*x^13 - 46687896*x^12 - 245393056*x^11 + 235875554*x^10 + 595179728*x^9 - 603277524*x^8 - 773906888*x^7 + 753516480*x^6 + 505329876*x^5 - 379624562*x^4 - 197885496*x^3 + 59502960*x^2 + 38091612*x + 4464473, 1)
 

Normalized defining polynomial

\( x^{24} - 8 x^{23} - 90 x^{22} + 812 x^{21} + 2884 x^{20} - 32480 x^{19} - 37878 x^{18} + 677528 x^{17} + 54533 x^{16} - 8131192 x^{15} + 4000252 x^{14} + 58030784 x^{13} - 46687896 x^{12} - 245393056 x^{11} + 235875554 x^{10} + 595179728 x^{9} - 603277524 x^{8} - 773906888 x^{7} + 753516480 x^{6} + 505329876 x^{5} - 379624562 x^{4} - 197885496 x^{3} + 59502960 x^{2} + 38091612 x + 4464473 \)

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

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[24, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(157788676378110831653078238647024412255068893478912=2^{36}\cdot 7^{16}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.48$
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:  $2, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(952=2^{3}\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{952}(1,·)$, $\chi_{952}(389,·)$, $\chi_{952}(897,·)$, $\chi_{952}(905,·)$, $\chi_{952}(501,·)$, $\chi_{952}(81,·)$, $\chi_{952}(597,·)$, $\chi_{952}(625,·)$, $\chi_{952}(169,·)$, $\chi_{952}(361,·)$, $\chi_{952}(93,·)$, $\chi_{952}(869,·)$, $\chi_{952}(225,·)$, $\chi_{952}(933,·)$, $\chi_{952}(849,·)$, $\chi_{952}(681,·)$, $\chi_{952}(485,·)$, $\chi_{952}(365,·)$, $\chi_{952}(893,·)$, $\chi_{952}(305,·)$, $\chi_{952}(53,·)$, $\chi_{952}(137,·)$, $\chi_{952}(253,·)$, $\chi_{952}(757,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{52} a^{12} - \frac{3}{13} a^{11} - \frac{3}{13} a^{10} - \frac{3}{13} a^{9} + \frac{1}{52} a^{8} - \frac{3}{13} a^{7} - \frac{3}{13} a^{6} - \frac{3}{13} a^{5} - \frac{3}{13} a^{4} - \frac{3}{13} a^{3} + \frac{7}{26} a^{2} - \frac{3}{13} a + \frac{1}{4}$, $\frac{1}{52} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{52} a$, $\frac{1}{52} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{4} - \frac{1}{52} a^{2} - \frac{1}{2}$, $\frac{1}{52} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{5} - \frac{1}{52} a^{3} - \frac{1}{2} a$, $\frac{1}{52} a^{16} - \frac{1}{4} a^{8} - \frac{1}{52} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{52} a^{17} - \frac{1}{4} a^{9} - \frac{1}{52} a^{5} - \frac{1}{2} a^{3} + \frac{1}{4} a$, $\frac{1}{416} a^{18} + \frac{1}{208} a^{17} + \frac{3}{416} a^{16} - \frac{1}{208} a^{15} - \frac{3}{416} a^{14} + \frac{1}{208} a^{13} - \frac{1}{16} a^{11} - \frac{1}{8} a^{9} + \frac{7}{32} a^{8} - \frac{1}{8} a^{7} - \frac{53}{416} a^{6} + \frac{25}{208} a^{5} + \frac{75}{416} a^{4} - \frac{103}{208} a^{3} - \frac{57}{208} a^{2} - \frac{33}{104} a - \frac{11}{32}$, $\frac{1}{416} a^{19} - \frac{1}{416} a^{17} + \frac{1}{416} a^{15} - \frac{1}{104} a^{13} - \frac{1}{208} a^{12} - \frac{7}{104} a^{11} - \frac{7}{104} a^{10} - \frac{93}{416} a^{9} + \frac{51}{208} a^{8} - \frac{29}{416} a^{7} + \frac{19}{104} a^{6} - \frac{105}{416} a^{5} + \frac{45}{104} a^{4} - \frac{99}{208} a^{3} + \frac{3}{52} a^{2} - \frac{167}{416} a + \frac{3}{16}$, $\frac{1}{5408} a^{20} - \frac{5}{5408} a^{19} - \frac{1}{2704} a^{18} - \frac{5}{5408} a^{17} + \frac{19}{2704} a^{16} - \frac{51}{5408} a^{15} + \frac{3}{416} a^{14} + \frac{5}{676} a^{13} + \frac{7}{2704} a^{12} - \frac{539}{2704} a^{11} - \frac{233}{5408} a^{10} + \frac{235}{5408} a^{9} + \frac{37}{208} a^{8} + \frac{721}{5408} a^{7} - \frac{25}{338} a^{6} + \frac{1527}{5408} a^{5} - \frac{1805}{5408} a^{4} + \frac{37}{104} a^{3} + \frac{235}{5408} a^{2} + \frac{73}{416} a + \frac{9}{32}$, $\frac{1}{70304} a^{21} + \frac{1}{17576} a^{20} - \frac{15}{17576} a^{19} + \frac{81}{70304} a^{18} + \frac{55}{35152} a^{17} + \frac{395}{70304} a^{16} - \frac{433}{70304} a^{15} + \frac{495}{70304} a^{14} + \frac{213}{35152} a^{13} + \frac{57}{35152} a^{12} + \frac{15805}{70304} a^{11} - \frac{4285}{35152} a^{10} - \frac{3421}{35152} a^{9} - \frac{371}{70304} a^{8} + \frac{3753}{35152} a^{7} + \frac{3283}{70304} a^{6} + \frac{16943}{70304} a^{5} + \frac{8611}{70304} a^{4} + \frac{34685}{70304} a^{3} + \frac{653}{4394} a^{2} + \frac{1749}{5408} a - \frac{29}{416}$, $\frac{1}{831502988767598460364035296} a^{22} + \frac{5045942465289403347691}{831502988767598460364035296} a^{21} - \frac{1141620436132855294833}{415751494383799230182017648} a^{20} - \frac{9685915606981313292269}{51968936797974903772752206} a^{19} - \frac{458769015315260907513975}{415751494383799230182017648} a^{18} + \frac{440605839472936391851443}{103937873595949807545504412} a^{17} + \frac{3428574851705474394711739}{831502988767598460364035296} a^{16} + \frac{5163724561420057796862021}{831502988767598460364035296} a^{15} + \frac{3525313743732512977176133}{415751494383799230182017648} a^{14} + \frac{608004099828343159453615}{415751494383799230182017648} a^{13} + \frac{5092562136996770421230165}{831502988767598460364035296} a^{12} - \frac{57752415819434764826241417}{831502988767598460364035296} a^{11} - \frac{78813595350346735055672477}{415751494383799230182017648} a^{10} - \frac{4943798862874262856115935}{103937873595949807545504412} a^{9} - \frac{12686670579227586519978691}{415751494383799230182017648} a^{8} - \frac{45554760037716316847716717}{415751494383799230182017648} a^{7} - \frac{169591320013783515392748717}{831502988767598460364035296} a^{6} + \frac{192562577464286250269297543}{831502988767598460364035296} a^{5} - \frac{74727762327134607446633805}{831502988767598460364035296} a^{4} - \frac{4706958203319256143843957}{63961768366738343104925792} a^{3} + \frac{163337006112318155633514593}{831502988767598460364035296} a^{2} + \frac{5382406301307845411687945}{63961768366738343104925792} a + \frac{441806104804038039413485}{2460068014105320888650992}$, $\frac{1}{1759255033627838520978820949297735264} a^{23} + \frac{977605671}{1759255033627838520978820949297735264} a^{22} + \frac{1387712006653516991122752841}{33831827569766125403438864409571832} a^{21} - \frac{22918513084326571274117417932257}{1759255033627838520978820949297735264} a^{20} + \frac{340797175052484828209528920101007}{1759255033627838520978820949297735264} a^{19} - \frac{202764196529211859512661693328093}{439813758406959630244705237324433816} a^{18} + \frac{975663812658364540098992686523083}{439813758406959630244705237324433816} a^{17} + \frac{401936990508794789510888623445901}{135327310279064501613755457638287328} a^{16} + \frac{946700136079854060200414063062083}{135327310279064501613755457638287328} a^{15} + \frac{13947156848950953260052840303503437}{1759255033627838520978820949297735264} a^{14} + \frac{56461995971818417832627012448703}{7360899722292211384848623218818976} a^{13} - \frac{3314694346738105588349964264953479}{1759255033627838520978820949297735264} a^{12} + \frac{207668846254403247030968276102161517}{879627516813919260489410474648867632} a^{11} + \frac{28709398778662514513251249645733553}{135327310279064501613755457638287328} a^{10} - \frac{61433054846124992687311853551266157}{1759255033627838520978820949297735264} a^{9} + \frac{67703649375244313494491196693228419}{879627516813919260489410474648867632} a^{8} + \frac{214801653877926077528689570135194359}{879627516813919260489410474648867632} a^{7} + \frac{72622905017303081762992835861260857}{1759255033627838520978820949297735264} a^{6} - \frac{414551982836392799147433420646038497}{879627516813919260489410474648867632} a^{5} + \frac{395495025596146186382648230372310811}{879627516813919260489410474648867632} a^{4} - \frac{607618742185955406490425453284381173}{1759255033627838520978820949297735264} a^{3} + \frac{68023778714135805352536379427079471}{879627516813919260489410474648867632} a^{2} - \frac{32371915076962355092052001078832505}{135327310279064501613755457638287328} a - \frac{2663314568902931671531310296087243}{10409793098389577047211958279868256}$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  $23$
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:  \( 613781865761664800 \) (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

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), 4.4.4913.1, 6.6.11796113.1, 8.8.1680747204608.1, 12.12.683635509017782097.1

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 $24$ $24$ R $24$ ${\href{/LocalNumberField/13.1.0.1}{1} }^{24}$ R ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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$2.12.18.28$x^{12} - 52 x^{10} + 1100 x^{8} - 12000 x^{6} - 61072 x^{4} + 62144 x^{2} - 62144$$2$$6$$18$$C_{12}$$[3]^{6}$
2.12.18.28$x^{12} - 52 x^{10} + 1100 x^{8} - 12000 x^{6} - 61072 x^{4} + 62144 x^{2} - 62144$$2$$6$$18$$C_{12}$$[3]^{6}$
7Data not computed
17Data not computed