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Properties

Label 24.0.15778867637...8912.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{36}\cdot 7^{16}\cdot 17^{21}$
Root discriminant $123.48$
Ramified primes $2, 7, 17$
Class number $21734244$ (GRH)
Class group $[4662, 4662]$ (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![5865361489, -9245671084, 12232348200, -4797698440, 3220405394, -1615397140, 1736251528, -1515252680, 1174873040, -739565888, 444585438, -222385392, 108451452, -43557136, 17355316, -5644568, 1873125, -486360, 129606, -24864, 5264, -684, 114, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^23 + 114*x^22 - 684*x^21 + 5264*x^20 - 24864*x^19 + 129606*x^18 - 486360*x^17 + 1873125*x^16 - 5644568*x^15 + 17355316*x^14 - 43557136*x^13 + 108451452*x^12 - 222385392*x^11 + 444585438*x^10 - 739565888*x^9 + 1174873040*x^8 - 1515252680*x^7 + 1736251528*x^6 - 1615397140*x^5 + 3220405394*x^4 - 4797698440*x^3 + 12232348200*x^2 - 9245671084*x + 5865361489)
 
gp: K = bnfinit(x^24 - 8*x^23 + 114*x^22 - 684*x^21 + 5264*x^20 - 24864*x^19 + 129606*x^18 - 486360*x^17 + 1873125*x^16 - 5644568*x^15 + 17355316*x^14 - 43557136*x^13 + 108451452*x^12 - 222385392*x^11 + 444585438*x^10 - 739565888*x^9 + 1174873040*x^8 - 1515252680*x^7 + 1736251528*x^6 - 1615397140*x^5 + 3220405394*x^4 - 4797698440*x^3 + 12232348200*x^2 - 9245671084*x + 5865361489, 1)
 

Normalized defining polynomial

\( x^{24} - 8 x^{23} + 114 x^{22} - 684 x^{21} + 5264 x^{20} - 24864 x^{19} + 129606 x^{18} - 486360 x^{17} + 1873125 x^{16} - 5644568 x^{15} + 17355316 x^{14} - 43557136 x^{13} + 108451452 x^{12} - 222385392 x^{11} + 444585438 x^{10} - 739565888 x^{9} + 1174873040 x^{8} - 1515252680 x^{7} + 1736251528 x^{6} - 1615397140 x^{5} + 3220405394 x^{4} - 4797698440 x^{3} + 12232348200 x^{2} - 9245671084 x + 5865361489 \)

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:  \(157788676378110831653078238647024412255068893478912=2^{36}\cdot 7^{16}\cdot 17^{21}\)
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$$\chi_{952}(1,·)$, $\chi_{952}(603,·)$, $\chi_{952}(835,·)$, $\chi_{952}(897,·)$, $\chi_{952}(905,·)$, $\chi_{952}(331,·)$, $\chi_{952}(81,·)$, $\chi_{952}(723,·)$, $\chi_{952}(739,·)$, $\chi_{952}(625,·)$, $\chi_{952}(169,·)$, $\chi_{952}(155,·)$, $\chi_{952}(491,·)$, $\chi_{952}(361,·)$, $\chi_{952}(627,·)$, $\chi_{952}(225,·)$, $\chi_{952}(291,·)$, $\chi_{952}(849,·)$, $\chi_{952}(681,·)$, $\chi_{952}(43,·)$, $\chi_{952}(305,·)$, $\chi_{952}(179,·)$, $\chi_{952}(219,·)$, $\chi_{952}(137,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{2} - \frac{1}{2}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{52} a^{12} + \frac{3}{26} a^{11} + \frac{5}{26} a^{10} + \frac{2}{13} a^{9} + \frac{9}{52} a^{8} + \frac{1}{26} a^{7} + \frac{3}{13} a^{6} - \frac{3}{26} a^{5} + \frac{4}{13} a^{4} + \frac{9}{26} a^{3} + \frac{1}{13} a^{2} + \frac{6}{13} a - \frac{1}{4}$, $\frac{1}{52} a^{13} - \frac{1}{4} a^{9} - \frac{1}{2} a^{5} - \frac{1}{52} a$, $\frac{1}{52} a^{14} - \frac{1}{4} a^{10} - \frac{1}{2} a^{4} - \frac{1}{52} a^{2} - \frac{1}{2}$, $\frac{1}{52} a^{15} - \frac{1}{4} a^{11} - \frac{1}{2} a^{5} - \frac{1}{52} a^{3} - \frac{1}{2} a$, $\frac{1}{52} a^{16} - \frac{1}{4} a^{8} - \frac{1}{52} a^{4} - \frac{1}{2} a^{2} + \frac{1}{4}$, $\frac{1}{676} a^{17} + \frac{1}{338} a^{16} - \frac{3}{338} a^{15} + \frac{1}{338} a^{14} - \frac{5}{676} a^{13} - \frac{1}{169} a^{12} + \frac{53}{338} a^{11} + \frac{45}{338} a^{10} + \frac{23}{338} a^{9} - \frac{9}{169} a^{8} - \frac{43}{338} a^{7} + \frac{41}{338} a^{6} + \frac{49}{676} a^{5} + \frac{123}{338} a^{4} + \frac{149}{338} a^{3} - \frac{63}{169} a^{2} - \frac{1}{26} a - \frac{1}{2}$, $\frac{1}{5408} a^{18} + \frac{1}{2704} a^{17} - \frac{45}{5408} a^{16} - \frac{25}{2704} a^{15} + \frac{21}{5408} a^{14} - \frac{15}{2704} a^{13} - \frac{3}{676} a^{12} + \frac{331}{2704} a^{11} + \frac{139}{676} a^{10} - \frac{269}{1352} a^{9} - \frac{749}{5408} a^{8} - \frac{129}{1352} a^{7} - \frac{1173}{5408} a^{6} + \frac{513}{2704} a^{5} + \frac{2651}{5408} a^{4} + \frac{1265}{2704} a^{3} + \frac{95}{208} a^{2} - \frac{1}{104} a + \frac{1}{32}$, $\frac{1}{5408} a^{19} - \frac{1}{5408} a^{17} + \frac{1}{169} a^{16} + \frac{41}{5408} a^{15} + \frac{3}{676} a^{14} + \frac{1}{1352} a^{13} - \frac{1}{2704} a^{12} - \frac{237}{1352} a^{11} + \frac{307}{1352} a^{10} - \frac{549}{5408} a^{9} + \frac{667}{2704} a^{8} + \frac{99}{5408} a^{7} + \frac{267}{1352} a^{6} - \frac{157}{416} a^{5} + \frac{205}{1352} a^{4} - \frac{279}{2704} a^{3} - \frac{15}{338} a^{2} - \frac{11}{416} a - \frac{1}{16}$, $\frac{1}{5408} a^{20} + \frac{1}{2704} a^{17} + \frac{9}{1352} a^{16} - \frac{21}{2704} a^{15} - \frac{3}{416} a^{14} + \frac{3}{676} a^{13} - \frac{3}{1352} a^{12} + \frac{393}{2704} a^{11} + \frac{595}{5408} a^{10} - \frac{659}{2704} a^{9} + \frac{615}{2704} a^{8} - \frac{55}{676} a^{7} - \frac{631}{2704} a^{6} + \frac{347}{2704} a^{5} + \frac{2021}{5408} a^{4} - \frac{87}{2704} a^{3} + \frac{199}{5408} a^{2} + \frac{61}{208} a + \frac{9}{32}$, $\frac{1}{5408} a^{21} - \frac{1}{338} a^{16} + \frac{45}{5408} a^{15} + \frac{11}{2704} a^{14} + \frac{1}{208} a^{12} + \frac{83}{416} a^{11} - \frac{35}{208} a^{10} - \frac{85}{2704} a^{9} - \frac{31}{208} a^{8} + \frac{25}{208} a^{7} + \frac{101}{416} a^{5} + \frac{355}{1352} a^{4} + \frac{1411}{5408} a^{3} + \frac{431}{2704} a^{2} + \frac{141}{416} a - \frac{5}{16}$, $\frac{1}{72454449611427702084772175235591197250208} a^{22} + \frac{3124155993671615768048672178192470101}{72454449611427702084772175235591197250208} a^{21} + \frac{3013904568163650313711844668644975185}{36227224805713851042386087617795598625104} a^{20} - \frac{2365094157853831277454886765480329733}{72454449611427702084772175235591197250208} a^{19} + \frac{563157167751836450700348651431303079}{72454449611427702084772175235591197250208} a^{18} + \frac{10148085660478204295281267053082974991}{72454449611427702084772175235591197250208} a^{17} - \frac{347851209453480223227513257336553823659}{36227224805713851042386087617795598625104} a^{16} + \frac{10213619831090170228443743786379837971}{4528403100714231380298260952224449828138} a^{15} + \frac{181687474357189256859204945542489426435}{72454449611427702084772175235591197250208} a^{14} - \frac{61228387714116754350966744846900538945}{18113612402856925521193043808897799312552} a^{13} - \frac{156631460910435595405895063544012449589}{72454449611427702084772175235591197250208} a^{12} - \frac{2595079246447840553660985696038247675889}{72454449611427702084772175235591197250208} a^{11} - \frac{4569778210444783129235351535278196197059}{36227224805713851042386087617795598625104} a^{10} - \frac{11292537381636091195188548353545544662695}{72454449611427702084772175235591197250208} a^{9} - \frac{4143115530678295967246316274394681432457}{72454449611427702084772175235591197250208} a^{8} - \frac{1312866722941041225166728033743391315977}{72454449611427702084772175235591197250208} a^{7} + \frac{3827962149664970915101434760766122541035}{36227224805713851042386087617795598625104} a^{6} - \frac{39522438044597389598226503246812572321}{2264201550357115690149130476112224914069} a^{5} + \frac{10147241724200891009244919519239244354721}{36227224805713851042386087617795598625104} a^{4} - \frac{24847113432284873654409637149458407065331}{72454449611427702084772175235591197250208} a^{3} - \frac{34787046370206404070630813192211955187553}{72454449611427702084772175235591197250208} a^{2} + \frac{966944292504363915647377826575563406327}{2786709600439527003260468278291969125008} a - \frac{11650843718153296415155810327239596199}{428724553913773385116995119737226019232}$, $\frac{1}{1229595031988211091344818188148849994209634167100717344} a^{23} + \frac{1847407569245}{307398757997052772836204547037212498552408541775179336} a^{22} - \frac{12238241075510307335683274722984642548913807086195}{153699378998526386418102273518606249276204270887589668} a^{21} + \frac{14662850047621624096571271278119612808113533894537}{307398757997052772836204547037212498552408541775179336} a^{20} + \frac{1380109344873489861024170112189413911412914528427}{307398757997052772836204547037212498552408541775179336} a^{19} + \frac{13609395285190611062946702965133177761609512894639}{153699378998526386418102273518606249276204270887589668} a^{18} + \frac{849379349615728523806693395204760816123129185165505}{1229595031988211091344818188148849994209634167100717344} a^{17} - \frac{4141648215169165035057797417190259317995251503081307}{614797515994105545672409094074424997104817083550358672} a^{16} + \frac{107970893232514852473330432438627641857688296995621}{23646058307465597910477272849016346042492964751936872} a^{15} + \frac{3352570561407014212623237936732159827741011643459875}{614797515994105545672409094074424997104817083550358672} a^{14} + \frac{5786365550293054696634037378331412526259519099462847}{1229595031988211091344818188148849994209634167100717344} a^{13} - \frac{2828093226043865786723991432819467137269059689766833}{614797515994105545672409094074424997104817083550358672} a^{12} - \frac{26906425329688974765786914239293832387944053079345609}{614797515994105545672409094074424997104817083550358672} a^{11} + \frac{70458290011770501204819914497479439867894898058723471}{614797515994105545672409094074424997104817083550358672} a^{10} - \frac{128139880320810995764874923481469534266097914556677677}{614797515994105545672409094074424997104817083550358672} a^{9} - \frac{474584309745080123858121571919252616710671377837097}{2955757288433199738809659106127043255311620593992109} a^{8} + \frac{263165639211863344994544682261716135182473251643052725}{1229595031988211091344818188148849994209634167100717344} a^{7} - \frac{5210578385843687354842284900935114882827971982721173}{38424844749631596604525568379651562319051067721897417} a^{6} - \frac{441033059568954306276292488242226037141481551378933553}{1229595031988211091344818188148849994209634167100717344} a^{5} + \frac{41908643499234799163246824953994887348180818160509359}{614797515994105545672409094074424997104817083550358672} a^{4} - \frac{3661042148245959450183107106166814843808143969703543}{1229595031988211091344818188148849994209634167100717344} a^{3} - \frac{47263352056721789195507400561427014172989899848897637}{614797515994105545672409094074424997104817083550358672} a^{2} + \frac{8871104762903702436998549686774276192381332341789635}{23646058307465597910477272849016346042492964751936872} a - \frac{104748199105487875526621943124190923549043757138399}{1818927562112738300805944065308949695576381903995144}$

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

Class group and class number

$C_{4662}\times C_{4662}$, which has order $21734244$ (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:  \( 250243842.68845215 \) (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, 8.0.1680747204608.1, 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 ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/41.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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$2.12.18.27$x^{12} - 156 x^{10} + 9900 x^{8} - 61856 x^{6} + 33904 x^{4} + 27712 x^{2} + 47936$$2$$6$$18$$C_{12}$$[3]^{6}$
2.12.18.27$x^{12} - 156 x^{10} + 9900 x^{8} - 61856 x^{6} + 33904 x^{4} + 27712 x^{2} + 47936$$2$$6$$18$$C_{12}$$[3]^{6}$
7Data not computed
17Data not computed