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Properties

Label 24.0.17490945783...5472.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{93}\cdot 3^{12}\cdot 7^{16}$
Root discriminant $92.99$
Ramified primes $2, 3, 7$
Class number $1419282$ (GRH)
Class group $[9, 157698]$ (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![1528760449, -2534548816, 3391259336, -1380779704, 906259624, -417111304, 455672172, -416648024, 335285658, -220741648, 136971996, -71363632, 35924714, -15129912, 6251172, -2139056, 733607, -203128, 57616, -12544, 2898, -464, 84, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^23 + 84*x^22 - 464*x^21 + 2898*x^20 - 12544*x^19 + 57616*x^18 - 203128*x^17 + 733607*x^16 - 2139056*x^15 + 6251172*x^14 - 15129912*x^13 + 35924714*x^12 - 71363632*x^11 + 136971996*x^10 - 220741648*x^9 + 335285658*x^8 - 416648024*x^7 + 455672172*x^6 - 417111304*x^5 + 906259624*x^4 - 1380779704*x^3 + 3391259336*x^2 - 2534548816*x + 1528760449)
 
gp: K = bnfinit(x^24 - 8*x^23 + 84*x^22 - 464*x^21 + 2898*x^20 - 12544*x^19 + 57616*x^18 - 203128*x^17 + 733607*x^16 - 2139056*x^15 + 6251172*x^14 - 15129912*x^13 + 35924714*x^12 - 71363632*x^11 + 136971996*x^10 - 220741648*x^9 + 335285658*x^8 - 416648024*x^7 + 455672172*x^6 - 417111304*x^5 + 906259624*x^4 - 1380779704*x^3 + 3391259336*x^2 - 2534548816*x + 1528760449, 1)
 

Normalized defining polynomial

\( x^{24} - 8 x^{23} + 84 x^{22} - 464 x^{21} + 2898 x^{20} - 12544 x^{19} + 57616 x^{18} - 203128 x^{17} + 733607 x^{16} - 2139056 x^{15} + 6251172 x^{14} - 15129912 x^{13} + 35924714 x^{12} - 71363632 x^{11} + 136971996 x^{10} - 220741648 x^{9} + 335285658 x^{8} - 416648024 x^{7} + 455672172 x^{6} - 417111304 x^{5} + 906259624 x^{4} - 1380779704 x^{3} + 3391259336 x^{2} - 2534548816 x + 1528760449 \)

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:  \(174909457836898599788885373561654160049383145472=2^{93}\cdot 3^{12}\cdot 7^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $92.99$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(672=2^{5}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{672}(1,·)$, $\chi_{672}(389,·)$, $\chi_{672}(193,·)$, $\chi_{672}(457,·)$, $\chi_{672}(653,·)$, $\chi_{672}(365,·)$, $\chi_{672}(529,·)$, $\chi_{672}(533,·)$, $\chi_{672}(121,·)$, $\chi_{672}(25,·)$, $\chi_{672}(29,·)$, $\chi_{672}(197,·)$, $\chi_{672}(289,·)$, $\chi_{672}(485,·)$, $\chi_{672}(337,·)$, $\chi_{672}(169,·)$, $\chi_{672}(557,·)$, $\chi_{672}(221,·)$, $\chi_{672}(625,·)$, $\chi_{672}(53,·)$, $\chi_{672}(361,·)$, $\chi_{672}(505,·)$, $\chi_{672}(317,·)$, $\chi_{672}(149,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6} + \frac{1}{3} a^{5} - \frac{1}{3} a^{2} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{7} - \frac{1}{3} a^{5} - \frac{1}{3} a^{3} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{8} + \frac{1}{3} a^{5} - \frac{1}{3} a^{4} + \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} - \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{3} a^{10} - \frac{1}{3} a^{5} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} - \frac{1}{3}$, $\frac{1}{3} a^{11} - \frac{1}{3} a^{5} - \frac{1}{3} a^{4} - \frac{1}{3} a^{2} + \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{9} a^{12} - \frac{1}{9} a^{11} + \frac{1}{9} a^{10} + \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{4}{9} a^{5} + \frac{1}{9} a^{4} - \frac{1}{9} a^{3} - \frac{4}{9} a^{2} + \frac{4}{9} a - \frac{2}{9}$, $\frac{1}{9} a^{13} + \frac{1}{9} a^{10} + \frac{1}{9} a^{9} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} + \frac{4}{9} a^{3} - \frac{1}{3} a^{2} - \frac{1}{9} a + \frac{1}{9}$, $\frac{1}{9} a^{14} + \frac{1}{9} a^{11} + \frac{1}{9} a^{10} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} + \frac{4}{9} a^{4} - \frac{1}{3} a^{3} - \frac{1}{9} a^{2} + \frac{1}{9} a$, $\frac{1}{9} a^{15} - \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{4}{9} a^{5} - \frac{4}{9} a^{4} + \frac{2}{9} a^{2} - \frac{4}{9} a + \frac{2}{9}$, $\frac{1}{9} a^{16} + \frac{1}{9} a^{11} + \frac{1}{9} a^{9} - \frac{1}{9} a^{8} - \frac{1}{9} a^{7} - \frac{1}{9} a^{6} - \frac{2}{9} a^{5} + \frac{1}{9} a^{4} + \frac{1}{9} a^{3} + \frac{1}{9} a^{2} + \frac{1}{3} a + \frac{1}{9}$, $\frac{1}{9} a^{17} + \frac{1}{9} a^{11} - \frac{1}{9} a^{9} + \frac{1}{9} a^{8} + \frac{1}{9} a^{6} + \frac{2}{9} a^{5} - \frac{1}{3} a^{4} + \frac{2}{9} a^{3} - \frac{2}{9} a^{2} - \frac{1}{3} a - \frac{1}{9}$, $\frac{1}{27} a^{18} + \frac{1}{27} a^{15} + \frac{1}{9} a^{11} - \frac{1}{9} a^{10} - \frac{1}{9} a^{8} + \frac{1}{9} a^{7} - \frac{4}{27} a^{6} + \frac{1}{9} a^{5} - \frac{1}{9} a^{4} - \frac{1}{27} a^{3} - \frac{2}{9} a^{2} - \frac{1}{9} a - \frac{2}{27}$, $\frac{1}{27} a^{19} + \frac{1}{27} a^{16} - \frac{1}{9} a^{10} - \frac{1}{9} a^{9} - \frac{1}{27} a^{7} + \frac{1}{9} a^{6} + \frac{1}{3} a^{5} - \frac{4}{27} a^{4} - \frac{1}{9} a^{3} + \frac{1}{3} a^{2} + \frac{13}{27} a + \frac{2}{9}$, $\frac{1}{2619} a^{20} - \frac{29}{2619} a^{19} + \frac{31}{2619} a^{18} + \frac{43}{2619} a^{17} + \frac{58}{2619} a^{16} - \frac{134}{2619} a^{15} + \frac{4}{97} a^{14} + \frac{7}{291} a^{13} - \frac{5}{873} a^{12} + \frac{37}{873} a^{11} + \frac{125}{873} a^{10} - \frac{15}{97} a^{9} + \frac{158}{2619} a^{8} + \frac{266}{2619} a^{7} - \frac{229}{2619} a^{6} + \frac{197}{2619} a^{5} + \frac{662}{2619} a^{4} + \frac{692}{2619} a^{3} - \frac{1196}{2619} a^{2} - \frac{494}{2619} a - \frac{2}{27}$, $\frac{1}{2619} a^{21} - \frac{34}{2619} a^{19} - \frac{28}{2619} a^{18} + \frac{47}{873} a^{17} - \frac{4}{2619} a^{16} - \frac{92}{2619} a^{15} - \frac{2}{873} a^{14} + \frac{22}{873} a^{13} - \frac{11}{873} a^{12} + \frac{131}{873} a^{11} - \frac{2}{873} a^{10} + \frac{344}{2619} a^{9} - \frac{130}{873} a^{8} - \frac{275}{2619} a^{7} + \frac{55}{2619} a^{6} + \frac{94}{291} a^{5} - \frac{674}{2619} a^{4} - \frac{176}{873} a^{3} + \frac{133}{291} a^{2} + \frac{515}{2619} a - \frac{11}{27}$, $\frac{1}{30465001870954918125722204139811396935963} a^{22} + \frac{1121398291933174179437076310393691428}{10155000623651639375240734713270465645321} a^{21} + \frac{1167512830022612671850003951033846434}{10155000623651639375240734713270465645321} a^{20} + \frac{51257078616251178488226933030900424324}{30465001870954918125722204139811396935963} a^{19} + \frac{142932996518868792795174785468236234943}{30465001870954918125722204139811396935963} a^{18} + \frac{62616444751504046829445041478438788019}{1128333402627959930582303857030051738369} a^{17} - \frac{76380240311696681261322106693874589320}{3385000207883879791746911571090155215107} a^{16} - \frac{816676670627475532346692996436498802391}{30465001870954918125722204139811396935963} a^{15} - \frac{520117663549147833308020727154543019777}{10155000623651639375240734713270465645321} a^{14} - \frac{407285561928082926150013375238848434005}{10155000623651639375240734713270465645321} a^{13} - \frac{454252227328262242468502445711045315719}{10155000623651639375240734713270465645321} a^{12} + \frac{1386806729024322254143118766656460115988}{10155000623651639375240734713270465645321} a^{11} - \frac{18126720137609245228518973778936717926}{239881904495708016737970111337097613669} a^{10} + \frac{857760732839364906809035880678238948529}{10155000623651639375240734713270465645321} a^{9} + \frac{900192846913693062856746341589622754965}{10155000623651639375240734713270465645321} a^{8} - \frac{3801046430495070098835856882775920035715}{30465001870954918125722204139811396935963} a^{7} + \frac{2961004530063075811316688342535711421584}{30465001870954918125722204139811396935963} a^{6} + \frac{3709623425116755371998505636197721525608}{10155000623651639375240734713270465645321} a^{5} + \frac{12794174904600565009552417451745049486264}{30465001870954918125722204139811396935963} a^{4} + \frac{5254286560385775120542570770708972741012}{30465001870954918125722204139811396935963} a^{3} + \frac{437663403923805174467078643026437920368}{10155000623651639375240734713270465645321} a^{2} - \frac{1394615900152645858011016898410978216138}{3385000207883879791746911571090155215107} a - \frac{39969193875742372536435877387798025353}{314072184236648640471362929276406153979}$, $\frac{1}{30811131808868429148338623003395839922808888861658619} a^{23} - \frac{310451130653}{30811131808868429148338623003395839922808888861658619} a^{22} + \frac{1355689340982628557968262451333446532584887542717}{30811131808868429148338623003395839922808888861658619} a^{21} - \frac{1497316837087924505280260986906130152534286961379}{30811131808868429148338623003395839922808888861658619} a^{20} - \frac{334906628190372692064130479884982452467941027711854}{30811131808868429148338623003395839922808888861658619} a^{19} - \frac{568428587782989882575444970505587618094704466417}{317640534112045661323078587663874638379473081048027} a^{18} - \frac{1462874050769894232226455221093473100092282659068}{40066491298918633482885075427042704711064874982651} a^{17} + \frac{484354453706857354890189986304224891516631117543301}{10270377269622809716112874334465279974269629620552873} a^{16} + \frac{645283242878364153556392593478133607275832984621686}{30811131808868429148338623003395839922808888861658619} a^{15} - \frac{40769866974929472425767125370989935255517523940149}{10270377269622809716112874334465279974269629620552873} a^{14} - \frac{56710692272391945600994229469458684529838753950428}{3423459089874269905370958111488426658089876540184291} a^{13} - \frac{510118944945460713540308294568612845084212265870063}{10270377269622809716112874334465279974269629620552873} a^{12} + \frac{4948812671471309178609520031045800598952680591102489}{30811131808868429148338623003395839922808888861658619} a^{11} - \frac{2750517200465797559917446363745094425915822627864561}{30811131808868429148338623003395839922808888861658619} a^{10} - \frac{3179674884089827628820617052175083429994138418443660}{30811131808868429148338623003395839922808888861658619} a^{9} - \frac{3057081350883532745760221915979446599561037012524394}{30811131808868429148338623003395839922808888861658619} a^{8} + \frac{3291006153100760338189253769426602509853366144281259}{30811131808868429148338623003395839922808888861658619} a^{7} + \frac{1988762185946762943583487582756150917616180647005793}{30811131808868429148338623003395839922808888861658619} a^{6} - \frac{2169792775974944746896950443708921806668060078059078}{10270377269622809716112874334465279974269629620552873} a^{5} - \frac{12455768136746039558174650695542424882937170648589418}{30811131808868429148338623003395839922808888861658619} a^{4} - \frac{3476795396695365614539593479334784743888426414087371}{10270377269622809716112874334465279974269629620552873} a^{3} + \frac{6007773850087540461787491149169512324402324625025102}{30811131808868429148338623003395839922808888861658619} a^{2} - \frac{644873942191610403116913472613334862526108762606854}{3423459089874269905370958111488426658089876540184291} a - \frac{143108738111988648358380523380064546976040814610789}{317640534112045661323078587663874638379473081048027}$

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

Class group and class number

$C_{9}\times C_{157698}$, which has order $1419282$ (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:  \( 39019312.21180779 \) (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

\(\Q(\sqrt{2}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{16})^+\), 6.6.1229312.1, 8.0.173946175488.1, 12.12.49519263525896192.1

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 $24$ R $24$ ${\href{/LocalNumberField/13.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/17.3.0.1}{3} }^{8}$ $24$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$ $24$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/47.3.0.1}{3} }^{8}$ $24$ $24$

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
2Data not computed
3Data not computed
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$