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Properties

Label 24.0.24706027099...7601.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{12}\cdot 7^{20}\cdot 17^{12}$
Root discriminant $36.14$
Ramified primes $3, 7, 17$
Class number $65$ (GRH)
Class group $[65]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16777216, 4194304, -4194304, -3407872, -851968, 851968, 954368, -479232, -418048, -67648, 84032, 98228, 28653, -24557, 5252, 1057, -1633, 468, 233, -52, -13, 13, -4, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 4*x^22 + 13*x^21 - 13*x^20 - 52*x^19 + 233*x^18 + 468*x^17 - 1633*x^16 + 1057*x^15 + 5252*x^14 - 24557*x^13 + 28653*x^12 + 98228*x^11 + 84032*x^10 - 67648*x^9 - 418048*x^8 - 479232*x^7 + 954368*x^6 + 851968*x^5 - 851968*x^4 - 3407872*x^3 - 4194304*x^2 + 4194304*x + 16777216)
 
gp: K = bnfinit(x^24 - x^23 - 4*x^22 + 13*x^21 - 13*x^20 - 52*x^19 + 233*x^18 + 468*x^17 - 1633*x^16 + 1057*x^15 + 5252*x^14 - 24557*x^13 + 28653*x^12 + 98228*x^11 + 84032*x^10 - 67648*x^9 - 418048*x^8 - 479232*x^7 + 954368*x^6 + 851968*x^5 - 851968*x^4 - 3407872*x^3 - 4194304*x^2 + 4194304*x + 16777216, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - 4 x^{22} + 13 x^{21} - 13 x^{20} - 52 x^{19} + 233 x^{18} + 468 x^{17} - 1633 x^{16} + 1057 x^{15} + 5252 x^{14} - 24557 x^{13} + 28653 x^{12} + 98228 x^{11} + 84032 x^{10} - 67648 x^{9} - 418048 x^{8} - 479232 x^{7} + 954368 x^{6} + 851968 x^{5} - 851968 x^{4} - 3407872 x^{3} - 4194304 x^{2} + 4194304 x + 16777216 \)

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:  \(24706027099974596656878326981466227601=3^{12}\cdot 7^{20}\cdot 17^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.14$
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:  $3, 7, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(357=3\cdot 7\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{357}(256,·)$, $\chi_{357}(1,·)$, $\chi_{357}(67,·)$, $\chi_{357}(137,·)$, $\chi_{357}(205,·)$, $\chi_{357}(271,·)$, $\chi_{357}(16,·)$, $\chi_{357}(341,·)$, $\chi_{357}(86,·)$, $\chi_{357}(152,·)$, $\chi_{357}(220,·)$, $\chi_{357}(290,·)$, $\chi_{357}(356,·)$, $\chi_{357}(101,·)$, $\chi_{357}(103,·)$, $\chi_{357}(169,·)$, $\chi_{357}(239,·)$, $\chi_{357}(305,·)$, $\chi_{357}(50,·)$, $\chi_{357}(307,·)$, $\chi_{357}(52,·)$, $\chi_{357}(118,·)$, $\chi_{357}(188,·)$, $\chi_{357}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{4} a^{13} - \frac{1}{4} a^{12} + \frac{1}{4} a^{10} - \frac{1}{4} a^{9} + \frac{1}{4} a^{7} - \frac{1}{4} a^{5} + \frac{1}{4} a^{4} - \frac{1}{4} a^{2} + \frac{1}{4} a$, $\frac{1}{2896} a^{14} - \frac{1}{16} a^{13} - \frac{7}{16} a^{11} + \frac{3}{16} a^{10} + \frac{5}{16} a^{8} - \frac{29}{181} a^{7} + \frac{3}{16} a^{6} + \frac{1}{16} a^{5} + \frac{7}{16} a^{3} - \frac{3}{16} a^{2} + \frac{62}{181}$, $\frac{1}{11584} a^{15} - \frac{1}{11584} a^{14} - \frac{1}{16} a^{13} + \frac{9}{64} a^{12} + \frac{7}{64} a^{11} + \frac{3}{16} a^{10} - \frac{11}{64} a^{9} + \frac{789}{2896} a^{8} - \frac{4785}{11584} a^{7} - \frac{3}{64} a^{6} + \frac{1}{16} a^{5} + \frac{23}{64} a^{4} + \frac{25}{64} a^{3} - \frac{3}{16} a^{2} + \frac{243}{724} a + \frac{75}{181}$, $\frac{1}{46336} a^{16} - \frac{1}{46336} a^{15} - \frac{1}{11584} a^{14} - \frac{7}{256} a^{13} + \frac{7}{256} a^{12} + \frac{7}{64} a^{11} - \frac{27}{256} a^{10} - \frac{5003}{11584} a^{9} - \frac{1889}{46336} a^{8} + \frac{1313}{46336} a^{7} - \frac{3}{64} a^{6} - \frac{89}{256} a^{5} + \frac{89}{256} a^{4} + \frac{25}{64} a^{3} - \frac{75}{724} a^{2} + \frac{75}{724} a + \frac{75}{181}$, $\frac{1}{185344} a^{17} - \frac{1}{185344} a^{16} - \frac{1}{46336} a^{15} + \frac{13}{185344} a^{14} + \frac{7}{1024} a^{13} - \frac{57}{256} a^{12} - \frac{27}{1024} a^{11} + \frac{6581}{46336} a^{10} - \frac{1889}{185344} a^{9} - \frac{45023}{185344} a^{8} + \frac{13153}{46336} a^{7} - \frac{345}{1024} a^{6} + \frac{89}{1024} a^{5} - \frac{103}{256} a^{4} - \frac{799}{2896} a^{3} + \frac{75}{2896} a^{2} + \frac{64}{181} a - \frac{27}{181}$, $\frac{1}{3706880} a^{18} - \frac{1}{741376} a^{17} + \frac{29}{3706880} a^{15} - \frac{13}{741376} a^{14} + \frac{4981}{20480} a^{12} - \frac{9273}{46336} a^{11} - \frac{160445}{741376} a^{10} + \frac{3889}{20480} a^{9} + \frac{145}{724} a^{8} + \frac{138343}{741376} a^{7} + \frac{701}{20480} a^{6} - \frac{3861}{46336} a^{4} - \frac{7589}{57920} a^{3} + \frac{233}{724} a + \frac{441}{905}$, $\frac{1}{1505838448640} a^{19} - \frac{52737}{1505838448640} a^{18} + \frac{185439}{75291922432} a^{17} - \frac{3445091}{1505838448640} a^{16} - \frac{15309773}{1505838448640} a^{15} + \frac{2410707}{75291922432} a^{14} + \frac{241065461}{8319549440} a^{13} + \frac{91223372357}{376459612160} a^{12} - \frac{140806668669}{301167689728} a^{11} - \frac{31759310111}{1505838448640} a^{10} - \frac{51035234107}{376459612160} a^{9} + \frac{148707794919}{301167689728} a^{8} - \frac{632045628499}{1505838448640} a^{7} - \frac{961947683}{2079887360} a^{6} + \frac{3916243907}{18822980608} a^{5} + \frac{1094592373}{11764362880} a^{4} - \frac{2122012293}{5882181440} a^{3} + \frac{37575869}{294109072} a^{2} - \frac{103206829}{367636340} a - \frac{26633148}{91909085}$, $\frac{1}{6023353794560} a^{20} - \frac{1}{6023353794560} a^{19} - \frac{113761}{1505838448640} a^{18} + \frac{244109}{6023353794560} a^{17} + \frac{2031091}{6023353794560} a^{16} - \frac{1267949}{1505838448640} a^{15} + \frac{3173481}{6023353794560} a^{14} - \frac{120347964939}{1505838448640} a^{13} + \frac{847383383071}{6023353794560} a^{12} + \frac{575948033569}{6023353794560} a^{11} + \frac{431924583809}{1505838448640} a^{10} + \frac{461010990739}{6023353794560} a^{9} - \frac{2283783278099}{6023353794560} a^{8} - \frac{581411717939}{1505838448640} a^{7} + \frac{18726197299}{94114903040} a^{6} - \frac{19628321209}{94114903040} a^{5} + \frac{4449438751}{23528725760} a^{4} - \frac{288320271}{1470545360} a^{3} + \frac{342278061}{1470545360} a^{2} - \frac{58490879}{367636340} a + \frac{17036261}{91909085}$, $\frac{1}{24093415178240} a^{21} - \frac{1}{24093415178240} a^{20} - \frac{1}{6023353794560} a^{19} - \frac{99991}{4818683035648} a^{18} - \frac{53317389}{24093415178240} a^{17} + \frac{13954291}{6023353794560} a^{16} + \frac{41754005}{4818683035648} a^{15} + \frac{105619381}{6023353794560} a^{14} + \frac{964677754271}{24093415178240} a^{13} - \frac{2376419400339}{4818683035648} a^{12} - \frac{870595349471}{6023353794560} a^{11} - \frac{907678375661}{24093415178240} a^{10} + \frac{708636729289}{4818683035648} a^{9} + \frac{2131332104941}{6023353794560} a^{8} - \frac{111056134571}{376459612160} a^{7} + \frac{28697975727}{75291922432} a^{6} + \frac{34465907131}{94114903040} a^{5} - \frac{1639637781}{5882181440} a^{4} + \frac{247094461}{1176436288} a^{3} - \frac{661682989}{1470545360} a^{2} + \frac{156397731}{367636340} a - \frac{27818801}{91909085}$, $\frac{1}{96373660712960} a^{22} - \frac{1}{96373660712960} a^{21} - \frac{1}{24093415178240} a^{20} + \frac{13}{96373660712960} a^{19} - \frac{5560333}{96373660712960} a^{18} + \frac{26888691}{24093415178240} a^{17} - \frac{79752983}{96373660712960} a^{16} + \frac{158835317}{24093415178240} a^{15} + \frac{282608031}{96373660712960} a^{14} - \frac{8924056202207}{96373660712960} a^{13} + \frac{6091886083873}{24093415178240} a^{12} - \frac{20279783731181}{96373660712960} a^{11} - \frac{436691422683}{2350577090560} a^{10} - \frac{6484074822163}{24093415178240} a^{9} + \frac{329570622689}{1505838448640} a^{8} - \frac{375726873153}{1505838448640} a^{7} + \frac{90512064087}{376459612160} a^{6} - \frac{1811713553}{11764362880} a^{5} + \frac{1366322897}{23528725760} a^{4} + \frac{2365573307}{5882181440} a^{3} - \frac{114251607}{735272680} a^{2} - \frac{1462331}{91909085} a + \frac{43413651}{91909085}$, $\frac{1}{385494642851840} a^{23} - \frac{1}{385494642851840} a^{22} - \frac{1}{96373660712960} a^{21} + \frac{13}{385494642851840} a^{20} - \frac{13}{385494642851840} a^{19} + \frac{8270579}{96373660712960} a^{18} + \frac{360654057}{385494642851840} a^{17} + \frac{147384437}{96373660712960} a^{16} - \frac{2260478561}{385494642851840} a^{15} + \frac{226088993}{385494642851840} a^{14} + \frac{1998607863073}{96373660712960} a^{13} + \frac{58009907940371}{385494642851840} a^{12} + \frac{78254224429037}{385494642851840} a^{11} - \frac{23793239003923}{96373660712960} a^{10} - \frac{1632126508399}{6023353794560} a^{9} - \frac{544074261153}{6023353794560} a^{8} + \frac{315725582147}{1505838448640} a^{7} + \frac{92235997399}{376459612160} a^{6} + \frac{19876329527}{94114903040} a^{5} - \frac{759405159}{11764362880} a^{4} + \frac{2262461499}{5882181440} a^{3} + \frac{586093281}{1470545360} a^{2} + \frac{15079261}{367636340} a - \frac{42503562}{91909085}$

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

Class group and class number

$C_{65}$, which has order $65$ (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:  \( -\frac{2053229}{96373660712960} a^{23} - \frac{2053229}{24093415178240} a^{22} + \frac{26691977}{96373660712960} a^{21} - \frac{26691977}{96373660712960} a^{20} - \frac{26691977}{24093415178240} a^{19} + \frac{478402357}{96373660712960} a^{18} - \frac{526391093}{96373660712960} a^{17} - \frac{3352922957}{96373660712960} a^{16} + \frac{2170263053}{96373660712960} a^{15} + \frac{2695889677}{24093415178240} a^{14} - \frac{50421144553}{96373660712960} a^{13} + \frac{58831170537}{96373660712960} a^{12} + \frac{50421144553}{24093415178240} a^{11} - \frac{911659695637}{96373660712960} a^{10} - \frac{2170263053}{1505838448640} a^{9} - \frac{3352922957}{376459612160} a^{8} - \frac{240227793}{23528725760} a^{7} + \frac{478402357}{23528725760} a^{6} + \frac{26691977}{1470545360} a^{5} - \frac{26691977}{1470545360} a^{4} - \frac{29691273}{367636340} a^{3} - \frac{8212916}{91909085} a^{2} + \frac{8212916}{91909085} a + \frac{32851664}{91909085} \) (order $42$)
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:  \( 213152765.0365848 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_6$ (as 24T3):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 24
The 24 conjugacy class representatives for $C_2^2\times C_6$
Character table for $C_2^2\times C_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{357}) \), \(\Q(\sqrt{-119}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-51}) \), \(\Q(\sqrt{17}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{-119})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{17})\), \(\Q(\sqrt{-7}, \sqrt{-51})\), \(\Q(\sqrt{17}, \sqrt{21})\), \(\Q(\sqrt{-7}, \sqrt{17})\), \(\Q(\sqrt{21}, \sqrt{-51})\), 6.0.64827.1, 6.6.2229465357.1, 6.0.82572791.1, \(\Q(\zeta_{7})\), \(\Q(\zeta_{21})^+\), 6.0.318495051.1, 6.6.11796113.1, 8.0.16243247601.1, 12.0.4970515778063137449.1, \(\Q(\zeta_{21})\), 12.0.101439097511492601.1, 12.0.4970515778063137449.2, 12.12.4970515778063137449.1, 12.0.6818265813529681.1, 12.0.4970515778063137449.3

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ R ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
$17$17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
17.12.6.1$x^{12} + 117912 x^{6} - 1419857 x^{2} + 3475809936$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$