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Properties

Label 24.0.71849836263...1417.1
Degree $24$
Signature $[0, 12]$
Discriminant $73^{23}$
Root discriminant $61.05$
Ramified prime $73$
Class number $89$ (GRH)
Class group $[89]$ (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![2264, -2620, 11808, 1635, 19195, 4983, -2247, 25522, 1136, 11999, 9467, -5456, 12938, 3352, -2494, 4426, 766, -471, 673, 67, -37, 44, 2, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 2*x^22 + 44*x^21 - 37*x^20 + 67*x^19 + 673*x^18 - 471*x^17 + 766*x^16 + 4426*x^15 - 2494*x^14 + 3352*x^13 + 12938*x^12 - 5456*x^11 + 9467*x^10 + 11999*x^9 + 1136*x^8 + 25522*x^7 - 2247*x^6 + 4983*x^5 + 19195*x^4 + 1635*x^3 + 11808*x^2 - 2620*x + 2264)
 
gp: K = bnfinit(x^24 - x^23 + 2*x^22 + 44*x^21 - 37*x^20 + 67*x^19 + 673*x^18 - 471*x^17 + 766*x^16 + 4426*x^15 - 2494*x^14 + 3352*x^13 + 12938*x^12 - 5456*x^11 + 9467*x^10 + 11999*x^9 + 1136*x^8 + 25522*x^7 - 2247*x^6 + 4983*x^5 + 19195*x^4 + 1635*x^3 + 11808*x^2 - 2620*x + 2264, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} + 2 x^{22} + 44 x^{21} - 37 x^{20} + 67 x^{19} + 673 x^{18} - 471 x^{17} + 766 x^{16} + 4426 x^{15} - 2494 x^{14} + 3352 x^{13} + 12938 x^{12} - 5456 x^{11} + 9467 x^{10} + 11999 x^{9} + 1136 x^{8} + 25522 x^{7} - 2247 x^{6} + 4983 x^{5} + 19195 x^{4} + 1635 x^{3} + 11808 x^{2} - 2620 x + 2264 \)

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:  \(7184983626352716099297100617536359330111417=73^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $61.05$
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:  $73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(73\)
Dirichlet character group:    $\lbrace$$\chi_{73}(64,·)$, $\chi_{73}(1,·)$, $\chi_{73}(66,·)$, $\chi_{73}(3,·)$, $\chi_{73}(70,·)$, $\chi_{73}(7,·)$, $\chi_{73}(8,·)$, $\chi_{73}(9,·)$, $\chi_{73}(10,·)$, $\chi_{73}(17,·)$, $\chi_{73}(21,·)$, $\chi_{73}(22,·)$, $\chi_{73}(24,·)$, $\chi_{73}(72,·)$, $\chi_{73}(27,·)$, $\chi_{73}(30,·)$, $\chi_{73}(65,·)$, $\chi_{73}(43,·)$, $\chi_{73}(46,·)$, $\chi_{73}(49,·)$, $\chi_{73}(51,·)$, $\chi_{73}(52,·)$, $\chi_{73}(56,·)$, $\chi_{73}(63,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2} a^{8} - \frac{1}{2} a$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{4}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{5}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{6}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{7}$, $\frac{1}{2} a^{15} - \frac{1}{2} a$, $\frac{1}{4} a^{16} - \frac{1}{4} a^{2}$, $\frac{1}{4} a^{17} - \frac{1}{4} a^{3}$, $\frac{1}{4} a^{18} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{19} - \frac{1}{4} a^{5}$, $\frac{1}{4} a^{20} - \frac{1}{4} a^{6}$, $\frac{1}{8} a^{21} - \frac{1}{8} a^{20} - \frac{1}{8} a^{19} - \frac{1}{8} a^{17} - \frac{1}{4} a^{15} - \frac{1}{4} a^{13} - \frac{1}{4} a^{10} - \frac{1}{4} a^{9} - \frac{1}{4} a^{8} + \frac{3}{8} a^{7} - \frac{1}{8} a^{6} - \frac{3}{8} a^{5} - \frac{1}{8} a^{3} + \frac{1}{4} a^{2} - \frac{1}{2} a$, $\frac{1}{92799416} a^{22} + \frac{5502803}{92799416} a^{21} + \frac{10324031}{92799416} a^{20} - \frac{1605537}{23199854} a^{19} + \frac{2071595}{92799416} a^{18} - \frac{3267251}{46399708} a^{17} - \frac{348467}{46399708} a^{16} - \frac{8695}{11599927} a^{15} + \frac{3704537}{46399708} a^{14} + \frac{840345}{23199854} a^{13} - \frac{5464483}{23199854} a^{12} - \frac{2673035}{46399708} a^{11} + \frac{774323}{46399708} a^{10} + \frac{2706993}{46399708} a^{9} + \frac{18503167}{92799416} a^{8} + \frac{125943}{677368} a^{7} - \frac{19499487}{92799416} a^{6} - \frac{2830541}{11599927} a^{5} + \frac{9535983}{92799416} a^{4} + \frac{3584199}{11599927} a^{3} - \frac{7224569}{23199854} a^{2} - \frac{2313799}{23199854} a - \frac{5132783}{11599927}$, $\frac{1}{136759612182205480462104134975819325224} a^{23} + \frac{612060021007226827900125309077}{136759612182205480462104134975819325224} a^{22} + \frac{1895605581214615716088940693369369065}{136759612182205480462104134975819325224} a^{21} + \frac{612233901549428019836205388361007565}{34189903045551370115526033743954831306} a^{20} + \frac{14925223098593370593770035389310898187}{136759612182205480462104134975819325224} a^{19} - \frac{3623448502700539240573230080916188511}{68379806091102740231052067487909662612} a^{18} - \frac{2724660335255589337913495051291864663}{68379806091102740231052067487909662612} a^{17} + \frac{5667432534259413298997089654787771505}{68379806091102740231052067487909662612} a^{16} - \frac{15597480949044133316559684307371712011}{68379806091102740231052067487909662612} a^{15} - \frac{1338496554659675558962541228531587441}{17094951522775685057763016871977415653} a^{14} - \frac{1646632495624102578084679278463140}{45830969229961622138774844160797361} a^{13} - \frac{16574256873291149285864427678832232263}{68379806091102740231052067487909662612} a^{12} - \frac{15677882929564206506729176939375801561}{68379806091102740231052067487909662612} a^{11} + \frac{7308184529293864392350321924297844101}{68379806091102740231052067487909662612} a^{10} - \frac{23480272043451750000072628529117127925}{136759612182205480462104134975819325224} a^{9} + \frac{9044298218661291711260076762793290237}{136759612182205480462104134975819325224} a^{8} + \frac{22034829232312604769763017024772990243}{136759612182205480462104134975819325224} a^{7} - \frac{7220120925002575579623099338483443169}{34189903045551370115526033743954831306} a^{6} - \frac{10569138540467121382428467663104631353}{136759612182205480462104134975819325224} a^{5} + \frac{617372935538628682593894641288098963}{17094951522775685057763016871977415653} a^{4} - \frac{7604234673312246766653221406638766061}{34189903045551370115526033743954831306} a^{3} - \frac{30062672993169578476208280898359633529}{68379806091102740231052067487909662612} a^{2} - \frac{4899051813309070561743741783557912292}{17094951522775685057763016871977415653} a - \frac{24745794144901556028652384678062378}{60406189126415848260646702727835391}$

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

Class group and class number

$C_{89}$, which has order $89$ (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:  \( 7851263416.155112 \) (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

\(\Q(\sqrt{73}) \), 3.3.5329.1, 4.4.389017.1, 6.6.2073071593.1, 8.0.11047398519097.1, 12.12.313726685568359708377.1

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 ${\href{/LocalNumberField/2.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/3.4.0.1}{4} }^{6}$ $24$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ $24$ $24$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{3}$ $24$ $24$ $24$

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
73Data not computed