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Properties

Label 24.24.6589966906...7072.1
Degree $24$
Signature $[24, 0]$
Discriminant $2^{93}\cdot 13^{16}$
Root discriminant $81.12$
Ramified primes $2, 13$
Class number $1$ (GRH)
Class group Trivial (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![961, -10912, 6920, 187912, -410504, -496600, 1861164, -30664, -3185590, 1258912, 2691436, -1565456, -1254214, 896744, 333348, -286368, -47961, 53592, 2656, -5808, 162, 336, -28, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 8*x^23 - 28*x^22 + 336*x^21 + 162*x^20 - 5808*x^19 + 2656*x^18 + 53592*x^17 - 47961*x^16 - 286368*x^15 + 333348*x^14 + 896744*x^13 - 1254214*x^12 - 1565456*x^11 + 2691436*x^10 + 1258912*x^9 - 3185590*x^8 - 30664*x^7 + 1861164*x^6 - 496600*x^5 - 410504*x^4 + 187912*x^3 + 6920*x^2 - 10912*x + 961)
 
gp: K = bnfinit(x^24 - 8*x^23 - 28*x^22 + 336*x^21 + 162*x^20 - 5808*x^19 + 2656*x^18 + 53592*x^17 - 47961*x^16 - 286368*x^15 + 333348*x^14 + 896744*x^13 - 1254214*x^12 - 1565456*x^11 + 2691436*x^10 + 1258912*x^9 - 3185590*x^8 - 30664*x^7 + 1861164*x^6 - 496600*x^5 - 410504*x^4 + 187912*x^3 + 6920*x^2 - 10912*x + 961, 1)
 

Normalized defining polynomial

\( x^{24} - 8 x^{23} - 28 x^{22} + 336 x^{21} + 162 x^{20} - 5808 x^{19} + 2656 x^{18} + 53592 x^{17} - 47961 x^{16} - 286368 x^{15} + 333348 x^{14} + 896744 x^{13} - 1254214 x^{12} - 1565456 x^{11} + 2691436 x^{10} + 1258912 x^{9} - 3185590 x^{8} - 30664 x^{7} + 1861164 x^{6} - 496600 x^{5} - 410504 x^{4} + 187912 x^{3} + 6920 x^{2} - 10912 x + 961 \)

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:  \(6589966906506961482901542164675154259132547072=2^{93}\cdot 13^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $81.12$
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:  $2, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(416=2^{5}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{416}(1,·)$, $\chi_{416}(133,·)$, $\chi_{416}(321,·)$, $\chi_{416}(9,·)$, $\chi_{416}(269,·)$, $\chi_{416}(365,·)$, $\chi_{416}(209,·)$, $\chi_{416}(341,·)$, $\chi_{416}(185,·)$, $\chi_{416}(217,·)$, $\chi_{416}(29,·)$, $\chi_{416}(261,·)$, $\chi_{416}(289,·)$, $\chi_{416}(165,·)$, $\chi_{416}(81,·)$, $\chi_{416}(105,·)$, $\chi_{416}(237,·)$, $\chi_{416}(157,·)$, $\chi_{416}(113,·)$, $\chi_{416}(53,·)$, $\chi_{416}(393,·)$, $\chi_{416}(313,·)$, $\chi_{416}(61,·)$, $\chi_{416}(373,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{31} a^{18} + \frac{12}{31} a^{17} - \frac{12}{31} a^{16} + \frac{6}{31} a^{15} - \frac{4}{31} a^{14} + \frac{6}{31} a^{13} - \frac{5}{31} a^{12} + \frac{3}{31} a^{11} + \frac{10}{31} a^{10} + \frac{10}{31} a^{9} - \frac{2}{31} a^{8} + \frac{10}{31} a^{7} - \frac{4}{31} a^{6} - \frac{4}{31} a^{4} + \frac{3}{31} a^{3} + \frac{15}{31} a^{2} - \frac{14}{31} a$, $\frac{1}{31} a^{19} - \frac{1}{31} a^{17} - \frac{5}{31} a^{16} - \frac{14}{31} a^{15} - \frac{8}{31} a^{14} - \frac{15}{31} a^{13} + \frac{1}{31} a^{12} + \frac{5}{31} a^{11} + \frac{14}{31} a^{10} + \frac{2}{31} a^{9} + \frac{3}{31} a^{8} - \frac{14}{31} a^{6} - \frac{4}{31} a^{5} - \frac{11}{31} a^{4} + \frac{10}{31} a^{3} - \frac{8}{31} a^{2} + \frac{13}{31} a$, $\frac{1}{31} a^{20} + \frac{7}{31} a^{17} + \frac{5}{31} a^{16} - \frac{2}{31} a^{15} + \frac{12}{31} a^{14} + \frac{7}{31} a^{13} - \frac{14}{31} a^{11} + \frac{12}{31} a^{10} + \frac{13}{31} a^{9} - \frac{2}{31} a^{8} - \frac{4}{31} a^{7} - \frac{8}{31} a^{6} - \frac{11}{31} a^{5} + \frac{6}{31} a^{4} - \frac{5}{31} a^{3} - \frac{3}{31} a^{2} - \frac{14}{31} a$, $\frac{1}{31} a^{21} + \frac{14}{31} a^{17} - \frac{11}{31} a^{16} + \frac{1}{31} a^{15} + \frac{4}{31} a^{14} - \frac{11}{31} a^{13} - \frac{10}{31} a^{12} - \frac{9}{31} a^{11} + \frac{5}{31} a^{10} - \frac{10}{31} a^{9} + \frac{10}{31} a^{8} + \frac{15}{31} a^{7} - \frac{14}{31} a^{6} + \frac{6}{31} a^{5} - \frac{8}{31} a^{4} + \frac{7}{31} a^{3} + \frac{5}{31} a^{2} + \frac{5}{31} a$, $\frac{1}{343085355401517213857} a^{22} - \frac{1061423076618196432}{343085355401517213857} a^{21} + \frac{5228524545630817377}{343085355401517213857} a^{20} + \frac{3381495452248125663}{343085355401517213857} a^{19} + \frac{4287430875361297358}{343085355401517213857} a^{18} + \frac{136778300441364194627}{343085355401517213857} a^{17} + \frac{28476574741605894012}{343085355401517213857} a^{16} - \frac{35577102692658343261}{343085355401517213857} a^{15} + \frac{134052574824600054663}{343085355401517213857} a^{14} + \frac{72362879129075069750}{343085355401517213857} a^{13} - \frac{56667471227966575890}{343085355401517213857} a^{12} + \frac{166881563175412251907}{343085355401517213857} a^{11} + \frac{15420107500821422500}{343085355401517213857} a^{10} + \frac{128037227916738879906}{343085355401517213857} a^{9} + \frac{3949101286339129604}{343085355401517213857} a^{8} + \frac{96701574693325741476}{343085355401517213857} a^{7} - \frac{45335956847967468287}{343085355401517213857} a^{6} - \frac{166697715313471981022}{343085355401517213857} a^{5} + \frac{162344537911752642741}{343085355401517213857} a^{4} + \frac{36887415105902921146}{343085355401517213857} a^{3} - \frac{105237575033543974491}{343085355401517213857} a^{2} - \frac{70398716965070997967}{343085355401517213857} a + \frac{1520925151283212489}{11067269529081200447}$, $\frac{1}{40629539784983919132284154143} a^{23} + \frac{19112882}{40629539784983919132284154143} a^{22} - \frac{528746179290203847200148470}{40629539784983919132284154143} a^{21} + \frac{88227533768803589373225023}{40629539784983919132284154143} a^{20} - \frac{117105676284595565487402335}{40629539784983919132284154143} a^{19} + \frac{301510126365906083451375066}{40629539784983919132284154143} a^{18} + \frac{17657243767111707207648594111}{40629539784983919132284154143} a^{17} + \frac{6228024388026326308991124022}{40629539784983919132284154143} a^{16} - \frac{9251049555956918915108229795}{40629539784983919132284154143} a^{15} - \frac{3500426252852796583274265365}{40629539784983919132284154143} a^{14} - \frac{9692587672727365969073867762}{40629539784983919132284154143} a^{13} - \frac{10851360687021205378284281808}{40629539784983919132284154143} a^{12} + \frac{13841610722631378950278561456}{40629539784983919132284154143} a^{11} + \frac{4514364542464301714338369324}{40629539784983919132284154143} a^{10} - \frac{2894774695763308611115422723}{40629539784983919132284154143} a^{9} + \frac{815917342128114705281108679}{40629539784983919132284154143} a^{8} - \frac{2540478801831906487241286292}{40629539784983919132284154143} a^{7} + \frac{10158658957730254820274277144}{40629539784983919132284154143} a^{6} - \frac{7717089357953457412623309539}{40629539784983919132284154143} a^{5} + \frac{17357131929996413935336434214}{40629539784983919132284154143} a^{4} + \frac{2665466380654974878232155908}{40629539784983919132284154143} a^{3} - \frac{10252966129462971310513796982}{40629539784983919132284154143} a^{2} - \frac{241343232639474142621402883}{1310630315644642552654327553} a + \frac{8650180776553455332566068}{42278397278859437182397663}$

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

Class group and class number

Trivial group, which has order $1$ (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:  \( 2183476337861903.2 \) (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

\(\Q(\sqrt{2}) \), 3.3.169.1, \(\Q(\zeta_{16})^+\), 6.6.14623232.1, \(\Q(\zeta_{32})^+\), 12.12.7007073538075000832.1

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 $24$ ${\href{/LocalNumberField/5.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ $24$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ $24$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/31.1.0.1}{1} }^{24}$ $24$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{3}$ $24$

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
13Data not computed