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Properties

Label 24.24.3818394883...1897.1
Degree $24$
Signature $[24, 0]$
Discriminant $3^{12}\cdot 73^{23}$
Root discriminant $105.74$
Ramified primes $3, 73$
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![-136217, -1561389, -5823958, -6667061, 9434735, 30078720, 14798722, -28704066, -36699395, 199317, 22903362, 10129791, -4976758, -4718726, -76005, 909918, 190055, -83107, -29841, 3352, 2080, -29, -71, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 71*x^22 - 29*x^21 + 2080*x^20 + 3352*x^19 - 29841*x^18 - 83107*x^17 + 190055*x^16 + 909918*x^15 - 76005*x^14 - 4718726*x^13 - 4976758*x^12 + 10129791*x^11 + 22903362*x^10 + 199317*x^9 - 36699395*x^8 - 28704066*x^7 + 14798722*x^6 + 30078720*x^5 + 9434735*x^4 - 6667061*x^3 - 5823958*x^2 - 1561389*x - 136217)
 
gp: K = bnfinit(x^24 - x^23 - 71*x^22 - 29*x^21 + 2080*x^20 + 3352*x^19 - 29841*x^18 - 83107*x^17 + 190055*x^16 + 909918*x^15 - 76005*x^14 - 4718726*x^13 - 4976758*x^12 + 10129791*x^11 + 22903362*x^10 + 199317*x^9 - 36699395*x^8 - 28704066*x^7 + 14798722*x^6 + 30078720*x^5 + 9434735*x^4 - 6667061*x^3 - 5823958*x^2 - 1561389*x - 136217, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - 71 x^{22} - 29 x^{21} + 2080 x^{20} + 3352 x^{19} - 29841 x^{18} - 83107 x^{17} + 190055 x^{16} + 909918 x^{15} - 76005 x^{14} - 4718726 x^{13} - 4976758 x^{12} + 10129791 x^{11} + 22903362 x^{10} + 199317 x^{9} - 36699395 x^{8} - 28704066 x^{7} + 14798722 x^{6} + 30078720 x^{5} + 9434735 x^{4} - 6667061 x^{3} - 5823958 x^{2} - 1561389 x - 136217 \)

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:  \(3818394883372513796526550449284140338753741561897=3^{12}\cdot 73^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $105.74$
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:  $3, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(219=3\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{219}(64,·)$, $\chi_{219}(1,·)$, $\chi_{219}(197,·)$, $\chi_{219}(70,·)$, $\chi_{219}(145,·)$, $\chi_{219}(209,·)$, $\chi_{219}(76,·)$, $\chi_{219}(80,·)$, $\chi_{219}(17,·)$, $\chi_{219}(82,·)$, $\chi_{219}(83,·)$, $\chi_{219}(212,·)$, $\chi_{219}(154,·)$, $\chi_{219}(95,·)$, $\chi_{219}(97,·)$, $\chi_{219}(100,·)$, $\chi_{219}(167,·)$, $\chi_{219}(46,·)$, $\chi_{219}(176,·)$, $\chi_{219}(49,·)$, $\chi_{219}(211,·)$, $\chi_{219}(116,·)$, $\chi_{219}(56,·)$, $\chi_{219}(125,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{3} a^{14} - \frac{1}{3} a^{13} + \frac{1}{3} a^{11} - \frac{1}{3} a^{10} + \frac{1}{3} a^{8} - \frac{1}{3} a^{7} + \frac{1}{3} a^{6} - \frac{1}{3} a^{4} + \frac{1}{3} a^{3} - \frac{1}{3} a + \frac{1}{3}$, $\frac{1}{3} a^{15} - \frac{1}{3} a^{13} + \frac{1}{3} a^{12} - \frac{1}{3} a^{10} + \frac{1}{3} a^{9} + \frac{1}{3} a^{6} - \frac{1}{3} a^{5} + \frac{1}{3} a^{3} - \frac{1}{3} a^{2} + \frac{1}{3}$, $\frac{1}{3} a^{16} + \frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{3} a^{17} + \frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{3} a^{18} + \frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{19} + \frac{1}{3} a^{11} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{20} + \frac{1}{3} a^{12} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{21} + \frac{1}{3} a^{13} + \frac{1}{3} a^{5}$, $\frac{1}{3} a^{22} + \frac{1}{3} a^{13} - \frac{1}{3} a^{11} + \frac{1}{3} a^{10} - \frac{1}{3} a^{8} + \frac{1}{3} a^{7} + \frac{1}{3} a^{4} - \frac{1}{3} a^{3} + \frac{1}{3} a - \frac{1}{3}$, $\frac{1}{18075987583669000396725472877237330073} a^{23} - \frac{684541751034479326643274764829371551}{6025329194556333465575157625745776691} a^{22} + \frac{2564446055556892211704764221994051632}{18075987583669000396725472877237330073} a^{21} - \frac{540516236630852393208701778594876958}{6025329194556333465575157625745776691} a^{20} + \frac{500316472973653358007865549656514654}{18075987583669000396725472877237330073} a^{19} - \frac{869235685815971840334226360163903750}{18075987583669000396725472877237330073} a^{18} + \frac{485468910406536897469356462871658220}{6025329194556333465575157625745776691} a^{17} + \frac{1119716253707130774291692825734688135}{18075987583669000396725472877237330073} a^{16} + \frac{404272954064904732660482934633481585}{18075987583669000396725472877237330073} a^{15} - \frac{220396980743957092345702879270513985}{18075987583669000396725472877237330073} a^{14} - \frac{653444232623692160499364471042490753}{18075987583669000396725472877237330073} a^{13} + \frac{50087081377128957454345925959648507}{6025329194556333465575157625745776691} a^{12} + \frac{8777498137623735737949401680445730344}{18075987583669000396725472877237330073} a^{11} + \frac{2422336850872011094169663626713911624}{6025329194556333465575157625745776691} a^{10} + \frac{496353527713227613936929538073713769}{6025329194556333465575157625745776691} a^{9} + \frac{72689049906290126294099471680037611}{6025329194556333465575157625745776691} a^{8} + \frac{719478921380077099095139299061456451}{6025329194556333465575157625745776691} a^{7} + \frac{4842903215820333594102655137009788005}{18075987583669000396725472877237330073} a^{6} + \frac{5835680017844991990708866095081929512}{18075987583669000396725472877237330073} a^{5} - \frac{8357552898561845320697048927164332166}{18075987583669000396725472877237330073} a^{4} + \frac{6848314537567252532124444740859406733}{18075987583669000396725472877237330073} a^{3} + \frac{4720756410830106402014838173629836172}{18075987583669000396725472877237330073} a^{2} - \frac{56978829899239955400435826281858769}{18075987583669000396725472877237330073} a + \frac{2466997204152723618628990515861959586}{6025329194556333465575157625745776691}$

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:  \( 57070082442997110 \) (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, 8.8.894839280046857.1, 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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{4}$ R $24$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ $24$ $24$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{3}$ $24$ $24$ $24$

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
$3$3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
3.4.2.2$x^{4} - 3 x^{2} + 18$$2$$2$$2$$C_4$$[\ ]_{2}^{2}$
73Data not computed