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Properties

Label 24.24.4963064143...1873.1
Degree $24$
Signature $[24, 0]$
Discriminant $97^{23}$
Root discriminant $80.17$
Ramified prime $97$
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![1, 6, -298, -1872, 10541, 16749, -82152, -62318, 264241, 124640, -410227, -136069, 345815, 82706, -168896, -28796, 49149, 5845, -8543, -677, 861, 41, -46, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 46*x^22 + 41*x^21 + 861*x^20 - 677*x^19 - 8543*x^18 + 5845*x^17 + 49149*x^16 - 28796*x^15 - 168896*x^14 + 82706*x^13 + 345815*x^12 - 136069*x^11 - 410227*x^10 + 124640*x^9 + 264241*x^8 - 62318*x^7 - 82152*x^6 + 16749*x^5 + 10541*x^4 - 1872*x^3 - 298*x^2 + 6*x + 1)
 
gp: K = bnfinit(x^24 - x^23 - 46*x^22 + 41*x^21 + 861*x^20 - 677*x^19 - 8543*x^18 + 5845*x^17 + 49149*x^16 - 28796*x^15 - 168896*x^14 + 82706*x^13 + 345815*x^12 - 136069*x^11 - 410227*x^10 + 124640*x^9 + 264241*x^8 - 62318*x^7 - 82152*x^6 + 16749*x^5 + 10541*x^4 - 1872*x^3 - 298*x^2 + 6*x + 1, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - 46 x^{22} + 41 x^{21} + 861 x^{20} - 677 x^{19} - 8543 x^{18} + 5845 x^{17} + 49149 x^{16} - 28796 x^{15} - 168896 x^{14} + 82706 x^{13} + 345815 x^{12} - 136069 x^{11} - 410227 x^{10} + 124640 x^{9} + 264241 x^{8} - 62318 x^{7} - 82152 x^{6} + 16749 x^{5} + 10541 x^{4} - 1872 x^{3} - 298 x^{2} + 6 x + 1 \)

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:  \(4963064143419831996986398968091856144361331873=97^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $80.17$
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:  $97$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(97\)
Dirichlet character group:    $\lbrace$$\chi_{97}(64,·)$, $\chi_{97}(1,·)$, $\chi_{97}(4,·)$, $\chi_{97}(6,·)$, $\chi_{97}(9,·)$, $\chi_{97}(75,·)$, $\chi_{97}(16,·)$, $\chi_{97}(81,·)$, $\chi_{97}(22,·)$, $\chi_{97}(24,·)$, $\chi_{97}(91,·)$, $\chi_{97}(93,·)$, $\chi_{97}(96,·)$, $\chi_{97}(33,·)$, $\chi_{97}(35,·)$, $\chi_{97}(36,·)$, $\chi_{97}(43,·)$, $\chi_{97}(47,·)$, $\chi_{97}(50,·)$, $\chi_{97}(54,·)$, $\chi_{97}(73,·)$, $\chi_{97}(88,·)$, $\chi_{97}(61,·)$, $\chi_{97}(62,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{401579} a^{22} - \frac{163649}{401579} a^{21} - \frac{34707}{401579} a^{20} - \frac{33257}{401579} a^{19} - \frac{148043}{401579} a^{18} + \frac{77852}{401579} a^{17} - \frac{46466}{401579} a^{16} - \frac{41391}{401579} a^{15} - \frac{177636}{401579} a^{14} + \frac{55241}{401579} a^{13} - \frac{157643}{401579} a^{12} - \frac{53337}{401579} a^{11} - \frac{35468}{401579} a^{10} + \frac{109718}{401579} a^{9} - \frac{5522}{401579} a^{8} + \frac{134209}{401579} a^{7} - \frac{72609}{401579} a^{6} - \frac{37178}{401579} a^{5} + \frac{50262}{401579} a^{4} - \frac{20000}{401579} a^{3} - \frac{99902}{401579} a^{2} + \frac{126996}{401579} a + \frac{6060}{401579}$, $\frac{1}{602107757552888416617470834580769} a^{23} + \frac{485870573994694554143799947}{602107757552888416617470834580769} a^{22} - \frac{290321139723157034591802907502448}{602107757552888416617470834580769} a^{21} - \frac{158415383305902690664909621586981}{602107757552888416617470834580769} a^{20} + \frac{180873647236376036438726473336057}{602107757552888416617470834580769} a^{19} - \frac{197714878455243678920168353402056}{602107757552888416617470834580769} a^{18} + \frac{129051660503981776535212584360797}{602107757552888416617470834580769} a^{17} + \frac{202296723749712713640577822064857}{602107757552888416617470834580769} a^{16} - \frac{150756920867361518594838720587445}{602107757552888416617470834580769} a^{15} + \frac{135326930495021978465985246401705}{602107757552888416617470834580769} a^{14} - \frac{146985846104732089235845122120668}{602107757552888416617470834580769} a^{13} - \frac{30539250182135167860537906968189}{602107757552888416617470834580769} a^{12} - \frac{143334515933073139522922620367884}{602107757552888416617470834580769} a^{11} - \frac{32924807673089632680995250328111}{602107757552888416617470834580769} a^{10} + \frac{211733703598660784307220061906494}{602107757552888416617470834580769} a^{9} + \frac{155612492331568026800795271910722}{602107757552888416617470834580769} a^{8} + \frac{194535216966450513456922175487770}{602107757552888416617470834580769} a^{7} - \frac{46836478076866204814641551280333}{602107757552888416617470834580769} a^{6} + \frac{128181906342315038489506051530816}{602107757552888416617470834580769} a^{5} + \frac{22589857625101257062329987548270}{602107757552888416617470834580769} a^{4} - \frac{96091240828597106147852579566397}{602107757552888416617470834580769} a^{3} - \frac{36892169468035958602779326306507}{602107757552888416617470834580769} a^{2} + \frac{60664490207825398256567459642037}{602107757552888416617470834580769} a + \frac{242180827022692753700189904595553}{602107757552888416617470834580769}$

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:  \( 1104768408470740.2 \) (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

\(\Q(\sqrt{97}) \), 3.3.9409.1, 4.4.912673.1, 6.6.8587340257.1, 8.8.80798284478113.1, 12.12.7153014030880804126753.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ $24$ $24$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ $24$ $24$ ${\href{/LocalNumberField/19.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{2}$ $24$ $24$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ $24$

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
97Data not computed