↑

Properties

Label 24.24.1739852432...0512.1
Degree $24$
Signature $[24, 0]$
Discriminant $2^{24}\cdot 3^{36}\cdot 17^{21}$
Root discriminant $123.98$
Ramified primes $2, 3, 17$
Class number $2$ (GRH)
Class group $[2]$ (GRH)
Galois group $C_{24}$ (as 24T1)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![397953, 0, -5969295, 0, 27856710, 0, -60356205, 0, 70929270, 0, -48441024, 0, 19879443, 0, -5001723, 0, 773874, 0, -72114, 0, 3825, 0, -102, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 102*x^22 + 3825*x^20 - 72114*x^18 + 773874*x^16 - 5001723*x^14 + 19879443*x^12 - 48441024*x^10 + 70929270*x^8 - 60356205*x^6 + 27856710*x^4 - 5969295*x^2 + 397953)
 
gp: K = bnfinit(x^24 - 102*x^22 + 3825*x^20 - 72114*x^18 + 773874*x^16 - 5001723*x^14 + 19879443*x^12 - 48441024*x^10 + 70929270*x^8 - 60356205*x^6 + 27856710*x^4 - 5969295*x^2 + 397953, 1)
 

Normalized defining polynomial

\( x^{24} - 102 x^{22} + 3825 x^{20} - 72114 x^{18} + 773874 x^{16} - 5001723 x^{14} + 19879443 x^{12} - 48441024 x^{10} + 70929270 x^{8} - 60356205 x^{6} + 27856710 x^{4} - 5969295 x^{2} + 397953 \)

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:  \(173985243292970336610023479950719759658537568960512=2^{24}\cdot 3^{36}\cdot 17^{21}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $123.98$
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:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(612=2^{2}\cdot 3^{2}\cdot 17\)
Dirichlet character group:    $\lbrace$$\chi_{612}(383,·)$, $\chi_{612}(1,·)$, $\chi_{612}(263,·)$, $\chi_{612}(587,·)$, $\chi_{612}(13,·)$, $\chi_{612}(467,·)$, $\chi_{612}(205,·)$, $\chi_{612}(563,·)$, $\chi_{612}(83,·)$, $\chi_{612}(217,·)$, $\chi_{612}(409,·)$, $\chi_{612}(155,·)$, $\chi_{612}(157,·)$, $\chi_{612}(287,·)$, $\chi_{612}(421,·)$, $\chi_{612}(359,·)$, $\chi_{612}(169,·)$, $\chi_{612}(491,·)$, $\chi_{612}(179,·)$, $\chi_{612}(565,·)$, $\chi_{612}(577,·)$, $\chi_{612}(361,·)$, $\chi_{612}(59,·)$, $\chi_{612}(373,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3} a^{6}$, $\frac{1}{3} a^{7}$, $\frac{1}{51} a^{8}$, $\frac{1}{51} a^{9}$, $\frac{1}{51} a^{10}$, $\frac{1}{51} a^{11}$, $\frac{1}{153} a^{12}$, $\frac{1}{153} a^{13}$, $\frac{1}{153} a^{14}$, $\frac{1}{153} a^{15}$, $\frac{1}{2601} a^{16}$, $\frac{1}{2601} a^{17}$, $\frac{1}{405756} a^{18} - \frac{5}{33813} a^{16} + \frac{5}{1989} a^{14} - \frac{1}{884} a^{12} + \frac{1}{663} a^{10} - \frac{1}{663} a^{8} + \frac{1}{78} a^{6} + \frac{1}{13} a^{4} - \frac{6}{13} a^{2} + \frac{7}{52}$, $\frac{1}{405756} a^{19} - \frac{5}{33813} a^{17} + \frac{5}{1989} a^{15} - \frac{1}{884} a^{13} + \frac{1}{663} a^{11} - \frac{1}{663} a^{9} + \frac{1}{78} a^{7} + \frac{1}{13} a^{5} - \frac{6}{13} a^{3} + \frac{7}{52} a$, $\frac{1}{36112284} a^{20} - \frac{1}{925956} a^{18} + \frac{149}{3009357} a^{16} - \frac{317}{708084} a^{14} + \frac{1279}{708084} a^{12} + \frac{5}{3471} a^{10} - \frac{1091}{118014} a^{8} - \frac{277}{2314} a^{6} + \frac{54}{1157} a^{4} + \frac{855}{4628} a^{2} + \frac{771}{4628}$, $\frac{1}{36112284} a^{21} - \frac{1}{925956} a^{19} + \frac{149}{3009357} a^{17} - \frac{317}{708084} a^{15} + \frac{1279}{708084} a^{13} + \frac{5}{3471} a^{11} - \frac{1091}{118014} a^{9} - \frac{277}{2314} a^{7} + \frac{54}{1157} a^{5} + \frac{855}{4628} a^{3} + \frac{771}{4628} a$, $\frac{1}{166419019003068} a^{22} + \frac{3175}{3263118019668} a^{20} + \frac{1099}{5543235594} a^{18} + \frac{792875}{362568668852} a^{16} + \frac{553678127}{191948118804} a^{14} - \frac{13408531}{31991353134} a^{12} + \frac{24056901}{10663784378} a^{10} - \frac{151441021}{31991353134} a^{8} + \frac{1502060611}{15995676567} a^{6} - \frac{476280449}{1254562868} a^{4} - \frac{386706241}{1254562868} a^{2} - \frac{156836189}{627281434}$, $\frac{1}{166419019003068} a^{23} + \frac{3175}{3263118019668} a^{21} + \frac{1099}{5543235594} a^{19} + \frac{792875}{362568668852} a^{17} + \frac{553678127}{191948118804} a^{15} - \frac{13408531}{31991353134} a^{13} + \frac{24056901}{10663784378} a^{11} - \frac{151441021}{31991353134} a^{9} + \frac{1502060611}{15995676567} a^{7} - \frac{476280449}{1254562868} a^{5} - \frac{386706241}{1254562868} a^{3} - \frac{156836189}{627281434} a$

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

Class group and class number

$C_{2}$, which has order $2$ (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:  \( 132447362854217420 \) (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

\(\Q(\sqrt{17}) \), \(\Q(\zeta_{9})^+\), 4.4.4913.1, 6.6.32234193.1, 8.8.8508782723328.1, 12.12.5104819233548816337.1

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 R $24$ $24$ $24$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/19.4.0.1}{4} }^{6}$ $24$ $24$ $24$ ${\href{/LocalNumberField/37.8.0.1}{8} }^{3}$ $24$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
$2$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
3Data not computed
$17$17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$
17.8.7.1$x^{8} - 1377$$8$$1$$7$$C_8$$[\ ]_{8}$