↑

Properties

Label 24.24.1882160448...3392.1
Degree $24$
Signature $[24, 0]$
Discriminant $2^{93}\cdot 13^{20}$
Root discriminant $124.39$
Ramified primes $2, 13$
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![228488, 0, -3655808, 0, 19192992, 0, -43377568, 0, 52112840, 0, -36909600, 0, 16152344, 0, -4421040, 0, 745290, 0, -74048, 0, 4004, 0, -104, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 104*x^22 + 4004*x^20 - 74048*x^18 + 745290*x^16 - 4421040*x^14 + 16152344*x^12 - 36909600*x^10 + 52112840*x^8 - 43377568*x^6 + 19192992*x^4 - 3655808*x^2 + 228488)
 
gp: K = bnfinit(x^24 - 104*x^22 + 4004*x^20 - 74048*x^18 + 745290*x^16 - 4421040*x^14 + 16152344*x^12 - 36909600*x^10 + 52112840*x^8 - 43377568*x^6 + 19192992*x^4 - 3655808*x^2 + 228488, 1)
 

Normalized defining polynomial

\( x^{24} - 104 x^{22} + 4004 x^{20} - 74048 x^{18} + 745290 x^{16} - 4421040 x^{14} + 16152344 x^{12} - 36909600 x^{10} + 52112840 x^{8} - 43377568 x^{6} + 19192992 x^{4} - 3655808 x^{2} + 228488 \)

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:  \(188216044816745326913150945765287080795084676923392=2^{93}\cdot 13^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $124.39$
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$$\chi_{416}(1,·)$, $\chi_{416}(389,·)$, $\chi_{416}(321,·)$, $\chi_{416}(393,·)$, $\chi_{416}(77,·)$, $\chi_{416}(205,·)$, $\chi_{416}(209,·)$, $\chi_{416}(277,·)$, $\chi_{416}(313,·)$, $\chi_{416}(217,·)$, $\chi_{416}(285,·)$, $\chi_{416}(69,·)$, $\chi_{416}(289,·)$, $\chi_{416}(101,·)$, $\chi_{416}(81,·)$, $\chi_{416}(105,·)$, $\chi_{416}(173,·)$, $\chi_{416}(413,·)$, $\chi_{416}(113,·)$, $\chi_{416}(181,·)$, $\chi_{416}(9,·)$, $\chi_{416}(185,·)$, $\chi_{416}(381,·)$, $\chi_{416}(309,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{13} a^{6}$, $\frac{1}{13} a^{7}$, $\frac{1}{26} a^{8}$, $\frac{1}{26} a^{9}$, $\frac{1}{26} a^{10}$, $\frac{1}{26} a^{11}$, $\frac{1}{338} a^{12}$, $\frac{1}{338} a^{13}$, $\frac{1}{338} a^{14}$, $\frac{1}{338} a^{15}$, $\frac{1}{3380} a^{16} + \frac{1}{845} a^{12} - \frac{1}{65} a^{8} + \frac{1}{5} a^{4} - \frac{1}{5}$, $\frac{1}{3380} a^{17} + \frac{1}{845} a^{13} - \frac{1}{65} a^{9} + \frac{1}{5} a^{5} - \frac{1}{5} a$, $\frac{1}{43940} a^{18} - \frac{1}{1690} a^{14} + \frac{1}{130} a^{10} + \frac{1}{65} a^{6} - \frac{2}{5} a^{2}$, $\frac{1}{43940} a^{19} - \frac{1}{1690} a^{15} + \frac{1}{130} a^{11} + \frac{1}{65} a^{7} - \frac{2}{5} a^{3}$, $\frac{1}{1362140} a^{20} + \frac{7}{1362140} a^{18} + \frac{7}{104780} a^{16} - \frac{16}{26195} a^{14} - \frac{22}{26195} a^{12} - \frac{63}{4030} a^{10} + \frac{3}{310} a^{8} + \frac{4}{155} a^{6} + \frac{52}{155} a^{4} + \frac{56}{155} a^{2} - \frac{4}{155}$, $\frac{1}{1362140} a^{21} + \frac{7}{1362140} a^{19} + \frac{7}{104780} a^{17} - \frac{16}{26195} a^{15} - \frac{22}{26195} a^{13} - \frac{63}{4030} a^{11} + \frac{3}{310} a^{9} + \frac{4}{155} a^{7} + \frac{52}{155} a^{5} + \frac{56}{155} a^{3} - \frac{4}{155} a$, $\frac{1}{1604041201690460} a^{22} - \frac{30407886}{401010300422615} a^{20} - \frac{1865563}{275230130695} a^{18} + \frac{2847963}{398025112082} a^{16} + \frac{662978783}{995062780205} a^{14} - \frac{39620395664}{30846946186355} a^{12} + \frac{66048142709}{4745684028670} a^{10} + \frac{414114833}{153086581570} a^{8} - \frac{1417999}{76543290785} a^{6} + \frac{175777965}{1177589089} a^{4} - \frac{34328675859}{182526308795} a^{2} + \frac{57657206828}{182526308795}$, $\frac{1}{1604041201690460} a^{23} - \frac{30407886}{401010300422615} a^{21} - \frac{1865563}{275230130695} a^{19} + \frac{2847963}{398025112082} a^{17} + \frac{662978783}{995062780205} a^{15} - \frac{39620395664}{30846946186355} a^{13} + \frac{66048142709}{4745684028670} a^{11} + \frac{414114833}{153086581570} a^{9} - \frac{1417999}{76543290785} a^{7} + \frac{175777965}{1177589089} a^{5} - \frac{34328675859}{182526308795} a^{3} + \frac{57657206828}{182526308795} a$

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:  \( 155322620827080350 \) (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, 8.8.61334280470528.1, 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 ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ $24$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ $24$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ $24$ ${\href{/LocalNumberField/47.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/53.8.0.1}{8} }^{3}$ $24$

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