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Properties

Label 24.24.1205440222...5072.1
Degree $24$
Signature $[24, 0]$
Discriminant $2^{24}\cdot 73^{23}$
Root discriminant $122.10$
Ramified primes $2, 73$
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![73, 0, -39128, 0, 634370, 0, -2893939, 0, 5868543, 0, -6199963, 0, 3704020, 0, -1295531, 0, 266304, 0, -31682, 0, 2117, 0, -73, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 73*x^22 + 2117*x^20 - 31682*x^18 + 266304*x^16 - 1295531*x^14 + 3704020*x^12 - 6199963*x^10 + 5868543*x^8 - 2893939*x^6 + 634370*x^4 - 39128*x^2 + 73)
 
gp: K = bnfinit(x^24 - 73*x^22 + 2117*x^20 - 31682*x^18 + 266304*x^16 - 1295531*x^14 + 3704020*x^12 - 6199963*x^10 + 5868543*x^8 - 2893939*x^6 + 634370*x^4 - 39128*x^2 + 73, 1)
 

Normalized defining polynomial

\( x^{24} - 73 x^{22} + 2117 x^{20} - 31682 x^{18} + 266304 x^{16} - 1295531 x^{14} + 3704020 x^{12} - 6199963 x^{10} + 5868543 x^{8} - 2893939 x^{6} + 634370 x^{4} - 39128 x^{2} + 73 \)

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:  \(120544022255782810184584905234140888334894547075072=2^{24}\cdot 73^{23}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $122.10$
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:  $2, 73$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(292=2^{2}\cdot 73\)
Dirichlet character group:    $\lbrace$$\chi_{292}(1,·)$, $\chi_{292}(83,·)$, $\chi_{292}(97,·)$, $\chi_{292}(81,·)$, $\chi_{292}(7,·)$, $\chi_{292}(9,·)$, $\chi_{292}(139,·)$, $\chi_{292}(271,·)$, $\chi_{292}(145,·)$, $\chi_{292}(275,·)$, $\chi_{292}(149,·)$, $\chi_{292}(137,·)$, $\chi_{292}(103,·)$, $\chi_{292}(95,·)$, $\chi_{292}(289,·)$, $\chi_{292}(163,·)$, $\chi_{292}(65,·)$, $\chi_{292}(167,·)$, $\chi_{292}(43,·)$, $\chi_{292}(173,·)$, $\chi_{292}(49,·)$, $\chi_{292}(51,·)$, $\chi_{292}(265,·)$, $\chi_{292}(63,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{3} a^{8} + \frac{1}{3}$, $\frac{1}{3} a^{9} + \frac{1}{3} a$, $\frac{1}{3} a^{10} + \frac{1}{3} a^{2}$, $\frac{1}{3} a^{11} + \frac{1}{3} a^{3}$, $\frac{1}{3} a^{12} + \frac{1}{3} a^{4}$, $\frac{1}{3} a^{13} + \frac{1}{3} a^{5}$, $\frac{1}{12} a^{14} + \frac{1}{3} a^{6} - \frac{1}{4}$, $\frac{1}{12} a^{15} + \frac{1}{3} a^{7} - \frac{1}{4} a$, $\frac{1}{36} a^{16} - \frac{1}{9} a^{8} + \frac{1}{4} a^{2} + \frac{1}{9}$, $\frac{1}{36} a^{17} - \frac{1}{9} a^{9} + \frac{1}{4} a^{3} + \frac{1}{9} a$, $\frac{1}{36} a^{18} - \frac{1}{9} a^{10} + \frac{1}{4} a^{4} + \frac{1}{9} a^{2}$, $\frac{1}{36} a^{19} - \frac{1}{9} a^{11} + \frac{1}{4} a^{5} + \frac{1}{9} a^{3}$, $\frac{1}{91692} a^{20} - \frac{209}{22923} a^{18} - \frac{595}{91692} a^{16} - \frac{11}{5094} a^{14} - \frac{3445}{22923} a^{12} + \frac{1361}{22923} a^{10} + \frac{3067}{22923} a^{8} - \frac{1097}{3396} a^{6} - \frac{9392}{22923} a^{4} - \frac{23087}{91692} a^{2} + \frac{18235}{45846}$, $\frac{1}{91692} a^{21} - \frac{209}{22923} a^{19} - \frac{595}{91692} a^{17} - \frac{11}{5094} a^{15} - \frac{3445}{22923} a^{13} + \frac{1361}{22923} a^{11} + \frac{3067}{22923} a^{9} - \frac{1097}{3396} a^{7} - \frac{9392}{22923} a^{5} - \frac{23087}{91692} a^{3} + \frac{18235}{45846} a$, $\frac{1}{416108055908530220112} a^{22} - \frac{429998500405297}{208054027954265110056} a^{20} + \frac{4992347354534977523}{416108055908530220112} a^{18} + \frac{840959891857733389}{138702685302843406704} a^{16} + \frac{14657370372269603621}{416108055908530220112} a^{14} + \frac{12256972186722548621}{104027013977132555028} a^{12} - \frac{898670204070325040}{26006753494283138757} a^{10} - \frac{5535821511570975769}{138702685302843406704} a^{8} + \frac{8125237418339566313}{208054027954265110056} a^{6} - \frac{102091996204400144345}{416108055908530220112} a^{4} + \frac{8578047514313012615}{416108055908530220112} a^{2} - \frac{43663756464916730993}{138702685302843406704}$, $\frac{1}{416108055908530220112} a^{23} - \frac{429998500405297}{208054027954265110056} a^{21} + \frac{4992347354534977523}{416108055908530220112} a^{19} + \frac{840959891857733389}{138702685302843406704} a^{17} + \frac{14657370372269603621}{416108055908530220112} a^{15} + \frac{12256972186722548621}{104027013977132555028} a^{13} - \frac{898670204070325040}{26006753494283138757} a^{11} - \frac{5535821511570975769}{138702685302843406704} a^{9} + \frac{8125237418339566313}{208054027954265110056} a^{7} - \frac{102091996204400144345}{416108055908530220112} a^{5} + \frac{8578047514313012615}{416108055908530220112} a^{3} - \frac{43663756464916730993}{138702685302843406704} a$

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:  \( 350191229019042400 \) (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, 8.8.2828134020888832.1, 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 R ${\href{/LocalNumberField/3.4.0.1}{4} }^{6}$ $24$ ${\href{/LocalNumberField/7.8.0.1}{8} }^{3}$ $24$ $24$ ${\href{/LocalNumberField/17.8.0.1}{8} }^{3}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ $24$ $24$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.8.0.1}{8} }^{3}$ $24$ $24$ $24$

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
$2$2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
2.6.6.3$x^{6} + 2 x^{4} + x^{2} - 7$$2$$3$$6$$C_6$$[2]^{3}$
73Data not computed