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Properties

Label 24.0.51067050430...0000.3
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 5^{18}\cdot 7^{20}$
Root discriminant $33.85$
Ramified primes $2, 5, 7$
Class number $26$ (GRH)
Class group $[26]$ (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15625, 0, 15625, 0, 12500, 0, 9375, 0, 6875, 0, 5000, 0, 3625, 0, 1000, 0, 275, 0, 75, 0, 20, 0, 5, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 5*x^22 + 20*x^20 + 75*x^18 + 275*x^16 + 1000*x^14 + 3625*x^12 + 5000*x^10 + 6875*x^8 + 9375*x^6 + 12500*x^4 + 15625*x^2 + 15625)
 
gp: K = bnfinit(x^24 + 5*x^22 + 20*x^20 + 75*x^18 + 275*x^16 + 1000*x^14 + 3625*x^12 + 5000*x^10 + 6875*x^8 + 9375*x^6 + 12500*x^4 + 15625*x^2 + 15625, 1)
 

Normalized defining polynomial

\( x^{24} + 5 x^{22} + 20 x^{20} + 75 x^{18} + 275 x^{16} + 1000 x^{14} + 3625 x^{12} + 5000 x^{10} + 6875 x^{8} + 9375 x^{6} + 12500 x^{4} + 15625 x^{2} + 15625 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 12]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(5106705043047168064000000000000000000=2^{24}\cdot 5^{18}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.85$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(140=2^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{140}(1,·)$, $\chi_{140}(3,·)$, $\chi_{140}(69,·)$, $\chi_{140}(129,·)$, $\chi_{140}(9,·)$, $\chi_{140}(87,·)$, $\chi_{140}(81,·)$, $\chi_{140}(67,·)$, $\chi_{140}(23,·)$, $\chi_{140}(89,·)$, $\chi_{140}(27,·)$, $\chi_{140}(29,·)$, $\chi_{140}(107,·)$, $\chi_{140}(101,·)$, $\chi_{140}(103,·)$, $\chi_{140}(41,·)$, $\chi_{140}(43,·)$, $\chi_{140}(109,·)$, $\chi_{140}(47,·)$, $\chi_{140}(83,·)$, $\chi_{140}(121,·)$, $\chi_{140}(123,·)$, $\chi_{140}(61,·)$, $\chi_{140}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{125} a^{12}$, $\frac{1}{125} a^{13}$, $\frac{1}{3625} a^{14} + \frac{11}{29}$, $\frac{1}{3625} a^{15} + \frac{11}{29} a$, $\frac{1}{18125} a^{16} + \frac{8}{29} a^{2}$, $\frac{1}{18125} a^{17} + \frac{8}{29} a^{3}$, $\frac{1}{18125} a^{18} + \frac{11}{145} a^{4}$, $\frac{1}{18125} a^{19} + \frac{11}{145} a^{5}$, $\frac{1}{90625} a^{20} + \frac{8}{145} a^{6}$, $\frac{1}{90625} a^{21} + \frac{8}{145} a^{7}$, $\frac{1}{90625} a^{22} + \frac{11}{725} a^{8}$, $\frac{1}{90625} a^{23} + \frac{11}{725} a^{9}$

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

Class group and class number

$C_{26}$, which has order $26$ (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:  $11$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -\frac{11}{90625} a^{22} - \frac{44}{90625} a^{20} - \frac{33}{18125} a^{18} - \frac{121}{18125} a^{16} - \frac{88}{3625} a^{14} - \frac{11}{125} a^{12} - \frac{8}{25} a^{10} - \frac{121}{725} a^{8} - \frac{33}{145} a^{6} - \frac{44}{145} a^{4} - \frac{11}{29} a^{2} - \frac{11}{29} \) (order $14$)
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:  \( 36584901.886295445 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{12}$ (as 24T2):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
An abelian group of order 24
The 24 conjugacy class representatives for $C_2\times C_{12}$
Character table for $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\zeta_{20})^+\), 4.0.98000.1, 6.0.2100875.1, 6.6.300125.1, \(\Q(\zeta_{7})\), 8.0.9604000000.2, 12.0.4413675765625.1, 12.12.46118408000000000.1, 12.0.2259801992000000000.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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}$
5Data not computed
7Data not computed