↑

Properties

Label 24.0.99537902621...7936.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{48}\cdot 3^{12}\cdot 13^{16}$
Root discriminant $38.30$
Ramified primes $2, 3, 13$
Class number $39$ (GRH)
Class group $[39]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, -178, 0, 0, 0, 31631, 0, 0, 0, -9432, 0, 0, 0, 2631, 0, 0, 0, -53, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 53*x^20 + 2631*x^16 - 9432*x^12 + 31631*x^8 - 178*x^4 + 1)
 
gp: K = bnfinit(x^24 - 53*x^20 + 2631*x^16 - 9432*x^12 + 31631*x^8 - 178*x^4 + 1, 1)
 

Normalized defining polynomial

\( x^{24} - 53 x^{20} + 2631 x^{16} - 9432 x^{12} + 31631 x^{8} - 178 x^{4} + 1 \)

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

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 12]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(99537902621050480929474090019571367936=2^{48}\cdot 3^{12}\cdot 13^{16}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.30$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(312=2^{3}\cdot 3\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{312}(1,·)$, $\chi_{312}(131,·)$, $\chi_{312}(133,·)$, $\chi_{312}(263,·)$, $\chi_{312}(139,·)$, $\chi_{312}(269,·)$, $\chi_{312}(79,·)$, $\chi_{312}(209,·)$, $\chi_{312}(211,·)$, $\chi_{312}(217,·)$, $\chi_{312}(157,·)$, $\chi_{312}(287,·)$, $\chi_{312}(107,·)$, $\chi_{312}(289,·)$, $\chi_{312}(35,·)$, $\chi_{312}(295,·)$, $\chi_{312}(235,·)$, $\chi_{312}(29,·)$, $\chi_{312}(113,·)$, $\chi_{312}(53,·)$, $\chi_{312}(55,·)$, $\chi_{312}(185,·)$, $\chi_{312}(61,·)$, $\chi_{312}(191,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{5} a^{8} - \frac{1}{5} a^{4} + \frac{1}{5}$, $\frac{1}{5} a^{9} - \frac{1}{5} a^{5} + \frac{1}{5} a$, $\frac{1}{5} a^{10} - \frac{1}{5} a^{6} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{11} - \frac{1}{5} a^{7} + \frac{1}{5} a^{3}$, $\frac{1}{5} a^{12} + \frac{1}{5}$, $\frac{1}{5} a^{13} + \frac{1}{5} a$, $\frac{1}{5} a^{14} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{15} + \frac{1}{5} a^{3}$, $\frac{1}{125} a^{16} + \frac{8}{125} a^{12} - \frac{2}{125} a^{8} - \frac{22}{125} a^{4} + \frac{31}{125}$, $\frac{1}{125} a^{17} + \frac{8}{125} a^{13} - \frac{2}{125} a^{9} - \frac{22}{125} a^{5} + \frac{31}{125} a$, $\frac{1}{125} a^{18} + \frac{8}{125} a^{14} - \frac{2}{125} a^{10} - \frac{22}{125} a^{6} + \frac{31}{125} a^{2}$, $\frac{1}{125} a^{19} + \frac{8}{125} a^{15} - \frac{2}{125} a^{11} - \frac{22}{125} a^{7} + \frac{31}{125} a^{3}$, $\frac{1}{10401465875} a^{20} - \frac{25120571}{10401465875} a^{16} + \frac{714782016}{10401465875} a^{12} + \frac{560501036}{10401465875} a^{8} - \frac{3686436506}{10401465875} a^{4} - \frac{2780113149}{10401465875}$, $\frac{1}{10401465875} a^{21} - \frac{25120571}{10401465875} a^{17} + \frac{714782016}{10401465875} a^{13} + \frac{560501036}{10401465875} a^{9} - \frac{3686436506}{10401465875} a^{5} - \frac{2780113149}{10401465875} a$, $\frac{1}{10401465875} a^{22} - \frac{25120571}{10401465875} a^{18} + \frac{714782016}{10401465875} a^{14} + \frac{560501036}{10401465875} a^{10} - \frac{3686436506}{10401465875} a^{6} - \frac{2780113149}{10401465875} a^{2}$, $\frac{1}{10401465875} a^{23} - \frac{25120571}{10401465875} a^{19} + \frac{714782016}{10401465875} a^{15} + \frac{560501036}{10401465875} a^{11} - \frac{3686436506}{10401465875} a^{7} - \frac{2780113149}{10401465875} a^{3}$

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

Class group and class number

$C_{39}$, which has order $39$ (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:  \( -\frac{2853609136}{10401465875} a^{23} + \frac{151225243418}{10401465875} a^{19} - \frac{1501399193336}{2080293175} a^{15} + \frac{5374612492536}{2080293175} a^{11} - \frac{90112408349173}{10401465875} a^{7} + \frac{849670136}{10401465875} a^{3} \) (order $24$)
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:  \( 381194856.24035984 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2\times C_6$ (as 24T3):

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 $C_2^2\times C_6$
Character table for $C_2^2\times C_6$ is not computed

Intermediate fields

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{6}) \), \(\Q(\sqrt{-6}) \), 3.3.169.1, \(\Q(\zeta_{12})\), \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{6})\), \(\Q(\sqrt{2}, \sqrt{3})\), \(\Q(\sqrt{-2}, \sqrt{3})\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-2}, \sqrt{-3})\), 6.0.1827904.1, 6.6.49353408.1, 6.0.771147.1, 6.6.14623232.1, 6.0.14623232.1, 6.6.394827264.1, 6.0.394827264.1, \(\Q(\zeta_{24})\), 12.0.2435758881214464.1, 12.0.13685690504052736.1, 12.0.9976868377454444544.2, 12.12.9976868377454444544.1, 12.0.9976868377454444544.1, 12.0.155888568397725696.1, 12.0.155888568397725696.2

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
2Data not computed
$3$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
13.6.4.3$x^{6} + 65 x^{3} + 1352$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$