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Properties

Label 24.0.53494217359...3456.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{48}\cdot 13^{20}$
Root discriminant $33.91$
Ramified primes $2, 13$
Class number $27$ (GRH)
Class group $[3, 9]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 301, 0, 0, 0, 1462, 0, 0, 0, 1216, 0, 0, 0, 317, 0, 0, 0, 31, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 31*x^20 + 317*x^16 + 1216*x^12 + 1462*x^8 + 301*x^4 + 1)
 
gp: K = bnfinit(x^24 + 31*x^20 + 317*x^16 + 1216*x^12 + 1462*x^8 + 301*x^4 + 1, 1)
 

Normalized defining polynomial

\( x^{24} + 31 x^{20} + 317 x^{16} + 1216 x^{12} + 1462 x^{8} + 301 x^{4} + 1 \)

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:  \(5349421735921433961299014349756563456=2^{48}\cdot 13^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $33.91$
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:  \(104=2^{3}\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{104}(1,·)$, $\chi_{104}(3,·)$, $\chi_{104}(69,·)$, $\chi_{104}(103,·)$, $\chi_{104}(9,·)$, $\chi_{104}(75,·)$, $\chi_{104}(77,·)$, $\chi_{104}(79,·)$, $\chi_{104}(17,·)$, $\chi_{104}(23,·)$, $\chi_{104}(25,·)$, $\chi_{104}(27,·)$, $\chi_{104}(29,·)$, $\chi_{104}(95,·)$, $\chi_{104}(35,·)$, $\chi_{104}(101,·)$, $\chi_{104}(81,·)$, $\chi_{104}(43,·)$, $\chi_{104}(49,·)$, $\chi_{104}(51,·)$, $\chi_{104}(53,·)$, $\chi_{104}(55,·)$, $\chi_{104}(87,·)$, $\chi_{104}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $\frac{1}{5} a^{16} + \frac{1}{5} a^{8} + \frac{1}{5}$, $\frac{1}{5} a^{17} + \frac{1}{5} a^{9} + \frac{1}{5} a$, $\frac{1}{5} a^{18} + \frac{1}{5} a^{10} + \frac{1}{5} a^{2}$, $\frac{1}{5} a^{19} + \frac{1}{5} a^{11} + \frac{1}{5} a^{3}$, $\frac{1}{5758715} a^{20} + \frac{22879}{1151743} a^{16} + \frac{2324846}{5758715} a^{12} - \frac{236478}{1151743} a^{8} + \frac{993646}{5758715} a^{4} + \frac{121670}{1151743}$, $\frac{1}{5758715} a^{21} + \frac{22879}{1151743} a^{17} + \frac{2324846}{5758715} a^{13} - \frac{236478}{1151743} a^{9} + \frac{993646}{5758715} a^{5} + \frac{121670}{1151743} a$, $\frac{1}{5758715} a^{22} + \frac{22879}{1151743} a^{18} + \frac{2324846}{5758715} a^{14} - \frac{236478}{1151743} a^{10} + \frac{993646}{5758715} a^{6} + \frac{121670}{1151743} a^{2}$, $\frac{1}{5758715} a^{23} + \frac{22879}{1151743} a^{19} + \frac{2324846}{5758715} a^{15} - \frac{236478}{1151743} a^{11} + \frac{993646}{5758715} a^{7} + \frac{121670}{1151743} a^{3}$

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

Class group and class number

$C_{3}\times C_{9}$, which has order $27$ (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:  \( \frac{125684}{5758715} a^{21} + \frac{773708}{1151743} a^{17} + \frac{39056569}{5758715} a^{13} + \frac{29172481}{1151743} a^{9} + \frac{163154394}{5758715} a^{5} + \frac{4887441}{1151743} a \) (order $8$)
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:  \( 34260599.803001165 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{26}) \), \(\Q(\sqrt{-26}) \), 3.3.169.1, \(\Q(\zeta_{8})\), \(\Q(i, \sqrt{13})\), \(\Q(i, \sqrt{26})\), \(\Q(\sqrt{-2}, \sqrt{-13})\), \(\Q(\sqrt{-2}, \sqrt{13})\), \(\Q(\sqrt{2}, \sqrt{-13})\), \(\Q(\sqrt{2}, \sqrt{13})\), 6.0.1827904.1, 6.0.14623232.1, 6.6.14623232.1, 6.0.23762752.1, \(\Q(\zeta_{13})^+\), 6.6.190102016.1, 6.0.190102016.1, 8.0.1871773696.1, 12.0.13685690504052736.1, 12.0.564668382613504.1, 12.0.2312881695184912384.1, 12.0.2312881695184912384.3, 12.0.36138776487264256.1, 12.0.2312881695184912384.2, 12.12.36138776487264256.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/5.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$13$13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
13.12.10.1$x^{12} - 117 x^{6} + 10816$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$