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Properties

Label 24.0.28637078059...8464.3
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 3^{12}\cdot 13^{22}$
Root discriminant $36.37$
Ramified primes $2, 3, 13$
Class number $104$ (GRH)
Class group $[2, 52]$ (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![169, 0, 1183, 0, 5915, 0, 12506, 0, 18083, 0, 16731, 0, 11349, 0, 5499, 0, 2015, 0, 533, 0, 104, 0, 13, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 13*x^22 + 104*x^20 + 533*x^18 + 2015*x^16 + 5499*x^14 + 11349*x^12 + 16731*x^10 + 18083*x^8 + 12506*x^6 + 5915*x^4 + 1183*x^2 + 169)
 
gp: K = bnfinit(x^24 + 13*x^22 + 104*x^20 + 533*x^18 + 2015*x^16 + 5499*x^14 + 11349*x^12 + 16731*x^10 + 18083*x^8 + 12506*x^6 + 5915*x^4 + 1183*x^2 + 169, 1)
 

Normalized defining polynomial

\( x^{24} + 13 x^{22} + 104 x^{20} + 533 x^{18} + 2015 x^{16} + 5499 x^{14} + 11349 x^{12} + 16731 x^{10} + 18083 x^{8} + 12506 x^{6} + 5915 x^{4} + 1183 x^{2} + 169 \)

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:  \(28637078059459331679625147758321598464=2^{24}\cdot 3^{12}\cdot 13^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $36.37$
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:  \(156=2^{2}\cdot 3\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{156}(1,·)$, $\chi_{156}(67,·)$, $\chi_{156}(133,·)$, $\chi_{156}(7,·)$, $\chi_{156}(11,·)$, $\chi_{156}(77,·)$, $\chi_{156}(115,·)$, $\chi_{156}(17,·)$, $\chi_{156}(19,·)$, $\chi_{156}(151,·)$, $\chi_{156}(25,·)$, $\chi_{156}(29,·)$, $\chi_{156}(31,·)$, $\chi_{156}(101,·)$, $\chi_{156}(113,·)$, $\chi_{156}(71,·)$, $\chi_{156}(47,·)$, $\chi_{156}(49,·)$, $\chi_{156}(83,·)$, $\chi_{156}(53,·)$, $\chi_{156}(119,·)$, $\chi_{156}(121,·)$, $\chi_{156}(59,·)$, $\chi_{156}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{13} a^{12}$, $\frac{1}{13} a^{13}$, $\frac{1}{13} a^{14}$, $\frac{1}{13} a^{15}$, $\frac{1}{13} a^{16}$, $\frac{1}{13} a^{17}$, $\frac{1}{13} a^{18}$, $\frac{1}{13} a^{19}$, $\frac{1}{6695} a^{20} + \frac{121}{6695} a^{18} + \frac{89}{6695} a^{16} - \frac{88}{6695} a^{14} - \frac{251}{6695} a^{12} - \frac{35}{103} a^{10} - \frac{106}{515} a^{8} - \frac{26}{515} a^{6} + \frac{31}{515} a^{4} - \frac{67}{515} a^{2} + \frac{161}{515}$, $\frac{1}{6695} a^{21} + \frac{121}{6695} a^{19} + \frac{89}{6695} a^{17} - \frac{88}{6695} a^{15} - \frac{251}{6695} a^{13} - \frac{35}{103} a^{11} - \frac{106}{515} a^{9} - \frac{26}{515} a^{7} + \frac{31}{515} a^{5} - \frac{67}{515} a^{3} + \frac{161}{515} a$, $\frac{1}{2473273595} a^{22} + \frac{7758}{190251815} a^{20} - \frac{4002321}{190251815} a^{18} - \frac{5779922}{190251815} a^{16} + \frac{1117145}{38050363} a^{14} - \frac{3352671}{190251815} a^{12} - \frac{5497556}{190251815} a^{10} - \frac{2275763}{14634755} a^{8} + \frac{3088396}{14634755} a^{6} - \frac{5535798}{14634755} a^{4} - \frac{388110}{2926951} a^{2} - \frac{6915369}{14634755}$, $\frac{1}{2473273595} a^{23} + \frac{7758}{190251815} a^{21} - \frac{4002321}{190251815} a^{19} - \frac{5779922}{190251815} a^{17} + \frac{1117145}{38050363} a^{15} - \frac{3352671}{190251815} a^{13} - \frac{5497556}{190251815} a^{11} - \frac{2275763}{14634755} a^{9} + \frac{3088396}{14634755} a^{7} - \frac{5535798}{14634755} a^{5} - \frac{388110}{2926951} a^{3} - \frac{6915369}{14634755} a$

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

Class group and class number

$C_{2}\times C_{52}$, which has order $104$ (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:  \( -\frac{838684}{494654719} a^{22} - \frac{4095549}{190251815} a^{20} - \frac{32345379}{190251815} a^{18} - \frac{12507737}{14634755} a^{16} - \frac{604368468}{190251815} a^{14} - \frac{1608302916}{190251815} a^{12} - \frac{647438924}{38050363} a^{10} - \frac{352051821}{14634755} a^{8} - \frac{366180111}{14634755} a^{6} - \frac{230910889}{14634755} a^{4} - \frac{113228487}{14634755} a^{2} - \frac{7855309}{14634755} \) (order $6$)
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:  \( 14479835.771174202 \) (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

\(\Q(\sqrt{-39}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-3}) \), 3.3.169.1, \(\Q(\sqrt{-3}, \sqrt{13})\), 4.0.316368.2, 4.4.35152.1, 6.0.10024911.1, \(\Q(\zeta_{13})^+\), 6.0.771147.1, 8.0.100088711424.2, 12.0.100498840557921.1, 12.0.5351362262028177408.1, \(\Q(\zeta_{52})^+\)

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 ${\href{/LocalNumberField/5.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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}$
$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$13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$
13.12.11.4$x^{12} - 832$$12$$1$$11$$C_{12}$$[\ ]_{12}$