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Properties

Label 24.0.41672433230...5625.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{12}\cdot 5^{12}\cdot 13^{22}$
Root discriminant $40.66$
Ramified primes $3, 5, 13$
Class number $256$ (GRH)
Class group $[2, 2, 8, 8]$ (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![67861, -15272, -13105, 18947, -869, 19186, -1326, 15854, 5663, 3087, 13614, -2482, 14455, -2863, 9699, -1709, 4301, -648, 1248, -150, 226, -19, 23, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 23*x^22 - 19*x^21 + 226*x^20 - 150*x^19 + 1248*x^18 - 648*x^17 + 4301*x^16 - 1709*x^15 + 9699*x^14 - 2863*x^13 + 14455*x^12 - 2482*x^11 + 13614*x^10 + 3087*x^9 + 5663*x^8 + 15854*x^7 - 1326*x^6 + 19186*x^5 - 869*x^4 + 18947*x^3 - 13105*x^2 - 15272*x + 67861)
 
gp: K = bnfinit(x^24 - x^23 + 23*x^22 - 19*x^21 + 226*x^20 - 150*x^19 + 1248*x^18 - 648*x^17 + 4301*x^16 - 1709*x^15 + 9699*x^14 - 2863*x^13 + 14455*x^12 - 2482*x^11 + 13614*x^10 + 3087*x^9 + 5663*x^8 + 15854*x^7 - 1326*x^6 + 19186*x^5 - 869*x^4 + 18947*x^3 - 13105*x^2 - 15272*x + 67861, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} + 23 x^{22} - 19 x^{21} + 226 x^{20} - 150 x^{19} + 1248 x^{18} - 648 x^{17} + 4301 x^{16} - 1709 x^{15} + 9699 x^{14} - 2863 x^{13} + 14455 x^{12} - 2482 x^{11} + 13614 x^{10} + 3087 x^{9} + 5663 x^{8} + 15854 x^{7} - 1326 x^{6} + 19186 x^{5} - 869 x^{4} + 18947 x^{3} - 13105 x^{2} - 15272 x + 67861 \)

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:  \(416724332309376501938461264338134765625=3^{12}\cdot 5^{12}\cdot 13^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $40.66$
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:  $3, 5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(195=3\cdot 5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{195}(1,·)$, $\chi_{195}(194,·)$, $\chi_{195}(134,·)$, $\chi_{195}(71,·)$, $\chi_{195}(74,·)$, $\chi_{195}(11,·)$, $\chi_{195}(14,·)$, $\chi_{195}(16,·)$, $\chi_{195}(19,·)$, $\chi_{195}(86,·)$, $\chi_{195}(154,·)$, $\chi_{195}(29,·)$, $\chi_{195}(161,·)$, $\chi_{195}(34,·)$, $\chi_{195}(166,·)$, $\chi_{195}(41,·)$, $\chi_{195}(109,·)$, $\chi_{195}(176,·)$, $\chi_{195}(179,·)$, $\chi_{195}(181,·)$, $\chi_{195}(184,·)$, $\chi_{195}(121,·)$, $\chi_{195}(124,·)$, $\chi_{195}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{233} a^{13} + \frac{13}{233} a^{11} + \frac{65}{233} a^{9} - \frac{77}{233} a^{7} - \frac{51}{233} a^{5} + \frac{91}{233} a^{3} + \frac{13}{233} a - \frac{89}{233}$, $\frac{1}{233} a^{14} + \frac{13}{233} a^{12} + \frac{65}{233} a^{10} - \frac{77}{233} a^{8} - \frac{51}{233} a^{6} + \frac{91}{233} a^{4} + \frac{13}{233} a^{2} - \frac{89}{233} a$, $\frac{1}{233} a^{15} - \frac{104}{233} a^{11} + \frac{10}{233} a^{9} + \frac{18}{233} a^{7} + \frac{55}{233} a^{5} - \frac{5}{233} a^{3} - \frac{89}{233} a^{2} + \frac{64}{233} a - \frac{8}{233}$, $\frac{1}{233} a^{16} - \frac{104}{233} a^{12} + \frac{10}{233} a^{10} + \frac{18}{233} a^{8} + \frac{55}{233} a^{6} - \frac{5}{233} a^{4} - \frac{89}{233} a^{3} + \frac{64}{233} a^{2} - \frac{8}{233} a$, $\frac{1}{233} a^{17} - \frac{36}{233} a^{11} + \frac{21}{233} a^{9} - \frac{31}{233} a^{7} + \frac{50}{233} a^{5} - \frac{89}{233} a^{4} - \frac{25}{233} a^{3} - \frac{8}{233} a^{2} - \frac{46}{233} a + \frac{64}{233}$, $\frac{1}{466} a^{18} - \frac{1}{466} a^{15} - \frac{1}{466} a^{14} - \frac{1}{466} a^{13} - \frac{49}{466} a^{12} - \frac{71}{233} a^{11} + \frac{189}{466} a^{10} + \frac{79}{233} a^{9} + \frac{23}{233} a^{8} + \frac{59}{466} a^{7} + \frac{101}{466} a^{6} - \frac{93}{466} a^{5} + \frac{117}{466} a^{4} + \frac{139}{466} a^{3} + \frac{15}{233} a^{2} - \frac{157}{466} a + \frac{97}{466}$, $\frac{1}{63245986} a^{19} + \frac{32561}{63245986} a^{18} + \frac{51427}{31622993} a^{17} - \frac{125263}{63245986} a^{16} + \frac{57531}{31622993} a^{15} - \frac{29786}{31622993} a^{14} + \frac{47032}{31622993} a^{13} + \frac{10108411}{63245986} a^{12} + \frac{13054653}{63245986} a^{11} + \frac{9990991}{63245986} a^{10} + \frac{9945876}{31622993} a^{9} - \frac{30310093}{63245986} a^{8} - \frac{6537833}{31622993} a^{7} - \frac{7460683}{31622993} a^{6} + \frac{10272698}{31622993} a^{5} - \frac{7976936}{31622993} a^{4} - \frac{6641993}{63245986} a^{3} + \frac{29993617}{63245986} a^{2} - \frac{3862333}{31622993} a - \frac{6973611}{63245986}$, $\frac{1}{63245986} a^{20} + \frac{432}{31622993} a^{18} - \frac{102961}{63245986} a^{17} + \frac{116113}{63245986} a^{16} - \frac{15149}{63245986} a^{15} - \frac{42297}{63245986} a^{14} + \frac{31480}{31622993} a^{13} - \frac{12041341}{63245986} a^{12} + \frac{4138862}{31622993} a^{11} - \frac{13026613}{31622993} a^{10} - \frac{2874747}{63245986} a^{9} - \frac{21019525}{63245986} a^{8} - \frac{24031047}{63245986} a^{7} - \frac{16215353}{63245986} a^{6} - \frac{25376865}{63245986} a^{5} + \frac{13439846}{31622993} a^{4} + \frac{20778093}{63245986} a^{3} + \frac{20328291}{63245986} a^{2} - \frac{11387528}{31622993} a + \frac{13640042}{31622993}$, $\frac{1}{63245986} a^{21} - \frac{5697}{63245986} a^{18} + \frac{11791}{63245986} a^{17} - \frac{93275}{63245986} a^{16} - \frac{108093}{63245986} a^{15} - \frac{20406}{31622993} a^{14} + \frac{63411}{63245986} a^{13} + \frac{14714723}{31622993} a^{12} - \frac{14107764}{31622993} a^{11} - \frac{18164681}{63245986} a^{10} + \frac{4643967}{63245986} a^{9} - \frac{11805639}{63245986} a^{8} + \frac{28803403}{63245986} a^{7} + \frac{10104471}{63245986} a^{6} + \frac{9503381}{31622993} a^{5} - \frac{27489377}{63245986} a^{4} + \frac{19369153}{63245986} a^{3} + \frac{4397434}{31622993} a^{2} - \frac{1070062}{31622993} a + \frac{3801103}{31622993}$, $\frac{1}{63245986} a^{22} - \frac{18799}{63245986} a^{18} + \frac{94127}{63245986} a^{17} - \frac{55193}{31622993} a^{16} + \frac{70693}{63245986} a^{15} + \frac{59974}{31622993} a^{14} + \frac{34089}{63245986} a^{13} + \frac{130780}{31622993} a^{12} + \frac{2707667}{31622993} a^{11} - \frac{24533301}{63245986} a^{10} + \frac{1299909}{63245986} a^{9} - \frac{49090}{31622993} a^{8} + \frac{4456641}{31622993} a^{7} + \frac{26669013}{63245986} a^{6} + \frac{3139277}{31622993} a^{5} + \frac{10917732}{31622993} a^{4} - \frac{11759444}{31622993} a^{3} - \frac{5857589}{63245986} a^{2} - \frac{13865385}{63245986} a + \frac{14113192}{31622993}$, $\frac{1}{63245986} a^{23} - \frac{29065}{63245986} a^{18} - \frac{19703}{31622993} a^{17} + \frac{5453}{31622993} a^{16} - \frac{86533}{63245986} a^{15} - \frac{12984}{31622993} a^{14} - \frac{9655}{63245986} a^{13} + \frac{7470208}{31622993} a^{12} - \frac{10954761}{31622993} a^{11} + \frac{8098335}{63245986} a^{10} + \frac{12453262}{31622993} a^{9} - \frac{13951759}{63245986} a^{8} + \frac{2416648}{31622993} a^{7} - \frac{21769537}{63245986} a^{6} + \frac{7019573}{63245986} a^{5} - \frac{2784619}{63245986} a^{4} - \frac{25857867}{63245986} a^{3} - \frac{505133}{31622993} a^{2} + \frac{22435407}{63245986} a + \frac{1325520}{31622993}$

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

Class group and class number

$C_{2}\times C_{2}\times C_{8}\times C_{8}$, which has order $256$ (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:  \( -1 \) (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:  \( 5703268.037899434 \) (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

\(\Q(\sqrt{-195}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-15}) \), 3.3.169.1, \(\Q(\sqrt{13}, \sqrt{-15})\), 4.0.54925.1, 4.4.19773.1, 6.0.1253113875.1, \(\Q(\zeta_{13})^+\), 6.0.96393375.1, 8.0.244357205625.2, 12.0.1570294383717515625.4, 12.0.28002506156828125.1, \(\Q(\zeta_{39})^+\)

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
$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}$
$5$5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
5.8.4.1$x^{8} + 10 x^{6} + 125 x^{4} + 2500$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$
13Data not computed