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Properties

Label 24.0.673...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $6.737\times 10^{31}$
Root discriminant $21.19$
Ramified primes $3, 5, 7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 2*x^22 + 5*x^21 - 4*x^20 + 8*x^19 + 15*x^18 - 59*x^17 + 26*x^16 + 114*x^15 + 34*x^14 - 119*x^13 - 10*x^12 - 196*x^11 - 198*x^10 + 289*x^9 + 559*x^8 - 307*x^7 + 22*x^6 + 46*x^5 - 22*x^4 + 12*x^3 - x^2 - 2*x + 1)
 
gp: K = bnfinit(x^24 - x^23 - 2*x^22 + 5*x^21 - 4*x^20 + 8*x^19 + 15*x^18 - 59*x^17 + 26*x^16 + 114*x^15 + 34*x^14 - 119*x^13 - 10*x^12 - 196*x^11 - 198*x^10 + 289*x^9 + 559*x^8 - 307*x^7 + 22*x^6 + 46*x^5 - 22*x^4 + 12*x^3 - x^2 - 2*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -2, -1, 12, -22, 46, 22, -307, 559, 289, -198, -196, -10, -119, 34, 114, 26, -59, 15, 8, -4, 5, -2, -1, 1]);
 

\( x^{24} - x^{23} - 2 x^{22} + 5 x^{21} - 4 x^{20} + 8 x^{19} + 15 x^{18} - 59 x^{17} + 26 x^{16} + 114 x^{15} + 34 x^{14} - 119 x^{13} - 10 x^{12} - 196 x^{11} - 198 x^{10} + 289 x^{9} + 559 x^{8} - 307 x^{7} + 22 x^{6} + 46 x^{5} - 22 x^{4} + 12 x^{3} - x^{2} - 2 x + 1 \)

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

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(67372672480923938907623291015625\)\(\medspace = 3^{12}\cdot 5^{18}\cdot 7^{16}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $21.19$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $3, 5, 7$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $24$
This field is Galois and abelian over $\Q$.
Conductor:  \(105=3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(2,·)$, $\chi_{105}(67,·)$, $\chi_{105}(4,·)$, $\chi_{105}(86,·)$, $\chi_{105}(71,·)$, $\chi_{105}(8,·)$, $\chi_{105}(74,·)$, $\chi_{105}(11,·)$, $\chi_{105}(79,·)$, $\chi_{105}(16,·)$, $\chi_{105}(22,·)$, $\chi_{105}(23,·)$, $\chi_{105}(88,·)$, $\chi_{105}(92,·)$, $\chi_{105}(29,·)$, $\chi_{105}(32,·)$, $\chi_{105}(37,·)$, $\chi_{105}(43,·)$, $\chi_{105}(44,·)$, $\chi_{105}(46,·)$, $\chi_{105}(53,·)$, $\chi_{105}(58,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{181} a^{18} + \frac{43}{181} a^{17} - \frac{48}{181} a^{15} - \frac{73}{181} a^{14} - \frac{23}{181} a^{13} + \frac{48}{181} a^{12} + \frac{65}{181} a^{11} + \frac{18}{181} a^{10} + \frac{49}{181} a^{9} + \frac{8}{181} a^{8} + \frac{62}{181} a^{7} + \frac{1}{181} a^{6} - \frac{22}{181} a^{5} - \frac{80}{181} a^{4} - \frac{1}{181} a^{3} + \frac{39}{181} a + \frac{48}{181}$, $\frac{1}{181} a^{19} - \frac{39}{181} a^{17} - \frac{48}{181} a^{16} + \frac{39}{181} a^{14} - \frac{49}{181} a^{13} - \frac{8}{181} a^{12} - \frac{62}{181} a^{11} - \frac{1}{181} a^{10} + \frac{73}{181} a^{9} + \frac{80}{181} a^{8} + \frac{50}{181} a^{7} - \frac{65}{181} a^{6} - \frac{39}{181} a^{5} + \frac{43}{181} a^{3} + \frac{39}{181} a^{2} - \frac{73}{181}$, $\frac{1}{2353} a^{20} - \frac{3}{2353} a^{19} + \frac{841}{2353} a^{17} + \frac{506}{2353} a^{16} - \frac{1109}{2353} a^{15} + \frac{969}{2353} a^{14} + \frac{509}{2353} a^{13} - \frac{881}{2353} a^{12} + \frac{186}{2353} a^{11} - \frac{127}{2353} a^{10} + \frac{505}{2353} a^{9} - \frac{59}{2353} a^{8} - \frac{1055}{2353} a^{7} + \frac{738}{2353} a^{6} - \frac{922}{2353} a^{5} + \frac{5}{13} a^{4} - \frac{310}{2353} a^{3} - \frac{9}{181} a^{2} + \frac{3}{13} a + \frac{100}{2353}$, $\frac{1}{184806973} a^{21} + \frac{26051}{184806973} a^{20} + \frac{32884}{184806973} a^{19} - \frac{111037}{184806973} a^{18} - \frac{28576544}{184806973} a^{17} + \frac{82830352}{184806973} a^{16} - \frac{79766532}{184806973} a^{15} - \frac{18675511}{184806973} a^{14} + \frac{31433071}{184806973} a^{13} - \frac{44362867}{184806973} a^{12} - \frac{45523675}{184806973} a^{11} + \frac{35106634}{184806973} a^{10} + \frac{28795990}{184806973} a^{9} + \frac{43507812}{184806973} a^{8} - \frac{9371057}{184806973} a^{7} + \frac{31125778}{184806973} a^{6} - \frac{54307542}{184806973} a^{5} + \frac{71515657}{184806973} a^{4} + \frac{22450640}{184806973} a^{3} + \frac{86998103}{184806973} a^{2} + \frac{86706103}{184806973} a + \frac{1004489}{184806973}$, $\frac{1}{184806973} a^{22} - \frac{27477}{184806973} a^{20} + \frac{31648}{184806973} a^{19} + \frac{3906}{14215921} a^{18} - \frac{7356750}{184806973} a^{17} + \frac{40972094}{184806973} a^{16} - \frac{41877687}{184806973} a^{15} - \frac{86032801}{184806973} a^{14} + \frac{63509435}{184806973} a^{13} + \frac{47239093}{184806973} a^{12} - \frac{75434023}{184806973} a^{11} + \frac{87678347}{184806973} a^{10} - \frac{74305633}{184806973} a^{9} + \frac{88509977}{184806973} a^{8} + \frac{8927168}{184806973} a^{7} + \frac{20183353}{184806973} a^{6} - \frac{42222595}{184806973} a^{5} - \frac{19460453}{184806973} a^{4} - \frac{1221923}{184806973} a^{3} - \frac{3667721}{14215921} a^{2} + \frac{15021030}{184806973} a + \frac{25166397}{184806973}$, $\frac{1}{184806973} a^{23} + \frac{12301}{184806973} a^{20} + \frac{225864}{184806973} a^{19} - \frac{326564}{184806973} a^{18} + \frac{60847156}{184806973} a^{17} + \frac{78740134}{184806973} a^{16} + \frac{34096128}{184806973} a^{15} - \frac{65004216}{184806973} a^{14} + \frac{91789795}{184806973} a^{13} + \frac{90467158}{184806973} a^{12} - \frac{62490772}{184806973} a^{11} - \frac{91927281}{184806973} a^{10} - \frac{91699176}{184806973} a^{9} + \frac{1223361}{184806973} a^{8} - \frac{53521058}{184806973} a^{7} - \frac{76466613}{184806973} a^{6} - \frac{92286394}{184806973} a^{5} - \frac{44212596}{184806973} a^{4} + \frac{5004964}{184806973} a^{3} + \frac{3851427}{14215921} a^{2} - \frac{64123171}{184806973} a + \frac{68503572}{184806973}$

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{1800399}{184806973} a^{23} - \frac{2416974}{184806973} a^{22} + \frac{7818171}{184806973} a^{21} - \frac{7596204}{184806973} a^{20} - \frac{9634676}{184806973} a^{19} + \frac{17930001}{184806973} a^{18} - \frac{88860789}{184806973} a^{17} + \frac{59832438}{184806973} a^{16} + \frac{150000366}{184806973} a^{15} - \frac{436806393}{184806973} a^{14} - \frac{180705801}{184806973} a^{13} + \frac{32900442}{184806973} a^{12} - \frac{271588956}{184806973} a^{11} - \frac{201003450}{184806973} a^{10} + \frac{1771395312}{184806973} a^{9} + \frac{671795457}{184806973} a^{8} - \frac{652163709}{184806973} a^{7} + \frac{152072058}{184806973} a^{6} + \frac{55985010}{184806973} a^{5} - \frac{4737629686}{184806973} a^{4} + \frac{25550868}{184806973} a^{3} - \frac{6141087}{184806973} a^{2} - \frac{2416974}{184806973} a + \frac{2195007}{184806973} \) (order $30$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 8057321.833968681 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 8057321.833968681 \cdot 1}{30\sqrt{67372672480923938907623291015625}}\approx 0.123875662215965$ (assuming GRH)

Galois group

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

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), 6.0.64827.1, 6.6.300125.1, 6.0.8103375.1, \(\Q(\zeta_{15})\), 12.0.65664686390625.1, 12.12.8208085798828125.1, 12.0.11259376953125.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
3Data not computed
5Data not computed
$7$7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
7.12.8.1$x^{12} - 63 x^{9} + 637 x^{6} + 6174 x^{3} + 300125$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$