Properties

Label 24.0.103...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $1.035\times 10^{31}$
Root discriminant $19.60$
Ramified primes $3, 5, 7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - x^22 + 4*x^21 - 4*x^20 - 4*x^19 + 17*x^18 + 12*x^17 - 46*x^16 + 43*x^15 + 44*x^14 - 188*x^13 + 189*x^12 + 188*x^11 + 44*x^10 - 43*x^9 - 46*x^8 - 12*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1)
 
gp: K = bnfinit(x^24 - x^23 - x^22 + 4*x^21 - 4*x^20 - 4*x^19 + 17*x^18 + 12*x^17 - 46*x^16 + 43*x^15 + 44*x^14 - 188*x^13 + 189*x^12 + 188*x^11 + 44*x^10 - 43*x^9 - 46*x^8 - 12*x^7 + 17*x^6 + 4*x^5 - 4*x^4 - 4*x^3 - x^2 + x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 1, -1, -4, -4, 4, 17, -12, -46, -43, 44, 188, 189, -188, 44, 43, -46, 12, 17, -4, -4, 4, -1, -1, 1]);
 

\(x^{24} - x^{23} - x^{22} + 4 x^{21} - 4 x^{20} - 4 x^{19} + 17 x^{18} + 12 x^{17} - 46 x^{16} + 43 x^{15} + 44 x^{14} - 188 x^{13} + 189 x^{12} + 188 x^{11} + 44 x^{10} - 43 x^{9} - 46 x^{8} - 12 x^{7} + 17 x^{6} + 4 x^{5} - 4 x^{4} - 4 x^{3} - x^{2} + x + 1\)  Toggle raw display

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:  \(10352754344108696148301025390625\)\(\medspace = 3^{12}\cdot 5^{12}\cdot 7^{20}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $19.60$
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$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(4,·)$, $\chi_{105}(71,·)$, $\chi_{105}(74,·)$, $\chi_{105}(11,·)$, $\chi_{105}(76,·)$, $\chi_{105}(79,·)$, $\chi_{105}(16,·)$, $\chi_{105}(19,·)$, $\chi_{105}(86,·)$, $\chi_{105}(89,·)$, $\chi_{105}(26,·)$, $\chi_{105}(29,·)$, $\chi_{105}(94,·)$, $\chi_{105}(31,·)$, $\chi_{105}(34,·)$, $\chi_{105}(101,·)$, $\chi_{105}(104,·)$, $\chi_{105}(41,·)$, $\chi_{105}(44,·)$, $\chi_{105}(46,·)$, $\chi_{105}(59,·)$, $\chi_{105}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{13} a^{14} - \frac{5}{13} a^{7} - \frac{1}{13}$, $\frac{1}{13} a^{15} - \frac{5}{13} a^{8} - \frac{1}{13} a$, $\frac{1}{13} a^{16} - \frac{5}{13} a^{9} - \frac{1}{13} a^{2}$, $\frac{1}{13} a^{17} - \frac{5}{13} a^{10} - \frac{1}{13} a^{3}$, $\frac{1}{26} a^{18} - \frac{1}{26} a^{15} - \frac{1}{2} a^{12} + \frac{4}{13} a^{11} - \frac{1}{2} a^{9} - \frac{4}{13} a^{8} - \frac{1}{2} a^{6} + \frac{6}{13} a^{4} - \frac{1}{2} a^{3} - \frac{6}{13} a - \frac{1}{2}$, $\frac{1}{10946} a^{19} + \frac{50}{5473} a^{18} + \frac{159}{5473} a^{17} + \frac{25}{842} a^{16} - \frac{10}{5473} a^{15} - \frac{206}{5473} a^{14} - \frac{159}{842} a^{13} + \frac{1694}{5473} a^{12} + \frac{1440}{5473} a^{11} + \frac{477}{10946} a^{10} - \frac{192}{421} a^{9} - \frac{340}{5473} a^{8} - \frac{2685}{10946} a^{7} + \frac{10}{421} a^{6} + \frac{2606}{5473} a^{5} + \frac{1525}{10946} a^{4} - \frac{1134}{5473} a^{3} + \frac{128}{421} a^{2} - \frac{3321}{10946} a + \frac{1909}{5473}$, $\frac{1}{10946} a^{20} + \frac{1}{10946} a^{18} - \frac{321}{10946} a^{17} + \frac{159}{5473} a^{16} + \frac{25}{842} a^{15} + \frac{401}{10946} a^{14} + \frac{1057}{5473} a^{13} - \frac{159}{842} a^{12} - \frac{2085}{10946} a^{11} - \frac{244}{5473} a^{10} + \frac{477}{10946} a^{9} + \frac{37}{842} a^{8} + \frac{1344}{5473} a^{7} - \frac{4369}{10946} a^{6} - \frac{401}{842} a^{5} + \frac{2606}{5473} a^{4} - \frac{3527}{10946} a^{3} + \frac{3205}{10946} a^{2} + \frac{128}{421} a + \frac{1731}{10946}$, $\frac{1}{10946} a^{21} + \frac{4181}{10946}$, $\frac{1}{10946} a^{22} + \frac{4181}{10946} a$, $\frac{1}{10946} a^{23} + \frac{4181}{10946} a^{2}$  Toggle raw display

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:  \( -\frac{6765}{10946} a^{23} + \frac{6765}{10946} a^{22} + \frac{6765}{10946} a^{21} - \frac{27061}{10946} a^{20} + \frac{13530}{5473} a^{19} + \frac{13530}{5473} a^{18} - \frac{115005}{10946} a^{17} - \frac{40590}{5473} a^{16} + \frac{155595}{5473} a^{15} - \frac{290895}{10946} a^{14} - \frac{148830}{5473} a^{13} + \frac{635910}{5473} a^{12} - \frac{1278585}{10946} a^{11} - \frac{635910}{5473} a^{10} - \frac{148830}{5473} a^{9} + \frac{290895}{10946} a^{8} + \frac{155595}{5473} a^{7} + \frac{40590}{5473} a^{6} - \frac{115005}{10946} a^{5} - \frac{13530}{5473} a^{4} + \frac{13530}{5473} a^{3} + \frac{13530}{5473} a^{2} + \frac{6765}{10946} a - \frac{6765}{10946} \) (order $42$)  Toggle raw display
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:  \( 4608312.303502872 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 4608312.303502872 \cdot 1}{42\sqrt{10352754344108696148301025390625}}\approx 0.129099059619775$ (assuming GRH)

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{5}, \sqrt{21})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), 6.0.2100875.1, 6.6.56723625.1, 6.0.64827.1, 6.6.300125.1, \(\Q(\zeta_{7})\), \(\Q(\zeta_{21})^+\), 6.0.8103375.1, 8.0.121550625.1, 12.0.3217569633140625.2, 12.0.4413675765625.1, 12.0.3217569633140625.1, 12.12.3217569633140625.1, 12.0.3217569633140625.3, 12.0.65664686390625.1, \(\Q(\zeta_{21})\)

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{4}$ R R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$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$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$