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Properties

Label 24.0.16176178662...5625.2
Degree $24$
Signature $[0, 12]$
Discriminant $3^{12}\cdot 5^{18}\cdot 7^{20}$
Root discriminant $29.31$
Ramified primes $3, 5, 7$
Class number $26$ (GRH)
Class group $[26]$ (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![1, 4, 20, 95, 455, 2176, 10409, -7509, 8969, -6745, 5996, -4456, 3936, -1331, 1106, -380, 299, -99, 74, -19, 20, -5, 5, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 + 5*x^22 - 5*x^21 + 20*x^20 - 19*x^19 + 74*x^18 - 99*x^17 + 299*x^16 - 380*x^15 + 1106*x^14 - 1331*x^13 + 3936*x^12 - 4456*x^11 + 5996*x^10 - 6745*x^9 + 8969*x^8 - 7509*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^24 - x^23 + 5*x^22 - 5*x^21 + 20*x^20 - 19*x^19 + 74*x^18 - 99*x^17 + 299*x^16 - 380*x^15 + 1106*x^14 - 1331*x^13 + 3936*x^12 - 4456*x^11 + 5996*x^10 - 6745*x^9 + 8969*x^8 - 7509*x^7 + 10409*x^6 + 2176*x^5 + 455*x^4 + 95*x^3 + 20*x^2 + 4*x + 1, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} + 5 x^{22} - 5 x^{21} + 20 x^{20} - 19 x^{19} + 74 x^{18} - 99 x^{17} + 299 x^{16} - 380 x^{15} + 1106 x^{14} - 1331 x^{13} + 3936 x^{12} - 4456 x^{11} + 5996 x^{10} - 6745 x^{9} + 8969 x^{8} - 7509 x^{7} + 10409 x^{6} + 2176 x^{5} + 455 x^{4} + 95 x^{3} + 20 x^{2} + 4 x + 1 \)

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:  \(161761786626698377317203521728515625=3^{12}\cdot 5^{18}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.31$
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:  $3, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(105=3\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{105}(64,·)$, $\chi_{105}(1,·)$, $\chi_{105}(2,·)$, $\chi_{105}(4,·)$, $\chi_{105}(8,·)$, $\chi_{105}(76,·)$, $\chi_{105}(79,·)$, $\chi_{105}(16,·)$, $\chi_{105}(17,·)$, $\chi_{105}(19,·)$, $\chi_{105}(23,·)$, $\chi_{105}(68,·)$, $\chi_{105}(92,·)$, $\chi_{105}(94,·)$, $\chi_{105}(31,·)$, $\chi_{105}(32,·)$, $\chi_{105}(34,·)$, $\chi_{105}(38,·)$, $\chi_{105}(46,·)$, $\chi_{105}(47,·)$, $\chi_{105}(83,·)$, $\chi_{105}(53,·)$, $\chi_{105}(61,·)$, $\chi_{105}(62,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $\frac{1}{2146586119} a^{19} - \frac{80671238}{2146586119} a^{18} - \frac{363514156}{2146586119} a^{17} + \frac{40829208}{2146586119} a^{16} + \frac{329015334}{2146586119} a^{15} + \frac{607502231}{2146586119} a^{14} - \frac{911196026}{2146586119} a^{13} - \frac{744303332}{2146586119} a^{12} + \frac{227825652}{2146586119} a^{11} + \frac{10477406}{2146586119} a^{10} + \frac{981394019}{2146586119} a^{9} + \frac{716121545}{2146586119} a^{8} + \frac{876952384}{2146586119} a^{7} - \frac{537125891}{2146586119} a^{6} + \frac{479576926}{2146586119} a^{5} + \frac{489657714}{2146586119} a^{4} - \frac{256694009}{2146586119} a^{3} + \frac{377586222}{2146586119} a^{2} + \frac{74691672}{2146586119} a - \frac{80780534}{2146586119}$, $\frac{1}{2146586119} a^{20} - \frac{935272120}{2146586119} a^{18} + \frac{674003709}{2146586119} a^{17} - \frac{121919952}{2146586119} a^{16} - \frac{923153692}{2146586119} a^{15} - \frac{226411398}{2146586119} a^{14} + \frac{744303298}{2146586119} a^{13} + \frac{151546479}{2146586119} a^{12} - \frac{642200301}{2146586119} a^{11} - \frac{576258079}{2146586119} a^{10} + \frac{15925612}{2146586119} a^{9} - \frac{748133373}{2146586119} a^{8} - \frac{850996194}{2146586119} a^{7} - \frac{510269076}{2146586119} a^{6} - \frac{617694871}{2146586119} a^{5} + \frac{572504157}{2146586119} a^{4} - \frac{907932223}{2146586119} a^{3} - \frac{909147676}{2146586119} a^{2} + \frac{185130283}{2146586119} a + \frac{560390440}{2146586119}$, $\frac{1}{2146586119} a^{21} - \frac{1067568172}{2146586119} a^{14} + \frac{991307695}{2146586119} a^{7} + \frac{401941670}{2146586119}$, $\frac{1}{2146586119} a^{22} - \frac{1067568172}{2146586119} a^{15} + \frac{991307695}{2146586119} a^{8} + \frac{401941670}{2146586119} a$, $\frac{1}{2146586119} a^{23} - \frac{1067568172}{2146586119} a^{16} + \frac{991307695}{2146586119} a^{9} + \frac{401941670}{2146586119} a^{2}$

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

Class group and class number

$C_{26}$, which has order $26$ (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:  \( -\frac{375299640}{2146586119} a^{23} + \frac{469124550}{2146586119} a^{22} - \frac{1970323110}{2146586119} a^{21} + \frac{2345622750}{2146586119} a^{20} - \frac{7975180155}{2146586119} a^{19} + \frac{9007191360}{2146586119} a^{18} - \frac{29554846650}{2146586119} a^{17} + \frac{44097707700}{2146586119} a^{16} - \frac{121503258450}{2146586119} a^{15} + \frac{170667511290}{2146586119} a^{14} - \frac{450734867640}{2146586119} a^{13} + \frac{603295786795}{2146586119} a^{12} - \frac{1602060338250}{2146586119} a^{11} + \frac{2041630041600}{2146586119} a^{10} - \frac{2668380440400}{2146586119} a^{9} + \frac{3093970232160}{2146586119} a^{8} - \frac{3998911489110}{2146586119} a^{7} + \frac{126194503950}{74020211} a^{6} - \frac{4610521190949}{2146586119} a^{5} + \frac{159971471550}{2146586119} a^{4} + \frac{33401667960}{2146586119} a^{3} + \frac{7036868250}{2146586119} a^{2} + \frac{1407373650}{2146586119} a + \frac{375299640}{2146586119} \) (order $14$)
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:  \( 16114643.667937363 \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\zeta_{15})^+\), 4.0.55125.1, 6.0.2100875.1, 6.6.300125.1, \(\Q(\zeta_{7})\), 8.0.3038765625.2, 12.0.4413675765625.1, 12.12.8208085798828125.1, 12.0.402196204142578125.1

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 ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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

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$3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
3.12.6.1$x^{12} - 243 x^{2} + 1458$$2$$6$$6$$C_{12}$$[\ ]_{2}^{6}$
5Data not computed
7Data not computed