↑

Properties

Label 24.0.14276491594...2256.2
Degree $24$
Signature $[0, 12]$
Discriminant $2^{36}\cdot 3^{36}\cdot 7^{12}$
Root discriminant $38.88$
Ramified primes $2, 3, 7$
Class number $39$ (GRH)
Class group $[39]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![64, 0, 0, 832, 0, 0, 7144, 0, 0, 48088, 0, 0, 212961, 0, 0, -10306, 0, 0, 25, 0, 0, -22, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 22*x^21 + 25*x^18 - 10306*x^15 + 212961*x^12 + 48088*x^9 + 7144*x^6 + 832*x^3 + 64)
 
gp: K = bnfinit(x^24 - 22*x^21 + 25*x^18 - 10306*x^15 + 212961*x^12 + 48088*x^9 + 7144*x^6 + 832*x^3 + 64, 1)
 

Normalized defining polynomial

\( x^{24} - 22 x^{21} + 25 x^{18} - 10306 x^{15} + 212961 x^{12} + 48088 x^{9} + 7144 x^{6} + 832 x^{3} + 64 \)

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:  \(142764915949024053774865343014036832256=2^{36}\cdot 3^{36}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.88$
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:  $2, 3, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(504=2^{3}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{504}(1,·)$, $\chi_{504}(323,·)$, $\chi_{504}(211,·)$, $\chi_{504}(449,·)$, $\chi_{504}(265,·)$, $\chi_{504}(139,·)$, $\chi_{504}(209,·)$, $\chi_{504}(83,·)$, $\chi_{504}(251,·)$, $\chi_{504}(281,·)$, $\chi_{504}(155,·)$, $\chi_{504}(491,·)$, $\chi_{504}(97,·)$, $\chi_{504}(419,·)$, $\chi_{504}(113,·)$, $\chi_{504}(337,·)$, $\chi_{504}(41,·)$, $\chi_{504}(43,·)$, $\chi_{504}(433,·)$, $\chi_{504}(307,·)$, $\chi_{504}(169,·)$, $\chi_{504}(377,·)$, $\chi_{504}(379,·)$, $\chi_{504}(475,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{13} + \frac{1}{4} a^{10} - \frac{1}{2} a^{7} + \frac{1}{4} a^{4}$, $\frac{1}{4} a^{17} + \frac{1}{4} a^{11} + \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{77720} a^{18} + \frac{647}{38860} a^{15} - \frac{28927}{77720} a^{12} - \frac{19311}{38860} a^{9} + \frac{1281}{77720} a^{6} - \frac{2131}{19430} a^{3} - \frac{1112}{9715}$, $\frac{1}{77720} a^{19} + \frac{647}{38860} a^{16} - \frac{28927}{77720} a^{13} - \frac{19311}{38860} a^{10} + \frac{1281}{77720} a^{7} - \frac{2131}{19430} a^{4} - \frac{1112}{9715} a$, $\frac{1}{155440} a^{20} + \frac{647}{77720} a^{17} - \frac{28927}{155440} a^{14} - \frac{19311}{77720} a^{11} - \frac{76439}{155440} a^{8} + \frac{17299}{38860} a^{5} + \frac{8603}{19430} a^{2}$, $\frac{1}{890826343575920} a^{21} - \frac{293978439}{89082634357592} a^{18} + \frac{166244692795097}{890826343575920} a^{15} + \frac{183874441564103}{445413171787960} a^{12} + \frac{264076791335009}{890826343575920} a^{9} + \frac{48009869551819}{111353292946990} a^{6} - \frac{3563514636893}{22270658589398} a^{3} + \frac{1005195594934}{55676646473495}$, $\frac{1}{1781652687151840} a^{22} - \frac{293978439}{178165268715184} a^{19} + \frac{166244692795097}{1781652687151840} a^{16} + \frac{183874441564103}{890826343575920} a^{13} + \frac{264076791335009}{1781652687151840} a^{10} - \frac{63343423395171}{222706585893980} a^{7} + \frac{18707143952505}{44541317178796} a^{4} + \frac{502597797467}{55676646473495} a$, $\frac{1}{1781652687151840} a^{23} - \frac{293978439}{178165268715184} a^{20} + \frac{166244692795097}{1781652687151840} a^{17} + \frac{183874441564103}{890826343575920} a^{14} + \frac{264076791335009}{1781652687151840} a^{11} - \frac{63343423395171}{222706585893980} a^{8} + \frac{18707143952505}{44541317178796} a^{5} + \frac{502597797467}{55676646473495} a^{2}$

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

Class group and class number

$C_{39}$, which has order $39$ (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:  \( -\frac{203320941819}{44541317178796} a^{22} + \frac{11221467104953}{111353292946990} a^{19} - \frac{13567418416093}{111353292946990} a^{16} + \frac{2619834157791652}{55676646473495} a^{13} - \frac{108648785957408681}{111353292946990} a^{10} - \frac{16116020821326977}{111353292946990} a^{7} - \frac{6127000972812919}{222706585893980} a^{4} - \frac{71786110120884}{55676646473495} a \) (order $18$)
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:  \( 221439677.80749813 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

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

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

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-42}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-2}, \sqrt{-3})\), \(\Q(\sqrt{6}, \sqrt{14})\), \(\Q(\sqrt{6}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{21})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{14})\), 6.6.10077696.1, 6.0.3359232.1, \(\Q(\zeta_{9})\), 6.6.6751269.1, 6.6.1152216576.1, 6.0.3456649728.1, 6.0.2250423.1, 8.0.796594176.1, 12.0.101559956668416.1, 12.12.11948427342082473984.1, 12.0.11948427342082473984.5, 12.0.11948427342082473984.6, 12.0.1327603038009163776.1, 12.0.45579633110361.1, 12.0.11948427342082473984.2

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$2$2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$3$3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
3.12.18.82$x^{12} - 9 x^{9} + 9 x^{8} - 9 x^{5} - 9 x^{4} - 9 x^{3} + 9$$6$$2$$18$$C_6\times C_2$$[2]_{2}^{2}$
$7$7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
7.12.6.1$x^{12} + 294 x^{8} + 3430 x^{6} + 21609 x^{4} + 487403 x^{2} + 2941225$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$