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Properties

Label 24.0.11500519106...0625.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{36}\cdot 5^{12}\cdot 11^{12}$
Root discriminant $38.54$
Ramified primes $3, 5, 11$
Class number $42$ (GRH)
Class group $[42]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16777216, 0, 0, 0, 0, 0, 479232, 0, 0, 0, 0, 0, 9593, 0, 0, 0, 0, 0, 117, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 117*x^18 + 9593*x^12 + 479232*x^6 + 16777216)
 
gp: K = bnfinit(x^24 + 117*x^18 + 9593*x^12 + 479232*x^6 + 16777216, 1)
 

Normalized defining polynomial

\( x^{24} + 117 x^{18} + 9593 x^{12} + 479232 x^{6} + 16777216 \)

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:  \(115005191066819204356102017148681640625=3^{36}\cdot 5^{12}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $38.54$
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:  $3, 5, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(495=3^{2}\cdot 5\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{495}(1,·)$, $\chi_{495}(386,·)$, $\chi_{495}(131,·)$, $\chi_{495}(199,·)$, $\chi_{495}(329,·)$, $\chi_{495}(331,·)$, $\chi_{495}(76,·)$, $\chi_{495}(461,·)$, $\chi_{495}(274,·)$, $\chi_{495}(406,·)$, $\chi_{495}(89,·)$, $\chi_{495}(221,·)$, $\chi_{495}(34,·)$, $\chi_{495}(419,·)$, $\chi_{495}(164,·)$, $\chi_{495}(166,·)$, $\chi_{495}(296,·)$, $\chi_{495}(364,·)$, $\chi_{495}(109,·)$, $\chi_{495}(494,·)$, $\chi_{495}(241,·)$, $\chi_{495}(439,·)$, $\chi_{495}(56,·)$, $\chi_{495}(254,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{10} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{2}$, $\frac{1}{14} a^{12} + \frac{3}{7} a^{6} - \frac{1}{2} a^{3} - \frac{3}{7}$, $\frac{1}{28} a^{13} - \frac{1}{28} a^{7} - \frac{1}{2} a^{4} - \frac{13}{28} a$, $\frac{1}{112} a^{14} + \frac{13}{112} a^{8} - \frac{1}{2} a^{5} + \frac{1}{112} a^{2}$, $\frac{1}{448} a^{15} - \frac{43}{448} a^{9} - \frac{1}{2} a^{6} - \frac{167}{448} a^{3}$, $\frac{1}{1792} a^{16} - \frac{267}{1792} a^{10} - \frac{391}{1792} a^{4} - \frac{1}{2} a$, $\frac{1}{7168} a^{17} + \frac{629}{7168} a^{11} + \frac{1401}{7168} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{550100992} a^{18} - \frac{1}{896} a^{15} - \frac{5}{8192} a^{12} - \frac{181}{896} a^{9} - \frac{585}{8192} a^{6} - \frac{57}{896} a^{3} - \frac{57441}{134302}$, $\frac{1}{2200403968} a^{19} - \frac{1}{3584} a^{16} - \frac{5}{32768} a^{13} - \frac{629}{3584} a^{10} + \frac{3511}{32768} a^{7} - \frac{1401}{3584} a^{4} + \frac{211163}{537208} a$, $\frac{1}{8801615872} a^{20} - \frac{1}{14336} a^{17} - \frac{5}{131072} a^{14} + \frac{2955}{14336} a^{11} + \frac{3511}{131072} a^{8} + \frac{5767}{14336} a^{5} + \frac{748371}{2148832} a^{2}$, $\frac{1}{35206463488} a^{21} + \frac{4061}{3670016} a^{15} - \frac{610303}{3670016} a^{9} - \frac{213415}{2148832} a^{3} - \frac{1}{2}$, $\frac{1}{140825853952} a^{22} + \frac{4061}{14680064} a^{16} - \frac{1}{56} a^{13} - \frac{2445311}{14680064} a^{10} - \frac{13}{56} a^{7} + \frac{1935417}{8595328} a^{4} - \frac{1}{56} a$, $\frac{1}{563303415808} a^{23} + \frac{4061}{58720256} a^{17} - \frac{1}{224} a^{14} + \frac{12234753}{58720256} a^{11} + \frac{43}{224} a^{8} - \frac{6659911}{34381312} a^{5} - \frac{57}{224} a^{2}$

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

Class group and class number

$C_{42}$, which has order $42$ (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:  \( -\frac{1173}{70412926976} a^{22} + \frac{95}{7340032} a^{16} - \frac{1173}{7340032} a^{10} - \frac{137241}{17190656} a^{4} \) (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:  \( 213596502.21615326 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{165}) \), \(\Q(\sqrt{-55}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-3}, \sqrt{-55})\), \(\Q(\sqrt{-11}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{-11})\), \(\Q(\sqrt{-15}, \sqrt{33})\), \(\Q(\sqrt{5}, \sqrt{33})\), \(\Q(\zeta_{9})\), 6.0.8732691.1, 6.6.26198073.1, 6.0.2460375.1, 6.6.820125.1, 6.6.3274759125.1, 6.0.1091586375.3, 8.0.741200625.1, 12.0.686339028913329.1, 12.0.6053445140625.1, 12.0.10724047326770765625.1, 12.0.10724047326770765625.2, 12.0.1191560814085640625.1, 12.0.10724047326770765625.3, 12.12.10724047326770765625.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{4}$ R R ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$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}$
$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}$
$11$11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
11.12.6.1$x^{12} + 242 x^{8} + 21296 x^{6} + 14641 x^{4} + 1932612 x^{2} + 113379904$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$