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Properties

Label 24.0.50720286974...0625.1
Degree $24$
Signature $[0, 12]$
Discriminant $3^{36}\cdot 5^{12}\cdot 7^{12}$
Root discriminant $30.74$
Ramified primes $3, 5, 7$
Class number $9$ (GRH)
Class group $[3, 3]$ (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![531441, 0, 0, 0, 0, 0, 18954, 0, 0, 0, 0, 0, -53, 0, 0, 0, 0, 0, 26, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 26*x^18 - 53*x^12 + 18954*x^6 + 531441)
 
gp: K = bnfinit(x^24 + 26*x^18 - 53*x^12 + 18954*x^6 + 531441, 1)
 

Normalized defining polynomial

\( x^{24} + 26 x^{18} - 53 x^{12} + 18954 x^{6} + 531441 \)

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:  \(507202869744901554493542558837890625=3^{36}\cdot 5^{12}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $30.74$
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:  \(315=3^{2}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{315}(64,·)$, $\chi_{315}(1,·)$, $\chi_{315}(134,·)$, $\chi_{315}(71,·)$, $\chi_{315}(139,·)$, $\chi_{315}(76,·)$, $\chi_{315}(209,·)$, $\chi_{315}(146,·)$, $\chi_{315}(211,·)$, $\chi_{315}(281,·)$, $\chi_{315}(29,·)$, $\chi_{315}(286,·)$, $\chi_{315}(34,·)$, $\chi_{315}(104,·)$, $\chi_{315}(41,·)$, $\chi_{315}(106,·)$, $\chi_{315}(274,·)$, $\chi_{315}(239,·)$, $\chi_{315}(176,·)$, $\chi_{315}(244,·)$, $\chi_{315}(181,·)$, $\chi_{315}(169,·)$, $\chi_{315}(314,·)$, $\chi_{315}(251,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{8} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{9} - \frac{1}{2}$, $\frac{1}{2} a^{10} - \frac{1}{2} a$, $\frac{1}{2} a^{11} - \frac{1}{2} a^{2}$, $\frac{1}{8} a^{12} + \frac{1}{8} a^{6} + \frac{1}{8}$, $\frac{1}{24} a^{13} + \frac{5}{24} a^{7} - \frac{1}{2} a^{4} - \frac{11}{24} a$, $\frac{1}{72} a^{14} + \frac{17}{72} a^{8} + \frac{1}{72} a^{2}$, $\frac{1}{216} a^{15} + \frac{53}{216} a^{9} + \frac{1}{216} a^{3} - \frac{1}{2}$, $\frac{1}{648} a^{16} - \frac{55}{648} a^{10} - \frac{215}{648} a^{4}$, $\frac{1}{1944} a^{17} + \frac{269}{1944} a^{11} + \frac{433}{1944} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{618192} a^{18} - \frac{1}{432} a^{15} + \frac{7}{2916} a^{12} + \frac{55}{432} a^{9} - \frac{547}{2916} a^{6} + \frac{215}{432} a^{3} + \frac{291}{848}$, $\frac{1}{1854576} a^{19} - \frac{1}{1296} a^{16} + \frac{7}{8748} a^{13} + \frac{55}{1296} a^{10} + \frac{911}{8748} a^{7} - \frac{433}{1296} a^{4} + \frac{715}{2544} a$, $\frac{1}{5563728} a^{20} - \frac{1}{3888} a^{17} + \frac{7}{26244} a^{14} + \frac{703}{3888} a^{11} + \frac{5285}{26244} a^{8} + \frac{1511}{3888} a^{5} + \frac{3259}{7632} a^{2}$, $\frac{1}{16691184} a^{21} - \frac{701}{314928} a^{15} - \frac{1}{16} a^{12} + \frac{61235}{314928} a^{9} - \frac{1}{16} a^{6} + \frac{3511}{11448} a^{3} - \frac{1}{16}$, $\frac{1}{50073552} a^{22} - \frac{701}{944784} a^{16} - \frac{1}{48} a^{13} - \frac{96229}{944784} a^{10} + \frac{7}{48} a^{7} - \frac{13661}{34344} a^{4} - \frac{1}{48} a$, $\frac{1}{150220656} a^{23} - \frac{701}{2834352} a^{17} - \frac{1}{144} a^{14} + \frac{376163}{2834352} a^{11} - \frac{17}{144} a^{8} - \frac{30833}{103032} a^{5} - \frac{1}{144} a^{2}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( -\frac{1}{34344} a^{22} - \frac{16039}{34344} a^{4} \) (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:  \( 70281481.48527941 \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{-35})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{-15}, \sqrt{21})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{-7}, \sqrt{-15})\), \(\Q(\sqrt{5}, \sqrt{21})\), 6.0.281302875.3, \(\Q(\zeta_{9})\), 6.6.843908625.1, 6.0.2250423.1, 6.6.820125.1, 6.6.6751269.1, 6.0.2460375.1, 8.0.121550625.1, 12.0.712181767349390625.1, 12.0.79131307483265625.1, 12.0.712181767349390625.3, 12.0.45579633110361.1, 12.0.6053445140625.1, 12.0.712181767349390625.2, 12.12.712181767349390625.1

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 R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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}$
$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}$