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Properties

Label 24.0.79030965520...1056.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 3^{36}\cdot 11^{12}$
Root discriminant $34.47$
Ramified primes $2, 3, 11$
Class number $39$ (GRH)
Class group $[39]$ (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, -7290, 0, 0, 0, 0, 0, -629, 0, 0, 0, 0, 0, -10, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 10*x^18 - 629*x^12 - 7290*x^6 + 531441)
 
gp: K = bnfinit(x^24 - 10*x^18 - 629*x^12 - 7290*x^6 + 531441, 1)
 

Normalized defining polynomial

\( x^{24} - 10 x^{18} - 629 x^{12} - 7290 x^{6} + 531441 \)

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:  \(7903096552035517335267182427091501056=2^{24}\cdot 3^{36}\cdot 11^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.47$
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:  $2, 3, 11$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(396=2^{2}\cdot 3^{2}\cdot 11\)
Dirichlet character group:    $\lbrace$$\chi_{396}(1,·)$, $\chi_{396}(131,·)$, $\chi_{396}(133,·)$, $\chi_{396}(263,·)$, $\chi_{396}(265,·)$, $\chi_{396}(395,·)$, $\chi_{396}(67,·)$, $\chi_{396}(23,·)$, $\chi_{396}(89,·)$, $\chi_{396}(331,·)$, $\chi_{396}(155,·)$, $\chi_{396}(221,·)$, $\chi_{396}(199,·)$, $\chi_{396}(287,·)$, $\chi_{396}(353,·)$, $\chi_{396}(197,·)$, $\chi_{396}(65,·)$, $\chi_{396}(43,·)$, $\chi_{396}(109,·)$, $\chi_{396}(175,·)$, $\chi_{396}(241,·)$, $\chi_{396}(307,·)$, $\chi_{396}(373,·)$, $\chi_{396}(329,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{16} a^{12} + \frac{3}{16} a^{6} - \frac{7}{16}$, $\frac{1}{48} a^{13} - \frac{13}{48} a^{7} - \frac{23}{48} a$, $\frac{1}{144} a^{14} + \frac{35}{144} a^{8} - \frac{71}{144} a^{2}$, $\frac{1}{432} a^{15} + \frac{179}{432} a^{9} + \frac{73}{432} a^{3}$, $\frac{1}{1296} a^{16} - \frac{253}{1296} a^{10} + \frac{505}{1296} a^{4}$, $\frac{1}{3888} a^{17} - \frac{253}{3888} a^{11} + \frac{1801}{3888} a^{5}$, $\frac{1}{7336656} a^{18} + \frac{73}{11664} a^{12} - \frac{2917}{11664} a^{6} + \frac{1253}{5032}$, $\frac{1}{22009968} a^{19} + \frac{73}{34992} a^{13} + \frac{8747}{34992} a^{7} - \frac{3779}{15096} a$, $\frac{1}{66029904} a^{20} + \frac{73}{104976} a^{14} - \frac{26245}{104976} a^{8} + \frac{11317}{45288} a^{2}$, $\frac{1}{198089712} a^{21} + \frac{73}{314928} a^{15} - \frac{131221}{314928} a^{9} + \frac{56605}{135864} a^{3}$, $\frac{1}{594269136} a^{22} + \frac{73}{944784} a^{16} - \frac{131221}{944784} a^{10} + \frac{56605}{407592} a^{4}$, $\frac{1}{1782807408} a^{23} + \frac{73}{2834352} a^{17} - \frac{1076005}{2834352} a^{11} + \frac{56605}{1222776} a^{5}$

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:  \( -\frac{1801}{1782807408} a^{23} - \frac{253}{2834352} a^{17} + \frac{1801}{2834352} a^{11} + \frac{9005}{1222776} a^{5} \) (order $36$)
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:  \( 211275649.8529983 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{11}) \), \(\Q(\sqrt{-11}) \), \(\Q(\sqrt{33}) \), \(\Q(\sqrt{-33}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{11})\), \(\Q(i, \sqrt{33})\), \(\Q(\sqrt{3}, \sqrt{11})\), \(\Q(\sqrt{3}, \sqrt{-11})\), \(\Q(\sqrt{-3}, \sqrt{11})\), \(\Q(\sqrt{-3}, \sqrt{-11})\), 6.0.419904.1, \(\Q(\zeta_{36})^+\), \(\Q(\zeta_{9})\), 6.6.558892224.1, 6.0.8732691.1, 6.6.26198073.1, 6.0.1676676672.2, 8.0.303595776.1, \(\Q(\zeta_{36})\), 12.0.312360518047666176.1, 12.0.2811244662428995584.3, 12.12.2811244662428995584.1, 12.0.2811244662428995584.2, 12.0.2811244662428995584.1, 12.0.686339028913329.1

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 R R ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$2$2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{6}$
2.12.12.26$x^{12} - 162 x^{10} + 26423 x^{8} + 125508 x^{6} - 64481 x^{4} - 122498 x^{2} - 86071$$2$$6$$12$$C_6\times C_2$$[2]^{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}$
$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}$