↑

Properties

Label 24.0.34854715807...6336.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 3^{36}\cdot 7^{12}$
Root discriminant $27.50$
Ramified primes $2, 3, 7$
Class number $7$ (GRH)
Class group $[7]$ (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![4096, 0, 0, 0, 0, 0, -576, 0, 0, 0, 0, 0, 17, 0, 0, 0, 0, 0, -9, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 9*x^18 + 17*x^12 - 576*x^6 + 4096)
 
gp: K = bnfinit(x^24 - 9*x^18 + 17*x^12 - 576*x^6 + 4096, 1)
 

Normalized defining polynomial

\( x^{24} - 9 x^{18} + 17 x^{12} - 576 x^{6} + 4096 \)

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:  \(34854715807867200628629234134286336=2^{24}\cdot 3^{36}\cdot 7^{12}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $27.50$
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:  \(252=2^{2}\cdot 3^{2}\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{252}(1,·)$, $\chi_{252}(197,·)$, $\chi_{252}(71,·)$, $\chi_{252}(139,·)$, $\chi_{252}(13,·)$, $\chi_{252}(209,·)$, $\chi_{252}(211,·)$, $\chi_{252}(85,·)$, $\chi_{252}(155,·)$, $\chi_{252}(29,·)$, $\chi_{252}(223,·)$, $\chi_{252}(97,·)$, $\chi_{252}(167,·)$, $\chi_{252}(169,·)$, $\chi_{252}(41,·)$, $\chi_{252}(43,·)$, $\chi_{252}(239,·)$, $\chi_{252}(113,·)$, $\chi_{252}(83,·)$, $\chi_{252}(181,·)$, $\chi_{252}(55,·)$, $\chi_{252}(251,·)$, $\chi_{252}(125,·)$, $\chi_{252}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5} a^{12} - \frac{2}{5} a^{6} - \frac{1}{5}$, $\frac{1}{10} a^{13} + \frac{3}{10} a^{7} - \frac{1}{10} a$, $\frac{1}{20} a^{14} + \frac{3}{20} a^{8} + \frac{9}{20} a^{2}$, $\frac{1}{40} a^{15} - \frac{17}{40} a^{9} + \frac{9}{40} a^{3}$, $\frac{1}{80} a^{16} + \frac{23}{80} a^{10} - \frac{31}{80} a^{4}$, $\frac{1}{160} a^{17} + \frac{23}{160} a^{11} + \frac{49}{160} a^{5}$, $\frac{1}{5440} a^{18} + \frac{7}{320} a^{12} + \frac{1}{320} a^{6} - \frac{9}{85}$, $\frac{1}{10880} a^{19} + \frac{7}{640} a^{13} + \frac{1}{640} a^{7} - \frac{9}{170} a$, $\frac{1}{21760} a^{20} + \frac{7}{1280} a^{14} - \frac{639}{1280} a^{8} + \frac{161}{340} a^{2}$, $\frac{1}{43520} a^{21} + \frac{7}{2560} a^{15} - \frac{639}{2560} a^{9} + \frac{161}{680} a^{3}$, $\frac{1}{87040} a^{22} + \frac{7}{5120} a^{16} + \frac{1921}{5120} a^{10} - \frac{519}{1360} a^{4}$, $\frac{1}{174080} a^{23} + \frac{7}{10240} a^{17} - \frac{3199}{10240} a^{11} - \frac{519}{2720} a^{5}$

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

Class group and class number

$C_{7}$, which has order $7$ (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:  \( \frac{1}{10880} a^{19} + \frac{7}{640} a^{13} + \frac{1}{640} a^{7} - \frac{9}{170} a \) (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:  \( 79907554.2564404 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-21}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{7})\), \(\Q(i, \sqrt{21})\), \(\Q(\sqrt{3}, \sqrt{7})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), 6.0.419904.1, \(\Q(\zeta_{36})^+\), \(\Q(\zeta_{9})\), 6.6.144027072.1, 6.0.2250423.1, 6.6.6751269.1, 6.0.432081216.1, 8.0.49787136.1, \(\Q(\zeta_{36})\), 12.0.20743797468893184.1, 12.0.186694177220038656.1, 12.12.186694177220038656.1, 12.0.186694177220038656.2, 12.0.186694177220038656.3, 12.0.45579633110361.1

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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}$
$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}$