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Properties

Label 24.0.14276491594...2256.1
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 $18$ (GRH)
Class group $[3, 6]$ (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![64, 0, 0, -10176, 0, 0, 434872, 0, 0, 293680, 0, 0, 105407, 0, 0, -13006, 0, 0, 993, 0, 0, -34, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 34*x^21 + 993*x^18 - 13006*x^15 + 105407*x^12 + 293680*x^9 + 434872*x^6 - 10176*x^3 + 64)
 
gp: K = bnfinit(x^24 - 34*x^21 + 993*x^18 - 13006*x^15 + 105407*x^12 + 293680*x^9 + 434872*x^6 - 10176*x^3 + 64, 1)
 

Normalized defining polynomial

\( x^{24} - 34 x^{21} + 993 x^{18} - 13006 x^{15} + 105407 x^{12} + 293680 x^{9} + 434872 x^{6} - 10176 x^{3} + 64 \)

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$$\chi_{504}(1,·)$, $\chi_{504}(197,·)$, $\chi_{504}(449,·)$, $\chi_{504}(265,·)$, $\chi_{504}(13,·)$, $\chi_{504}(461,·)$, $\chi_{504}(209,·)$, $\chi_{504}(85,·)$, $\chi_{504}(281,·)$, $\chi_{504}(29,·)$, $\chi_{504}(421,·)$, $\chi_{504}(97,·)$, $\chi_{504}(113,·)$, $\chi_{504}(293,·)$, $\chi_{504}(337,·)$, $\chi_{504}(169,·)$, $\chi_{504}(125,·)$, $\chi_{504}(365,·)$, $\chi_{504}(349,·)$, $\chi_{504}(433,·)$, $\chi_{504}(181,·)$, $\chi_{504}(41,·)$, $\chi_{504}(377,·)$, $\chi_{504}(253,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{8} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{9} - \frac{1}{2} a^{3}$, $\frac{1}{4} a^{16} - \frac{1}{2} a^{13} + \frac{1}{4} a^{10} - \frac{1}{2} a^{7} - \frac{1}{4} a^{4}$, $\frac{1}{4} a^{17} + \frac{1}{4} a^{11} - \frac{1}{4} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{504} a^{18} + \frac{17}{84} a^{15} - \frac{25}{72} a^{12} + \frac{41}{252} a^{9} + \frac{247}{504} a^{6} + \frac{23}{63} a^{3} - \frac{19}{63}$, $\frac{1}{504} a^{19} - \frac{1}{21} a^{16} + \frac{11}{72} a^{13} - \frac{11}{126} a^{10} - \frac{5}{504} a^{7} - \frac{97}{252} a^{4} - \frac{19}{63} a$, $\frac{1}{1008} a^{20} + \frac{17}{168} a^{17} - \frac{25}{144} a^{14} - \frac{211}{504} a^{11} - \frac{257}{1008} a^{8} + \frac{23}{126} a^{5} - \frac{19}{126} a^{2}$, $\frac{1}{458375514617492909136} a^{21} - \frac{18473275848008579}{114593878654373227284} a^{18} + \frac{34238716911221299469}{458375514617492909136} a^{15} + \frac{64194559534154243}{1005209461880466906} a^{12} - \frac{482113786995877933}{458375514617492909136} a^{9} - \frac{54126367884111913307}{229187757308746454568} a^{6} - \frac{11831811001614823801}{28648469663593306821} a^{3} - \frac{8055811426797356861}{28648469663593306821}$, $\frac{1}{916751029234985818272} a^{22} + \frac{417791062011813109}{458375514617492909136} a^{19} - \frac{102182567201127780631}{916751029234985818272} a^{16} + \frac{8644475499140175013}{24125027085131205744} a^{13} - \frac{51697634149219389571}{305583676411661939424} a^{10} + \frac{9698970450287027107}{76395919102915484856} a^{7} - \frac{3488240159584751349}{12732653183819247476} a^{4} - \frac{8347913043623065967}{28648469663593306821} a$, $\frac{1}{916751029234985818272} a^{23} - \frac{18473275848008579}{229187757308746454568} a^{20} + \frac{34238716911221299469}{916751029234985818272} a^{17} + \frac{64194559534154243}{2010418923760933812} a^{14} + \frac{457893400830497031203}{916751029234985818272} a^{11} - \frac{54126367884111913307}{458375514617492909136} a^{8} + \frac{8408329330989241510}{28648469663593306821} a^{5} - \frac{8055811426797356861}{57296939327186613642} a^{2}$

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

Class group and class number

$C_{3}\times C_{6}$, which has order $18$ (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:  \( -\frac{20688780229076371}{1723216220366514696} a^{23} + \frac{9844153577373316819}{24125027085131205744} a^{20} - \frac{35936192399599479791}{3015628385641400718} a^{17} + \frac{179210477883886484695}{1148810813577676464} a^{14} - \frac{3810288983946377891753}{3015628385641400718} a^{11} - \frac{85451321576773344915607}{24125027085131205744} a^{8} - \frac{7941232678616854547446}{1507814192820700359} a^{5} + \frac{82368160836748013236}{1507814192820700359} a^{2} \) (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:  \( 513325248.7446666 \) (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{2}) \), \(\Q(\sqrt{-6}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{-14}) \), \(\Q(\sqrt{42}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{2}, \sqrt{-3})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{-14})\), \(\Q(\sqrt{2}, \sqrt{-7})\), \(\Q(\sqrt{2}, \sqrt{21})\), \(\Q(\sqrt{-6}, \sqrt{-7})\), \(\Q(\sqrt{-6}, \sqrt{-14})\), \(\Q(\zeta_{9})\), 6.6.3359232.1, 6.0.10077696.1, 6.0.2250423.1, 6.6.6751269.1, 6.0.1152216576.2, 6.6.3456649728.1, 8.0.796594176.2, 12.0.101559956668416.2, 12.0.45579633110361.1, 12.0.11948427342082473984.1, 12.0.1327603038009163776.2, 12.12.11948427342082473984.2, 12.0.11948427342082473984.3, 12.0.11948427342082473984.4

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

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$2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.23$x^{12} + 52 x^{10} - 28 x^{8} + 8 x^{6} + 64 x^{4} - 32 x^{2} + 64$$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$7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$
7.6.3.2$x^{6} - 49 x^{2} + 686$$2$$3$$3$$C_6$$[\ ]_{2}^{3}$