↑

Properties

Label 24.0.16945016603...1056.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 3^{12}\cdot 13^{20}$
Root discriminant $29.37$
Ramified primes $2, 3, 13$
Class number $6$ (GRH)
Class group $[6]$ (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![1, 0, -21, 0, 371, 0, -1302, 0, 3091, 0, -4001, 0, 3677, 0, -2261, 0, 1031, 0, -327, 0, 76, 0, -11, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 11*x^22 + 76*x^20 - 327*x^18 + 1031*x^16 - 2261*x^14 + 3677*x^12 - 4001*x^10 + 3091*x^8 - 1302*x^6 + 371*x^4 - 21*x^2 + 1)
 
gp: K = bnfinit(x^24 - 11*x^22 + 76*x^20 - 327*x^18 + 1031*x^16 - 2261*x^14 + 3677*x^12 - 4001*x^10 + 3091*x^8 - 1302*x^6 + 371*x^4 - 21*x^2 + 1, 1)
 

Normalized defining polynomial

\( x^{24} - 11 x^{22} + 76 x^{20} - 327 x^{18} + 1031 x^{16} - 2261 x^{14} + 3677 x^{12} - 4001 x^{10} + 3091 x^{8} - 1302 x^{6} + 371 x^{4} - 21 x^{2} + 1 \)

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:  \(169450166032303737749261229339181056=2^{24}\cdot 3^{12}\cdot 13^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $29.37$
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:  $2, 3, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(156=2^{2}\cdot 3\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{156}(1,·)$, $\chi_{156}(131,·)$, $\chi_{156}(133,·)$, $\chi_{156}(113,·)$, $\chi_{156}(139,·)$, $\chi_{156}(77,·)$, $\chi_{156}(79,·)$, $\chi_{156}(17,·)$, $\chi_{156}(107,·)$, $\chi_{156}(23,·)$, $\chi_{156}(25,·)$, $\chi_{156}(155,·)$, $\chi_{156}(29,·)$, $\chi_{156}(95,·)$, $\chi_{156}(35,·)$, $\chi_{156}(101,·)$, $\chi_{156}(103,·)$, $\chi_{156}(43,·)$, $\chi_{156}(49,·)$, $\chi_{156}(53,·)$, $\chi_{156}(55,·)$, $\chi_{156}(121,·)$, $\chi_{156}(61,·)$, $\chi_{156}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5} a^{18} - \frac{1}{5} a^{12} + \frac{1}{5} a^{6} - \frac{1}{5}$, $\frac{1}{5} a^{19} - \frac{1}{5} a^{13} + \frac{1}{5} a^{7} - \frac{1}{5} a$, $\frac{1}{5} a^{20} - \frac{1}{5} a^{14} + \frac{1}{5} a^{8} - \frac{1}{5} a^{2}$, $\frac{1}{5} a^{21} - \frac{1}{5} a^{15} + \frac{1}{5} a^{9} - \frac{1}{5} a^{3}$, $\frac{1}{3140120548525} a^{22} + \frac{51460653714}{628024109705} a^{20} + \frac{174816507931}{3140120548525} a^{18} + \frac{1453912814519}{3140120548525} a^{16} + \frac{206892109419}{628024109705} a^{14} + \frac{1123625754299}{3140120548525} a^{12} + \frac{803708014511}{3140120548525} a^{10} - \frac{73095569372}{628024109705} a^{8} - \frac{411144212009}{3140120548525} a^{6} - \frac{1000770905871}{3140120548525} a^{4} - \frac{252578346066}{628024109705} a^{2} - \frac{1543772195661}{3140120548525}$, $\frac{1}{3140120548525} a^{23} + \frac{51460653714}{628024109705} a^{21} + \frac{174816507931}{3140120548525} a^{19} + \frac{1453912814519}{3140120548525} a^{17} + \frac{206892109419}{628024109705} a^{15} + \frac{1123625754299}{3140120548525} a^{13} + \frac{803708014511}{3140120548525} a^{11} - \frac{73095569372}{628024109705} a^{9} - \frac{411144212009}{3140120548525} a^{7} - \frac{1000770905871}{3140120548525} a^{5} - \frac{252578346066}{628024109705} a^{3} - \frac{1543772195661}{3140120548525} a$

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

Class group and class number

$C_{6}$, which has order $6$ (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:  \( -\frac{245903431384}{3140120548525} a^{23} + \frac{105667011281}{125604821941} a^{21} - \frac{17975844073174}{3140120548525} a^{19} + \frac{75426341030079}{3140120548525} a^{17} - \frac{46337912069137}{628024109705} a^{15} + \frac{486228806334304}{3140120548525} a^{13} - \frac{748357514123674}{3140120548525} a^{11} + \frac{145577986639644}{628024109705} a^{9} - \frac{477964007210989}{3140120548525} a^{7} + \frac{107298786752664}{3140120548525} a^{5} - \frac{1220293417317}{628024109705} a^{3} - \frac{12756519451156}{3140120548525} a \) (order $12$)
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:  \( 67326381.43157153 \) (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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{13}) \), \(\Q(\sqrt{-13}) \), \(\Q(\sqrt{39}) \), \(\Q(\sqrt{-39}) \), 3.3.169.1, \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{13})\), \(\Q(i, \sqrt{39})\), \(\Q(\sqrt{3}, \sqrt{13})\), \(\Q(\sqrt{3}, \sqrt{-13})\), \(\Q(\sqrt{-3}, \sqrt{13})\), \(\Q(\sqrt{-3}, \sqrt{-13})\), 6.0.1827904.1, 6.6.49353408.1, 6.0.771147.1, \(\Q(\zeta_{13})^+\), 6.0.23762752.1, 6.6.641594304.1, 6.0.10024911.1, 8.0.592240896.1, 12.0.2435758881214464.1, 12.0.564668382613504.1, 12.0.411643250925244416.3, 12.12.411643250925244416.1, 12.0.411643250925244416.2, 12.0.100498840557921.1, 12.0.411643250925244416.1

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 ${\href{/LocalNumberField/5.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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$3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
3.12.6.2$x^{12} + 108 x^{6} - 243 x^{2} + 2916$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
$13$13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$
13.6.5.2$x^{6} - 13$$6$$1$$5$$C_6$$[\ ]_{6}$