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Properties

Label 24.0.96060566590...0000.3
Degree $24$
Signature $[0, 12]$
Discriminant $2^{24}\cdot 3^{36}\cdot 5^{18}$
Root discriminant $34.75$
Ramified primes $2, 3, 5$
Class number $50$ (GRH)
Class group $[5, 10]$ (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![15625, 0, 0, 0, 0, 0, 6250, 0, 0, 0, 0, 0, 2375, 0, 0, 0, 0, 0, 50, 0, 0, 0, 0, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 + 50*x^18 + 2375*x^12 + 6250*x^6 + 15625)
 
gp: K = bnfinit(x^24 + 50*x^18 + 2375*x^12 + 6250*x^6 + 15625, 1)
 

Normalized defining polynomial

\( x^{24} + 50 x^{18} + 2375 x^{12} + 6250 x^{6} + 15625 \)

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:  \(9606056659007943744000000000000000000=2^{24}\cdot 3^{36}\cdot 5^{18}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $34.75$
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:  $2, 3, 5$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(180=2^{2}\cdot 3^{2}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{180}(1,·)$, $\chi_{180}(67,·)$, $\chi_{180}(7,·)$, $\chi_{180}(143,·)$, $\chi_{180}(83,·)$, $\chi_{180}(149,·)$, $\chi_{180}(23,·)$, $\chi_{180}(89,·)$, $\chi_{180}(103,·)$, $\chi_{180}(29,·)$, $\chi_{180}(107,·)$, $\chi_{180}(161,·)$, $\chi_{180}(163,·)$, $\chi_{180}(101,·)$, $\chi_{180}(167,·)$, $\chi_{180}(41,·)$, $\chi_{180}(43,·)$, $\chi_{180}(109,·)$, $\chi_{180}(47,·)$, $\chi_{180}(49,·)$, $\chi_{180}(169,·)$, $\chi_{180}(121,·)$, $\chi_{180}(61,·)$, $\chi_{180}(127,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{5} a^{4}$, $\frac{1}{5} a^{5}$, $\frac{1}{5} a^{6}$, $\frac{1}{5} a^{7}$, $\frac{1}{25} a^{8}$, $\frac{1}{25} a^{9}$, $\frac{1}{25} a^{10}$, $\frac{1}{25} a^{11}$, $\frac{1}{500} a^{12} - \frac{1}{20} a^{6} + \frac{1}{4}$, $\frac{1}{500} a^{13} - \frac{1}{20} a^{7} + \frac{1}{4} a$, $\frac{1}{500} a^{14} - \frac{1}{100} a^{8} + \frac{1}{4} a^{2}$, $\frac{1}{500} a^{15} - \frac{1}{100} a^{9} + \frac{1}{4} a^{3}$, $\frac{1}{2500} a^{16} - \frac{1}{100} a^{10} + \frac{1}{20} a^{4}$, $\frac{1}{2500} a^{17} - \frac{1}{100} a^{11} + \frac{1}{20} a^{5}$, $\frac{1}{47500} a^{18} + \frac{29}{76}$, $\frac{1}{47500} a^{19} + \frac{29}{76} a$, $\frac{1}{237500} a^{20} + \frac{21}{76} a^{2}$, $\frac{1}{237500} a^{21} + \frac{21}{76} a^{3}$, $\frac{1}{237500} a^{22} + \frac{29}{380} a^{4}$, $\frac{1}{237500} a^{23} + \frac{29}{380} a^{5}$

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

Class group and class number

$C_{5}\times C_{10}$, which has order $50$ (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:  \( \frac{1}{237500} a^{22} - \frac{199}{380} a^{4} \) (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:  \( 136791199.54421148 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2\times C_{12}$ (as 24T2):

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 $C_2\times C_{12}$
Character table for $C_2\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{9})^+\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\zeta_{20})^+\), 4.0.18000.1, \(\Q(\zeta_{9})\), 6.6.820125.1, 6.0.2460375.1, 8.0.324000000.2, 12.0.6053445140625.1, 12.12.344373768000000000.1, 12.0.3099363912000000000.1

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 R ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/19.1.0.1}{1} }^{24}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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$2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
2.12.12.25$x^{12} - 78 x^{10} - 1621 x^{8} + 460 x^{6} - 1977 x^{4} + 866 x^{2} + 749$$2$$6$$12$$C_{12}$$[2]^{6}$
3Data not computed
5Data not computed