↑

Properties

Label 24.24.1225219298...0625.1
Degree $24$
Signature $[24, 0]$
Discriminant $5^{18}\cdot 13^{22}$
Root discriminant $35.10$
Ramified primes $5, 13$
Class number $1$ (GRH)
Class group Trivial (GRH)
Galois group $C_2\times C_{12}$ (as 24T2)

Related objects

Downloads

Learn more about

Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -12, -180, -101, 2085, 1802, -9126, -7168, 20886, 13653, -28667, -15001, 25284, 10282, -14822, -4540, 5832, 1292, -1521, -229, 252, 23, -24, -1, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - x^23 - 24*x^22 + 23*x^21 + 252*x^20 - 229*x^19 - 1521*x^18 + 1292*x^17 + 5832*x^16 - 4540*x^15 - 14822*x^14 + 10282*x^13 + 25284*x^12 - 15001*x^11 - 28667*x^10 + 13653*x^9 + 20886*x^8 - 7168*x^7 - 9126*x^6 + 1802*x^5 + 2085*x^4 - 101*x^3 - 180*x^2 - 12*x + 1)
 
gp: K = bnfinit(x^24 - x^23 - 24*x^22 + 23*x^21 + 252*x^20 - 229*x^19 - 1521*x^18 + 1292*x^17 + 5832*x^16 - 4540*x^15 - 14822*x^14 + 10282*x^13 + 25284*x^12 - 15001*x^11 - 28667*x^10 + 13653*x^9 + 20886*x^8 - 7168*x^7 - 9126*x^6 + 1802*x^5 + 2085*x^4 - 101*x^3 - 180*x^2 - 12*x + 1, 1)
 

Normalized defining polynomial

\( x^{24} - x^{23} - 24 x^{22} + 23 x^{21} + 252 x^{20} - 229 x^{19} - 1521 x^{18} + 1292 x^{17} + 5832 x^{16} - 4540 x^{15} - 14822 x^{14} + 10282 x^{13} + 25284 x^{12} - 15001 x^{11} - 28667 x^{10} + 13653 x^{9} + 20886 x^{8} - 7168 x^{7} - 9126 x^{6} + 1802 x^{5} + 2085 x^{4} - 101 x^{3} - 180 x^{2} - 12 x + 1 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $24$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[24, 0]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(12252192985362453861836887359619140625=5^{18}\cdot 13^{22}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $35.10$
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:  $5, 13$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(65=5\cdot 13\)
Dirichlet character group:    $\lbrace$$\chi_{65}(64,·)$, $\chi_{65}(1,·)$, $\chi_{65}(2,·)$, $\chi_{65}(4,·)$, $\chi_{65}(7,·)$, $\chi_{65}(8,·)$, $\chi_{65}(9,·)$, $\chi_{65}(14,·)$, $\chi_{65}(16,·)$, $\chi_{65}(18,·)$, $\chi_{65}(28,·)$, $\chi_{65}(29,·)$, $\chi_{65}(32,·)$, $\chi_{65}(33,·)$, $\chi_{65}(36,·)$, $\chi_{65}(37,·)$, $\chi_{65}(47,·)$, $\chi_{65}(49,·)$, $\chi_{65}(51,·)$, $\chi_{65}(56,·)$, $\chi_{65}(57,·)$, $\chi_{65}(58,·)$, $\chi_{65}(61,·)$, $\chi_{65}(63,·)$$\rbrace$
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$

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

Class group and class number

Trivial group, which has order $1$ (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:  $23$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 58142934011.05266 \) (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{65}) \), \(\Q(\sqrt{13}) \), 3.3.169.1, \(\Q(\sqrt{5}, \sqrt{13})\), 4.4.274625.1, 4.4.274625.2, 6.6.3570125.1, 6.6.46411625.1, \(\Q(\zeta_{13})^+\), 8.8.75418890625.1, 12.12.2154038935140625.1, 12.12.3500313269603515625.2, 12.12.3500313269603515625.1

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 ${\href{/LocalNumberField/2.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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
$5$5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.2$x^{8} + 15 x^{4} + 100$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
13Data not computed