Properties

Label 24.0.614...000.1
Degree $24$
Signature $[0, 12]$
Discriminant $6.148\times 10^{32}$
Root discriminant $23.24$
Ramified primes $2, 3, 5$
Class number $3$ (GRH)
Class group $[3]$ (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

Related objects

Downloads

Learn more about

Show commands for: SageMath / Pari/GP / Magma

Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 18*x^18 + 323*x^12 - 18*x^6 + 1)
 
gp: K = bnfinit(x^24 - 18*x^18 + 323*x^12 - 18*x^6 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 0, 0, -18, 0, 0, 0, 0, 0, 323, 0, 0, 0, 0, 0, -18, 0, 0, 0, 0, 0, 1]);
 

\( x^{24} - 18 x^{18} + 323 x^{12} - 18 x^{6} + 1 \)

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

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[0, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(614787626176508399616000000000000\)\(\medspace = 2^{24}\cdot 3^{36}\cdot 5^{12}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $23.24$
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
Ramified primes:  $2, 3, 5$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Gal(K/\Q)|$:  $24$
This field is Galois and abelian over $\Q$.
Conductor:  \(180=2^{2}\cdot 3^{2}\cdot 5\)
Dirichlet character group:    $\lbrace$$\chi_{180}(1,·)$, $\chi_{180}(131,·)$, $\chi_{180}(71,·)$, $\chi_{180}(11,·)$, $\chi_{180}(79,·)$, $\chi_{180}(19,·)$, $\chi_{180}(139,·)$, $\chi_{180}(149,·)$, $\chi_{180}(151,·)$, $\chi_{180}(89,·)$, $\chi_{180}(91,·)$, $\chi_{180}(29,·)$, $\chi_{180}(31,·)$, $\chi_{180}(161,·)$, $\chi_{180}(101,·)$, $\chi_{180}(119,·)$, $\chi_{180}(41,·)$, $\chi_{180}(109,·)$, $\chi_{180}(49,·)$, $\chi_{180}(179,·)$, $\chi_{180}(169,·)$, $\chi_{180}(121,·)$, $\chi_{180}(59,·)$, $\chi_{180}(61,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{8} a^{12} + \frac{3}{8} a^{6} + \frac{1}{8}$, $\frac{1}{8} a^{13} + \frac{3}{8} a^{7} + \frac{1}{8} a$, $\frac{1}{8} a^{14} + \frac{3}{8} a^{8} + \frac{1}{8} a^{2}$, $\frac{1}{8} a^{15} + \frac{3}{8} a^{9} + \frac{1}{8} a^{3}$, $\frac{1}{8} a^{16} + \frac{3}{8} a^{10} + \frac{1}{8} a^{4}$, $\frac{1}{8} a^{17} + \frac{3}{8} a^{11} + \frac{1}{8} a^{5}$, $\frac{1}{2584} a^{18} - \frac{987}{2584}$, $\frac{1}{2584} a^{19} - \frac{987}{2584} a$, $\frac{1}{2584} a^{20} - \frac{987}{2584} a^{2}$, $\frac{1}{2584} a^{21} - \frac{987}{2584} a^{3}$, $\frac{1}{2584} a^{22} - \frac{987}{2584} a^{4}$, $\frac{1}{2584} a^{23} - \frac{987}{2584} a^{5}$

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

Class group and class number

$C_{3}$, which has order $3$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, f := UnitGroup(K);
 
Rank:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -\frac{233}{2584} a^{19} + \frac{13}{8} a^{13} - \frac{233}{8} a^{7} + \frac{2097}{1292} a \) (order $36$)
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
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)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 13636610.630449586 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 13636610.630449586 \cdot 3}{36\sqrt{614787626176508399616000000000000}}\approx 0.173508619931334$ (assuming GRH)

Galois group

$C_2^2\times C_6$ (as 24T3):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
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

\(\Q(\sqrt{-1}) \), \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{15}) \), \(\Q(\sqrt{-15}) \), \(\Q(\zeta_{9})^+\), \(\Q(\zeta_{12})\), \(\Q(i, \sqrt{5})\), \(\Q(i, \sqrt{15})\), \(\Q(\sqrt{-3}, \sqrt{-5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), \(\Q(\sqrt{3}, \sqrt{-5})\), \(\Q(\sqrt{3}, \sqrt{5})\), 6.0.419904.1, \(\Q(\zeta_{9})\), \(\Q(\zeta_{36})^+\), 6.0.52488000.1, 6.6.820125.1, 6.6.157464000.1, 6.0.2460375.1, 8.0.12960000.1, \(\Q(\zeta_{36})\), 12.0.2754990144000000.1, 12.0.24794911296000000.2, 12.0.24794911296000000.3, 12.0.6053445140625.1, 12.0.24794911296000000.1, 12.12.24794911296000000.1

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 ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/17.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

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]])
 
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];
 

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}$
$5$5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$
5.12.6.1$x^{12} + 500 x^{6} - 3125 x^{2} + 62500$$2$$6$$6$$C_6\times C_2$$[\ ]_{2}^{6}$