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Properties

Label 24.0.13386920868...0000.1
Degree $24$
Signature $[0, 12]$
Discriminant $2^{36}\cdot 5^{12}\cdot 7^{20}$
Root discriminant $32.01$
Ramified primes $2, 5, 7$
Class number $9$ (GRH)
Class group $[3, 3]$ (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![4096, 0, -6144, 0, 8192, 0, -10752, 0, 14080, 0, -18432, 0, 24128, 0, -4608, 0, 880, 0, -168, 0, 32, 0, -6, 0, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^22 + 32*x^20 - 168*x^18 + 880*x^16 - 4608*x^14 + 24128*x^12 - 18432*x^10 + 14080*x^8 - 10752*x^6 + 8192*x^4 - 6144*x^2 + 4096)
 
gp: K = bnfinit(x^24 - 6*x^22 + 32*x^20 - 168*x^18 + 880*x^16 - 4608*x^14 + 24128*x^12 - 18432*x^10 + 14080*x^8 - 10752*x^6 + 8192*x^4 - 6144*x^2 + 4096, 1)
 

Normalized defining polynomial

\( x^{24} - 6 x^{22} + 32 x^{20} - 168 x^{18} + 880 x^{16} - 4608 x^{14} + 24128 x^{12} - 18432 x^{10} + 14080 x^{8} - 10752 x^{6} + 8192 x^{4} - 6144 x^{2} + 4096 \)

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:  \(1338692086804556824969216000000000000=2^{36}\cdot 5^{12}\cdot 7^{20}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $32.01$
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:  $2, 5, 7$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is Galois and abelian over $\Q$.
Conductor:  \(280=2^{3}\cdot 5\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{280}(1,·)$, $\chi_{280}(131,·)$, $\chi_{280}(179,·)$, $\chi_{280}(211,·)$, $\chi_{280}(129,·)$, $\chi_{280}(9,·)$, $\chi_{280}(11,·)$, $\chi_{280}(209,·)$, $\chi_{280}(19,·)$, $\chi_{280}(139,·)$, $\chi_{280}(121,·)$, $\chi_{280}(89,·)$, $\chi_{280}(219,·)$, $\chi_{280}(99,·)$, $\chi_{280}(251,·)$, $\chi_{280}(81,·)$, $\chi_{280}(41,·)$, $\chi_{280}(171,·)$, $\chi_{280}(241,·)$, $\chi_{280}(51,·)$, $\chi_{280}(201,·)$, $\chi_{280}(249,·)$, $\chi_{280}(59,·)$, $\chi_{280}(169,·)$$\rbrace$
This is a CM field.

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

$1$, $a$, $\frac{1}{2} a^{2}$, $\frac{1}{2} a^{3}$, $\frac{1}{4} a^{4}$, $\frac{1}{4} a^{5}$, $\frac{1}{8} a^{6}$, $\frac{1}{8} a^{7}$, $\frac{1}{16} a^{8}$, $\frac{1}{16} a^{9}$, $\frac{1}{32} a^{10}$, $\frac{1}{32} a^{11}$, $\frac{1}{64} a^{12}$, $\frac{1}{64} a^{13}$, $\frac{1}{48256} a^{14} - \frac{144}{377}$, $\frac{1}{48256} a^{15} - \frac{144}{377} a$, $\frac{1}{96512} a^{16} - \frac{72}{377} a^{2}$, $\frac{1}{96512} a^{17} - \frac{72}{377} a^{3}$, $\frac{1}{193024} a^{18} - \frac{36}{377} a^{4}$, $\frac{1}{193024} a^{19} - \frac{36}{377} a^{5}$, $\frac{1}{386048} a^{20} - \frac{18}{377} a^{6}$, $\frac{1}{386048} a^{21} - \frac{18}{377} a^{7}$, $\frac{1}{772096} a^{22} - \frac{9}{377} a^{8}$, $\frac{1}{772096} a^{23} - \frac{9}{377} a^{9}$

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

Class group and class number

$C_{3}\times C_{3}$, which has order $9$ (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:  \( -\frac{9}{48256} a^{22} + \frac{27}{24128} a^{20} - \frac{9}{1508} a^{18} + \frac{189}{6032} a^{16} - \frac{495}{3016} a^{14} + \frac{55}{64} a^{12} - \frac{9}{2} a^{10} + \frac{1296}{377} a^{8} - \frac{990}{377} a^{6} + \frac{756}{377} a^{4} - \frac{576}{377} a^{2} + \frac{432}{377} \) (order $14$)
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:  \( 69523271.83090398 \) (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

\(\Q(\sqrt{-35}) \), \(\Q(\sqrt{70}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{14}) \), \(\Q(\sqrt{-10}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\zeta_{7})^+\), \(\Q(\sqrt{-2}, \sqrt{-35})\), \(\Q(\sqrt{-10}, \sqrt{14})\), \(\Q(\sqrt{5}, \sqrt{-7})\), \(\Q(\sqrt{5}, \sqrt{14})\), \(\Q(\sqrt{-7}, \sqrt{-10})\), \(\Q(\sqrt{-2}, \sqrt{-7})\), \(\Q(\sqrt{-2}, \sqrt{5})\), 6.0.2100875.1, 6.6.1075648000.1, 6.0.1229312.1, 6.6.8605184.1, 6.0.153664000.1, 6.6.300125.1, \(\Q(\zeta_{7})\), 8.0.6146560000.1, 12.0.1157018619904000000.1, 12.0.1157018619904000000.3, 12.0.4413675765625.1, 12.12.1157018619904000000.1, 12.0.1157018619904000000.6, 12.0.74049191673856.1, 12.0.23612624896000000.1

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 ${\href{/LocalNumberField/3.6.0.1}{6} }^{4}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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$2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
2.12.18.15$x^{12} - 16 x^{10} + 24 x^{6} + 64 x^{4} + 64$$2$$6$$18$$C_6\times C_2$$[3]^{6}$
$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}$
$7$7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$
7.12.10.1$x^{12} - 70 x^{6} + 35721$$6$$2$$10$$C_6\times C_2$$[\ ]_{6}^{2}$