Properties

Label 24.0.711...256.1
Degree $24$
Signature $[0, 12]$
Discriminant $7.114\times 10^{29}$
Root discriminant $17.53$
Ramified primes $2, 3, 7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_2^2\times C_6$ (as 24T3)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 + x^22 - x^18 - x^16 + x^12 - x^8 - x^6 + x^2 + 1)
 
gp: K = bnfinit(x^24 + x^22 - x^18 - x^16 + x^12 - x^8 - x^6 + x^2 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 0, 0, -1, 0, -1, 0, 0, 0, 1, 0, 1]);
 

\(x^{24} + x^{22} - x^{18} - x^{16} + x^{12} - x^{8} - x^{6} + x^{2} + 1\)  Toggle raw display

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:  \(711435861303500483618465120256\)\(\medspace = 2^{24}\cdot 3^{12}\cdot 7^{20}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $17.53$
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:  $2, 3, 7$
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:  \(84=2^{2}\cdot 3\cdot 7\)
Dirichlet character group:    $\lbrace$$\chi_{84}(1,·)$, $\chi_{84}(67,·)$, $\chi_{84}(5,·)$, $\chi_{84}(65,·)$, $\chi_{84}(73,·)$, $\chi_{84}(11,·)$, $\chi_{84}(13,·)$, $\chi_{84}(79,·)$, $\chi_{84}(17,·)$, $\chi_{84}(19,·)$, $\chi_{84}(23,·)$, $\chi_{84}(25,·)$, $\chi_{84}(71,·)$, $\chi_{84}(29,·)$, $\chi_{84}(31,·)$, $\chi_{84}(37,·)$, $\chi_{84}(41,·)$, $\chi_{84}(43,·)$, $\chi_{84}(47,·)$, $\chi_{84}(83,·)$, $\chi_{84}(53,·)$, $\chi_{84}(55,·)$, $\chi_{84}(59,·)$, $\chi_{84}(61,·)$$\rbrace$
This is a CM field.

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

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

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

Class group and class number

Trivial group, which has order $1$ (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:  \( a \) (order $84$)  Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Fundamental units:  \( a^{12} + 1 \),  \( a^{22} + a^{20} + a^{10} - a^{6} - a^{4} \),  \( a^{8} - 1 \),  \( a^{18} - a^{4} - 1 \),  \( a^{16} - 1 \),  \( a^{9} - 1 \),  \( a^{15} - 1 \),  \( a^{5} - 1 \),  \( a^{23} - a^{9} - a^{2} \),  \( a^{11} - 1 \),  \( a^{23} + a^{22} - a^{18} - a^{16} + a^{12} + a^{10} - a^{9} - a^{8} - a^{6} + a^{2} + 1 \)  Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K!f(g): g in Generators(UK)];
 
Regulator:  \( 2172613.5864137784 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 2172613.5864137784 \cdot 1}{84\sqrt{711435861303500483618465120256}}\approx 0.116089716033712$ (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

\(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-1}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{21}) \), \(\Q(\sqrt{7}) \), \(\Q(\sqrt{-21}) \), \(\Q(\zeta_{7})^+\), \(\Q(\zeta_{12})\), \(\Q(\sqrt{-3}, \sqrt{-7})\), \(\Q(\sqrt{-3}, \sqrt{7})\), \(\Q(i, \sqrt{7})\), \(\Q(i, \sqrt{21})\), \(\Q(\sqrt{3}, \sqrt{-7})\), \(\Q(\sqrt{3}, \sqrt{7})\), 6.0.64827.1, 6.0.153664.1, 6.6.4148928.1, \(\Q(\zeta_{7})\), \(\Q(\zeta_{21})^+\), \(\Q(\zeta_{28})^+\), 6.0.29042496.1, 8.0.49787136.1, 12.0.17213603549184.1, \(\Q(\zeta_{21})\), 12.0.843466573910016.2, \(\Q(\zeta_{28})\), 12.0.843466573910016.1, 12.0.843466573910016.3, \(\Q(\zeta_{84})^+\)

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 ${\href{/LocalNumberField/5.6.0.1}{6} }^{4}$ R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/17.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/41.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/47.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/53.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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

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$3.12.6.2$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}$
$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}$