Properties

Label 24.0.385...816.1
Degree $24$
Signature $[0, 12]$
Discriminant $3.852\times 10^{30}$
Root discriminant $18.81$
Ramified primes $2, 13$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_{12}\times S_3$ (as 24T65)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 1)
 
gp: K = bnfinit(x^24 - 3*x^16 - 8*x^12 + 18*x^8 + 8*x^4 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, 0, 8, 0, 0, 0, 18, 0, 0, 0, -8, 0, 0, 0, -3, 0, 0, 0, 0, 0, 0, 0, 1]);
 

\( x^{24} - 3 x^{16} - 8 x^{12} + 18 x^{8} + 8 x^{4} + 1 \)

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:  \(3852179415897489839182437154816\)\(\medspace = 2^{72}\cdot 13^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $18.81$
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:  $2, 13$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $12$
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{899} a^{20} + \frac{287}{899} a^{16} - \frac{342}{899} a^{12} - \frac{171}{899} a^{8} + \frac{386}{899} a^{4} + \frac{213}{899}$, $\frac{1}{899} a^{21} + \frac{287}{899} a^{17} - \frac{342}{899} a^{13} - \frac{171}{899} a^{9} + \frac{386}{899} a^{5} + \frac{213}{899} a$, $\frac{1}{899} a^{22} + \frac{287}{899} a^{18} - \frac{342}{899} a^{14} - \frac{171}{899} a^{10} + \frac{386}{899} a^{6} + \frac{213}{899} a^{2}$, $\frac{1}{899} a^{23} + \frac{287}{899} a^{19} - \frac{342}{899} a^{15} - \frac{171}{899} a^{11} + \frac{386}{899} a^{7} + \frac{213}{899} a^{3}$

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:  \( \frac{952}{899} a^{23} - \frac{72}{899} a^{19} - \frac{2843}{899} a^{15} - \frac{7265}{899} a^{11} + \frac{17761}{899} a^{7} + \frac{5895}{899} a^{3} \) (order $16$)
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:  \( 2446825.407448486 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 2446825.407448486 \cdot 1}{16\sqrt{3852179415897489839182437154816}}\approx 0.294977007875353$ (assuming GRH)

Galois group

$C_{12}\times S_3$ (as 24T65):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A solvable group of order 72
The 36 conjugacy class representatives for $C_{12}\times S_3$
Character table for $C_{12}\times S_3$ is not computed

Intermediate fields

\(\Q(\sqrt{2}) \), \(\Q(\sqrt{-2}) \), \(\Q(\sqrt{-1}) \), 4.0.2048.2, \(\Q(\zeta_{16})^+\), \(\Q(\zeta_{8})\), 6.0.10816.1, \(\Q(\zeta_{16})\), 12.0.479174066176.4

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 36 siblings: data not computed

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/5.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/7.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ R ${\href{/LocalNumberField/17.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/17.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/19.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/23.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/29.12.0.1}{12} }{,}\,{\href{/LocalNumberField/29.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/37.12.0.1}{12} }{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/41.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.12.0.1}{12} }^{2}$

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

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
2Data not computed
$13$13.12.8.2$x^{12} + 169 x^{6} - 2197 x^{3} + 57122$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$
13.12.0.1$x^{12} + x^{2} - x + 2$$1$$12$$0$$C_{12}$$[\ ]^{12}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.8.2t1.b.a$1$ $ 2^{3}$ \(\Q(\sqrt{-2}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.8.2t1.a.a$1$ $ 2^{3}$ \(\Q(\sqrt{2}) \) $C_2$ (as 2T1) $1$ $1$
1.52.6t1.b.a$1$ $ 2^{2} \cdot 13 $ 6.0.1827904.1 $C_6$ (as 6T1) $0$ $-1$
1.52.6t1.b.b$1$ $ 2^{2} \cdot 13 $ 6.0.1827904.1 $C_6$ (as 6T1) $0$ $-1$
1.104.6t1.d.a$1$ $ 2^{3} \cdot 13 $ 6.0.14623232.1 $C_6$ (as 6T1) $0$ $-1$
1.13.3t1.a.a$1$ $ 13 $ 3.3.169.1 $C_3$ (as 3T1) $0$ $1$
1.13.3t1.a.b$1$ $ 13 $ 3.3.169.1 $C_3$ (as 3T1) $0$ $1$
1.104.6t1.a.a$1$ $ 2^{3} \cdot 13 $ 6.6.14623232.1 $C_6$ (as 6T1) $0$ $1$
1.104.6t1.a.b$1$ $ 2^{3} \cdot 13 $ 6.6.14623232.1 $C_6$ (as 6T1) $0$ $1$
1.104.6t1.d.b$1$ $ 2^{3} \cdot 13 $ 6.0.14623232.1 $C_6$ (as 6T1) $0$ $-1$
* 1.16.4t1.a.a$1$ $ 2^{4}$ \(\Q(\zeta_{16})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.16.4t1.b.a$1$ $ 2^{4}$ 4.0.2048.2 $C_4$ (as 4T1) $0$ $-1$
* 1.16.4t1.b.b$1$ $ 2^{4}$ 4.0.2048.2 $C_4$ (as 4T1) $0$ $-1$
* 1.16.4t1.a.b$1$ $ 2^{4}$ \(\Q(\zeta_{16})^+\) $C_4$ (as 4T1) $0$ $1$
1.208.12t1.b.a$1$ $ 2^{4} \cdot 13 $ 12.12.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $1$
1.208.12t1.a.a$1$ $ 2^{4} \cdot 13 $ 12.0.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $-1$
1.208.12t1.a.b$1$ $ 2^{4} \cdot 13 $ 12.0.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $-1$
1.208.12t1.b.b$1$ $ 2^{4} \cdot 13 $ 12.12.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $1$
1.208.12t1.b.c$1$ $ 2^{4} \cdot 13 $ 12.12.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $1$
1.208.12t1.b.d$1$ $ 2^{4} \cdot 13 $ 12.12.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $1$
1.208.12t1.a.c$1$ $ 2^{4} \cdot 13 $ 12.0.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $-1$
1.208.12t1.a.d$1$ $ 2^{4} \cdot 13 $ 12.0.7007073538075000832.1 $C_{12}$ (as 12T1) $0$ $-1$
2.676.3t2.b.a$2$ $ 2^{2} \cdot 13^{2}$ 3.1.676.1 $S_3$ (as 3T2) $1$ $0$
2.10816.6t3.d.a$2$ $ 2^{6} \cdot 13^{2}$ 6.0.58492928.3 $D_{6}$ (as 6T3) $1$ $0$
* 2.832.12t18.b.a$2$ $ 2^{6} \cdot 13 $ 12.0.479174066176.4 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.832.12t18.b.b$2$ $ 2^{6} \cdot 13 $ 12.0.479174066176.4 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.52.6t5.b.a$2$ $ 2^{2} \cdot 13 $ 6.0.10816.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.52.6t5.b.b$2$ $ 2^{2} \cdot 13 $ 6.0.10816.1 $S_3\times C_3$ (as 6T5) $0$ $0$
2.43264.12t11.b.a$2$ $ 2^{8} \cdot 13^{2}$ 12.4.28028294152300003328.26 $S_3 \times C_4$ (as 12T11) $0$ $0$
2.43264.12t11.b.b$2$ $ 2^{8} \cdot 13^{2}$ 12.4.28028294152300003328.26 $S_3 \times C_4$ (as 12T11) $0$ $0$
* 2.3328.24t65.a.a$2$ $ 2^{8} \cdot 13 $ 24.0.3852179415897489839182437154816.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.3328.24t65.a.b$2$ $ 2^{8} \cdot 13 $ 24.0.3852179415897489839182437154816.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.3328.24t65.a.c$2$ $ 2^{8} \cdot 13 $ 24.0.3852179415897489839182437154816.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.3328.24t65.a.d$2$ $ 2^{8} \cdot 13 $ 24.0.3852179415897489839182437154816.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.