Properties

Label 24.0.368...000.1
Degree $24$
Signature $[0, 12]$
Discriminant $3.689\times 10^{26}$
Root discriminant $12.79$
Ramified primes $2, 5, 7$
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 - 6*x^23 + 21*x^22 - 54*x^21 + 107*x^20 - 178*x^19 + 265*x^18 - 376*x^17 + 537*x^16 - 744*x^15 + 966*x^14 - 1140*x^13 + 1221*x^12 - 1228*x^11 + 1176*x^10 - 1074*x^9 + 914*x^8 - 696*x^7 + 471*x^6 - 280*x^5 + 140*x^4 - 58*x^3 + 21*x^2 - 6*x + 1)
 
gp: K = bnfinit(x^24 - 6*x^23 + 21*x^22 - 54*x^21 + 107*x^20 - 178*x^19 + 265*x^18 - 376*x^17 + 537*x^16 - 744*x^15 + 966*x^14 - 1140*x^13 + 1221*x^12 - 1228*x^11 + 1176*x^10 - 1074*x^9 + 914*x^8 - 696*x^7 + 471*x^6 - 280*x^5 + 140*x^4 - 58*x^3 + 21*x^2 - 6*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -6, 21, -58, 140, -280, 471, -696, 914, -1074, 1176, -1228, 1221, -1140, 966, -744, 537, -376, 265, -178, 107, -54, 21, -6, 1]);
 

\( x^{24} - 6 x^{23} + 21 x^{22} - 54 x^{21} + 107 x^{20} - 178 x^{19} + 265 x^{18} - 376 x^{17} + 537 x^{16} - 744 x^{15} + 966 x^{14} - 1140 x^{13} + 1221 x^{12} - 1228 x^{11} + 1176 x^{10} - 1074 x^{9} + 914 x^{8} - 696 x^{7} + 471 x^{6} - 280 x^{5} + 140 x^{4} - 58 x^{3} + 21 x^{2} - 6 x + 1 \)

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:  \(368947264000000000000000000\)\(\medspace = 2^{24}\cdot 5^{18}\cdot 7^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $12.79$
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:  $2, 5, 7$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{191679830229179} a^{23} + \frac{8422786950576}{191679830229179} a^{22} - \frac{22694145349783}{191679830229179} a^{21} - \frac{6161595013249}{191679830229179} a^{20} - \frac{73723363546300}{191679830229179} a^{19} + \frac{55063250317610}{191679830229179} a^{18} + \frac{33976018607284}{191679830229179} a^{17} - \frac{85199078093611}{191679830229179} a^{16} + \frac{33893792053320}{191679830229179} a^{15} + \frac{31808053967254}{191679830229179} a^{14} + \frac{54734987496610}{191679830229179} a^{13} + \frac{5199015835275}{191679830229179} a^{12} + \frac{59348984355966}{191679830229179} a^{11} - \frac{32238076066833}{191679830229179} a^{10} - \frac{79659112112203}{191679830229179} a^{9} + \frac{42917911640006}{191679830229179} a^{8} + \frac{15963632334858}{191679830229179} a^{7} - \frac{40481349346422}{191679830229179} a^{6} + \frac{21920184363788}{191679830229179} a^{5} - \frac{54317148824536}{191679830229179} a^{4} - \frac{64837609145563}{191679830229179} a^{3} - \frac{21523508422520}{191679830229179} a^{2} - \frac{69505981625860}{191679830229179} a + \frac{95071079886986}{191679830229179}$

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:  \( -\frac{246938601708342}{191679830229179} a^{23} + \frac{1390625883620859}{191679830229179} a^{22} - \frac{4690132406743003}{191679830229179} a^{21} + \frac{11693310212992961}{191679830229179} a^{20} - \frac{22407405942837677}{191679830229179} a^{19} + \frac{36436954373691654}{191679830229179} a^{18} - \frac{53417981694349363}{191679830229179} a^{17} + \frac{75464163619737171}{191679830229179} a^{16} - \frac{108114681538802938}{191679830229179} a^{15} + \frac{148486908248637245}{191679830229179} a^{14} - \frac{190461567438894078}{191679830229179} a^{13} + \frac{220408936121823028}{191679830229179} a^{12} - \frac{232075014107646222}{191679830229179} a^{11} + \frac{231190167248330130}{191679830229179} a^{10} - \frac{219003463204885278}{191679830229179} a^{9} + \frac{198179686228003322}{191679830229179} a^{8} - \frac{165442350017954611}{191679830229179} a^{7} + \frac{122617740183232305}{191679830229179} a^{6} - \frac{80797338352179268}{191679830229179} a^{5} + \frac{46132174266051938}{191679830229179} a^{4} - \frac{21969283656399143}{191679830229179} a^{3} + \frac{8687105857380701}{191679830229179} a^{2} - \frac{3018529100074687}{191679830229179} a + \frac{731375154086694}{191679830229179} \) (order $20$)
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:  \( 15456.373586053287 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 15456.373586053287 \cdot 1}{20\sqrt{368947264000000000000000000}}\approx 0.152318931258119$ (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), \(\Q(i, \sqrt{5})\), 6.0.392000.1, \(\Q(\zeta_{20})\), 12.0.153664000000.1

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 ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R R ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
2Data not computed
$5$5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.2$x^{12} - 10 x^{8} + 25 x^{4} - 500$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
$7$7.12.0.1$x^{12} + 3 x^{2} - 2 x + 3$$1$$12$$0$$C_{12}$$[\ ]^{12}$
7.12.8.3$x^{12} + 14 x^{9} + 539 x^{6} + 343 x^{3} + 60025$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.20.2t1.a.a$1$ $ 2^{2} \cdot 5 $ \(\Q(\sqrt{-5}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.140.6t1.b.a$1$ $ 2^{2} \cdot 5 \cdot 7 $ 6.0.19208000.1 $C_6$ (as 6T1) $0$ $-1$
1.7.3t1.a.a$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
1.140.6t1.b.b$1$ $ 2^{2} \cdot 5 \cdot 7 $ 6.0.19208000.1 $C_6$ (as 6T1) $0$ $-1$
1.28.6t1.a.a$1$ $ 2^{2} \cdot 7 $ 6.0.153664.1 $C_6$ (as 6T1) $0$ $-1$
1.35.6t1.b.a$1$ $ 5 \cdot 7 $ 6.6.300125.1 $C_6$ (as 6T1) $0$ $1$
1.35.6t1.b.b$1$ $ 5 \cdot 7 $ 6.6.300125.1 $C_6$ (as 6T1) $0$ $1$
1.28.6t1.a.b$1$ $ 2^{2} \cdot 7 $ 6.0.153664.1 $C_6$ (as 6T1) $0$ $-1$
1.7.3t1.a.b$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.20.4t1.a.a$1$ $ 2^{2} \cdot 5 $ \(\Q(\zeta_{20})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.20.4t1.a.b$1$ $ 2^{2} \cdot 5 $ \(\Q(\zeta_{20})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
1.140.12t1.a.a$1$ $ 2^{2} \cdot 5 \cdot 7 $ 12.12.46118408000000000.1 $C_{12}$ (as 12T1) $0$ $1$
1.35.12t1.a.a$1$ $ 5 \cdot 7 $ 12.0.11259376953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.140.12t1.a.b$1$ $ 2^{2} \cdot 5 \cdot 7 $ 12.12.46118408000000000.1 $C_{12}$ (as 12T1) $0$ $1$
1.35.12t1.a.b$1$ $ 5 \cdot 7 $ 12.0.11259376953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.35.12t1.a.c$1$ $ 5 \cdot 7 $ 12.0.11259376953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.35.12t1.a.d$1$ $ 5 \cdot 7 $ 12.0.11259376953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.140.12t1.a.c$1$ $ 2^{2} \cdot 5 \cdot 7 $ 12.12.46118408000000000.1 $C_{12}$ (as 12T1) $0$ $1$
1.140.12t1.a.d$1$ $ 2^{2} \cdot 5 \cdot 7 $ 12.12.46118408000000000.1 $C_{12}$ (as 12T1) $0$ $1$
2.980.3t2.a.a$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ 3.1.980.1 $S_3$ (as 3T2) $1$ $0$
2.980.6t3.d.a$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ 6.2.4802000.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.140.12t18.a.a$2$ $ 2^{2} \cdot 5 \cdot 7 $ 12.0.153664000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.140.6t5.a.a$2$ $ 2^{2} \cdot 5 \cdot 7 $ 6.0.392000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.140.12t18.a.b$2$ $ 2^{2} \cdot 5 \cdot 7 $ 12.0.153664000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.140.6t5.a.b$2$ $ 2^{2} \cdot 5 \cdot 7 $ 6.0.392000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
2.4900.12t11.a.a$2$ $ 2^{2} \cdot 5^{2} \cdot 7^{2}$ 12.4.46118408000000000.2 $S_3 \times C_4$ (as 12T11) $0$ $0$
2.4900.12t11.a.b$2$ $ 2^{2} \cdot 5^{2} \cdot 7^{2}$ 12.4.46118408000000000.2 $S_3 \times C_4$ (as 12T11) $0$ $0$
* 2.700.24t65.a.a$2$ $ 2^{2} \cdot 5^{2} \cdot 7 $ 24.0.368947264000000000000000000.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.700.24t65.a.b$2$ $ 2^{2} \cdot 5^{2} \cdot 7 $ 24.0.368947264000000000000000000.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.700.24t65.a.c$2$ $ 2^{2} \cdot 5^{2} \cdot 7 $ 24.0.368947264000000000000000000.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.700.24t65.a.d$2$ $ 2^{2} \cdot 5^{2} \cdot 7 $ 24.0.368947264000000000000000000.1 $C_{12}\times S_3$ (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 *.