Properties

Label 24.0.572...625.2
Degree $24$
Signature $[0, 12]$
Discriminant $5.726\times 10^{29}$
Root discriminant $17.37$
Ramified primes $3, 5$
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 - 2*x^21 - 2*x^15 + 9*x^12 - 8*x^9 + 5*x^6 - 3*x^3 + 1)
 
gp: K = bnfinit(x^24 - 2*x^21 - 2*x^15 + 9*x^12 - 8*x^9 + 5*x^6 - 3*x^3 + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 0, 0, -3, 0, 0, 5, 0, 0, -8, 0, 0, 9, 0, 0, -2, 0, 0, 0, 0, 0, -2, 0, 0, 1]);
 

\( x^{24} - 2 x^{21} - 2 x^{15} + 9 x^{12} - 8 x^{9} + 5 x^{6} - 3 x^{3} + 1 \)

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:  \(572565594852444156646728515625\)\(\medspace = 3^{36}\cdot 5^{18}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $17.37$
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:  $3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{61} a^{21} - \frac{26}{61} a^{18} + \frac{14}{61} a^{15} + \frac{28}{61} a^{12} + \frac{8}{61} a^{9} - \frac{17}{61} a^{6} - \frac{14}{61} a^{3} + \frac{28}{61}$, $\frac{1}{61} a^{22} - \frac{26}{61} a^{19} + \frac{14}{61} a^{16} + \frac{28}{61} a^{13} + \frac{8}{61} a^{10} - \frac{17}{61} a^{7} - \frac{14}{61} a^{4} + \frac{28}{61} a$, $\frac{1}{61} a^{23} - \frac{26}{61} a^{20} + \frac{14}{61} a^{17} + \frac{28}{61} a^{14} + \frac{8}{61} a^{11} - \frac{17}{61} a^{8} - \frac{14}{61} a^{5} + \frac{28}{61} a^{2}$

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:  \( -\frac{19}{61} a^{21} + \frac{6}{61} a^{18} + \frac{39}{61} a^{15} + \frac{78}{61} a^{12} - \frac{91}{61} a^{9} - \frac{104}{61} a^{6} - \frac{39}{61} a^{3} + \frac{17}{61} \) (order $30$)
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:  \( 1414385.0473488418 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 1414385.0473488418 \cdot 1}{30\sqrt{572565594852444156646728515625}}\approx 0.235880585985437$ (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

\(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{15})^+\), \(\Q(\zeta_{5})\), \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.2460375.2, \(\Q(\zeta_{15})\), 12.0.6053445140625.2

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{3}$ R R ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/31.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
3Data not computed
5Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.3.2t1.a.a$1$ $ 3 $ \(\Q(\sqrt{-3}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.15.2t1.a.a$1$ $ 3 \cdot 5 $ \(\Q(\sqrt{-15}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.45.6t1.b.a$1$ $ 3^{2} \cdot 5 $ 6.0.2460375.1 $C_6$ (as 6T1) $0$ $-1$
1.45.6t1.a.a$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
1.9.6t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})\) $C_6$ (as 6T1) $0$ $-1$
1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.9.6t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})\) $C_6$ (as 6T1) $0$ $-1$
1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.45.6t1.b.b$1$ $ 3^{2} \cdot 5 $ 6.0.2460375.1 $C_6$ (as 6T1) $0$ $-1$
1.45.6t1.a.b$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
* 1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.15.4t1.a.a$1$ $ 3 \cdot 5 $ \(\Q(\zeta_{15})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.15.4t1.a.b$1$ $ 3 \cdot 5 $ \(\Q(\zeta_{15})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
1.45.12t1.b.a$1$ $ 3^{2} \cdot 5 $ \(\Q(\zeta_{45})^+\) $C_{12}$ (as 12T1) $0$ $1$
1.45.12t1.b.b$1$ $ 3^{2} \cdot 5 $ \(\Q(\zeta_{45})^+\) $C_{12}$ (as 12T1) $0$ $1$
1.45.12t1.a.a$1$ $ 3^{2} \cdot 5 $ 12.0.84075626953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.45.12t1.a.b$1$ $ 3^{2} \cdot 5 $ 12.0.84075626953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.45.12t1.b.c$1$ $ 3^{2} \cdot 5 $ \(\Q(\zeta_{45})^+\) $C_{12}$ (as 12T1) $0$ $1$
1.45.12t1.b.d$1$ $ 3^{2} \cdot 5 $ \(\Q(\zeta_{45})^+\) $C_{12}$ (as 12T1) $0$ $1$
1.45.12t1.a.c$1$ $ 3^{2} \cdot 5 $ 12.0.84075626953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.45.12t1.a.d$1$ $ 3^{2} \cdot 5 $ 12.0.84075626953125.1 $C_{12}$ (as 12T1) $0$ $-1$
2.135.3t2.b.a$2$ $ 3^{3} \cdot 5 $ 3.1.135.1 $S_3$ (as 3T2) $1$ $0$
2.135.6t3.a.a$2$ $ 3^{3} \cdot 5 $ 6.2.91125.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.405.6t5.a.a$2$ $ 3^{4} \cdot 5 $ 6.0.2460375.2 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.405.12t18.a.a$2$ $ 3^{4} \cdot 5 $ 12.0.6053445140625.2 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.405.6t5.a.b$2$ $ 3^{4} \cdot 5 $ 6.0.2460375.2 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.405.12t18.a.b$2$ $ 3^{4} \cdot 5 $ 12.0.6053445140625.2 $C_6\times S_3$ (as 12T18) $0$ $0$
2.675.12t11.a.a$2$ $ 3^{3} \cdot 5^{2}$ 12.0.1037970703125.1 $S_3 \times C_4$ (as 12T11) $0$ $0$
2.675.12t11.a.b$2$ $ 3^{3} \cdot 5^{2}$ 12.0.1037970703125.1 $S_3 \times C_4$ (as 12T11) $0$ $0$
* 2.2025.24t65.a.a$2$ $ 3^{4} \cdot 5^{2}$ 24.0.572565594852444156646728515625.2 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.2025.24t65.a.b$2$ $ 3^{4} \cdot 5^{2}$ 24.0.572565594852444156646728515625.2 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.2025.24t65.a.c$2$ $ 3^{4} \cdot 5^{2}$ 24.0.572565594852444156646728515625.2 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.2025.24t65.a.d$2$ $ 3^{4} \cdot 5^{2}$ 24.0.572565594852444156646728515625.2 $C_{12}\times S_3$ (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 *.