Properties

Label 24.0.275...000.1
Degree $24$
Signature $[0, 12]$
Discriminant $2.755\times 10^{27}$
Root discriminant $13.91$
Ramified primes $2, 3, 5$
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^23 + 5*x^22 - 12*x^21 + 16*x^20 - 30*x^19 + 35*x^18 - 36*x^17 + 47*x^16 - 36*x^15 + 9*x^14 - 16*x^13 + 39*x^12 + 80*x^11 + 53*x^10 - 46*x^8 - 20*x^7 + 34*x^6 + 50*x^5 + 38*x^4 + 18*x^3 + 9*x^2 + 4*x + 1)
 
gp: K = bnfinit(x^24 - 2*x^23 + 5*x^22 - 12*x^21 + 16*x^20 - 30*x^19 + 35*x^18 - 36*x^17 + 47*x^16 - 36*x^15 + 9*x^14 - 16*x^13 + 39*x^12 + 80*x^11 + 53*x^10 - 46*x^8 - 20*x^7 + 34*x^6 + 50*x^5 + 38*x^4 + 18*x^3 + 9*x^2 + 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, 4, 9, 18, 38, 50, 34, -20, -46, 0, 53, 80, 39, -16, 9, -36, 47, -36, 35, -30, 16, -12, 5, -2, 1]);
 

\( x^{24} - 2 x^{23} + 5 x^{22} - 12 x^{21} + 16 x^{20} - 30 x^{19} + 35 x^{18} - 36 x^{17} + 47 x^{16} - 36 x^{15} + 9 x^{14} - 16 x^{13} + 39 x^{12} + 80 x^{11} + 53 x^{10} - 46 x^{8} - 20 x^{7} + 34 x^{6} + 50 x^{5} + 38 x^{4} + 18 x^{3} + 9 x^{2} + 4 x + 1 \)

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:  \(2754990144000000000000000000\)\(\medspace = 2^{24}\cdot 3^{16}\cdot 5^{18}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $13.91$
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:  $2, 3, 5$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{1627978215053041} a^{23} + \frac{125004402368765}{1627978215053041} a^{22} - \frac{181872483737775}{1627978215053041} a^{21} - \frac{692289305945831}{1627978215053041} a^{20} - \frac{344318626782306}{1627978215053041} a^{19} - \frac{666441772493729}{1627978215053041} a^{18} - \frac{216584206881134}{1627978215053041} a^{17} - \frac{715052720422932}{1627978215053041} a^{16} + \frac{333893682608393}{1627978215053041} a^{15} - \frac{282478792253345}{1627978215053041} a^{14} + \frac{209562288719590}{1627978215053041} a^{13} + \frac{14579344490490}{1627978215053041} a^{12} - \frac{647605619942813}{1627978215053041} a^{11} - \frac{298397729799972}{1627978215053041} a^{10} + \frac{50380689934097}{1627978215053041} a^{9} - \frac{163402279732272}{1627978215053041} a^{8} - \frac{154289618103356}{1627978215053041} a^{7} - \frac{515290496409285}{1627978215053041} a^{6} - \frac{610745727853033}{1627978215053041} a^{5} + \frac{713158735293030}{1627978215053041} a^{4} + \frac{83841989991487}{1627978215053041} a^{3} + \frac{231051949626161}{1627978215053041} a^{2} - \frac{718099794527682}{1627978215053041} a - \frac{754456509957561}{1627978215053041}$

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:  \( -\frac{182726643215655}{1627978215053041} a^{23} + \frac{962616722473751}{1627978215053041} a^{22} - \frac{2683818657405245}{1627978215053041} a^{21} + \frac{6455789481222930}{1627978215053041} a^{20} - \frac{12748207408726678}{1627978215053041} a^{19} + \frac{21221675082736354}{1627978215053041} a^{18} - \frac{32307569887879579}{1627978215053041} a^{17} + \frac{40882368026131925}{1627978215053041} a^{16} - \frac{45509506867823154}{1627978215053041} a^{15} + \frac{47167863732701051}{1627978215053041} a^{14} - \frac{39145484118482017}{1627978215053041} a^{13} + \frac{21803020531739926}{1627978215053041} a^{12} - \frac{12721052992563022}{1627978215053041} a^{11} + \frac{3465405237312246}{1627978215053041} a^{10} + \frac{17419680328333945}{1627978215053041} a^{9} - \frac{4770319527126960}{1627978215053041} a^{8} + \frac{6724773344866219}{1627978215053041} a^{7} - \frac{9567093455679535}{1627978215053041} a^{6} - \frac{2646793271721465}{1627978215053041} a^{5} + \frac{4931217970856872}{1627978215053041} a^{4} + \frac{2708519680365961}{1627978215053041} a^{3} + \frac{7387044184720634}{1627978215053041} a^{2} + \frac{986346911660636}{1627978215053041} a + \frac{536769987693331}{1627978215053041} \) (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:  \( 48570.384925004604 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 48570.384925004604 \cdot 1}{20\sqrt{2754990144000000000000000000}}\approx 0.175162020115116$ (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

\(\Q(\sqrt{-5}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), \(\Q(i, \sqrt{5})\), 6.0.648000.1, \(\Q(\zeta_{20})\), 12.0.419904000000.1

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 R R ${\href{/LocalNumberField/7.12.0.1}{12} }{,}\,{\href{/LocalNumberField/7.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/11.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/17.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/19.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/23.12.0.1}{12} }{,}\,{\href{/LocalNumberField/23.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/41.3.0.1}{3} }^{4}{,}\,{\href{/LocalNumberField/41.1.0.1}{1} }^{12}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$3$3.12.16.25$x^{12} + 93 x^{11} - 36 x^{10} + 357 x^{9} + 270 x^{8} + 324 x^{7} + 207 x^{6} - 216 x^{5} - 324 x^{4} - 54 x^{3} - 81 x^{2} - 324$$3$$4$$16$$C_{12}$$[2]^{4}$
3.12.0.1$x^{12} - x^{4} - x^{3} - x^{2} + x - 1$$1$$12$$0$$C_{12}$$[\ ]^{12}$
$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}$

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) $1$ $1$
1.9.3t1.a.a$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
1.36.6t1.b.a$1$ $ 2^{2} \cdot 3^{2}$ 6.0.419904.1 $C_6$ (as 6T1) $0$ $-1$
1.45.6t1.a.a$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
1.45.6t1.a.b$1$ $ 3^{2} \cdot 5 $ 6.6.820125.1 $C_6$ (as 6T1) $0$ $1$
1.180.6t1.b.a$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 6.0.52488000.1 $C_6$ (as 6T1) $0$ $-1$
1.36.6t1.b.b$1$ $ 2^{2} \cdot 3^{2}$ 6.0.419904.1 $C_6$ (as 6T1) $0$ $-1$
1.180.6t1.b.b$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 6.0.52488000.1 $C_6$ (as 6T1) $0$ $-1$
1.9.3t1.a.b$1$ $ 3^{2}$ \(\Q(\zeta_{9})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.20.4t1.a.a$1$ $ 2^{2} \cdot 5 $ \(\Q(\zeta_{20})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.20.4t1.a.b$1$ $ 2^{2} \cdot 5 $ \(\Q(\zeta_{20})^+\) $C_4$ (as 4T1) $0$ $1$
1.45.12t1.a.a$1$ $ 3^{2} \cdot 5 $ 12.0.84075626953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.180.12t1.b.a$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 12.12.344373768000000000.1 $C_{12}$ (as 12T1) $0$ $1$
1.45.12t1.a.b$1$ $ 3^{2} \cdot 5 $ 12.0.84075626953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.180.12t1.b.b$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 12.12.344373768000000000.1 $C_{12}$ (as 12T1) $0$ $1$
1.180.12t1.b.c$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 12.12.344373768000000000.1 $C_{12}$ (as 12T1) $0$ $1$
1.180.12t1.b.d$1$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 12.12.344373768000000000.1 $C_{12}$ (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) $0$ $-1$
2.1620.3t2.b.a$2$ $ 2^{2} \cdot 3^{4} \cdot 5 $ 3.1.1620.1 $S_3$ (as 3T2) $1$ $0$
2.1620.6t3.g.a$2$ $ 2^{2} \cdot 3^{4} \cdot 5 $ 6.2.13122000.3 $D_{6}$ (as 6T3) $1$ $0$
* 2.180.6t5.b.a$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 6.0.648000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.180.12t18.a.a$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 12.0.419904000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.180.6t5.b.b$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 6.0.648000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.180.12t18.a.b$2$ $ 2^{2} \cdot 3^{2} \cdot 5 $ 12.0.419904000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$
2.8100.12t11.c.a$2$ $ 2^{2} \cdot 3^{4} \cdot 5^{2}$ 12.0.21523360500000000.3 $S_3 \times C_4$ (as 12T11) $0$ $0$
2.8100.12t11.c.b$2$ $ 2^{2} \cdot 3^{4} \cdot 5^{2}$ 12.0.21523360500000000.3 $S_3 \times C_4$ (as 12T11) $0$ $0$
* 2.900.24t65.b.a$2$ $ 2^{2} \cdot 3^{2} \cdot 5^{2}$ 24.0.2754990144000000000000000000.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.900.24t65.b.b$2$ $ 2^{2} \cdot 3^{2} \cdot 5^{2}$ 24.0.2754990144000000000000000000.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.900.24t65.b.c$2$ $ 2^{2} \cdot 3^{2} \cdot 5^{2}$ 24.0.2754990144000000000000000000.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.900.24t65.b.d$2$ $ 2^{2} \cdot 3^{2} \cdot 5^{2}$ 24.0.2754990144000000000000000000.1 $C_{12}\times S_3$ (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 *.