Properties

Label 24.0.430...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $4.307\times 10^{31}$
Root discriminant $20.80$
Ramified primes $5, 7, 13$
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 - x^23 + 3*x^22 + x^21 + 7*x^20 + 20*x^19 + 16*x^18 - 5*x^17 + 80*x^16 + 120*x^15 - 277*x^14 + 191*x^13 + 511*x^12 - 356*x^11 - 47*x^10 + 255*x^9 - 75*x^8 - 130*x^7 + 96*x^6 + 20*x^5 - 43*x^4 + 14*x^3 + 3*x^2 - 4*x + 1)
 
gp: K = bnfinit(x^24 - x^23 + 3*x^22 + x^21 + 7*x^20 + 20*x^19 + 16*x^18 - 5*x^17 + 80*x^16 + 120*x^15 - 277*x^14 + 191*x^13 + 511*x^12 - 356*x^11 - 47*x^10 + 255*x^9 - 75*x^8 - 130*x^7 + 96*x^6 + 20*x^5 - 43*x^4 + 14*x^3 + 3*x^2 - 4*x + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1, -4, 3, 14, -43, 20, 96, -130, -75, 255, -47, -356, 511, 191, -277, 120, 80, -5, 16, 20, 7, 1, 3, -1, 1]);
 

\( x^{24} - x^{23} + 3 x^{22} + x^{21} + 7 x^{20} + 20 x^{19} + 16 x^{18} - 5 x^{17} + 80 x^{16} + 120 x^{15} - 277 x^{14} + 191 x^{13} + 511 x^{12} - 356 x^{11} - 47 x^{10} + 255 x^{9} - 75 x^{8} - 130 x^{7} + 96 x^{6} + 20 x^{5} - 43 x^{4} + 14 x^{3} + 3 x^{2} - 4 x + 1 \)

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:  \(43070843460234840091705322265625\)\(\medspace = 5^{18}\cdot 7^{12}\cdot 13^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $20.80$
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:  $5, 7, 13$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{5} a^{20} + \frac{2}{5} a^{19} - \frac{2}{5} a^{15} - \frac{2}{5} a^{13} - \frac{1}{5} a^{10} + \frac{2}{5} a^{7} + \frac{2}{5} a^{5} - \frac{2}{5} a + \frac{1}{5}$, $\frac{1}{25} a^{21} + \frac{11}{25} a^{19} + \frac{2}{5} a^{18} - \frac{7}{25} a^{16} - \frac{11}{25} a^{15} + \frac{3}{25} a^{14} + \frac{9}{25} a^{13} + \frac{2}{5} a^{12} - \frac{1}{25} a^{11} + \frac{12}{25} a^{10} - \frac{1}{5} a^{9} + \frac{12}{25} a^{8} + \frac{6}{25} a^{7} - \frac{8}{25} a^{6} - \frac{9}{25} a^{5} - \frac{2}{5} a^{3} - \frac{7}{25} a^{2} - \frac{1}{5} a - \frac{2}{25}$, $\frac{1}{48579775} a^{22} + \frac{929372}{48579775} a^{21} + \frac{3117066}{48579775} a^{20} - \frac{24048838}{48579775} a^{19} - \frac{1365806}{9715955} a^{18} + \frac{21188693}{48579775} a^{17} - \frac{2297098}{9715955} a^{16} + \frac{9906676}{48579775} a^{15} + \frac{898481}{1943191} a^{14} + \frac{19022948}{48579775} a^{13} - \frac{10156881}{48579775} a^{12} + \frac{2825313}{9715955} a^{11} - \frac{20718171}{48579775} a^{10} - \frac{21859023}{48579775} a^{9} - \frac{3245356}{9715955} a^{8} + \frac{14409459}{48579775} a^{7} - \frac{2074297}{9715955} a^{6} - \frac{23899363}{48579775} a^{5} - \frac{2957667}{9715955} a^{4} + \frac{20623173}{48579775} a^{3} - \frac{20094159}{48579775} a^{2} - \frac{23026697}{48579775} a + \frac{1108186}{48579775}$, $\frac{1}{6185449344219184525} a^{23} + \frac{3891036209}{6185449344219184525} a^{22} + \frac{116498224961890979}{6185449344219184525} a^{21} + \frac{21644549374648754}{6185449344219184525} a^{20} + \frac{1408651668929942753}{6185449344219184525} a^{19} - \frac{603034350403729752}{6185449344219184525} a^{18} - \frac{1823178848810600949}{6185449344219184525} a^{17} - \frac{2129640756472300922}{6185449344219184525} a^{16} - \frac{2757114644170548352}{6185449344219184525} a^{15} - \frac{491529043765241361}{1237089868843836905} a^{14} - \frac{75928633062217539}{6185449344219184525} a^{13} + \frac{222408099220454083}{6185449344219184525} a^{12} + \frac{225705750380705847}{1237089868843836905} a^{11} + \frac{823287576665849463}{6185449344219184525} a^{10} + \frac{2245508767068038299}{6185449344219184525} a^{9} + \frac{2589788354561987012}{6185449344219184525} a^{8} + \frac{2753097424859937392}{6185449344219184525} a^{7} + \frac{17427322852597458}{247417973768767381} a^{6} - \frac{1404456415827245532}{6185449344219184525} a^{5} - \frac{2854421532648389322}{6185449344219184525} a^{4} + \frac{2760461889558071727}{6185449344219184525} a^{3} - \frac{2626818170710297123}{6185449344219184525} a^{2} - \frac{153552802669533373}{6185449344219184525} a + \frac{1651734680988137034}{6185449344219184525}$

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:  \( -\frac{1457897166849640138}{1237089868843836905} a^{23} + \frac{639520975469448258}{1237089868843836905} a^{22} - \frac{756379969124315973}{247417973768767381} a^{21} - \frac{3858059012597590842}{1237089868843836905} a^{20} - \frac{11523994884991911402}{1237089868843836905} a^{19} - \frac{35650695603798790534}{1237089868843836905} a^{18} - \frac{41390760923071746673}{1237089868843836905} a^{17} - \frac{11605642499717166083}{1237089868843836905} a^{16} - \frac{119480271356447902036}{1237089868843836905} a^{15} - \frac{241992695946705777594}{1237089868843836905} a^{14} + \frac{57671368561121023977}{247417973768767381} a^{13} - \frac{92608538521353306109}{1237089868843836905} a^{12} - \frac{858213682288180728847}{1237089868843836905} a^{11} + \frac{20901943470918526935}{247417973768767381} a^{10} + \frac{209116674464092941903}{1237089868843836905} a^{9} - \frac{332864747274619755427}{1237089868843836905} a^{8} - \frac{33487518560139406134}{1237089868843836905} a^{7} + \frac{196869994074594769109}{1237089868843836905} a^{6} - \frac{46453301483365639537}{1237089868843836905} a^{5} - \frac{72706267410129211654}{1237089868843836905} a^{4} + \frac{42637110868590830109}{1237089868843836905} a^{3} - \frac{649116610022975024}{247417973768767381} a^{2} - \frac{8900225595830356236}{1237089868843836905} a + \frac{4761840995787026854}{1237089868843836905} \) (order $10$)
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:  \( 3136393.8562446074 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 3136393.8562446074 \cdot 1}{10\sqrt{43070843460234840091705322265625}}\approx 0.180924477225979$ (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-7}) \), \(\Q(\sqrt{-35}) \), 4.4.6125.1, \(\Q(\zeta_{5})\), \(\Q(\sqrt{5}, \sqrt{-7})\), 6.0.7245875.1, 8.0.37515625.1, 12.0.52502704515625.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/3.12.0.1}{12} }{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }^{3}$ R R ${\href{/LocalNumberField/11.3.0.1}{3} }^{8}$ R ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{2}{,}\,{\href{/LocalNumberField/29.2.0.1}{2} }^{6}$ ${\href{/LocalNumberField/31.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/37.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$5$5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
5.8.6.1$x^{8} - 5 x^{4} + 400$$4$$2$$6$$C_4\times C_2$$[\ ]_{4}^{2}$
7Data not computed
$13$13.12.8.3$x^{12} + 26 x^{9} + 845 x^{6} + 6591 x^{3} + 114244$$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$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.7.2t1.a.a$1$ $ 7 $ \(\Q(\sqrt{-7}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.35.2t1.a.a$1$ $ 5 \cdot 7 $ \(\Q(\sqrt{-35}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.91.6t1.g.a$1$ $ 7 \cdot 13 $ 6.0.9796423.1 $C_6$ (as 6T1) $0$ $-1$
1.13.3t1.a.a$1$ $ 13 $ 3.3.169.1 $C_3$ (as 3T1) $0$ $1$
1.91.6t1.g.b$1$ $ 7 \cdot 13 $ 6.0.9796423.1 $C_6$ (as 6T1) $0$ $-1$
1.65.6t1.b.a$1$ $ 5 \cdot 13 $ 6.6.3570125.1 $C_6$ (as 6T1) $0$ $1$
1.455.6t1.b.a$1$ $ 5 \cdot 7 \cdot 13 $ 6.0.1224552875.2 $C_6$ (as 6T1) $0$ $-1$
1.65.6t1.b.b$1$ $ 5 \cdot 13 $ 6.6.3570125.1 $C_6$ (as 6T1) $0$ $1$
1.13.3t1.a.b$1$ $ 13 $ 3.3.169.1 $C_3$ (as 3T1) $0$ $1$
1.455.6t1.b.b$1$ $ 5 \cdot 7 \cdot 13 $ 6.0.1224552875.2 $C_6$ (as 6T1) $0$ $-1$
* 1.35.4t1.a.a$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (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) $0$ $-1$
* 1.35.4t1.a.b$1$ $ 5 \cdot 7 $ 4.4.6125.1 $C_4$ (as 4T1) $0$ $1$
1.455.12t1.a.a$1$ $ 5 \cdot 7 \cdot 13 $ 12.12.187441217958845703125.1 $C_{12}$ (as 12T1) $0$ $1$
1.455.12t1.a.b$1$ $ 5 \cdot 7 \cdot 13 $ 12.12.187441217958845703125.1 $C_{12}$ (as 12T1) $0$ $1$
1.455.12t1.a.c$1$ $ 5 \cdot 7 \cdot 13 $ 12.12.187441217958845703125.1 $C_{12}$ (as 12T1) $0$ $1$
1.455.12t1.a.d$1$ $ 5 \cdot 7 \cdot 13 $ 12.12.187441217958845703125.1 $C_{12}$ (as 12T1) $0$ $1$
1.65.12t1.a.a$1$ $ 5 \cdot 13 $ 12.0.1593224064453125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.65.12t1.a.b$1$ $ 5 \cdot 13 $ 12.0.1593224064453125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.65.12t1.a.c$1$ $ 5 \cdot 13 $ 12.0.1593224064453125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.65.12t1.a.d$1$ $ 5 \cdot 13 $ 12.0.1593224064453125.1 $C_{12}$ (as 12T1) $0$ $-1$
2.5915.3t2.a.a$2$ $ 5 \cdot 7 \cdot 13^{2}$ 3.1.5915.1 $S_3$ (as 3T2) $1$ $0$
2.5915.6t3.f.a$2$ $ 5 \cdot 7 \cdot 13^{2}$ 6.2.174936125.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.455.12t18.a.a$2$ $ 5 \cdot 7 \cdot 13 $ 12.0.52502704515625.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.455.6t5.a.a$2$ $ 5 \cdot 7 \cdot 13 $ 6.0.7245875.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.455.12t18.a.b$2$ $ 5 \cdot 7 \cdot 13 $ 12.0.52502704515625.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.455.6t5.a.b$2$ $ 5 \cdot 7 \cdot 13 $ 6.0.7245875.1 $S_3\times C_3$ (as 6T5) $0$ $0$
2.29575.12t11.b.a$2$ $ 5^{2} \cdot 7 \cdot 13^{2}$ 12.4.187441217958845703125.1 $S_3 \times C_4$ (as 12T11) $0$ $0$
2.29575.12t11.b.b$2$ $ 5^{2} \cdot 7 \cdot 13^{2}$ 12.4.187441217958845703125.1 $S_3 \times C_4$ (as 12T11) $0$ $0$
* 2.2275.24t65.a.a$2$ $ 5^{2} \cdot 7 \cdot 13 $ 24.0.43070843460234840091705322265625.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.2275.24t65.a.b$2$ $ 5^{2} \cdot 7 \cdot 13 $ 24.0.43070843460234840091705322265625.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.2275.24t65.a.c$2$ $ 5^{2} \cdot 7 \cdot 13 $ 24.0.43070843460234840091705322265625.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.2275.24t65.a.d$2$ $ 5^{2} \cdot 7 \cdot 13 $ 24.0.43070843460234840091705322265625.1 $C_{12}\times S_3$ (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 *.