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Properties

Label 24.0.344...625.1
Degree $24$
Signature $[0, 12]$
Discriminant $3.443\times 10^{28}$
Root discriminant $15.45$
Ramified primes $3, 5, 19$
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 - 4*x^22 + x^21 + x^20 - 5*x^19 + 37*x^18 + 33*x^17 - 39*x^16 - 57*x^15 - 111*x^14 - 16*x^13 + 118*x^12 + 13*x^11 + 47*x^9 + 156*x^8 + 319*x^7 + 175*x^6 - 76*x^5 - 127*x^4 - 44*x^3 + 48*x^2 + 56*x + 16)
 
gp: K = bnfinit(x^24 - x^23 - 4*x^22 + x^21 + x^20 - 5*x^19 + 37*x^18 + 33*x^17 - 39*x^16 - 57*x^15 - 111*x^14 - 16*x^13 + 118*x^12 + 13*x^11 + 47*x^9 + 156*x^8 + 319*x^7 + 175*x^6 - 76*x^5 - 127*x^4 - 44*x^3 + 48*x^2 + 56*x + 16, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![16, 56, 48, -44, -127, -76, 175, 319, 156, 47, 0, 13, 118, -16, -111, -57, -39, 33, 37, -5, 1, 1, -4, -1, 1]);
 

\( x^{24} - x^{23} - 4 x^{22} + x^{21} + x^{20} - 5 x^{19} + 37 x^{18} + 33 x^{17} - 39 x^{16} - 57 x^{15} - 111 x^{14} - 16 x^{13} + 118 x^{12} + 13 x^{11} + 47 x^{9} + 156 x^{8} + 319 x^{7} + 175 x^{6} - 76 x^{5} - 127 x^{4} - 44 x^{3} + 48 x^{2} + 56 x + 16 \)

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:  \(34430548576629947662353515625\)\(\medspace = 3^{12}\cdot 5^{18}\cdot 19^{8}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $15.45$
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:  $3, 5, 19$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{11} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{16} - \frac{1}{2} a^{14} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{5} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2} a$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{16} - \frac{1}{2} a^{15} - \frac{1}{2} a^{14} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{15} - \frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{16} - \frac{1}{2} a^{13} - \frac{1}{2} a^{12} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{1425556} a^{22} + \frac{41357}{1425556} a^{21} - \frac{15043}{712778} a^{20} - \frac{196965}{1425556} a^{19} - \frac{317161}{1425556} a^{18} + \frac{60521}{1425556} a^{17} + \frac{44291}{129596} a^{16} + \frac{682817}{1425556} a^{15} - \frac{353827}{1425556} a^{14} - \frac{55911}{1425556} a^{13} - \frac{519895}{1425556} a^{12} + \frac{135136}{356389} a^{11} + \frac{219189}{712778} a^{10} - \frac{51293}{129596} a^{9} + \frac{324799}{712778} a^{8} - \frac{18529}{1425556} a^{7} + \frac{3869}{32399} a^{6} - \frac{365647}{1425556} a^{5} + \frac{165805}{1425556} a^{4} + \frac{274965}{712778} a^{3} - \frac{405607}{1425556} a^{2} - \frac{29541}{356389} a + \frac{46623}{356389}$, $\frac{1}{85703606366693128} a^{23} + \frac{18163868979}{85703606366693128} a^{22} + \frac{1048338166402372}{10712950795836641} a^{21} + \frac{15906265505534113}{85703606366693128} a^{20} + \frac{12743854618137629}{85703606366693128} a^{19} - \frac{878378445445419}{7791236942426648} a^{18} + \frac{7913804544221361}{85703606366693128} a^{17} - \frac{28970327234607371}{85703606366693128} a^{16} + \frac{23854512785139669}{85703606366693128} a^{15} + \frac{36354610084564603}{85703606366693128} a^{14} + \frac{3661253358170429}{85703606366693128} a^{13} + \frac{906506742036311}{21425901591673282} a^{12} - \frac{3819911632942621}{42851803183346564} a^{11} - \frac{29553639811128595}{85703606366693128} a^{10} + \frac{2650181397954751}{21425901591673282} a^{9} + \frac{7273315788756047}{85703606366693128} a^{8} + \frac{5059110145331208}{10712950795836641} a^{7} + \frac{28804495284019239}{85703606366693128} a^{6} - \frac{1322431666416461}{85703606366693128} a^{5} + \frac{3976816316270953}{10712950795836641} a^{4} + \frac{27951577922856681}{85703606366693128} a^{3} + \frac{2755054537062789}{10712950795836641} a^{2} - \frac{3158720930017212}{10712950795836641} a + \frac{2636870551896277}{10712950795836641}$

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:  \( -\frac{117915814377341653}{42851803183346564} a^{23} + \frac{197207171645330503}{42851803183346564} a^{22} + \frac{85006782973489926}{10712950795836641} a^{21} - \frac{348478004387897971}{42851803183346564} a^{20} + \frac{113679103893968149}{42851803183346564} a^{19} + \frac{518245734125825657}{42851803183346564} a^{18} - \frac{4713653758914993037}{42851803183346564} a^{17} - \frac{727035149700994613}{42851803183346564} a^{16} + \frac{5129588610391762855}{42851803183346564} a^{15} + \frac{3265932441257164717}{42851803183346564} a^{14} + \frac{10829050500302674923}{42851803183346564} a^{13} - \frac{1346668237222617290}{10712950795836641} a^{12} - \frac{2592770900643767350}{10712950795836641} a^{11} + \frac{5523067453432221815}{42851803183346564} a^{10} - \frac{1809883378337436571}{21425901591673282} a^{9} - \frac{17747524028759473}{236750293830644} a^{8} - \frac{8086893335090639107}{21425901591673282} a^{7} - \frac{26743766156315915693}{42851803183346564} a^{6} - \frac{2551875730292042513}{42851803183346564} a^{5} + \frac{2704264347260549967}{10712950795836641} a^{4} + \frac{691362455020807511}{3895618471213324} a^{3} - \frac{25998071151352511}{21425901591673282} a^{2} - \frac{258156184163192927}{1947809235606662} a - \frac{691973979704513672}{10712950795836641} \) (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:  \( 242614.53860678803 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{0}\cdot(2\pi)^{12}\cdot 242614.53860678803 \cdot 1}{30\sqrt{34430548576629947662353515625}}\approx 0.164999267690649$ (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

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{-3}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{15})^+\), \(\Q(\sqrt{-3}, \sqrt{5})\), 6.0.1218375.1, \(\Q(\zeta_{15})\), 12.0.1484437640625.1

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 ${\href{/LocalNumberField/2.12.0.1}{12} }{,}\,{\href{/LocalNumberField/2.4.0.1}{4} }^{3}$ R R ${\href{/LocalNumberField/7.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/11.2.0.1}{2} }^{12}$ ${\href{/LocalNumberField/13.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/17.12.0.1}{12} }{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }^{3}$ R ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/31.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/41.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }{,}\,{\href{/LocalNumberField/47.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/53.12.0.1}{12} }{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }^{3}$ ${\href{/LocalNumberField/59.6.0.1}{6} }^{4}$

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
$19$19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
19.6.4.1$x^{6} + 57 x^{3} + 1444$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$
19.6.0.1$x^{6} - x + 3$$1$$6$$0$$C_6$$[\ ]^{6}$
19.6.4.1$x^{6} + 57 x^{3} + 1444$$3$$2$$4$$C_6$$[\ ]_{3}^{2}$

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) $1$ $1$
1.285.6t1.a.a$1$ $ 3 \cdot 5 \cdot 19 $ 6.0.439833375.1 $C_6$ (as 6T1) $0$ $-1$
1.95.6t1.a.a$1$ $ 5 \cdot 19 $ 6.6.16290125.1 $C_6$ (as 6T1) $0$ $1$
1.57.6t1.a.a$1$ $ 3 \cdot 19 $ 6.0.3518667.1 $C_6$ (as 6T1) $0$ $-1$
1.19.3t1.a.a$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
1.95.6t1.a.b$1$ $ 5 \cdot 19 $ 6.6.16290125.1 $C_6$ (as 6T1) $0$ $1$
1.19.3t1.a.b$1$ $ 19 $ 3.3.361.1 $C_3$ (as 3T1) $0$ $1$
1.285.6t1.a.b$1$ $ 3 \cdot 5 \cdot 19 $ 6.0.439833375.1 $C_6$ (as 6T1) $0$ $-1$
1.57.6t1.a.b$1$ $ 3 \cdot 19 $ 6.0.3518667.1 $C_6$ (as 6T1) $0$ $-1$
* 1.15.4t1.a.a$1$ $ 3 \cdot 5 $ \(\Q(\zeta_{15})^+\) $C_4$ (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) $0$ $-1$
* 1.15.4t1.a.b$1$ $ 3 \cdot 5 $ \(\Q(\zeta_{15})^+\) $C_4$ (as 4T1) $0$ $1$
1.95.12t1.a.a$1$ $ 5 \cdot 19 $ 12.0.33171021564453125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.95.12t1.a.b$1$ $ 5 \cdot 19 $ 12.0.33171021564453125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.95.12t1.a.c$1$ $ 5 \cdot 19 $ 12.0.33171021564453125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.95.12t1.a.d$1$ $ 5 \cdot 19 $ 12.0.33171021564453125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.285.12t1.a.a$1$ $ 3 \cdot 5 \cdot 19 $ 12.12.24181674720486328125.1 $C_{12}$ (as 12T1) $0$ $1$
1.285.12t1.a.b$1$ $ 3 \cdot 5 \cdot 19 $ 12.12.24181674720486328125.1 $C_{12}$ (as 12T1) $0$ $1$
1.285.12t1.a.c$1$ $ 3 \cdot 5 \cdot 19 $ 12.12.24181674720486328125.1 $C_{12}$ (as 12T1) $0$ $1$
1.285.12t1.a.d$1$ $ 3 \cdot 5 \cdot 19 $ 12.12.24181674720486328125.1 $C_{12}$ (as 12T1) $0$ $1$
2.5415.3t2.a.a$2$ $ 3 \cdot 5 \cdot 19^{2}$ 3.1.5415.1 $S_3$ (as 3T2) $1$ $0$
2.5415.6t3.d.a$2$ $ 3 \cdot 5 \cdot 19^{2}$ 6.2.146611125.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.285.6t5.c.a$2$ $ 3 \cdot 5 \cdot 19 $ 6.0.1218375.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.285.12t18.a.a$2$ $ 3 \cdot 5 \cdot 19 $ 12.0.1484437640625.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.285.12t18.a.b$2$ $ 3 \cdot 5 \cdot 19 $ 12.0.1484437640625.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.285.6t5.c.b$2$ $ 3 \cdot 5 \cdot 19 $ 6.0.1218375.1 $S_3\times C_3$ (as 6T5) $0$ $0$
2.27075.12t11.b.a$2$ $ 3 \cdot 5^{2} \cdot 19^{2}$ 12.4.24181674720486328125.1 $S_3 \times C_4$ (as 12T11) $0$ $0$
2.27075.12t11.b.b$2$ $ 3 \cdot 5^{2} \cdot 19^{2}$ 12.4.24181674720486328125.1 $S_3 \times C_4$ (as 12T11) $0$ $0$
* 2.1425.24t65.a.a$2$ $ 3 \cdot 5^{2} \cdot 19 $ 24.0.34430548576629947662353515625.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.1425.24t65.a.b$2$ $ 3 \cdot 5^{2} \cdot 19 $ 24.0.34430548576629947662353515625.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.1425.24t65.a.c$2$ $ 3 \cdot 5^{2} \cdot 19 $ 24.0.34430548576629947662353515625.1 $C_{12}\times S_3$ (as 24T65) $0$ $0$
* 2.1425.24t65.a.d$2$ $ 3 \cdot 5^{2} \cdot 19 $ 24.0.34430548576629947662353515625.1 $C_{12}\times S_3$ (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 *.