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Properties

Label 24.4.491...064.2
Degree $24$
Signature $[4, 10]$
Discriminant $4.918\times 10^{31}$
Root discriminant $20.92$
Ramified primes $2, 487$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $C_4.A_5$ (as 24T576)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 4*x^23 + 7*x^22 - 10*x^21 + 23*x^20 - 32*x^19 - 21*x^18 + 156*x^17 - 359*x^16 + 654*x^15 - 876*x^14 + 646*x^13 + 61*x^12 - 1012*x^11 + 2338*x^10 - 3178*x^9 + 2897*x^8 - 2474*x^7 + 1335*x^6 - 52*x^5 - 44*x^4 + 454*x^3 - 524*x^2 + 62*x - 197)
 
gp: K = bnfinit(x^24 - 4*x^23 + 7*x^22 - 10*x^21 + 23*x^20 - 32*x^19 - 21*x^18 + 156*x^17 - 359*x^16 + 654*x^15 - 876*x^14 + 646*x^13 + 61*x^12 - 1012*x^11 + 2338*x^10 - 3178*x^9 + 2897*x^8 - 2474*x^7 + 1335*x^6 - 52*x^5 - 44*x^4 + 454*x^3 - 524*x^2 + 62*x - 197, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-197, 62, -524, 454, -44, -52, 1335, -2474, 2897, -3178, 2338, -1012, 61, 646, -876, 654, -359, 156, -21, -32, 23, -10, 7, -4, 1]);
 

\( x^{24} - 4 x^{23} + 7 x^{22} - 10 x^{21} + 23 x^{20} - 32 x^{19} - 21 x^{18} + 156 x^{17} - 359 x^{16} + 654 x^{15} - 876 x^{14} + 646 x^{13} + 61 x^{12} - 1012 x^{11} + 2338 x^{10} - 3178 x^{9} + 2897 x^{8} - 2474 x^{7} + 1335 x^{6} - 52 x^{5} - 44 x^{4} + 454 x^{3} - 524 x^{2} + 62 x - 197 \)

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[4, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
Discriminant:  \(49177850545349555386638457176064\)\(\medspace = 2^{16}\cdot 487^{10}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $20.92$
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:  $2, 487$
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(Integers(K)));
 
$|\Aut(K/\Q)|$:  $4$
This field is not Galois over $\Q$.
This is not a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2} a^{12} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{13} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a - \frac{1}{2}$, $\frac{1}{2} a^{14} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{2} - \frac{1}{2} a$, $\frac{1}{2} a^{15} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{16} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{17} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{5} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{18} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{2} a^{19} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{5} - \frac{1}{2} a^{2}$, $\frac{1}{2} a^{20} - \frac{1}{2} a^{11} - \frac{1}{2} a^{10} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3}$, $\frac{1}{2} a^{21} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{6} - \frac{1}{2} a^{3} - \frac{1}{2}$, $\frac{1}{2} a^{22} - \frac{1}{2} a^{9} - \frac{1}{2} a^{8} - \frac{1}{2} a^{7} - \frac{1}{2} a^{4} - \frac{1}{2} a$, $\frac{1}{115185014073587688626044229896894882} a^{23} + \frac{13196870324992556049598572122405941}{115185014073587688626044229896894882} a^{22} - \frac{11671535596225596628078109391054221}{115185014073587688626044229896894882} a^{21} + \frac{23744713858496308676233787942642429}{115185014073587688626044229896894882} a^{20} - \frac{7530635156796957321962582417836355}{57592507036793844313022114948447441} a^{19} - \frac{9952557122952966122951643586578086}{57592507036793844313022114948447441} a^{18} - \frac{16497674490836624917515616632409013}{115185014073587688626044229896894882} a^{17} - \frac{26648920765431735264368239823907797}{115185014073587688626044229896894882} a^{16} + \frac{24639246379131273432660992799893513}{115185014073587688626044229896894882} a^{15} + \frac{8709625889055557196854135294074418}{57592507036793844313022114948447441} a^{14} + \frac{567001852376675415387248568428411}{115185014073587688626044229896894882} a^{13} + \frac{11760654826935316502380432883244974}{57592507036793844313022114948447441} a^{12} - \frac{13811697744241437944776138472011542}{57592507036793844313022114948447441} a^{11} + \frac{1736383686560729479278551873265387}{57592507036793844313022114948447441} a^{10} + \frac{20554503329569114793407113851770087}{115185014073587688626044229896894882} a^{9} - \frac{26383724191937117223132792792896514}{57592507036793844313022114948447441} a^{8} - \frac{18064222699371268733661761602401621}{115185014073587688626044229896894882} a^{7} - \frac{56763120701848791646755797650509427}{115185014073587688626044229896894882} a^{6} + \frac{6784424253851806116067608127518928}{57592507036793844313022114948447441} a^{5} - \frac{26030788305178578455692534578659120}{57592507036793844313022114948447441} a^{4} + \frac{22462599389480056929401766682526854}{57592507036793844313022114948447441} a^{3} - \frac{21314950002666189578013652663796383}{57592507036793844313022114948447441} a^{2} + \frac{12608012883362772250574401591255960}{57592507036793844313022114948447441} a + \frac{38903284968337995668744984436211425}{115185014073587688626044229896894882}$

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:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
Torsion generator:  \( -1 \) (order $2$)
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:  \( 1432261.5638336123 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{10}\cdot 1432261.5638336123 \cdot 1}{2\sqrt{49177850545349555386638457176064}}\approx 0.156684572333305$ (assuming GRH)

Galois group

$C_4.A_5$ (as 24T576):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: GaloisGroup(K);
 
A non-solvable group of order 240
The 18 conjugacy class representatives for $C_4.A_5$
Character table for $C_4.A_5$

Intermediate fields

6.2.3794704.1, Deg 12

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Degree 40 siblings: data not computed
Arithmetically equvalently sibling: 24.4.49177850545349555386638457176064.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R $20{,}\,{\href{/LocalNumberField/3.4.0.1}{4} }$ $20{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/11.12.0.1}{12} }^{2}$ $20{,}\,{\href{/LocalNumberField/13.4.0.1}{4} }$ ${\href{/LocalNumberField/17.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/19.6.0.1}{6} }^{4}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{6}$ $20{,}\,{\href{/LocalNumberField/37.4.0.1}{4} }$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/43.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/47.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/53.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/53.2.0.1}{2} }^{2}$ $20{,}\,{\href{/LocalNumberField/59.4.0.1}{4} }$

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
$2$2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
2.12.8.1$x^{12} - 6 x^{9} + 12 x^{6} - 8 x^{3} + 16$$3$$4$$8$$C_3 : C_4$$[\ ]_{3}^{4}$
487Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.487.2t1.a.a$1$ $ 487 $ \(\Q(\sqrt{-487}) \) $C_2$ (as 2T1) $1$ $-1$
2.1948.120.a.a$2$ $ 2^{2} \cdot 487 $ 24.4.49177850545349555386638457176064.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.1948.120.a.b$2$ $ 2^{2} \cdot 487 $ 24.4.49177850545349555386638457176064.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.1948.120.a.c$2$ $ 2^{2} \cdot 487 $ 24.4.49177850545349555386638457176064.2 $C_4.A_5$ (as 24T576) $0$ $0$
2.1948.120.a.d$2$ $ 2^{2} \cdot 487 $ 24.4.49177850545349555386638457176064.2 $C_4.A_5$ (as 24T576) $0$ $0$
* 3.1948.12t76.a.a$3$ $ 2^{2} \cdot 487 $ 10.0.438293256499312.1 $A_5\times C_2$ (as 10T11) $1$ $1$
* 3.1948.12t76.a.b$3$ $ 2^{2} \cdot 487 $ 10.0.438293256499312.1 $A_5\times C_2$ (as 10T11) $1$ $1$
3.948676.12t33.a.a$3$ $ 2^{2} \cdot 487^{2}$ 5.1.948676.1 $A_5$ (as 5T4) $1$ $-1$
3.948676.12t33.a.b$3$ $ 2^{2} \cdot 487^{2}$ 5.1.948676.1 $A_5$ (as 5T4) $1$ $-1$
4.948676.10t11.a.a$4$ $ 2^{2} \cdot 487^{2}$ 10.0.438293256499312.1 $A_5\times C_2$ (as 10T11) $1$ $0$
4.948676.5t4.a.a$4$ $ 2^{2} \cdot 487^{2}$ 5.1.948676.1 $A_5$ (as 5T4) $1$ $0$
4.948676.40t188.a.a$4$ $ 2^{2} \cdot 487^{2}$ 24.4.49177850545349555386638457176064.2 $C_4.A_5$ (as 24T576) $0$ $0$
4.948676.40t188.a.b$4$ $ 2^{2} \cdot 487^{2}$ 24.4.49177850545349555386638457176064.2 $C_4.A_5$ (as 24T576) $0$ $0$
5.1848020848.12t75.a.a$5$ $ 2^{4} \cdot 487^{3}$ 10.0.438293256499312.1 $A_5\times C_2$ (as 10T11) $1$ $-1$
* 5.3794704.6t12.a.a$5$ $ 2^{4} \cdot 487^{2}$ 5.1.948676.1 $A_5$ (as 5T4) $1$ $1$
* 6.1848020848.24t576.a.a$6$ $ 2^{4} \cdot 487^{3}$ 24.4.49177850545349555386638457176064.2 $C_4.A_5$ (as 24T576) $0$ $0$
* 6.1848020848.24t576.a.b$6$ $ 2^{4} \cdot 487^{3}$ 24.4.49177850545349555386638457176064.2 $C_4.A_5$ (as 24T576) $0$ $0$

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 *.