Properties

Label 24.4.491...064.1
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 - 5*x^23 + x^22 + 40*x^21 - 89*x^20 + 10*x^19 + 240*x^18 - 427*x^17 + 266*x^16 + 215*x^15 - 599*x^14 + 533*x^13 - 153*x^12 - 191*x^11 + 195*x^10 + 9*x^9 + 160*x^8 - 37*x^7 - 186*x^6 - 202*x^5 - 123*x^4 - 10*x^3 + x^2 + 5*x - 1)
 
gp: K = bnfinit(x^24 - 5*x^23 + x^22 + 40*x^21 - 89*x^20 + 10*x^19 + 240*x^18 - 427*x^17 + 266*x^16 + 215*x^15 - 599*x^14 + 533*x^13 - 153*x^12 - 191*x^11 + 195*x^10 + 9*x^9 + 160*x^8 - 37*x^7 - 186*x^6 - 202*x^5 - 123*x^4 - 10*x^3 + x^2 + 5*x - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-1, 5, 1, -10, -123, -202, -186, -37, 160, 9, 195, -191, -153, 533, -599, 215, 266, -427, 240, 10, -89, 40, 1, -5, 1]);
 

\( x^{24} - 5 x^{23} + x^{22} + 40 x^{21} - 89 x^{20} + 10 x^{19} + 240 x^{18} - 427 x^{17} + 266 x^{16} + 215 x^{15} - 599 x^{14} + 533 x^{13} - 153 x^{12} - 191 x^{11} + 195 x^{10} + 9 x^{9} + 160 x^{8} - 37 x^{7} - 186 x^{6} - 202 x^{5} - 123 x^{4} - 10 x^{3} + x^{2} + 5 x - 1 \)

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

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
Signature:  $[4, 10]$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{315064364685741209723776499939} a^{23} + \frac{127227427372761740961648285609}{315064364685741209723776499939} a^{22} + \frac{127528204227689410973558795102}{315064364685741209723776499939} a^{21} - \frac{24027862481510578762915595725}{315064364685741209723776499939} a^{20} + \frac{104700716296533251074023812355}{315064364685741209723776499939} a^{19} - \frac{41263925950790348732437607412}{315064364685741209723776499939} a^{18} - \frac{77012251767472614553607669401}{315064364685741209723776499939} a^{17} - \frac{36036652165766886455567124068}{315064364685741209723776499939} a^{16} - \frac{118002368984000926279410158617}{315064364685741209723776499939} a^{15} + \frac{85284753883745755674472439821}{315064364685741209723776499939} a^{14} - \frac{17959947497855809133657710046}{315064364685741209723776499939} a^{13} + \frac{5448185446643411918420352437}{315064364685741209723776499939} a^{12} + \frac{113585942117720024715220877505}{315064364685741209723776499939} a^{11} + \frac{53526883373664935267366762650}{315064364685741209723776499939} a^{10} - \frac{56361263565064110621683871872}{315064364685741209723776499939} a^{9} + \frac{147520388831076715364209901717}{315064364685741209723776499939} a^{8} + \frac{42774955343890185042902899883}{315064364685741209723776499939} a^{7} + \frac{105577889837455392308838232647}{315064364685741209723776499939} a^{6} + \frac{37890370083535137470558558044}{315064364685741209723776499939} a^{5} - \frac{46364540681712400272828258404}{315064364685741209723776499939} a^{4} + \frac{92837264245951052422903098132}{315064364685741209723776499939} a^{3} - \frac{75707701848181097722110104609}{315064364685741209723776499939} a^{2} + \frac{88950089428184987943983937575}{315064364685741209723776499939} a - \frac{59073476168793427198299187766}{315064364685741209723776499939}$

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: 24.4.49177850545349555386638457176064.2

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.b.a$2$ $ 2^{2} \cdot 487 $ 24.4.49177850545349555386638457176064.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.1948.120.b.b$2$ $ 2^{2} \cdot 487 $ 24.4.49177850545349555386638457176064.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.1948.120.b.c$2$ $ 2^{2} \cdot 487 $ 24.4.49177850545349555386638457176064.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.1948.120.b.d$2$ $ 2^{2} \cdot 487 $ 24.4.49177850545349555386638457176064.1 $C_4.A_5$ (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.b.a$4$ $ 2^{2} \cdot 487^{2}$ 24.4.49177850545349555386638457176064.1 $C_4.A_5$ (as 24T576) $0$ $0$
4.948676.40t188.b.b$4$ $ 2^{2} \cdot 487^{2}$ 24.4.49177850545349555386638457176064.1 $C_4.A_5$ (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.b.a$6$ $ 2^{4} \cdot 487^{3}$ 24.4.49177850545349555386638457176064.1 $C_4.A_5$ (as 24T576) $0$ $0$
* 6.1848020848.24t576.b.b$6$ $ 2^{4} \cdot 487^{3}$ 24.4.49177850545349555386638457176064.1 $C_4.A_5$ (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 *.