Properties

Label 24.4.197...064.1
Degree $24$
Signature $[4, 10]$
Discriminant $1.976\times 10^{34}$
Root discriminant $26.85$
Ramified primes $2, 887$
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 - 2*x^23 - 8*x^22 + 22*x^21 + 51*x^20 - 228*x^19 - 3*x^18 + 1321*x^17 - 2220*x^16 - 1748*x^15 + 9722*x^14 - 7588*x^13 - 14572*x^12 + 34524*x^11 - 16026*x^10 - 34263*x^9 + 60097*x^8 - 26110*x^7 - 33112*x^6 + 60245*x^5 - 45651*x^4 + 20521*x^3 - 6288*x^2 + 1547*x - 241)
 
gp: K = bnfinit(x^24 - 2*x^23 - 8*x^22 + 22*x^21 + 51*x^20 - 228*x^19 - 3*x^18 + 1321*x^17 - 2220*x^16 - 1748*x^15 + 9722*x^14 - 7588*x^13 - 14572*x^12 + 34524*x^11 - 16026*x^10 - 34263*x^9 + 60097*x^8 - 26110*x^7 - 33112*x^6 + 60245*x^5 - 45651*x^4 + 20521*x^3 - 6288*x^2 + 1547*x - 241, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![-241, 1547, -6288, 20521, -45651, 60245, -33112, -26110, 60097, -34263, -16026, 34524, -14572, -7588, 9722, -1748, -2220, 1321, -3, -228, 51, 22, -8, -2, 1]);
 

\( x^{24} - 2 x^{23} - 8 x^{22} + 22 x^{21} + 51 x^{20} - 228 x^{19} - 3 x^{18} + 1321 x^{17} - 2220 x^{16} - 1748 x^{15} + 9722 x^{14} - 7588 x^{13} - 14572 x^{12} + 34524 x^{11} - 16026 x^{10} - 34263 x^{9} + 60097 x^{8} - 26110 x^{7} - 33112 x^{6} + 60245 x^{5} - 45651 x^{4} + 20521 x^{3} - 6288 x^{2} + 1547 x - 241 \)

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:  \(19756778413055716819205133752664064\)\(\medspace = 2^{16}\cdot 887^{10}\)
sage: K.disc()
 
gp: K.disc
 
magma: Discriminant(Integers(K));
 
Root discriminant:  $26.85$
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:  $2, 887$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $\frac{1}{13} a^{22} - \frac{6}{13} a^{21} - \frac{1}{13} a^{20} - \frac{2}{13} a^{19} - \frac{2}{13} a^{18} - \frac{4}{13} a^{17} - \frac{5}{13} a^{16} + \frac{5}{13} a^{15} + \frac{3}{13} a^{14} + \frac{1}{13} a^{13} - \frac{5}{13} a^{12} - \frac{6}{13} a^{11} + \frac{6}{13} a^{10} - \frac{4}{13} a^{9} - \frac{5}{13} a^{8} + \frac{2}{13} a^{7} - \frac{3}{13} a^{6} - \frac{2}{13} a^{5} + \frac{6}{13} a^{4} - \frac{6}{13} a^{2} + \frac{5}{13} a - \frac{5}{13}$, $\frac{1}{16068817258523918389666518589958249370582973311} a^{23} + \frac{438989549681352212740570768467042321167681042}{16068817258523918389666518589958249370582973311} a^{22} + \frac{111381325219956805826393889377408744709619956}{412020955346767138196577399742519214630332649} a^{21} + \frac{4780568869732589170016740557295718173099347112}{16068817258523918389666518589958249370582973311} a^{20} + \frac{609407019047541077886555339680319761844416602}{16068817258523918389666518589958249370582973311} a^{19} + \frac{913198563956554066803459799561417636581605884}{16068817258523918389666518589958249370582973311} a^{18} - \frac{193225182760522793809929161419694557890601619}{1236062866040301414589732199227557643890997947} a^{17} + \frac{151353102308697776541863576278716852916323881}{16068817258523918389666518589958249370582973311} a^{16} - \frac{69485050235081446744367588504691808617742645}{1236062866040301414589732199227557643890997947} a^{15} + \frac{307443271143831992948819690538940393710761508}{5356272419507972796555506196652749790194324437} a^{14} - \frac{6199965098718612100762676902604338300342541140}{16068817258523918389666518589958249370582973311} a^{13} + \frac{4838478913330701587556704255544266650026544075}{16068817258523918389666518589958249370582973311} a^{12} - \frac{1876929633923420443815456590744068483345949897}{5356272419507972796555506196652749790194324437} a^{11} + \frac{214463314155434362948287189427288154268563608}{5356272419507972796555506196652749790194324437} a^{10} - \frac{68406661096528839790406807880815893499516300}{412020955346767138196577399742519214630332649} a^{9} - \frac{1875590306892151936985312236021032935476962035}{5356272419507972796555506196652749790194324437} a^{8} + \frac{1767767295571500536592349752527157984471378346}{16068817258523918389666518589958249370582973311} a^{7} - \frac{364287907870205235135812963447791938729535232}{5356272419507972796555506196652749790194324437} a^{6} + \frac{7977691909438592724682495268199226648305672008}{16068817258523918389666518589958249370582973311} a^{5} + \frac{6153092812174578471036194417486173261348251847}{16068817258523918389666518589958249370582973311} a^{4} + \frac{6750937018786041427041245348624030242680098476}{16068817258523918389666518589958249370582973311} a^{3} - \frac{3351422904772026836708566005192779938471207036}{16068817258523918389666518589958249370582973311} a^{2} - \frac{7964278552763654815349188066067565480950161123}{16068817258523918389666518589958249370582973311} a - \frac{7844127489182787447635036584527259128929129420}{16068817258523918389666518589958249370582973311}$

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:  \( 33313220.242250312 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 

Class number formula

$\displaystyle\lim_{s\to 1} (s-1)\zeta_K(s) \approx\frac{2^{4}\cdot(2\pi)^{10}\cdot 33313220.242250312 \cdot 1}{2\sqrt{19756778413055716819205133752664064}}\approx 0.181822331396691$ (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

6.2.12588304.1, 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: 24.4.19756778413055716819205133752664064.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type R ${\href{/LocalNumberField/3.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/3.2.0.1}{2} }^{2}$ $20{,}\,{\href{/LocalNumberField/5.4.0.1}{4} }$ ${\href{/LocalNumberField/7.3.0.1}{3} }^{8}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/13.3.0.1}{3} }^{8}$ $20{,}\,{\href{/LocalNumberField/17.4.0.1}{4} }$ ${\href{/LocalNumberField/19.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/23.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/29.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/31.12.0.1}{12} }^{2}$ ${\href{/LocalNumberField/37.6.0.1}{6} }^{4}$ $20{,}\,{\href{/LocalNumberField/41.4.0.1}{4} }$ ${\href{/LocalNumberField/43.4.0.1}{4} }^{6}$ ${\href{/LocalNumberField/47.5.0.1}{5} }^{4}{,}\,{\href{/LocalNumberField/47.1.0.1}{1} }^{4}$ $20{,}\,{\href{/LocalNumberField/53.4.0.1}{4} }$ ${\href{/LocalNumberField/59.10.0.1}{10} }^{2}{,}\,{\href{/LocalNumberField/59.2.0.1}{2} }^{2}$

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}$
887Data not computed

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.887.2t1.a.a$1$ $ 887 $ \(\Q(\sqrt{-887}) \) $C_2$ (as 2T1) $1$ $-1$
2.3548.120.a.a$2$ $ 2^{2} \cdot 887 $ 24.4.19756778413055716819205133752664064.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.3548.120.a.b$2$ $ 2^{2} \cdot 887 $ 24.4.19756778413055716819205133752664064.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.3548.120.a.c$2$ $ 2^{2} \cdot 887 $ 24.4.19756778413055716819205133752664064.1 $C_4.A_5$ (as 24T576) $0$ $0$
2.3548.120.a.d$2$ $ 2^{2} \cdot 887 $ 24.4.19756778413055716819205133752664064.1 $C_4.A_5$ (as 24T576) $0$ $0$
3.3147076.12t33.a.a$3$ $ 2^{2} \cdot 887^{2}$ 5.1.3147076.1 $A_5$ (as 5T4) $1$ $-1$
3.3147076.12t33.a.b$3$ $ 2^{2} \cdot 887^{2}$ 5.1.3147076.1 $A_5$ (as 5T4) $1$ $-1$
* 3.3548.12t76.a.a$3$ $ 2^{2} \cdot 887 $ 10.0.8784925479251312.1 $A_5\times C_2$ (as 10T11) $1$ $1$
* 3.3548.12t76.a.b$3$ $ 2^{2} \cdot 887 $ 10.0.8784925479251312.1 $A_5\times C_2$ (as 10T11) $1$ $1$
4.3147076.10t11.a.a$4$ $ 2^{2} \cdot 887^{2}$ 10.0.8784925479251312.1 $A_5\times C_2$ (as 10T11) $1$ $0$
4.3147076.5t4.a.a$4$ $ 2^{2} \cdot 887^{2}$ 5.1.3147076.1 $A_5$ (as 5T4) $1$ $0$
4.3147076.40t188.a.a$4$ $ 2^{2} \cdot 887^{2}$ 24.4.19756778413055716819205133752664064.1 $C_4.A_5$ (as 24T576) $0$ $0$
4.3147076.40t188.a.b$4$ $ 2^{2} \cdot 887^{2}$ 24.4.19756778413055716819205133752664064.1 $C_4.A_5$ (as 24T576) $0$ $0$
5.11165825648.12t75.a.a$5$ $ 2^{4} \cdot 887^{3}$ 10.0.8784925479251312.1 $A_5\times C_2$ (as 10T11) $1$ $-1$
* 5.12588304.6t12.a.a$5$ $ 2^{4} \cdot 887^{2}$ 5.1.3147076.1 $A_5$ (as 5T4) $1$ $1$
* 6.11165825648.24t576.a.a$6$ $ 2^{4} \cdot 887^{3}$ 24.4.19756778413055716819205133752664064.1 $C_4.A_5$ (as 24T576) $0$ $0$
* 6.11165825648.24t576.a.b$6$ $ 2^{4} \cdot 887^{3}$ 24.4.19756778413055716819205133752664064.1 $C_4.A_5$ (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 *.