Properties

Label 24.4.918...125.7
Degree $24$
Signature $(4, 10)$
Discriminant $9.181\times 10^{58}$
Root discriminant \(286.28\)
Ramified primes $5,89$
Class number $4$ (GRH)
Class group [4] (GRH)
Galois group $\GL(2,5)$ (as 24T1353)

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Normalized defining polynomial

Copy content comment:Define the number field
 
Copy content sage:x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 + 63*x^22 - 827*x^21 - 1196*x^20 - 68669*x^19 - 124547*x^18 - 1743022*x^17 - 3317052*x^16 - 20887621*x^15 - 93267574*x^14 - 124652522*x^13 + 2308036563*x^12 + 27608980468*x^11 + 216820490264*x^10 + 1095953073726*x^9 + 4305306446543*x^8 + 13985340544188*x^7 + 34336140894428*x^6 + 56766412919919*x^5 + 63217215313641*x^4 + 58841762063278*x^3 + 47391301297208*x^2 + 32252356357608*x + 25979013994259)
 
Copy content gp:K = bnfinit(y^24 - 2*y^23 + 63*y^22 - 827*y^21 - 1196*y^20 - 68669*y^19 - 124547*y^18 - 1743022*y^17 - 3317052*y^16 - 20887621*y^15 - 93267574*y^14 - 124652522*y^13 + 2308036563*y^12 + 27608980468*y^11 + 216820490264*y^10 + 1095953073726*y^9 + 4305306446543*y^8 + 13985340544188*y^7 + 34336140894428*y^6 + 56766412919919*y^5 + 63217215313641*y^4 + 58841762063278*y^3 + 47391301297208*y^2 + 32252356357608*y + 25979013994259, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^23 + 63*x^22 - 827*x^21 - 1196*x^20 - 68669*x^19 - 124547*x^18 - 1743022*x^17 - 3317052*x^16 - 20887621*x^15 - 93267574*x^14 - 124652522*x^13 + 2308036563*x^12 + 27608980468*x^11 + 216820490264*x^10 + 1095953073726*x^9 + 4305306446543*x^8 + 13985340544188*x^7 + 34336140894428*x^6 + 56766412919919*x^5 + 63217215313641*x^4 + 58841762063278*x^3 + 47391301297208*x^2 + 32252356357608*x + 25979013994259);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - 2*x^23 + 63*x^22 - 827*x^21 - 1196*x^20 - 68669*x^19 - 124547*x^18 - 1743022*x^17 - 3317052*x^16 - 20887621*x^15 - 93267574*x^14 - 124652522*x^13 + 2308036563*x^12 + 27608980468*x^11 + 216820490264*x^10 + 1095953073726*x^9 + 4305306446543*x^8 + 13985340544188*x^7 + 34336140894428*x^6 + 56766412919919*x^5 + 63217215313641*x^4 + 58841762063278*x^3 + 47391301297208*x^2 + 32252356357608*x + 25979013994259)
 

\( x^{24} - 2 x^{23} + 63 x^{22} - 827 x^{21} - 1196 x^{20} - 68669 x^{19} - 124547 x^{18} + \cdots + 25979013994259 \) Copy content Toggle raw display

Copy content comment:Defining polynomial
 
Copy content sage:K.defining_polynomial()
 
Copy content gp:K.pol
 
Copy content magma:DefiningPolynomial(K);
 
Copy content oscar:defining_polynomial(K)
 

Invariants

Degree:  $24$
Copy content comment:Degree over Q
 
Copy content sage:K.degree()
 
Copy content gp:poldegree(K.pol)
 
Copy content magma:Degree(K);
 
Copy content oscar:degree(K)
 
Signature:  $(4, 10)$
Copy content comment:Signature
 
Copy content sage:K.signature()
 
Copy content gp:K.sign
 
Copy content magma:Signature(K);
 
Copy content oscar:signature(K)
 
Discriminant:   \(91810054652229848026521481130092527303208363056182861328125\) \(\medspace = 5^{23}\cdot 89^{22}\) Copy content Toggle raw display
Copy content comment:Discriminant
 
Copy content sage:K.disc()
 
Copy content gp:K.disc
 
Copy content magma:OK := Integers(K); Discriminant(OK);
 
Copy content oscar:OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(286.28\)
Copy content comment:Root discriminant
 
Copy content sage:(K.disc().abs())^(1./K.degree())
 
Copy content gp:abs(K.disc)^(1/poldegree(K.pol))
 
Copy content magma:Abs(Discriminant(OK))^(1/Degree(K));
 
Copy content oscar:OK = ring_of_integers(K); (1.0 * abs(discriminant(OK)))^(1/degree(K))
 
Galois root discriminant:  $5^{23/20}89^{19/20}\approx 452.62236684313837$
Ramified primes:   \(5\), \(89\) Copy content Toggle raw display
Copy content comment:Ramified primes
 
Copy content sage:K.disc().support()
 
Copy content gp:factor(abs(K.disc))[,1]~
 
Copy content magma:PrimeDivisors(Discriminant(OK));
 
Copy content oscar:prime_divisors(discriminant(OK))
 
Discriminant root field:  \(\Q(\sqrt{5}) \)
$\Aut(K/\Q)$:   $C_4$
Copy content comment:Automorphisms
 
Copy content sage:K.automorphisms()
 
Copy content magma:Automorphisms(K);
 
Copy content oscar:automorphism_group(K)
 
This field is not Galois over $\Q$.
This is not a CM field.
This field has no CM subfields.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{5}a^{11}+\frac{1}{5}a^{10}-\frac{2}{5}a^{9}+\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{4}-\frac{1}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{13}-\frac{2}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{14}-\frac{2}{5}a^{9}+\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{1}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{5}a^{15}-\frac{2}{5}a^{10}+\frac{2}{5}a^{9}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}$, $\frac{1}{5}a^{16}-\frac{1}{5}a^{10}+\frac{2}{5}a^{9}+\frac{2}{5}a^{8}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{2}{5}$, $\frac{1}{5}a^{17}-\frac{2}{5}a^{10}+\frac{2}{5}a^{8}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}+\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{25}a^{18}-\frac{1}{25}a^{17}+\frac{2}{25}a^{16}-\frac{1}{25}a^{14}+\frac{1}{25}a^{13}+\frac{2}{25}a^{12}-\frac{2}{25}a^{11}+\frac{9}{25}a^{10}-\frac{3}{25}a^{9}+\frac{2}{5}a^{8}+\frac{9}{25}a^{7}+\frac{3}{25}a^{6}+\frac{7}{25}a^{5}+\frac{4}{25}a^{4}-\frac{12}{25}a^{3}-\frac{2}{5}a^{2}-\frac{3}{25}a+\frac{9}{25}$, $\frac{1}{25}a^{19}+\frac{1}{25}a^{17}+\frac{2}{25}a^{16}-\frac{1}{25}a^{15}-\frac{2}{25}a^{13}+\frac{2}{25}a^{11}+\frac{1}{25}a^{10}-\frac{8}{25}a^{9}+\frac{4}{25}a^{8}-\frac{3}{25}a^{7}+\frac{2}{5}a^{6}+\frac{11}{25}a^{5}+\frac{7}{25}a^{4}-\frac{2}{25}a^{3}+\frac{2}{25}a^{2}+\frac{11}{25}a-\frac{11}{25}$, $\frac{1}{25}a^{20}-\frac{2}{25}a^{17}+\frac{2}{25}a^{16}-\frac{1}{25}a^{14}-\frac{1}{25}a^{13}-\frac{2}{25}a^{11}+\frac{8}{25}a^{10}+\frac{2}{25}a^{9}+\frac{12}{25}a^{8}+\frac{6}{25}a^{7}-\frac{7}{25}a^{6}-\frac{1}{25}a^{4}-\frac{11}{25}a^{3}-\frac{4}{25}a^{2}+\frac{7}{25}a-\frac{9}{25}$, $\frac{1}{25}a^{21}-\frac{1}{25}a^{16}-\frac{1}{25}a^{15}+\frac{2}{25}a^{14}+\frac{2}{25}a^{13}+\frac{2}{25}a^{12}-\frac{1}{25}a^{11}-\frac{1}{5}a^{10}-\frac{4}{25}a^{9}+\frac{1}{25}a^{8}+\frac{1}{25}a^{7}+\frac{1}{25}a^{6}+\frac{3}{25}a^{5}+\frac{2}{25}a^{4}-\frac{3}{25}a^{3}-\frac{3}{25}a^{2}+\frac{3}{25}$, $\frac{1}{25}a^{22}-\frac{1}{25}a^{17}-\frac{1}{25}a^{16}+\frac{2}{25}a^{15}+\frac{2}{25}a^{14}+\frac{2}{25}a^{13}-\frac{1}{25}a^{12}+\frac{1}{25}a^{10}-\frac{9}{25}a^{9}+\frac{1}{25}a^{8}+\frac{6}{25}a^{7}-\frac{2}{25}a^{6}-\frac{8}{25}a^{5}+\frac{7}{25}a^{4}-\frac{3}{25}a^{3}-\frac{1}{5}a^{2}+\frac{3}{25}a+\frac{1}{5}$, $\frac{1}{72\cdots 25}a^{23}+\frac{10\cdots 36}{72\cdots 25}a^{22}+\frac{76\cdots 54}{72\cdots 25}a^{21}+\frac{47\cdots 48}{72\cdots 25}a^{20}+\frac{28\cdots 66}{72\cdots 25}a^{19}-\frac{45\cdots 24}{72\cdots 25}a^{18}-\frac{71\cdots 94}{72\cdots 25}a^{17}-\frac{49\cdots 31}{72\cdots 25}a^{16}-\frac{59\cdots 21}{72\cdots 25}a^{15}+\frac{20\cdots 27}{72\cdots 25}a^{14}-\frac{63\cdots 99}{72\cdots 25}a^{13}+\frac{44\cdots 86}{72\cdots 25}a^{12}+\frac{35\cdots 74}{72\cdots 25}a^{11}+\frac{60\cdots 43}{14\cdots 65}a^{10}-\frac{31\cdots 87}{72\cdots 25}a^{9}-\frac{80\cdots 44}{72\cdots 25}a^{8}+\frac{22\cdots 21}{72\cdots 25}a^{7}-\frac{25\cdots 86}{72\cdots 25}a^{6}+\frac{23\cdots 46}{72\cdots 25}a^{5}+\frac{27\cdots 94}{72\cdots 25}a^{4}-\frac{29\cdots 34}{72\cdots 25}a^{3}+\frac{32\cdots 01}{72\cdots 25}a^{2}-\frac{14\cdots 61}{72\cdots 25}a-\frac{21\cdots 38}{72\cdots 25}$ Copy content Toggle raw display

Copy content comment:Integral basis
 
Copy content sage:K.integral_basis()
 
Copy content gp:K.zk
 
Copy content magma:IntegralBasis(K);
 
Copy content oscar:basis(OK)
 

Monogenic:  Not computed
Index:  $1$
Inessential primes:  None

Class group and class number

Ideal class group:  $C_{4}$, which has order $4$ (assuming GRH)
Copy content comment:Class group
 
Copy content sage:K.class_group().invariants()
 
Copy content gp:K.clgp
 
Copy content magma:ClassGroup(K);
 
Copy content oscar:class_group(K)
 
Narrow class group:  $C_{4}$, which has order $4$ (assuming GRH)
Copy content comment:Narrow class group
 
Copy content sage:K.narrow_class_group().invariants()
 
Copy content gp:bnfnarrow(K)
 
Copy content magma:NarrowClassGroup(K);
 

Unit group

Copy content comment:Unit group
 
Copy content sage:UK = K.unit_group()
 
Copy content magma:UK, fUK := UnitGroup(K);
 
Copy content oscar:UK, fUK = unit_group(OK)
 
Rank:  $13$
Copy content comment:Unit rank
 
Copy content sage:UK.rank()
 
Copy content gp:K.fu
 
Copy content magma:UnitRank(K);
 
Copy content oscar:rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
Copy content comment:Generator for roots of unity
 
Copy content sage:UK.torsion_generator()
 
Copy content gp:K.tu[2]
 
Copy content magma:K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
Copy content oscar:torsion_units_generator(OK)
 
Fundamental units:   $\frac{76\cdots 71}{25\cdots 25}a^{23}-\frac{31\cdots 93}{25\cdots 25}a^{22}+\frac{54\cdots 12}{25\cdots 25}a^{21}-\frac{74\cdots 69}{25\cdots 25}a^{20}+\frac{13\cdots 29}{51\cdots 05}a^{19}-\frac{53\cdots 62}{25\cdots 25}a^{18}+\frac{19\cdots 66}{25\cdots 25}a^{17}-\frac{13\cdots 12}{25\cdots 25}a^{16}+\frac{37\cdots 69}{25\cdots 25}a^{15}-\frac{33\cdots 68}{51\cdots 05}a^{14}-\frac{70\cdots 76}{51\cdots 05}a^{13}-\frac{77\cdots 01}{10\cdots 21}a^{12}+\frac{17\cdots 99}{25\cdots 25}a^{11}+\frac{17\cdots 92}{25\cdots 25}a^{10}+\frac{25\cdots 99}{51\cdots 05}a^{9}+\frac{56\cdots 12}{25\cdots 25}a^{8}+\frac{20\cdots 99}{25\cdots 25}a^{7}+\frac{12\cdots 93}{51\cdots 05}a^{6}+\frac{25\cdots 22}{51\cdots 05}a^{5}+\frac{62\cdots 78}{10\cdots 21}a^{4}+\frac{14\cdots 09}{25\cdots 25}a^{3}+\frac{13\cdots 78}{25\cdots 25}a^{2}+\frac{72\cdots 36}{25\cdots 25}a+\frac{91\cdots 48}{25\cdots 25}$, $\frac{14\cdots 76}{11\cdots 75}a^{23}+\frac{49\cdots 54}{11\cdots 75}a^{22}-\frac{96\cdots 79}{11\cdots 75}a^{21}+\frac{13\cdots 19}{11\cdots 75}a^{20}-\frac{11\cdots 64}{47\cdots 51}a^{19}+\frac{19\cdots 19}{23\cdots 55}a^{18}+\frac{11\cdots 65}{47\cdots 51}a^{17}+\frac{24\cdots 94}{11\cdots 75}a^{16}+\frac{22\cdots 56}{23\cdots 55}a^{15}+\frac{54\cdots 44}{23\cdots 55}a^{14}+\frac{93\cdots 41}{11\cdots 75}a^{13}+\frac{22\cdots 86}{11\cdots 75}a^{12}-\frac{32\cdots 58}{11\cdots 75}a^{11}-\frac{34\cdots 64}{11\cdots 75}a^{10}-\frac{50\cdots 53}{23\cdots 55}a^{9}-\frac{11\cdots 58}{11\cdots 75}a^{8}-\frac{43\cdots 03}{11\cdots 75}a^{7}-\frac{26\cdots 07}{23\cdots 55}a^{6}-\frac{28\cdots 18}{11\cdots 75}a^{5}-\frac{73\cdots 01}{23\cdots 55}a^{4}-\frac{32\cdots 02}{11\cdots 75}a^{3}-\frac{33\cdots 12}{11\cdots 75}a^{2}-\frac{14\cdots 47}{11\cdots 75}a-\frac{23\cdots 67}{11\cdots 75}$, $\frac{12\cdots 26}{25\cdots 25}a^{23}+\frac{96\cdots 84}{25\cdots 25}a^{22}-\frac{10\cdots 71}{25\cdots 25}a^{21}+\frac{14\cdots 39}{25\cdots 25}a^{20}-\frac{45\cdots 42}{25\cdots 25}a^{19}+\frac{80\cdots 13}{25\cdots 25}a^{18}-\frac{48\cdots 06}{51\cdots 05}a^{17}+\frac{15\cdots 33}{25\cdots 25}a^{16}-\frac{17\cdots 81}{25\cdots 25}a^{15}-\frac{63\cdots 82}{25\cdots 25}a^{14}+\frac{18\cdots 64}{25\cdots 25}a^{13}-\frac{11\cdots 02}{25\cdots 25}a^{12}+\frac{71\cdots 34}{25\cdots 25}a^{11}-\frac{26\cdots 29}{25\cdots 25}a^{10}-\frac{16\cdots 79}{51\cdots 05}a^{9}-\frac{23\cdots 23}{25\cdots 25}a^{8}-\frac{58\cdots 02}{25\cdots 25}a^{7}-\frac{13\cdots 93}{25\cdots 25}a^{6}+\frac{89\cdots 67}{51\cdots 05}a^{5}+\frac{10\cdots 19}{25\cdots 25}a^{4}+\frac{67\cdots 47}{25\cdots 25}a^{3}+\frac{57\cdots 56}{10\cdots 21}a^{2}-\frac{76\cdots 28}{25\cdots 25}a+\frac{13\cdots 61}{25\cdots 25}$, $\frac{10\cdots 09}{11\cdots 75}a^{23}+\frac{21\cdots 61}{11\cdots 75}a^{22}-\frac{63\cdots 74}{11\cdots 75}a^{21}+\frac{16\cdots 28}{23\cdots 55}a^{20}+\frac{12\cdots 86}{11\cdots 75}a^{19}+\frac{68\cdots 88}{11\cdots 75}a^{18}+\frac{23\cdots 23}{23\cdots 55}a^{17}+\frac{33\cdots 06}{23\cdots 55}a^{16}+\frac{30\cdots 52}{11\cdots 75}a^{15}+\frac{18\cdots 12}{11\cdots 75}a^{14}+\frac{17\cdots 56}{23\cdots 55}a^{13}+\frac{88\cdots 74}{11\cdots 75}a^{12}-\frac{25\cdots 16}{11\cdots 75}a^{11}-\frac{28\cdots 36}{11\cdots 75}a^{10}-\frac{21\cdots 82}{11\cdots 75}a^{9}-\frac{10\cdots 23}{11\cdots 75}a^{8}-\frac{39\cdots 32}{11\cdots 75}a^{7}-\frac{12\cdots 92}{11\cdots 75}a^{6}-\frac{27\cdots 84}{11\cdots 75}a^{5}-\frac{36\cdots 91}{11\cdots 75}a^{4}-\frac{19\cdots 86}{11\cdots 75}a^{3}-\frac{30\cdots 18}{11\cdots 75}a^{2}-\frac{50\cdots 98}{11\cdots 75}a+\frac{11\cdots 28}{11\cdots 75}$, $\frac{43\cdots 41}{25\cdots 25}a^{23}+\frac{19\cdots 94}{25\cdots 25}a^{22}-\frac{31\cdots 76}{25\cdots 25}a^{21}+\frac{86\cdots 61}{51\cdots 05}a^{20}-\frac{40\cdots 98}{25\cdots 25}a^{19}+\frac{11\cdots 87}{10\cdots 21}a^{18}-\frac{24\cdots 97}{51\cdots 05}a^{17}+\frac{71\cdots 17}{25\cdots 25}a^{16}-\frac{72\cdots 72}{51\cdots 05}a^{15}+\frac{14\cdots 93}{51\cdots 05}a^{14}+\frac{29\cdots 77}{25\cdots 25}a^{13}-\frac{41\cdots 98}{25\cdots 25}a^{12}-\frac{91\cdots 39}{25\cdots 25}a^{11}-\frac{98\cdots 49}{25\cdots 25}a^{10}-\frac{70\cdots 16}{25\cdots 25}a^{9}-\frac{59\cdots 18}{51\cdots 05}a^{8}-\frac{21\cdots 79}{51\cdots 05}a^{7}-\frac{32\cdots 09}{25\cdots 25}a^{6}-\frac{59\cdots 77}{25\cdots 25}a^{5}-\frac{68\cdots 51}{25\cdots 25}a^{4}-\frac{66\cdots 46}{25\cdots 25}a^{3}-\frac{53\cdots 66}{25\cdots 25}a^{2}-\frac{36\cdots 68}{25\cdots 25}a-\frac{33\cdots 94}{25\cdots 25}$, $\frac{27\cdots 16}{72\cdots 25}a^{23}+\frac{41\cdots 61}{72\cdots 25}a^{22}-\frac{16\cdots 08}{72\cdots 25}a^{21}+\frac{21\cdots 73}{72\cdots 25}a^{20}+\frac{89\cdots 03}{14\cdots 65}a^{19}+\frac{18\cdots 06}{72\cdots 25}a^{18}+\frac{42\cdots 94}{72\cdots 25}a^{17}+\frac{47\cdots 81}{72\cdots 25}a^{16}+\frac{11\cdots 38}{72\cdots 25}a^{15}+\frac{56\cdots 51}{72\cdots 25}a^{14}+\frac{27\cdots 24}{72\cdots 25}a^{13}+\frac{40\cdots 43}{72\cdots 25}a^{12}-\frac{62\cdots 44}{72\cdots 25}a^{11}-\frac{78\cdots 71}{72\cdots 25}a^{10}-\frac{62\cdots 42}{72\cdots 25}a^{9}-\frac{31\cdots 72}{72\cdots 25}a^{8}-\frac{12\cdots 27}{72\cdots 25}a^{7}-\frac{41\cdots 78}{72\cdots 25}a^{6}-\frac{10\cdots 29}{72\cdots 25}a^{5}-\frac{16\cdots 99}{72\cdots 25}a^{4}-\frac{16\cdots 92}{72\cdots 25}a^{3}-\frac{96\cdots 96}{72\cdots 25}a^{2}-\frac{93\cdots 76}{72\cdots 25}a+\frac{33\cdots 89}{72\cdots 25}$, $\frac{83\cdots 36}{14\cdots 65}a^{23}-\frac{17\cdots 31}{72\cdots 25}a^{22}+\frac{29\cdots 97}{72\cdots 25}a^{21}-\frac{40\cdots 37}{72\cdots 25}a^{20}+\frac{36\cdots 73}{72\cdots 25}a^{19}-\frac{29\cdots 97}{72\cdots 25}a^{18}+\frac{20\cdots 48}{14\cdots 65}a^{17}-\frac{74\cdots 23}{72\cdots 25}a^{16}+\frac{20\cdots 18}{72\cdots 25}a^{15}-\frac{91\cdots 69}{72\cdots 25}a^{14}-\frac{19\cdots 99}{72\cdots 25}a^{13}-\frac{10\cdots 09}{72\cdots 25}a^{12}+\frac{98\cdots 87}{72\cdots 25}a^{11}+\frac{94\cdots 13}{72\cdots 25}a^{10}+\frac{70\cdots 39}{72\cdots 25}a^{9}+\frac{30\cdots 74}{72\cdots 25}a^{8}+\frac{11\cdots 87}{72\cdots 25}a^{7}+\frac{34\cdots 82}{72\cdots 25}a^{6}+\frac{71\cdots 28}{72\cdots 25}a^{5}+\frac{86\cdots 32}{72\cdots 25}a^{4}+\frac{80\cdots 17}{72\cdots 25}a^{3}+\frac{74\cdots 63}{72\cdots 25}a^{2}+\frac{39\cdots 07}{72\cdots 25}a+\frac{51\cdots 28}{72\cdots 25}$, $\frac{85\cdots 27}{25\cdots 25}a^{23}+\frac{36\cdots 24}{25\cdots 25}a^{22}-\frac{59\cdots 88}{25\cdots 25}a^{21}+\frac{81\cdots 09}{25\cdots 25}a^{20}-\frac{56\cdots 02}{25\cdots 25}a^{19}+\frac{56\cdots 39}{25\cdots 25}a^{18}-\frac{68\cdots 09}{25\cdots 25}a^{17}+\frac{13\cdots 58}{25\cdots 25}a^{16}+\frac{68\cdots 69}{25\cdots 25}a^{15}+\frac{10\cdots 82}{25\cdots 25}a^{14}+\frac{77\cdots 91}{25\cdots 25}a^{13}-\frac{16\cdots 06}{25\cdots 25}a^{12}-\frac{29\cdots 14}{51\cdots 05}a^{11}-\frac{40\cdots 86}{51\cdots 05}a^{10}-\frac{13\cdots 13}{25\cdots 25}a^{9}-\frac{59\cdots 91}{25\cdots 25}a^{8}-\frac{21\cdots 52}{25\cdots 25}a^{7}-\frac{63\cdots 47}{25\cdots 25}a^{6}-\frac{12\cdots 93}{25\cdots 25}a^{5}-\frac{14\cdots 92}{25\cdots 25}a^{4}-\frac{13\cdots 37}{25\cdots 25}a^{3}-\frac{11\cdots 57}{25\cdots 25}a^{2}-\frac{71\cdots 03}{25\cdots 25}a-\frac{12\cdots 06}{51\cdots 05}$, $\frac{12\cdots 64}{72\cdots 25}a^{23}+\frac{52\cdots 03}{72\cdots 25}a^{22}-\frac{91\cdots 19}{72\cdots 25}a^{21}+\frac{24\cdots 47}{14\cdots 65}a^{20}-\frac{23\cdots 47}{14\cdots 65}a^{19}+\frac{89\cdots 04}{72\cdots 25}a^{18}-\frac{31\cdots 24}{72\cdots 25}a^{17}+\frac{23\cdots 63}{72\cdots 25}a^{16}-\frac{55\cdots 53}{72\cdots 25}a^{15}+\frac{56\cdots 13}{14\cdots 65}a^{14}+\frac{60\cdots 57}{72\cdots 25}a^{13}+\frac{40\cdots 64}{72\cdots 25}a^{12}-\frac{11\cdots 22}{28\cdots 13}a^{11}-\frac{28\cdots 76}{72\cdots 25}a^{10}-\frac{43\cdots 94}{14\cdots 65}a^{9}-\frac{18\cdots 09}{14\cdots 65}a^{8}-\frac{35\cdots 38}{72\cdots 25}a^{7}-\frac{10\cdots 43}{72\cdots 25}a^{6}-\frac{22\cdots 84}{72\cdots 25}a^{5}-\frac{57\cdots 12}{14\cdots 65}a^{4}-\frac{27\cdots 02}{72\cdots 25}a^{3}-\frac{49\cdots 71}{14\cdots 65}a^{2}-\frac{13\cdots 26}{72\cdots 25}a-\frac{18\cdots 47}{72\cdots 25}$, $\frac{90\cdots 66}{72\cdots 25}a^{23}-\frac{37\cdots 77}{72\cdots 25}a^{22}+\frac{26\cdots 05}{28\cdots 13}a^{21}-\frac{88\cdots 82}{72\cdots 25}a^{20}+\frac{83\cdots 71}{72\cdots 25}a^{19}-\frac{63\cdots 78}{72\cdots 25}a^{18}+\frac{23\cdots 68}{72\cdots 25}a^{17}-\frac{16\cdots 97}{72\cdots 25}a^{16}+\frac{40\cdots 92}{72\cdots 25}a^{15}-\frac{20\cdots 63}{72\cdots 25}a^{14}-\frac{44\cdots 87}{72\cdots 25}a^{13}-\frac{23\cdots 37}{72\cdots 25}a^{12}+\frac{20\cdots 41}{72\cdots 25}a^{11}+\frac{20\cdots 46}{72\cdots 25}a^{10}+\frac{15\cdots 13}{72\cdots 25}a^{9}+\frac{66\cdots 89}{72\cdots 25}a^{8}+\frac{25\cdots 43}{72\cdots 25}a^{7}+\frac{76\cdots 19}{72\cdots 25}a^{6}+\frac{33\cdots 98}{14\cdots 65}a^{5}+\frac{23\cdots 79}{72\cdots 25}a^{4}+\frac{59\cdots 97}{14\cdots 65}a^{3}+\frac{37\cdots 78}{72\cdots 25}a^{2}+\frac{27\cdots 87}{72\cdots 25}a+\frac{24\cdots 54}{72\cdots 25}$, $\frac{98\cdots 80}{10\cdots 21}a^{23}-\frac{15\cdots 22}{25\cdots 25}a^{22}+\frac{17\cdots 22}{25\cdots 25}a^{21}-\frac{25\cdots 46}{25\cdots 25}a^{20}+\frac{10\cdots 31}{51\cdots 05}a^{19}-\frac{14\cdots 58}{25\cdots 25}a^{18}+\frac{25\cdots 12}{25\cdots 25}a^{17}-\frac{31\cdots 18}{25\cdots 25}a^{16}+\frac{35\cdots 19}{25\cdots 25}a^{15}-\frac{16\cdots 86}{25\cdots 25}a^{14}-\frac{15\cdots 22}{25\cdots 25}a^{13}+\frac{17\cdots 92}{10\cdots 21}a^{12}+\frac{55\cdots 91}{25\cdots 25}a^{11}+\frac{36\cdots 48}{25\cdots 25}a^{10}+\frac{24\cdots 07}{25\cdots 25}a^{9}+\frac{10\cdots 53}{25\cdots 25}a^{8}+\frac{32\cdots 97}{25\cdots 25}a^{7}+\frac{96\cdots 89}{25\cdots 25}a^{6}+\frac{18\cdots 21}{25\cdots 25}a^{5}+\frac{21\cdots 74}{25\cdots 25}a^{4}+\frac{21\cdots 27}{25\cdots 25}a^{3}+\frac{17\cdots 63}{25\cdots 25}a^{2}+\frac{11\cdots 91}{25\cdots 25}a+\frac{11\cdots 08}{25\cdots 25}$, $\frac{92\cdots 13}{14\cdots 65}a^{23}+\frac{23\cdots 12}{72\cdots 25}a^{22}-\frac{35\cdots 87}{72\cdots 25}a^{21}+\frac{48\cdots 12}{72\cdots 25}a^{20}-\frac{83\cdots 72}{72\cdots 25}a^{19}+\frac{32\cdots 62}{72\cdots 25}a^{18}-\frac{15\cdots 98}{28\cdots 13}a^{17}+\frac{82\cdots 29}{72\cdots 25}a^{16}-\frac{76\cdots 57}{72\cdots 25}a^{15}+\frac{91\cdots 26}{72\cdots 25}a^{14}+\frac{20\cdots 24}{72\cdots 25}a^{13}-\frac{45\cdots 42}{72\cdots 25}a^{12}-\frac{37\cdots 32}{28\cdots 13}a^{11}-\frac{96\cdots 76}{72\cdots 25}a^{10}-\frac{67\cdots 01}{72\cdots 25}a^{9}-\frac{27\cdots 79}{72\cdots 25}a^{8}-\frac{97\cdots 84}{72\cdots 25}a^{7}-\frac{28\cdots 49}{72\cdots 25}a^{6}-\frac{19\cdots 03}{28\cdots 13}a^{5}-\frac{55\cdots 78}{72\cdots 25}a^{4}-\frac{53\cdots 42}{72\cdots 25}a^{3}-\frac{41\cdots 61}{72\cdots 25}a^{2}-\frac{30\cdots 18}{72\cdots 25}a-\frac{23\cdots 29}{72\cdots 25}$, $\frac{87\cdots 86}{72\cdots 25}a^{23}+\frac{25\cdots 77}{72\cdots 25}a^{22}+\frac{51\cdots 01}{72\cdots 25}a^{21}-\frac{66\cdots 67}{72\cdots 25}a^{20}-\frac{31\cdots 29}{72\cdots 25}a^{19}-\frac{69\cdots 98}{72\cdots 25}a^{18}-\frac{22\cdots 11}{72\cdots 25}a^{17}-\frac{13\cdots 07}{72\cdots 25}a^{16}+\frac{15\cdots 74}{72\cdots 25}a^{15}+\frac{35\cdots 12}{72\cdots 25}a^{14}+\frac{26\cdots 53}{72\cdots 25}a^{13}+\frac{12\cdots 26}{72\cdots 25}a^{12}+\frac{75\cdots 13}{14\cdots 65}a^{11}+\frac{79\cdots 36}{14\cdots 65}a^{10}-\frac{12\cdots 77}{72\cdots 25}a^{9}-\frac{11\cdots 56}{72\cdots 25}a^{8}-\frac{37\cdots 62}{72\cdots 25}a^{7}-\frac{15\cdots 08}{72\cdots 25}a^{6}+\frac{32\cdots 81}{72\cdots 25}a^{5}+\frac{10\cdots 19}{72\cdots 25}a^{4}+\frac{25\cdots 98}{14\cdots 65}a^{3}+\frac{17\cdots 84}{14\cdots 65}a^{2}+\frac{12\cdots 99}{72\cdots 25}a-\frac{45\cdots 43}{72\cdots 25}$ Copy content Toggle raw display (assuming GRH)
Copy content comment:Fundamental units
 
Copy content sage:UK.fundamental_units()
 
Copy content gp:K.fu
 
Copy content magma:[K|fUK(g): g in Generators(UK)];
 
Copy content oscar:[K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 36799407855653930000 \) (assuming GRH)
Copy content comment:Regulator
 
Copy content sage:K.regulator()
 
Copy content gp:K.reg
 
Copy content magma:Regulator(K);
 
Copy content oscar:regulator(K)
 
Unit signature rank:  \( 4 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 36799407855653930000 \cdot 4}{2\cdot\sqrt{91810054652229848026521481130092527303208363056182861328125}}\cr\approx \mathstrut & 0.372687107562349 \end{aligned}\] (assuming GRH)

Copy content comment:Analytic class number formula
 
Copy content sage:# self-contained SageMath code snippet to compute the analytic class number formula x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 + 63*x^22 - 827*x^21 - 1196*x^20 - 68669*x^19 - 124547*x^18 - 1743022*x^17 - 3317052*x^16 - 20887621*x^15 - 93267574*x^14 - 124652522*x^13 + 2308036563*x^12 + 27608980468*x^11 + 216820490264*x^10 + 1095953073726*x^9 + 4305306446543*x^8 + 13985340544188*x^7 + 34336140894428*x^6 + 56766412919919*x^5 + 63217215313641*x^4 + 58841762063278*x^3 + 47391301297208*x^2 + 32252356357608*x + 25979013994259) DK = K.disc(); r1,r2 = K.signature(); RK = K.regulator(); RR = RK.parent() hK = K.class_number(); wK = K.unit_group().torsion_generator().order(); 2^r1 * (2*RR(pi))^r2 * RK * hK / (wK * RR(sqrt(abs(DK))))
 
Copy content gp:\\ self-contained Pari/GP code snippet to compute the analytic class number formula K = bnfinit(x^24 - 2*x^23 + 63*x^22 - 827*x^21 - 1196*x^20 - 68669*x^19 - 124547*x^18 - 1743022*x^17 - 3317052*x^16 - 20887621*x^15 - 93267574*x^14 - 124652522*x^13 + 2308036563*x^12 + 27608980468*x^11 + 216820490264*x^10 + 1095953073726*x^9 + 4305306446543*x^8 + 13985340544188*x^7 + 34336140894428*x^6 + 56766412919919*x^5 + 63217215313641*x^4 + 58841762063278*x^3 + 47391301297208*x^2 + 32252356357608*x + 25979013994259, 1); [polcoeff (lfunrootres (lfuncreate (K))[1][1][2], -1), 2^K.r1 * (2*Pi)^K.r2 * K.reg * K.no / (K.tu[1] * sqrt (abs (K.disc)))]
 
Copy content magma:/* self-contained Magma code snippet to compute the analytic class number formula */ Qx<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^23 + 63*x^22 - 827*x^21 - 1196*x^20 - 68669*x^19 - 124547*x^18 - 1743022*x^17 - 3317052*x^16 - 20887621*x^15 - 93267574*x^14 - 124652522*x^13 + 2308036563*x^12 + 27608980468*x^11 + 216820490264*x^10 + 1095953073726*x^9 + 4305306446543*x^8 + 13985340544188*x^7 + 34336140894428*x^6 + 56766412919919*x^5 + 63217215313641*x^4 + 58841762063278*x^3 + 47391301297208*x^2 + 32252356357608*x + 25979013994259); OK := Integers(K); DK := Discriminant(OK); UK, fUK := UnitGroup(OK); clK, fclK := ClassGroup(OK); r1,r2 := Signature(K); RK := Regulator(K); RR := Parent(RK); hK := #clK; wK := #TorsionSubgroup(UK); 2^r1 * (2*Pi(RR))^r2 * RK * hK / (wK * Sqrt(RR!Abs(DK)));
 
Copy content oscar:# self-contained Oscar code snippet to compute the analytic class number formula Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - 2*x^23 + 63*x^22 - 827*x^21 - 1196*x^20 - 68669*x^19 - 124547*x^18 - 1743022*x^17 - 3317052*x^16 - 20887621*x^15 - 93267574*x^14 - 124652522*x^13 + 2308036563*x^12 + 27608980468*x^11 + 216820490264*x^10 + 1095953073726*x^9 + 4305306446543*x^8 + 13985340544188*x^7 + 34336140894428*x^6 + 56766412919919*x^5 + 63217215313641*x^4 + 58841762063278*x^3 + 47391301297208*x^2 + 32252356357608*x + 25979013994259); OK = ring_of_integers(K); DK = discriminant(OK); UK, fUK = unit_group(OK); clK, fclK = class_group(OK); r1,r2 = signature(K); RK = regulator(K); RR = parent(RK); hK = order(clK); wK = torsion_units_order(K); 2^r1 * (2*pi)^r2 * RK * hK / (wK * sqrt(RR(abs(DK))))
 

Galois group

$\GL(2,5)$ (as 24T1353):

Copy content comment:Galois group
 
Copy content sage:K.galois_group()
 
Copy content gp:polgalois(K.pol)
 
Copy content magma:GaloisGroup(K);
 
Copy content oscar:G, Gtx = galois_group(K); degree(K) > 1 ? (G, transitive_group_identification(G)) : (G, nothing)
 
A non-solvable group of order 480
The 24 conjugacy class representatives for $\GL(2,5)$
Character table for $\GL(2,5)$

Intermediate fields

6.2.196069503125.2, 12.4.1522544918455380058642578125.2

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

Copy content comment:Intermediate fields
 
Copy content sage:K.subfields()[1:-1]
 
Copy content gp:L = nfsubfields(K); L[2..length(L)]
 
Copy content magma:L := Subfields(K); L[2..#L];
 
Copy content oscar:subfields(K)[2:end-1]
 

Sibling fields

Degree 24 siblings: 24.4.2295251366305746200663037028252313182580209076404571533203125.7, 24.4.2295251366305746200663037028252313182580209076404571533203125.8
Arithmetically equivalent sibling: 24.4.91810054652229848026521481130092527303208363056182861328125.2
Minimal sibling: 24.4.91810054652229848026521481130092527303208363056182861328125.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $24$ $24$ R ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ ${\href{/padicField/11.6.0.1}{6} }^{4}$ $24$ $24$ ${\href{/padicField/19.3.0.1}{3} }^{8}$ $24$ ${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ ${\href{/padicField/31.12.0.1}{12} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{3}$ ${\href{/padicField/41.12.0.1}{12} }^{2}$ $24$ ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.4.0.1}{4} }^{6}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

Copy content comment:Frobenius cycle types
 
Copy content sage:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Sage: p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
Copy content gp:\\ to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Pari: p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
Copy content magma:// to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Magma: p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
Copy content oscar:# to obtain a list of [e_i,f_i] for the factorization of the ideal pO_K for p=7 in Oscar: p = 7; pfac = factor(ideal(ring_of_integers(K), p)); [(e, valuation(norm(pr),p)) for (pr,e) in pfac]
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
\(5\) Copy content Toggle raw display 5.4.1.0a1.1$x^{4} + 4 x^{2} + 4 x + 2$$1$$4$$0$$C_4$$$[\ ]^{4}$$
5.1.20.23a3.3$x^{20} + 20 x^{4} + 15$$20$$1$$23$20T20$not computed$
\(89\) Copy content Toggle raw display 89.1.4.3a1.1$x^{4} + 89$$4$$1$$3$$C_4$$$[\ ]_{4}$$
89.1.20.19a1.3$x^{20} + 801$$20$$1$$19$20T6$$[\ ]_{20}^{2}$$

Spectrum of ring of integers

(0)(0)(2)(3)(5)(7)(11)(13)(17)(19)(23)(29)(31)(37)(41)(43)(47)(53)(59)