Properties

Label 24.4.676...125.3
Degree $24$
Signature $(4, 10)$
Discriminant $6.769\times 10^{60}$
Root discriminant \(342.46\)
Ramified primes $5,29$
Class number $32$ (GRH)
Class group [2, 4, 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 - 4*x^23 + 6*x^22 - 4*x^21 + x^20 - 1305*x^19 - 10005*x^18 + 60320*x^17 - 237655*x^16 + 297395*x^15 + 227650*x^14 + 48632275*x^13 - 58519100*x^12 - 162429725*x^11 - 1167334100*x^10 + 1650665500*x^9 - 3500169500*x^8 + 30497763000*x^7 + 43848203000*x^6 - 146123238875*x^5 - 523765574375*x^4 + 1137682488125*x^3 - 286773405625*x^2 - 99248258750*x + 56010581875)
 
Copy content gp:K = bnfinit(y^24 - 4*y^23 + 6*y^22 - 4*y^21 + y^20 - 1305*y^19 - 10005*y^18 + 60320*y^17 - 237655*y^16 + 297395*y^15 + 227650*y^14 + 48632275*y^13 - 58519100*y^12 - 162429725*y^11 - 1167334100*y^10 + 1650665500*y^9 - 3500169500*y^8 + 30497763000*y^7 + 43848203000*y^6 - 146123238875*y^5 - 523765574375*y^4 + 1137682488125*y^3 - 286773405625*y^2 - 99248258750*y + 56010581875, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 4*x^23 + 6*x^22 - 4*x^21 + x^20 - 1305*x^19 - 10005*x^18 + 60320*x^17 - 237655*x^16 + 297395*x^15 + 227650*x^14 + 48632275*x^13 - 58519100*x^12 - 162429725*x^11 - 1167334100*x^10 + 1650665500*x^9 - 3500169500*x^8 + 30497763000*x^7 + 43848203000*x^6 - 146123238875*x^5 - 523765574375*x^4 + 1137682488125*x^3 - 286773405625*x^2 - 99248258750*x + 56010581875);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - 4*x^23 + 6*x^22 - 4*x^21 + x^20 - 1305*x^19 - 10005*x^18 + 60320*x^17 - 237655*x^16 + 297395*x^15 + 227650*x^14 + 48632275*x^13 - 58519100*x^12 - 162429725*x^11 - 1167334100*x^10 + 1650665500*x^9 - 3500169500*x^8 + 30497763000*x^7 + 43848203000*x^6 - 146123238875*x^5 - 523765574375*x^4 + 1137682488125*x^3 - 286773405625*x^2 - 99248258750*x + 56010581875)
 

\( x^{24} - 4 x^{23} + 6 x^{22} - 4 x^{21} + x^{20} - 1305 x^{19} - 10005 x^{18} + 60320 x^{17} + \cdots + 56010581875 \) 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:   \(6769025210045733840131040987091228089411742985248565673828125\) \(\medspace = 5^{41}\cdot 29^{22}\) 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:  \(342.46\)
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:  $5^{39/20}29^{19/20}\approx 565.287791942645$
Ramified primes:   \(5\), \(29\) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}$, $\frac{1}{5}a^{8}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}$, $\frac{1}{5}a^{9}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}$, $\frac{1}{5}a^{10}-\frac{1}{5}a^{5}$, $\frac{1}{25}a^{11}+\frac{2}{25}a^{10}-\frac{2}{25}a^{9}-\frac{1}{25}a^{8}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}$, $\frac{1}{25}a^{12}-\frac{1}{25}a^{10}-\frac{2}{25}a^{9}+\frac{2}{25}a^{8}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{3}$, $\frac{1}{25}a^{13}-\frac{1}{25}a^{8}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{3}$, $\frac{1}{125}a^{14}+\frac{1}{125}a^{13}+\frac{1}{125}a^{12}+\frac{1}{125}a^{11}-\frac{9}{125}a^{10}+\frac{2}{25}a^{9}-\frac{2}{25}a^{8}+\frac{2}{25}a^{7}+\frac{7}{25}a^{6}+\frac{12}{25}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{125}a^{15}-\frac{11}{125}a^{10}+\frac{2}{25}a^{9}+\frac{2}{25}a^{8}+\frac{8}{25}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}$, $\frac{1}{125}a^{16}-\frac{1}{125}a^{11}+\frac{1}{25}a^{10}-\frac{2}{25}a^{9}-\frac{2}{25}a^{8}+\frac{8}{25}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{125}a^{17}-\frac{1}{125}a^{12}+\frac{1}{25}a^{10}+\frac{1}{25}a^{8}-\frac{2}{25}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}$, $\frac{1}{125}a^{18}-\frac{1}{125}a^{13}-\frac{2}{25}a^{10}-\frac{2}{25}a^{9}-\frac{1}{25}a^{8}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}$, $\frac{1}{3125}a^{19}-\frac{4}{3125}a^{18}+\frac{6}{3125}a^{17}-\frac{4}{3125}a^{16}+\frac{1}{3125}a^{15}-\frac{2}{625}a^{14}-\frac{12}{625}a^{13}-\frac{7}{625}a^{12}+\frac{8}{625}a^{11}+\frac{13}{625}a^{10}+\frac{8}{125}a^{9}-\frac{12}{125}a^{8}-\frac{2}{125}a^{7}-\frac{17}{125}a^{6}-\frac{47}{125}a^{5}+\frac{11}{25}a^{4}-\frac{4}{25}a^{3}+\frac{11}{25}a^{2}-\frac{4}{25}a-\frac{9}{25}$, $\frac{1}{3125}a^{20}-\frac{2}{625}a^{18}-\frac{1}{625}a^{17}+\frac{2}{625}a^{16}-\frac{6}{3125}a^{15}-\frac{2}{125}a^{13}+\frac{1}{125}a^{12}+\frac{2}{125}a^{11}+\frac{62}{625}a^{10}-\frac{1}{25}a^{9}-\frac{7}{25}a^{6}+\frac{7}{125}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{9}{25}$, $\frac{1}{3125}a^{21}+\frac{1}{625}a^{18}-\frac{1}{625}a^{17}+\frac{4}{3125}a^{16}+\frac{2}{625}a^{15}+\frac{1}{125}a^{13}+\frac{2}{125}a^{12}+\frac{12}{625}a^{11}+\frac{2}{125}a^{10}-\frac{1}{25}a^{8}+\frac{2}{25}a^{7}+\frac{52}{125}a^{6}+\frac{8}{25}a^{5}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{11}{25}a-\frac{2}{5}$, $\frac{1}{3125}a^{22}-\frac{2}{625}a^{18}-\frac{1}{3125}a^{17}+\frac{1}{625}a^{16}-\frac{1}{625}a^{15}+\frac{2}{125}a^{13}+\frac{2}{625}a^{12}+\frac{2}{125}a^{11}-\frac{1}{125}a^{10}+\frac{2}{25}a^{9}+\frac{2}{25}a^{8}-\frac{3}{125}a^{7}+\frac{6}{25}a^{6}+\frac{6}{25}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{11}{25}a^{2}+\frac{1}{5}$, $\frac{1}{19\cdots 75}a^{23}-\frac{29\cdots 94}{19\cdots 75}a^{22}+\frac{15\cdots 66}{19\cdots 75}a^{21}+\frac{43\cdots 48}{28\cdots 25}a^{20}-\frac{34\cdots 19}{19\cdots 75}a^{19}+\frac{82\cdots 99}{39\cdots 75}a^{18}-\frac{15\cdots 88}{39\cdots 75}a^{17}-\frac{12\cdots 39}{56\cdots 25}a^{16}-\frac{13\cdots 29}{39\cdots 75}a^{15}+\frac{10\cdots 68}{39\cdots 75}a^{14}+\frac{96\cdots 66}{79\cdots 75}a^{13}+\frac{12\cdots 86}{22\cdots 25}a^{12}+\frac{10\cdots 03}{63\cdots 95}a^{11}+\frac{25\cdots 72}{79\cdots 75}a^{10}+\frac{15\cdots 48}{79\cdots 75}a^{9}-\frac{12\cdots 51}{15\cdots 75}a^{8}-\frac{89\cdots 67}{15\cdots 75}a^{7}-\frac{71\cdots 77}{15\cdots 75}a^{6}+\frac{22\cdots 77}{31\cdots 75}a^{5}-\frac{73\cdots 39}{15\cdots 75}a^{4}-\frac{90\cdots 19}{31\cdots 75}a^{3}-\frac{13\cdots 46}{31\cdots 75}a^{2}+\frac{18\cdots 84}{31\cdots 75}a-\frac{88\cdots 52}{31\cdots 75}$ 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:  $C_{2}\times C_{4}\times C_{4}$, which has order $32$ (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:  $C_{2}\times C_{2}\times C_{4}\times C_{8}$, which has order $128$ (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:   $\frac{12\cdots 98}{13\cdots 25}a^{23}+\frac{81\cdots 56}{27\cdots 25}a^{22}-\frac{41\cdots 52}{13\cdots 25}a^{21}+\frac{17\cdots 48}{27\cdots 25}a^{20}+\frac{65\cdots 24}{27\cdots 25}a^{19}+\frac{16\cdots 53}{13\cdots 25}a^{18}+\frac{27\cdots 11}{27\cdots 25}a^{17}-\frac{65\cdots 28}{13\cdots 25}a^{16}+\frac{49\cdots 36}{27\cdots 25}a^{15}-\frac{67\cdots 99}{54\cdots 25}a^{14}-\frac{95\cdots 96}{27\cdots 25}a^{13}-\frac{24\cdots 53}{54\cdots 25}a^{12}+\frac{51\cdots 26}{27\cdots 25}a^{11}+\frac{95\cdots 78}{54\cdots 25}a^{10}+\frac{13\cdots 19}{10\cdots 25}a^{9}-\frac{34\cdots 46}{54\cdots 25}a^{8}+\frac{27\cdots 71}{10\cdots 25}a^{7}-\frac{14\cdots 19}{54\cdots 25}a^{6}-\frac{67\cdots 02}{10\cdots 25}a^{5}+\frac{20\cdots 32}{21\cdots 45}a^{4}+\frac{62\cdots 18}{10\cdots 25}a^{3}-\frac{27\cdots 84}{43\cdots 49}a^{2}-\frac{39\cdots 63}{10\cdots 25}a-\frac{64\cdots 14}{21\cdots 45}$, $\frac{80\cdots 81}{14\cdots 75}a^{23}+\frac{72\cdots 79}{72\cdots 75}a^{22}-\frac{80\cdots 86}{72\cdots 75}a^{21}-\frac{15\cdots 86}{72\cdots 75}a^{20}-\frac{77\cdots 26}{72\cdots 75}a^{19}+\frac{52\cdots 74}{72\cdots 75}a^{18}+\frac{41\cdots 89}{57\cdots 75}a^{17}-\frac{25\cdots 41}{14\cdots 75}a^{16}+\frac{13\cdots 82}{14\cdots 75}a^{15}+\frac{55\cdots 57}{14\cdots 75}a^{14}-\frac{66\cdots 23}{14\cdots 75}a^{13}-\frac{78\cdots 44}{28\cdots 75}a^{12}-\frac{78\cdots 32}{28\cdots 75}a^{11}+\frac{93\cdots 47}{28\cdots 75}a^{10}+\frac{20\cdots 37}{28\cdots 75}a^{9}+\frac{19\cdots 02}{28\cdots 75}a^{8}+\frac{19\cdots 23}{57\cdots 75}a^{7}-\frac{22\cdots 82}{23\cdots 91}a^{6}-\frac{26\cdots 16}{57\cdots 75}a^{5}-\frac{10\cdots 96}{57\cdots 75}a^{4}+\frac{14\cdots 49}{57\cdots 75}a^{3}-\frac{85\cdots 87}{11\cdots 55}a^{2}-\frac{17\cdots 23}{11\cdots 55}a+\frac{33\cdots 75}{23\cdots 91}$, $\frac{59\cdots 17}{72\cdots 75}a^{23}-\frac{11\cdots 82}{14\cdots 75}a^{22}+\frac{27\cdots 38}{72\cdots 75}a^{21}-\frac{95\cdots 07}{72\cdots 75}a^{20}+\frac{25\cdots 83}{72\cdots 75}a^{19}-\frac{13\cdots 94}{72\cdots 75}a^{18}-\frac{86\cdots 42}{72\cdots 75}a^{17}+\frac{10\cdots 57}{14\cdots 75}a^{16}-\frac{16\cdots 69}{28\cdots 75}a^{15}+\frac{36\cdots 44}{14\cdots 75}a^{14}-\frac{11\cdots 27}{14\cdots 75}a^{13}+\frac{86\cdots 24}{14\cdots 75}a^{12}-\frac{87\cdots 92}{28\cdots 75}a^{11}+\frac{53\cdots 59}{57\cdots 75}a^{10}-\frac{88\cdots 46}{28\cdots 75}a^{9}+\frac{29\cdots 48}{28\cdots 75}a^{8}-\frac{88\cdots 86}{28\cdots 75}a^{7}+\frac{58\cdots 37}{57\cdots 75}a^{6}-\frac{14\cdots 43}{57\cdots 75}a^{5}+\frac{19\cdots 68}{57\cdots 75}a^{4}-\frac{12\cdots 29}{57\cdots 75}a^{3}+\frac{13\cdots 73}{57\cdots 75}a^{2}+\frac{20\cdots 98}{11\cdots 55}a-\frac{66\cdots 31}{11\cdots 55}$, $\frac{10\cdots 22}{72\cdots 75}a^{23}+\frac{43\cdots 64}{72\cdots 75}a^{22}-\frac{12\cdots 36}{14\cdots 75}a^{21}+\frac{38\cdots 94}{72\cdots 75}a^{20}-\frac{38\cdots 86}{72\cdots 75}a^{19}+\frac{14\cdots 16}{72\cdots 75}a^{18}+\frac{21\cdots 24}{14\cdots 75}a^{17}-\frac{66\cdots 51}{72\cdots 75}a^{16}+\frac{51\cdots 27}{14\cdots 75}a^{15}-\frac{64\cdots 18}{14\cdots 75}a^{14}-\frac{56\cdots 22}{14\cdots 75}a^{13}-\frac{20\cdots 78}{28\cdots 75}a^{12}+\frac{13\cdots 97}{14\cdots 75}a^{11}+\frac{74\cdots 04}{28\cdots 75}a^{10}+\frac{49\cdots 07}{28\cdots 75}a^{9}-\frac{76\cdots 27}{28\cdots 75}a^{8}+\frac{28\cdots 73}{57\cdots 75}a^{7}-\frac{13\cdots 33}{28\cdots 75}a^{6}-\frac{37\cdots 07}{57\cdots 75}a^{5}+\frac{13\cdots 89}{57\cdots 75}a^{4}+\frac{46\cdots 46}{57\cdots 75}a^{3}-\frac{20\cdots 89}{11\cdots 55}a^{2}+\frac{17\cdots 19}{57\cdots 75}a+\frac{41\cdots 52}{11\cdots 55}$, $\frac{35\cdots 18}{13\cdots 25}a^{23}+\frac{42\cdots 52}{13\cdots 25}a^{22}+\frac{11\cdots 07}{13\cdots 25}a^{21}-\frac{59\cdots 24}{13\cdots 25}a^{20}+\frac{18\cdots 31}{13\cdots 25}a^{19}+\frac{43\cdots 44}{13\cdots 25}a^{18}+\frac{49\cdots 49}{13\cdots 25}a^{17}-\frac{10\cdots 91}{13\cdots 25}a^{16}+\frac{73\cdots 46}{27\cdots 25}a^{15}+\frac{26\cdots 08}{27\cdots 25}a^{14}-\frac{98\cdots 88}{27\cdots 25}a^{13}-\frac{33\cdots 98}{27\cdots 25}a^{12}-\frac{60\cdots 08}{27\cdots 25}a^{11}+\frac{33\cdots 82}{54\cdots 25}a^{10}+\frac{16\cdots 68}{54\cdots 25}a^{9}+\frac{40\cdots 27}{54\cdots 25}a^{8}+\frac{42\cdots 17}{54\cdots 25}a^{7}-\frac{10\cdots 58}{54\cdots 25}a^{6}-\frac{74\cdots 16}{21\cdots 45}a^{5}+\frac{16\cdots 66}{10\cdots 25}a^{4}+\frac{13\cdots 74}{10\cdots 25}a^{3}-\frac{54\cdots 21}{10\cdots 25}a^{2}-\frac{14\cdots 01}{10\cdots 25}a+\frac{14\cdots 88}{21\cdots 45}$, $\frac{37\cdots 46}{72\cdots 75}a^{23}-\frac{11\cdots 74}{72\cdots 75}a^{22}-\frac{21\cdots 86}{72\cdots 75}a^{21}+\frac{25\cdots 92}{72\cdots 75}a^{20}-\frac{93\cdots 58}{72\cdots 75}a^{19}-\frac{35\cdots 64}{72\cdots 75}a^{18}-\frac{36\cdots 14}{72\cdots 75}a^{17}+\frac{19\cdots 93}{72\cdots 75}a^{16}-\frac{81\cdots 33}{14\cdots 75}a^{15}-\frac{51\cdots 54}{14\cdots 75}a^{14}+\frac{37\cdots 08}{14\cdots 75}a^{13}+\frac{27\cdots 48}{14\cdots 75}a^{12}-\frac{11\cdots 11}{14\cdots 75}a^{11}-\frac{69\cdots 16}{28\cdots 75}a^{10}-\frac{23\cdots 89}{28\cdots 75}a^{9}-\frac{19\cdots 67}{28\cdots 75}a^{8}+\frac{90\cdots 63}{28\cdots 75}a^{7}+\frac{53\cdots 19}{28\cdots 75}a^{6}+\frac{18\cdots 61}{11\cdots 55}a^{5}-\frac{65\cdots 93}{57\cdots 75}a^{4}-\frac{38\cdots 94}{57\cdots 75}a^{3}+\frac{71\cdots 71}{57\cdots 75}a^{2}+\frac{12\cdots 03}{57\cdots 75}a+\frac{46\cdots 66}{11\cdots 55}$, $\frac{26\cdots 24}{19\cdots 75}a^{23}-\frac{13\cdots 26}{19\cdots 75}a^{22}+\frac{15\cdots 79}{19\cdots 75}a^{21}+\frac{57\cdots 77}{28\cdots 25}a^{20}-\frac{49\cdots 86}{19\cdots 75}a^{19}-\frac{15\cdots 73}{79\cdots 75}a^{18}-\frac{44\cdots 94}{39\cdots 75}a^{17}+\frac{57\cdots 54}{56\cdots 25}a^{16}-\frac{14\cdots 27}{39\cdots 75}a^{15}+\frac{70\cdots 42}{39\cdots 75}a^{14}+\frac{12\cdots 16}{79\cdots 75}a^{13}+\frac{72\cdots 17}{11\cdots 25}a^{12}-\frac{23\cdots 27}{15\cdots 75}a^{11}-\frac{61\cdots 99}{15\cdots 75}a^{10}-\frac{22\cdots 38}{79\cdots 75}a^{9}+\frac{10\cdots 03}{15\cdots 75}a^{8}-\frac{18\cdots 19}{15\cdots 75}a^{7}-\frac{12\cdots 28}{15\cdots 75}a^{6}+\frac{13\cdots 07}{15\cdots 75}a^{5}+\frac{79\cdots 59}{15\cdots 75}a^{4}-\frac{23\cdots 77}{31\cdots 75}a^{3}+\frac{24\cdots 44}{31\cdots 75}a^{2}+\frac{48\cdots 56}{31\cdots 75}a-\frac{23\cdots 14}{31\cdots 75}$, $\frac{86\cdots 69}{19\cdots 75}a^{23}+\frac{13\cdots 36}{19\cdots 75}a^{22}-\frac{18\cdots 34}{19\cdots 75}a^{21}-\frac{14\cdots 12}{28\cdots 25}a^{20}-\frac{32\cdots 69}{19\cdots 75}a^{19}+\frac{22\cdots 78}{39\cdots 75}a^{18}+\frac{22\cdots 06}{39\cdots 75}a^{17}-\frac{71\cdots 59}{56\cdots 25}a^{16}+\frac{58\cdots 41}{79\cdots 75}a^{15}+\frac{18\cdots 93}{39\cdots 75}a^{14}+\frac{84\cdots 73}{79\cdots 75}a^{13}-\frac{24\cdots 39}{11\cdots 25}a^{12}-\frac{40\cdots 83}{15\cdots 75}a^{11}+\frac{84\cdots 64}{79\cdots 75}a^{10}+\frac{42\cdots 48}{79\cdots 75}a^{9}+\frac{89\cdots 51}{15\cdots 75}a^{8}+\frac{36\cdots 85}{12\cdots 19}a^{7}-\frac{10\cdots 62}{15\cdots 75}a^{6}-\frac{55\cdots 93}{15\cdots 75}a^{5}-\frac{30\cdots 39}{15\cdots 75}a^{4}+\frac{23\cdots 84}{12\cdots 19}a^{3}-\frac{18\cdots 57}{31\cdots 75}a^{2}-\frac{44\cdots 51}{31\cdots 75}a+\frac{32\cdots 32}{31\cdots 75}$, $\frac{39\cdots 47}{19\cdots 75}a^{23}+\frac{13\cdots 43}{19\cdots 75}a^{22}-\frac{62\cdots 87}{19\cdots 75}a^{21}+\frac{42\cdots 54}{28\cdots 25}a^{20}+\frac{75\cdots 43}{19\cdots 75}a^{19}+\frac{10\cdots 42}{39\cdots 75}a^{18}+\frac{85\cdots 06}{39\cdots 75}a^{17}-\frac{67\cdots 06}{56\cdots 25}a^{16}+\frac{11\cdots 09}{39\cdots 75}a^{15}-\frac{21\cdots 21}{39\cdots 75}a^{14}-\frac{18\cdots 27}{79\cdots 75}a^{13}-\frac{23\cdots 33}{22\cdots 25}a^{12}+\frac{43\cdots 46}{79\cdots 75}a^{11}+\frac{57\cdots 94}{79\cdots 75}a^{10}+\frac{32\cdots 44}{79\cdots 75}a^{9}+\frac{31\cdots 42}{15\cdots 75}a^{8}+\frac{83\cdots 44}{15\cdots 75}a^{7}-\frac{49\cdots 66}{63\cdots 95}a^{6}-\frac{36\cdots 92}{15\cdots 75}a^{5}+\frac{37\cdots 83}{15\cdots 75}a^{4}+\frac{42\cdots 33}{31\cdots 75}a^{3}-\frac{14\cdots 63}{31\cdots 75}a^{2}-\frac{30\cdots 76}{31\cdots 75}a+\frac{48\cdots 79}{63\cdots 95}$, $\frac{23\cdots 06}{19\cdots 75}a^{23}-\frac{10\cdots 74}{19\cdots 75}a^{22}+\frac{17\cdots 31}{19\cdots 75}a^{21}-\frac{22\cdots 67}{28\cdots 25}a^{20}+\frac{79\cdots 46}{19\cdots 75}a^{19}-\frac{62\cdots 84}{39\cdots 75}a^{18}-\frac{45\cdots 34}{39\cdots 75}a^{17}+\frac{43\cdots 86}{56\cdots 25}a^{16}-\frac{24\cdots 34}{79\cdots 75}a^{15}+\frac{18\cdots 38}{39\cdots 75}a^{14}+\frac{87\cdots 37}{79\cdots 75}a^{13}+\frac{66\cdots 16}{11\cdots 25}a^{12}-\frac{14\cdots 24}{15\cdots 75}a^{11}-\frac{13\cdots 31}{79\cdots 75}a^{10}-\frac{10\cdots 32}{79\cdots 75}a^{9}+\frac{78\cdots 57}{31\cdots 75}a^{8}-\frac{32\cdots 92}{63\cdots 95}a^{7}+\frac{61\cdots 68}{15\cdots 75}a^{6}+\frac{62\cdots 27}{15\cdots 75}a^{5}-\frac{30\cdots 74}{15\cdots 75}a^{4}-\frac{17\cdots 17}{31\cdots 75}a^{3}+\frac{49\cdots 53}{31\cdots 75}a^{2}-\frac{28\cdots 66}{31\cdots 75}a+\frac{61\cdots 57}{31\cdots 75}$, $\frac{29\cdots 23}{19\cdots 75}a^{23}-\frac{18\cdots 13}{19\cdots 75}a^{22}+\frac{11\cdots 02}{19\cdots 75}a^{21}+\frac{72\cdots 71}{28\cdots 25}a^{20}-\frac{95\cdots 13}{19\cdots 75}a^{19}+\frac{64\cdots 23}{39\cdots 75}a^{18}+\frac{15\cdots 29}{39\cdots 75}a^{17}+\frac{68\cdots 89}{11\cdots 25}a^{16}-\frac{33\cdots 73}{39\cdots 75}a^{15}+\frac{84\cdots 86}{39\cdots 75}a^{14}+\frac{31\cdots 67}{79\cdots 75}a^{13}-\frac{21\cdots 14}{22\cdots 25}a^{12}-\frac{55\cdots 77}{79\cdots 75}a^{11}+\frac{12\cdots 19}{79\cdots 75}a^{10}+\frac{88\cdots 71}{79\cdots 75}a^{9}+\frac{61\cdots 58}{15\cdots 75}a^{8}-\frac{18\cdots 44}{15\cdots 75}a^{7}-\frac{12\cdots 31}{15\cdots 75}a^{6}+\frac{16\cdots 43}{31\cdots 75}a^{5}+\frac{78\cdots 47}{15\cdots 75}a^{4}-\frac{54\cdots 63}{31\cdots 75}a^{3}+\frac{15\cdots 13}{31\cdots 75}a^{2}+\frac{44\cdots 09}{31\cdots 75}a-\frac{27\cdots 74}{31\cdots 75}$, $\frac{90\cdots 11}{39\cdots 75}a^{23}-\frac{63\cdots 22}{39\cdots 75}a^{22}+\frac{21\cdots 43}{39\cdots 75}a^{21}-\frac{25\cdots 32}{56\cdots 25}a^{20}-\frac{24\cdots 61}{39\cdots 75}a^{19}+\frac{59\cdots 68}{39\cdots 75}a^{18}-\frac{12\cdots 99}{39\cdots 75}a^{17}+\frac{12\cdots 53}{56\cdots 25}a^{16}-\frac{42\cdots 67}{39\cdots 75}a^{15}+\frac{16\cdots 27}{79\cdots 75}a^{14}+\frac{47\cdots 54}{79\cdots 75}a^{13}+\frac{47\cdots 84}{11\cdots 25}a^{12}-\frac{11\cdots 82}{79\cdots 75}a^{11}-\frac{14\cdots 01}{79\cdots 75}a^{10}+\frac{26\cdots 12}{15\cdots 75}a^{9}-\frac{31\cdots 26}{15\cdots 75}a^{8}+\frac{84\cdots 63}{15\cdots 75}a^{7}+\frac{58\cdots 13}{15\cdots 75}a^{6}-\frac{10\cdots 91}{15\cdots 75}a^{5}-\frac{49\cdots 76}{31\cdots 75}a^{4}+\frac{84\cdots 58}{31\cdots 75}a^{3}+\frac{22\cdots 86}{31\cdots 75}a^{2}-\frac{36\cdots 99}{31\cdots 75}a+\frac{10\cdots 88}{31\cdots 75}$, $\frac{26\cdots 14}{72\cdots 75}a^{23}+\frac{92\cdots 27}{72\cdots 75}a^{22}-\frac{10\cdots 53}{72\cdots 75}a^{21}+\frac{37\cdots 21}{72\cdots 75}a^{20}+\frac{41\cdots 16}{72\cdots 75}a^{19}+\frac{70\cdots 53}{14\cdots 75}a^{18}+\frac{28\cdots 44}{72\cdots 75}a^{17}-\frac{14\cdots 01}{72\cdots 75}a^{16}+\frac{11\cdots 42}{14\cdots 75}a^{15}-\frac{92\cdots 77}{14\cdots 75}a^{14}-\frac{37\cdots 62}{28\cdots 75}a^{13}-\frac{26\cdots 33}{14\cdots 75}a^{12}+\frac{16\cdots 42}{14\cdots 75}a^{11}+\frac{19\cdots 22}{28\cdots 75}a^{10}+\frac{13\cdots 48}{28\cdots 75}a^{9}-\frac{20\cdots 19}{57\cdots 75}a^{8}+\frac{30\cdots 62}{28\cdots 75}a^{7}-\frac{30\cdots 03}{28\cdots 75}a^{6}-\frac{13\cdots 88}{57\cdots 75}a^{5}+\frac{24\cdots 91}{57\cdots 75}a^{4}+\frac{51\cdots 48}{23\cdots 91}a^{3}-\frac{17\cdots 46}{57\cdots 75}a^{2}-\frac{52\cdots 31}{57\cdots 75}a+\frac{15\cdots 66}{11\cdots 55}$ 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:  \( 35389578082944830000 \) (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:  \( 2 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 35389578082944830000 \cdot 32}{2\cdot\sqrt{6769025210045733840131040987091228089411742985248565673828125}}\cr\approx \mathstrut & 0.333926563504685 \end{aligned}\] (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 - 4*x^23 + 6*x^22 - 4*x^21 + x^20 - 1305*x^19 - 10005*x^18 + 60320*x^17 - 237655*x^16 + 297395*x^15 + 227650*x^14 + 48632275*x^13 - 58519100*x^12 - 162429725*x^11 - 1167334100*x^10 + 1650665500*x^9 - 3500169500*x^8 + 30497763000*x^7 + 43848203000*x^6 - 146123238875*x^5 - 523765574375*x^4 + 1137682488125*x^3 - 286773405625*x^2 - 99248258750*x + 56010581875) 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 - 4*x^23 + 6*x^22 - 4*x^21 + x^20 - 1305*x^19 - 10005*x^18 + 60320*x^17 - 237655*x^16 + 297395*x^15 + 227650*x^14 + 48632275*x^13 - 58519100*x^12 - 162429725*x^11 - 1167334100*x^10 + 1650665500*x^9 - 3500169500*x^8 + 30497763000*x^7 + 43848203000*x^6 - 146123238875*x^5 - 523765574375*x^4 + 1137682488125*x^3 - 286773405625*x^2 - 99248258750*x + 56010581875, 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 - 4*x^23 + 6*x^22 - 4*x^21 + x^20 - 1305*x^19 - 10005*x^18 + 60320*x^17 - 237655*x^16 + 297395*x^15 + 227650*x^14 + 48632275*x^13 - 58519100*x^12 - 162429725*x^11 - 1167334100*x^10 + 1650665500*x^9 - 3500169500*x^8 + 30497763000*x^7 + 43848203000*x^6 - 146123238875*x^5 - 523765574375*x^4 + 1137682488125*x^3 - 286773405625*x^2 - 99248258750*x + 56010581875); 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 - 4*x^23 + 6*x^22 - 4*x^21 + x^20 - 1305*x^19 - 10005*x^18 + 60320*x^17 - 237655*x^16 + 297395*x^15 + 227650*x^14 + 48632275*x^13 - 58519100*x^12 - 162429725*x^11 - 1167334100*x^10 + 1650665500*x^9 - 3500169500*x^8 + 30497763000*x^7 + 43848203000*x^6 - 146123238875*x^5 - 523765574375*x^4 + 1137682488125*x^3 - 286773405625*x^2 - 99248258750*x + 56010581875); 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

6.2.1381408203125.1, 12.4.8024353662494678497314453125.5

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: 24.4.270761008401829353605241639483649123576469719409942626953125.8, 24.4.270761008401829353605241639483649123576469719409942626953125.5
Arithmetically equivalent sibling: 24.4.6769025210045733840131040987091228089411742985248565673828125.4
Minimal sibling: 24.4.270761008401829353605241639483649123576469719409942626953125.5

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $24$ ${\href{/padicField/3.8.0.1}{8} }^{3}$ R ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }^{2}$ $24$ ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.3.0.1}{3} }^{8}$ $24$ R $20{,}\,{\href{/padicField/31.4.0.1}{4} }$ ${\href{/padicField/37.8.0.1}{8} }^{3}$ $20{,}\,{\href{/padicField/41.4.0.1}{4} }$ $24$ ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }^{2}$

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 5.1.4.3a1.1$x^{4} + 5$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.1.10.19a2.2$x^{10} + 25 x^{2} + 5$$10$$1$$19$$F_5$$$[\frac{9}{4}]_{4}$$
5.1.10.19a2.2$x^{10} + 25 x^{2} + 5$$10$$1$$19$$F_5$$$[\frac{9}{4}]_{4}$$
\(29\) Copy content Toggle raw display 29.1.4.3a1.1$x^{4} + 29$$4$$1$$3$$C_4$$$[\ ]_{4}$$
29.1.20.19a1.3$x^{20} + 116$$20$$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)