Properties

Label 24.4.676...125.5
Degree $24$
Signature $(4, 10)$
Discriminant $6.769\times 10^{60}$
Root discriminant \(342.46\)
Ramified primes $5,29$
Class number $160$ (GRH)
Class group [2, 4, 20] (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 - 30305*x^17 - 52780*x^16 + 1127520*x^15 + 2873900*x^14 + 19324150*x^13 + 65927150*x^12 + 57825275*x^11 - 1875785975*x^10 - 3653343875*x^9 + 4226336750*x^8 + 31324081750*x^7 - 91930159500*x^6 - 128610048250*x^5 + 436357363125*x^4 + 79569910000*x^3 - 882473280625*x^2 + 125130178750*x + 751938081875)
 
Copy content gp:K = bnfinit(y^24 - 4*y^23 + 6*y^22 - 4*y^21 + y^20 - 1305*y^19 - 10005*y^18 - 30305*y^17 - 52780*y^16 + 1127520*y^15 + 2873900*y^14 + 19324150*y^13 + 65927150*y^12 + 57825275*y^11 - 1875785975*y^10 - 3653343875*y^9 + 4226336750*y^8 + 31324081750*y^7 - 91930159500*y^6 - 128610048250*y^5 + 436357363125*y^4 + 79569910000*y^3 - 882473280625*y^2 + 125130178750*y + 751938081875, 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 - 30305*x^17 - 52780*x^16 + 1127520*x^15 + 2873900*x^14 + 19324150*x^13 + 65927150*x^12 + 57825275*x^11 - 1875785975*x^10 - 3653343875*x^9 + 4226336750*x^8 + 31324081750*x^7 - 91930159500*x^6 - 128610048250*x^5 + 436357363125*x^4 + 79569910000*x^3 - 882473280625*x^2 + 125130178750*x + 751938081875);
 
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 - 30305*x^17 - 52780*x^16 + 1127520*x^15 + 2873900*x^14 + 19324150*x^13 + 65927150*x^12 + 57825275*x^11 - 1875785975*x^10 - 3653343875*x^9 + 4226336750*x^8 + 31324081750*x^7 - 91930159500*x^6 - 128610048250*x^5 + 436357363125*x^4 + 79569910000*x^3 - 882473280625*x^2 + 125130178750*x + 751938081875)
 

\( x^{24} - 4 x^{23} + 6 x^{22} - 4 x^{21} + x^{20} - 1305 x^{19} - 10005 x^{18} - 30305 x^{17} + \cdots + 751938081875 \) 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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{5}a^{7}-\frac{2}{5}a^{6}+\frac{1}{5}a^{5}$, $\frac{1}{5}a^{8}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}$, $\frac{1}{5}a^{9}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}$, $\frac{1}{5}a^{10}-\frac{1}{5}a^{5}$, $\frac{1}{25}a^{11}+\frac{2}{25}a^{10}-\frac{2}{25}a^{9}-\frac{1}{25}a^{8}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{1}{5}a^{3}$, $\frac{1}{25}a^{12}-\frac{1}{25}a^{10}-\frac{2}{25}a^{9}+\frac{2}{25}a^{8}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}+\frac{2}{5}a^{3}$, $\frac{1}{25}a^{13}-\frac{1}{25}a^{8}+\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{3}$, $\frac{1}{125}a^{14}+\frac{1}{125}a^{13}+\frac{1}{125}a^{12}+\frac{1}{125}a^{11}-\frac{9}{125}a^{10}+\frac{2}{25}a^{9}-\frac{2}{25}a^{8}+\frac{2}{25}a^{7}+\frac{7}{25}a^{6}+\frac{12}{25}a^{5}+\frac{1}{5}a^{4}+\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{125}a^{15}-\frac{11}{125}a^{10}+\frac{2}{25}a^{9}+\frac{2}{25}a^{8}+\frac{8}{25}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}$, $\frac{1}{125}a^{16}-\frac{1}{125}a^{11}+\frac{1}{25}a^{10}-\frac{2}{25}a^{9}-\frac{2}{25}a^{8}+\frac{8}{25}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a$, $\frac{1}{125}a^{17}-\frac{1}{125}a^{12}+\frac{1}{25}a^{10}+\frac{1}{25}a^{8}-\frac{2}{25}a^{7}-\frac{2}{5}a^{6}+\frac{2}{5}a^{5}+\frac{1}{5}a^{3}-\frac{1}{5}a^{2}$, $\frac{1}{125}a^{18}-\frac{1}{125}a^{13}-\frac{2}{25}a^{10}-\frac{2}{25}a^{9}-\frac{1}{25}a^{8}+\frac{2}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{4}$, $\frac{1}{3125}a^{19}-\frac{4}{3125}a^{18}+\frac{6}{3125}a^{17}-\frac{4}{3125}a^{16}+\frac{1}{3125}a^{15}-\frac{2}{625}a^{14}-\frac{12}{625}a^{13}-\frac{7}{625}a^{12}+\frac{8}{625}a^{11}-\frac{37}{625}a^{10}-\frac{2}{125}a^{9}+\frac{8}{125}a^{8}-\frac{2}{125}a^{7}+\frac{33}{125}a^{6}-\frac{47}{125}a^{5}+\frac{1}{25}a^{4}-\frac{9}{25}a^{3}+\frac{11}{25}a^{2}-\frac{4}{25}a-\frac{9}{25}$, $\frac{1}{3125}a^{20}-\frac{2}{625}a^{18}-\frac{1}{625}a^{17}+\frac{2}{625}a^{16}-\frac{6}{3125}a^{15}-\frac{2}{125}a^{13}+\frac{1}{125}a^{12}+\frac{2}{125}a^{11}+\frac{37}{625}a^{10}+\frac{1}{25}a^{9}-\frac{1}{25}a^{8}+\frac{8}{25}a^{6}+\frac{32}{125}a^{5}+\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{9}{25}$, $\frac{1}{3125}a^{21}+\frac{1}{625}a^{18}-\frac{1}{625}a^{17}+\frac{4}{3125}a^{16}+\frac{2}{625}a^{15}+\frac{1}{125}a^{13}+\frac{2}{125}a^{12}+\frac{12}{625}a^{11}-\frac{3}{125}a^{10}+\frac{2}{25}a^{9}-\frac{2}{25}a^{8}+\frac{2}{25}a^{7}-\frac{48}{125}a^{6}-\frac{2}{25}a^{5}+\frac{2}{5}a^{4}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{11}{25}a-\frac{2}{5}$, $\frac{1}{3125}a^{22}-\frac{2}{625}a^{18}-\frac{1}{3125}a^{17}+\frac{1}{625}a^{16}-\frac{1}{625}a^{15}+\frac{2}{125}a^{13}+\frac{2}{625}a^{12}+\frac{2}{125}a^{11}-\frac{6}{125}a^{10}-\frac{1}{25}a^{9}+\frac{1}{25}a^{8}-\frac{3}{125}a^{7}+\frac{11}{25}a^{6}+\frac{1}{25}a^{5}-\frac{1}{5}a^{4}+\frac{2}{5}a^{3}-\frac{11}{25}a^{2}+\frac{1}{5}$, $\frac{1}{76\cdots 75}a^{23}-\frac{11\cdots 24}{76\cdots 75}a^{22}-\frac{93\cdots 24}{76\cdots 75}a^{21}-\frac{60\cdots 19}{76\cdots 75}a^{20}-\frac{10\cdots 44}{76\cdots 75}a^{19}+\frac{36\cdots 84}{15\cdots 75}a^{18}+\frac{68\cdots 08}{15\cdots 75}a^{17}+\frac{29\cdots 48}{30\cdots 75}a^{16}+\frac{63\cdots 57}{15\cdots 75}a^{15}+\frac{94\cdots 43}{15\cdots 75}a^{14}-\frac{53\cdots 59}{30\cdots 75}a^{13}+\frac{21\cdots 38}{30\cdots 75}a^{12}-\frac{12\cdots 16}{30\cdots 75}a^{11}-\frac{35\cdots 87}{60\cdots 75}a^{10}+\frac{28\cdots 48}{30\cdots 75}a^{9}+\frac{30\cdots 19}{60\cdots 75}a^{8}-\frac{54\cdots 89}{60\cdots 75}a^{7}+\frac{19\cdots 32}{60\cdots 75}a^{6}+\frac{37\cdots 73}{60\cdots 75}a^{5}-\frac{36\cdots 39}{60\cdots 75}a^{4}-\frac{59\cdots 14}{12\cdots 75}a^{3}-\frac{15\cdots 54}{24\cdots 55}a^{2}-\frac{40\cdots 13}{12\cdots 75}a+\frac{25\cdots 27}{71\cdots 75}$ 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:  $C_{2}\times C_{4}\times C_{20}$, which has order $160$ (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:  $C_{40}\times C_{4}\times C_{2}\times C_{2}$, which has order $640$ (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:   $\frac{11\cdots 98}{97\cdots 75}a^{23}-\frac{35\cdots 17}{48\cdots 75}a^{22}+\frac{77\cdots 87}{48\cdots 75}a^{21}-\frac{18\cdots 58}{77\cdots 55}a^{20}+\frac{36\cdots 94}{97\cdots 75}a^{19}-\frac{15\cdots 83}{97\cdots 75}a^{18}-\frac{40\cdots 48}{48\cdots 75}a^{17}-\frac{43\cdots 92}{48\cdots 75}a^{16}+\frac{77\cdots 58}{19\cdots 75}a^{15}+\frac{30\cdots 79}{19\cdots 75}a^{14}+\frac{35\cdots 96}{38\cdots 75}a^{13}+\frac{14\cdots 46}{97\cdots 75}a^{12}+\frac{15\cdots 89}{97\cdots 75}a^{11}-\frac{58\cdots 84}{38\cdots 75}a^{10}-\frac{10\cdots 03}{38\cdots 75}a^{9}-\frac{13\cdots 41}{38\cdots 75}a^{8}+\frac{29\cdots 81}{19\cdots 75}a^{7}+\frac{88\cdots 59}{19\cdots 75}a^{6}-\frac{12\cdots 63}{77\cdots 55}a^{5}+\frac{44\cdots 62}{77\cdots 55}a^{4}+\frac{58\cdots 66}{77\cdots 55}a^{3}+\frac{21\cdots 72}{38\cdots 75}a^{2}-\frac{29\cdots 82}{38\cdots 75}a-\frac{28\cdots 67}{15\cdots 51}$, $\frac{39\cdots 07}{38\cdots 25}a^{23}-\frac{56\cdots 79}{19\cdots 25}a^{22}+\frac{66\cdots 06}{19\cdots 25}a^{21}-\frac{51\cdots 84}{19\cdots 25}a^{20}+\frac{55\cdots 86}{19\cdots 25}a^{19}-\frac{25\cdots 94}{19\cdots 25}a^{18}-\frac{45\cdots 98}{38\cdots 25}a^{17}-\frac{17\cdots 66}{38\cdots 25}a^{16}-\frac{43\cdots 06}{38\cdots 25}a^{15}+\frac{38\cdots 73}{38\cdots 25}a^{14}+\frac{15\cdots 38}{38\cdots 25}a^{13}+\frac{38\cdots 47}{15\cdots 25}a^{12}+\frac{75\cdots 36}{76\cdots 25}a^{11}+\frac{14\cdots 36}{76\cdots 25}a^{10}-\frac{12\cdots 32}{76\cdots 25}a^{9}-\frac{42\cdots 62}{76\cdots 25}a^{8}-\frac{47\cdots 87}{15\cdots 25}a^{7}+\frac{40\cdots 99}{15\cdots 25}a^{6}-\frac{98\cdots 21}{15\cdots 25}a^{5}-\frac{29\cdots 19}{15\cdots 25}a^{4}+\frac{27\cdots 31}{15\cdots 25}a^{3}+\frac{68\cdots 09}{30\cdots 45}a^{2}-\frac{14\cdots 59}{30\cdots 45}a-\frac{14\cdots 59}{30\cdots 45}$, $\frac{99\cdots 14}{76\cdots 75}a^{23}+\frac{53\cdots 21}{76\cdots 75}a^{22}-\frac{13\cdots 54}{76\cdots 75}a^{21}+\frac{22\cdots 81}{76\cdots 75}a^{20}-\frac{30\cdots 99}{76\cdots 75}a^{19}+\frac{26\cdots 51}{15\cdots 75}a^{18}+\frac{16\cdots 18}{15\cdots 75}a^{17}+\frac{74\cdots 42}{30\cdots 75}a^{16}+\frac{51\cdots 56}{15\cdots 75}a^{15}-\frac{23\cdots 22}{15\cdots 75}a^{14}-\frac{48\cdots 88}{30\cdots 75}a^{13}-\frac{69\cdots 17}{30\cdots 75}a^{12}-\frac{16\cdots 21}{30\cdots 75}a^{11}+\frac{87\cdots 72}{30\cdots 75}a^{10}+\frac{74\cdots 83}{30\cdots 75}a^{9}+\frac{31\cdots 32}{24\cdots 55}a^{8}-\frac{46\cdots 49}{60\cdots 75}a^{7}-\frac{18\cdots 83}{60\cdots 75}a^{6}+\frac{19\cdots 57}{12\cdots 75}a^{5}-\frac{36\cdots 19}{60\cdots 75}a^{4}-\frac{61\cdots 22}{12\cdots 75}a^{3}+\frac{15\cdots 68}{24\cdots 55}a^{2}+\frac{34\cdots 02}{12\cdots 75}a-\frac{46\cdots 86}{71\cdots 75}$, $\frac{68\cdots 46}{76\cdots 75}a^{23}-\frac{56\cdots 14}{76\cdots 75}a^{22}+\frac{24\cdots 36}{76\cdots 75}a^{21}-\frac{75\cdots 09}{76\cdots 75}a^{20}+\frac{13\cdots 41}{76\cdots 75}a^{19}-\frac{14\cdots 38}{15\cdots 75}a^{18}-\frac{11\cdots 07}{15\cdots 75}a^{17}+\frac{52\cdots 76}{30\cdots 75}a^{16}-\frac{16\cdots 98}{15\cdots 75}a^{15}+\frac{20\cdots 98}{15\cdots 75}a^{14}-\frac{15\cdots 82}{60\cdots 75}a^{13}+\frac{61\cdots 98}{30\cdots 75}a^{12}+\frac{28\cdots 44}{30\cdots 75}a^{11}-\frac{66\cdots 31}{60\cdots 75}a^{10}-\frac{27\cdots 72}{30\cdots 75}a^{9}+\frac{15\cdots 68}{60\cdots 75}a^{8}+\frac{47\cdots 16}{60\cdots 75}a^{7}-\frac{96\cdots 78}{60\cdots 75}a^{6}-\frac{15\cdots 37}{60\cdots 75}a^{5}+\frac{45\cdots 71}{60\cdots 75}a^{4}+\frac{13\cdots 94}{12\cdots 75}a^{3}-\frac{66\cdots 33}{48\cdots 11}a^{2}+\frac{34\cdots 97}{12\cdots 75}a+\frac{69\cdots 12}{71\cdots 75}$, $\frac{32\cdots 39}{76\cdots 25}a^{23}+\frac{17\cdots 47}{19\cdots 25}a^{22}+\frac{14\cdots 74}{19\cdots 25}a^{21}-\frac{12\cdots 26}{19\cdots 25}a^{20}-\frac{22\cdots 56}{19\cdots 25}a^{19}+\frac{12\cdots 84}{19\cdots 25}a^{18}+\frac{72\cdots 77}{19\cdots 25}a^{17}-\frac{80\cdots 11}{38\cdots 25}a^{16}-\frac{71\cdots 89}{38\cdots 25}a^{15}-\frac{17\cdots 23}{38\cdots 25}a^{14}+\frac{19\cdots 87}{38\cdots 25}a^{13}+\frac{10\cdots 71}{38\cdots 25}a^{12}-\frac{24\cdots 57}{76\cdots 25}a^{11}-\frac{41\cdots 39}{76\cdots 25}a^{10}-\frac{73\cdots 08}{76\cdots 25}a^{9}+\frac{34\cdots 92}{76\cdots 25}a^{8}+\frac{80\cdots 36}{76\cdots 25}a^{7}-\frac{55\cdots 87}{30\cdots 45}a^{6}-\frac{22\cdots 07}{30\cdots 45}a^{5}+\frac{34\cdots 59}{15\cdots 25}a^{4}+\frac{33\cdots 39}{15\cdots 25}a^{3}-\frac{11\cdots 93}{15\cdots 25}a^{2}+\frac{15\cdots 12}{30\cdots 45}a+\frac{29\cdots 96}{30\cdots 45}$, $\frac{53\cdots 78}{19\cdots 25}a^{23}+\frac{21\cdots 61}{19\cdots 25}a^{22}-\frac{47\cdots 56}{38\cdots 25}a^{21}-\frac{32\cdots 86}{19\cdots 25}a^{20}+\frac{16\cdots 69}{19\cdots 25}a^{19}+\frac{65\cdots 22}{19\cdots 25}a^{18}+\frac{54\cdots 38}{19\cdots 25}a^{17}+\frac{14\cdots 34}{19\cdots 25}a^{16}+\frac{18\cdots 82}{15\cdots 25}a^{15}-\frac{12\cdots 03}{38\cdots 25}a^{14}-\frac{30\cdots 04}{38\cdots 25}a^{13}-\frac{18\cdots 56}{38\cdots 25}a^{12}-\frac{70\cdots 23}{38\cdots 25}a^{11}-\frac{64\cdots 91}{76\cdots 25}a^{10}+\frac{40\cdots 72}{76\cdots 25}a^{9}+\frac{77\cdots 01}{76\cdots 25}a^{8}-\frac{15\cdots 41}{76\cdots 25}a^{7}-\frac{63\cdots 78}{76\cdots 25}a^{6}+\frac{42\cdots 69}{15\cdots 25}a^{5}+\frac{67\cdots 44}{15\cdots 25}a^{4}-\frac{28\cdots 08}{15\cdots 25}a^{3}+\frac{94\cdots 48}{15\cdots 25}a^{2}+\frac{50\cdots 79}{15\cdots 25}a-\frac{10\cdots 09}{30\cdots 45}$, $\frac{10\cdots 88}{76\cdots 75}a^{23}+\frac{56\cdots 72}{76\cdots 75}a^{22}-\frac{15\cdots 43}{76\cdots 75}a^{21}+\frac{28\cdots 87}{76\cdots 75}a^{20}-\frac{48\cdots 98}{76\cdots 75}a^{19}+\frac{27\cdots 29}{15\cdots 75}a^{18}+\frac{31\cdots 72}{30\cdots 75}a^{17}+\frac{35\cdots 67}{15\cdots 75}a^{16}+\frac{48\cdots 07}{15\cdots 75}a^{15}-\frac{23\cdots 19}{15\cdots 75}a^{14}-\frac{78\cdots 24}{60\cdots 75}a^{13}-\frac{71\cdots 32}{30\cdots 75}a^{12}-\frac{14\cdots 71}{30\cdots 75}a^{11}+\frac{10\cdots 49}{30\cdots 75}a^{10}+\frac{75\cdots 91}{30\cdots 75}a^{9}+\frac{49\cdots 96}{60\cdots 75}a^{8}-\frac{41\cdots 01}{60\cdots 75}a^{7}-\frac{36\cdots 54}{12\cdots 75}a^{6}+\frac{20\cdots 89}{12\cdots 75}a^{5}-\frac{65\cdots 63}{60\cdots 75}a^{4}-\frac{48\cdots 42}{12\cdots 75}a^{3}+\frac{66\cdots 44}{12\cdots 75}a^{2}+\frac{33\cdots 11}{12\cdots 75}a-\frac{43\cdots 02}{71\cdots 75}$, $\frac{11\cdots 52}{76\cdots 75}a^{23}+\frac{62\cdots 43}{76\cdots 75}a^{22}-\frac{16\cdots 22}{76\cdots 75}a^{21}+\frac{31\cdots 58}{76\cdots 75}a^{20}-\frac{51\cdots 57}{76\cdots 75}a^{19}+\frac{31\cdots 63}{15\cdots 75}a^{18}+\frac{18\cdots 51}{15\cdots 75}a^{17}+\frac{89\cdots 41}{30\cdots 75}a^{16}+\frac{67\cdots 78}{15\cdots 75}a^{15}-\frac{26\cdots 96}{15\cdots 75}a^{14}-\frac{53\cdots 09}{30\cdots 75}a^{13}-\frac{33\cdots 77}{12\cdots 75}a^{12}-\frac{18\cdots 88}{30\cdots 75}a^{11}-\frac{40\cdots 39}{30\cdots 75}a^{10}+\frac{85\cdots 69}{30\cdots 75}a^{9}+\frac{16\cdots 38}{12\cdots 75}a^{8}-\frac{43\cdots 31}{60\cdots 75}a^{7}-\frac{21\cdots 89}{60\cdots 75}a^{6}+\frac{46\cdots 94}{24\cdots 55}a^{5}-\frac{59\cdots 42}{60\cdots 75}a^{4}-\frac{54\cdots 86}{12\cdots 75}a^{3}+\frac{69\cdots 52}{12\cdots 75}a^{2}+\frac{41\cdots 26}{12\cdots 75}a-\frac{43\cdots 23}{71\cdots 75}$, $\frac{98\cdots 17}{48\cdots 75}a^{23}-\frac{22\cdots 26}{48\cdots 75}a^{22}+\frac{24\cdots 24}{48\cdots 75}a^{21}-\frac{41\cdots 67}{48\cdots 75}a^{20}-\frac{73\cdots 02}{48\cdots 75}a^{19}-\frac{11\cdots 64}{48\cdots 75}a^{18}-\frac{12\cdots 66}{48\cdots 75}a^{17}-\frac{50\cdots 11}{48\cdots 75}a^{16}-\frac{26\cdots 62}{97\cdots 75}a^{15}+\frac{17\cdots 09}{97\cdots 75}a^{14}+\frac{90\cdots 43}{97\cdots 75}a^{13}+\frac{54\cdots 47}{97\cdots 75}a^{12}+\frac{21\cdots 97}{97\cdots 75}a^{11}+\frac{91\cdots 91}{19\cdots 75}a^{10}-\frac{57\cdots 96}{19\cdots 75}a^{9}-\frac{26\cdots 77}{19\cdots 75}a^{8}-\frac{32\cdots 33}{19\cdots 75}a^{7}+\frac{97\cdots 27}{19\cdots 75}a^{6}-\frac{39\cdots 98}{77\cdots 55}a^{5}-\frac{19\cdots 07}{38\cdots 75}a^{4}-\frac{50\cdots 19}{38\cdots 75}a^{3}+\frac{40\cdots 84}{38\cdots 75}a^{2}+\frac{14\cdots 99}{38\cdots 75}a-\frac{28\cdots 46}{77\cdots 55}$, $\frac{82\cdots 81}{19\cdots 25}a^{23}-\frac{30\cdots 53}{19\cdots 25}a^{22}+\frac{30\cdots 49}{19\cdots 25}a^{21}-\frac{41\cdots 29}{19\cdots 25}a^{20}-\frac{18\cdots 49}{19\cdots 25}a^{19}-\frac{21\cdots 07}{38\cdots 25}a^{18}-\frac{86\cdots 76}{19\cdots 25}a^{17}-\frac{26\cdots 83}{19\cdots 25}a^{16}-\frac{80\cdots 72}{38\cdots 25}a^{15}+\frac{19\cdots 98}{38\cdots 25}a^{14}+\frac{11\cdots 33}{76\cdots 25}a^{13}+\frac{32\cdots 92}{38\cdots 25}a^{12}+\frac{11\cdots 76}{38\cdots 25}a^{11}+\frac{15\cdots 38}{76\cdots 25}a^{10}-\frac{66\cdots 87}{76\cdots 25}a^{9}-\frac{30\cdots 79}{15\cdots 25}a^{8}+\frac{14\cdots 72}{76\cdots 25}a^{7}+\frac{13\cdots 66}{76\cdots 25}a^{6}-\frac{47\cdots 04}{15\cdots 25}a^{5}-\frac{11\cdots 34}{15\cdots 25}a^{4}+\frac{12\cdots 90}{60\cdots 89}a^{3}+\frac{40\cdots 69}{15\cdots 25}a^{2}-\frac{40\cdots 13}{15\cdots 25}a-\frac{19\cdots 01}{60\cdots 89}$, $\frac{58\cdots 24}{76\cdots 75}a^{23}-\frac{13\cdots 46}{76\cdots 75}a^{22}-\frac{58\cdots 36}{76\cdots 75}a^{21}+\frac{35\cdots 29}{76\cdots 75}a^{20}-\frac{59\cdots 36}{76\cdots 75}a^{19}-\frac{30\cdots 43}{30\cdots 75}a^{18}-\frac{28\cdots 19}{30\cdots 75}a^{17}-\frac{54\cdots 14}{15\cdots 75}a^{16}-\frac{45\cdots 32}{60\cdots 75}a^{15}+\frac{12\cdots 92}{15\cdots 75}a^{14}+\frac{11\cdots 66}{30\cdots 75}a^{13}+\frac{56\cdots 16}{30\cdots 75}a^{12}+\frac{22\cdots 04}{30\cdots 75}a^{11}+\frac{34\cdots 76}{30\cdots 75}a^{10}-\frac{43\cdots 13}{30\cdots 75}a^{9}-\frac{33\cdots 77}{60\cdots 75}a^{8}-\frac{94\cdots 32}{60\cdots 75}a^{7}+\frac{18\cdots 56}{60\cdots 75}a^{6}-\frac{10\cdots 37}{60\cdots 75}a^{5}-\frac{11\cdots 41}{60\cdots 75}a^{4}+\frac{23\cdots 93}{12\cdots 75}a^{3}+\frac{69\cdots 78}{12\cdots 75}a^{2}-\frac{28\cdots 76}{12\cdots 75}a-\frac{38\cdots 76}{71\cdots 75}$, $\frac{13\cdots 19}{76\cdots 75}a^{23}+\frac{14\cdots 04}{76\cdots 75}a^{22}-\frac{58\cdots 91}{76\cdots 75}a^{21}-\frac{18\cdots 46}{76\cdots 75}a^{20}+\frac{10\cdots 19}{76\cdots 75}a^{19}-\frac{35\cdots 53}{15\cdots 75}a^{18}-\frac{45\cdots 59}{15\cdots 75}a^{17}-\frac{25\cdots 32}{15\cdots 75}a^{16}-\frac{84\cdots 29}{15\cdots 75}a^{15}+\frac{16\cdots 32}{15\cdots 75}a^{14}+\frac{46\cdots 82}{30\cdots 75}a^{13}+\frac{25\cdots 39}{30\cdots 75}a^{12}+\frac{21\cdots 73}{60\cdots 75}a^{11}+\frac{30\cdots 14}{30\cdots 75}a^{10}-\frac{61\cdots 98}{30\cdots 75}a^{9}-\frac{15\cdots 91}{60\cdots 75}a^{8}-\frac{40\cdots 64}{60\cdots 75}a^{7}+\frac{55\cdots 22}{60\cdots 75}a^{6}+\frac{17\cdots 06}{60\cdots 75}a^{5}+\frac{54\cdots 14}{60\cdots 75}a^{4}-\frac{18\cdots 54}{24\cdots 55}a^{3}-\frac{50\cdots 91}{12\cdots 75}a^{2}+\frac{10\cdots 41}{12\cdots 75}a+\frac{50\cdots 02}{71\cdots 75}$, $\frac{46\cdots 36}{19\cdots 25}a^{23}-\frac{81\cdots 06}{19\cdots 25}a^{22}+\frac{71\cdots 64}{19\cdots 25}a^{21}-\frac{27\cdots 57}{19\cdots 25}a^{20}+\frac{40\cdots 78}{19\cdots 25}a^{19}-\frac{59\cdots 63}{19\cdots 25}a^{18}-\frac{62\cdots 96}{19\cdots 25}a^{17}-\frac{26\cdots 01}{19\cdots 25}a^{16}-\frac{13\cdots 52}{30\cdots 45}a^{15}+\frac{79\cdots 19}{38\cdots 25}a^{14}+\frac{46\cdots 46}{38\cdots 25}a^{13}+\frac{28\cdots 37}{38\cdots 25}a^{12}+\frac{11\cdots 32}{38\cdots 25}a^{11}+\frac{56\cdots 59}{76\cdots 25}a^{10}-\frac{24\cdots 61}{76\cdots 25}a^{9}-\frac{14\cdots 39}{76\cdots 25}a^{8}-\frac{21\cdots 38}{76\cdots 25}a^{7}+\frac{32\cdots 17}{76\cdots 25}a^{6}-\frac{27\cdots 75}{60\cdots 89}a^{5}-\frac{89\cdots 77}{15\cdots 25}a^{4}-\frac{69\cdots 28}{15\cdots 25}a^{3}+\frac{14\cdots 34}{15\cdots 25}a^{2}+\frac{19\cdots 39}{15\cdots 25}a+\frac{52\cdots 28}{30\cdots 45}$ 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:  \( 8743760410650709000 \) (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

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 8743760410650709000 \cdot 160}{2\cdot\sqrt{6769025210045733840131040987091228089411742985248565673828125}}\cr\approx \mathstrut & 0.412518886096020 \end{aligned}\] (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 - 30305*x^17 - 52780*x^16 + 1127520*x^15 + 2873900*x^14 + 19324150*x^13 + 65927150*x^12 + 57825275*x^11 - 1875785975*x^10 - 3653343875*x^9 + 4226336750*x^8 + 31324081750*x^7 - 91930159500*x^6 - 128610048250*x^5 + 436357363125*x^4 + 79569910000*x^3 - 882473280625*x^2 + 125130178750*x + 751938081875) 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 - 30305*x^17 - 52780*x^16 + 1127520*x^15 + 2873900*x^14 + 19324150*x^13 + 65927150*x^12 + 57825275*x^11 - 1875785975*x^10 - 3653343875*x^9 + 4226336750*x^8 + 31324081750*x^7 - 91930159500*x^6 - 128610048250*x^5 + 436357363125*x^4 + 79569910000*x^3 - 882473280625*x^2 + 125130178750*x + 751938081875, 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 - 30305*x^17 - 52780*x^16 + 1127520*x^15 + 2873900*x^14 + 19324150*x^13 + 65927150*x^12 + 57825275*x^11 - 1875785975*x^10 - 3653343875*x^9 + 4226336750*x^8 + 31324081750*x^7 - 91930159500*x^6 - 128610048250*x^5 + 436357363125*x^4 + 79569910000*x^3 - 882473280625*x^2 + 125130178750*x + 751938081875); 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 - 30305*x^17 - 52780*x^16 + 1127520*x^15 + 2873900*x^14 + 19324150*x^13 + 65927150*x^12 + 57825275*x^11 - 1875785975*x^10 - 3653343875*x^9 + 4226336750*x^8 + 31324081750*x^7 - 91930159500*x^6 - 128610048250*x^5 + 436357363125*x^4 + 79569910000*x^3 - 882473280625*x^2 + 125130178750*x + 751938081875); 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: 24.4.270761008401829353605241639483649123576469719409942626953125.2, 24.4.270761008401829353605241639483649123576469719409942626953125.7
Arithmetically equivalent sibling: 24.4.6769025210045733840131040987091228089411742985248565673828125.2
Minimal sibling: 24.4.270761008401829353605241639483649123576469719409942626953125.2

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $24$ ${\href{/padicField/3.8.0.1}{8} }^{3}$ R ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ ${\href{/padicField/11.12.0.1}{12} }^{2}$ $24$ ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ $24$ R $20{,}\,{\href{/padicField/31.4.0.1}{4} }$ ${\href{/padicField/37.8.0.1}{8} }^{3}$ $20{,}\,{\href{/padicField/41.4.0.1}{4} }$ $24$ ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ ${\href{/padicField/59.12.0.1}{12} }^{2}$

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)