Properties

Label 24.4.358...125.8
Degree $24$
Signature $(4, 10)$
Discriminant $3.586\times 10^{64}$
Root discriminant \(489.53\)
Ramified primes $5,89$
Class number $8$ (GRH)
Class group [2, 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 - 191*x^22 + 152*x^21 + 20801*x^20 + 145552*x^19 - 1176554*x^18 - 16138302*x^17 + 15768809*x^16 + 1059872352*x^15 + 7747419046*x^14 + 3640653858*x^13 - 202726936711*x^12 - 1134655611298*x^11 - 2506437023344*x^10 + 18528375557180*x^9 + 138487686799265*x^8 - 8715047372200*x^7 - 866046602082015*x^6 - 4747675198206540*x^5 - 13439138362502051*x^4 + 97786901456032952*x^3 + 139064771037891016*x^2 - 927383945402843232*x + 858450394618182224)
 
Copy content gp:K = bnfinit(y^24 - 2*y^23 - 191*y^22 + 152*y^21 + 20801*y^20 + 145552*y^19 - 1176554*y^18 - 16138302*y^17 + 15768809*y^16 + 1059872352*y^15 + 7747419046*y^14 + 3640653858*y^13 - 202726936711*y^12 - 1134655611298*y^11 - 2506437023344*y^10 + 18528375557180*y^9 + 138487686799265*y^8 - 8715047372200*y^7 - 866046602082015*y^6 - 4747675198206540*y^5 - 13439138362502051*y^4 + 97786901456032952*y^3 + 139064771037891016*y^2 - 927383945402843232*y + 858450394618182224, 1)
 
Copy content magma:R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^23 - 191*x^22 + 152*x^21 + 20801*x^20 + 145552*x^19 - 1176554*x^18 - 16138302*x^17 + 15768809*x^16 + 1059872352*x^15 + 7747419046*x^14 + 3640653858*x^13 - 202726936711*x^12 - 1134655611298*x^11 - 2506437023344*x^10 + 18528375557180*x^9 + 138487686799265*x^8 - 8715047372200*x^7 - 866046602082015*x^6 - 4747675198206540*x^5 - 13439138362502051*x^4 + 97786901456032952*x^3 + 139064771037891016*x^2 - 927383945402843232*x + 858450394618182224);
 
Copy content oscar:Qx, x = polynomial_ring(QQ); K, a = number_field(x^24 - 2*x^23 - 191*x^22 + 152*x^21 + 20801*x^20 + 145552*x^19 - 1176554*x^18 - 16138302*x^17 + 15768809*x^16 + 1059872352*x^15 + 7747419046*x^14 + 3640653858*x^13 - 202726936711*x^12 - 1134655611298*x^11 - 2506437023344*x^10 + 18528375557180*x^9 + 138487686799265*x^8 - 8715047372200*x^7 - 866046602082015*x^6 - 4747675198206540*x^5 - 13439138362502051*x^4 + 97786901456032952*x^3 + 139064771037891016*x^2 - 927383945402843232*x + 858450394618182224)
 

\( x^{24} - 2 x^{23} - 191 x^{22} + 152 x^{21} + 20801 x^{20} + 145552 x^{19} - 1176554 x^{18} + \cdots + 85\!\cdots\!24 \) 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:   \(35863302598527284385359953566442393477815766818821430206298828125\) \(\medspace = 5^{31}\cdot 89^{22}\) 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:  \(489.53\)
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:  $5^{31/20}89^{19/20}\approx 861.6363513918435$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{178}a^{11}+\frac{21}{178}a^{10}+\frac{41}{178}a^{9}-\frac{29}{89}a^{8}-\frac{45}{178}a^{7}+\frac{19}{89}a^{6}+\frac{11}{178}a^{5}-\frac{1}{89}a^{4}-\frac{3}{178}a^{3}-\frac{51}{178}a^{2}-\frac{1}{2}a+\frac{40}{89}$, $\frac{1}{178}a^{12}-\frac{22}{89}a^{10}-\frac{29}{178}a^{9}-\frac{73}{178}a^{8}-\frac{85}{178}a^{7}-\frac{75}{178}a^{6}-\frac{55}{178}a^{5}+\frac{39}{178}a^{4}+\frac{6}{89}a^{3}-\frac{43}{89}a^{2}-\frac{9}{178}a-\frac{39}{89}$, $\frac{1}{178}a^{13}+\frac{5}{178}a^{10}-\frac{49}{178}a^{9}+\frac{33}{178}a^{8}+\frac{81}{178}a^{7}+\frac{15}{178}a^{6}-\frac{11}{178}a^{5}-\frac{38}{89}a^{4}-\frac{20}{89}a^{3}+\frac{61}{178}a^{2}-\frac{39}{89}a-\frac{20}{89}$, $\frac{1}{178}a^{14}+\frac{12}{89}a^{10}+\frac{3}{89}a^{9}+\frac{15}{178}a^{8}+\frac{31}{89}a^{7}-\frac{23}{178}a^{6}+\frac{47}{178}a^{5}-\frac{15}{89}a^{4}+\frac{38}{89}a^{3}-\frac{1}{178}a^{2}+\frac{49}{178}a-\frac{22}{89}$, $\frac{1}{178}a^{15}+\frac{18}{89}a^{10}-\frac{79}{178}a^{9}+\frac{15}{89}a^{8}-\frac{11}{178}a^{7}+\frac{25}{178}a^{6}+\frac{31}{89}a^{5}-\frac{27}{89}a^{4}+\frac{71}{178}a^{3}+\frac{27}{178}a^{2}-\frac{22}{89}a+\frac{19}{89}$, $\frac{1}{534}a^{16}-\frac{1}{534}a^{15}+\frac{1}{534}a^{14}-\frac{1}{534}a^{13}-\frac{1}{534}a^{12}-\frac{1}{534}a^{11}-\frac{39}{178}a^{10}-\frac{13}{89}a^{9}+\frac{4}{89}a^{8}-\frac{13}{534}a^{7}+\frac{46}{267}a^{6}-\frac{116}{267}a^{5}+\frac{103}{267}a^{4}-\frac{185}{534}a^{3}-\frac{59}{267}a^{2}-\frac{227}{534}a-\frac{38}{267}$, $\frac{1}{534}a^{17}+\frac{1}{534}a^{13}+\frac{1}{534}a^{12}-\frac{1}{534}a^{11}+\frac{3}{178}a^{10}+\frac{79}{178}a^{9}+\frac{47}{534}a^{8}+\frac{71}{267}a^{7}-\frac{73}{267}a^{6}-\frac{5}{534}a^{5}+\frac{35}{89}a^{4}-\frac{34}{89}a^{3}+\frac{7}{178}a^{2}+\frac{79}{178}a-\frac{74}{267}$, $\frac{1}{570312}a^{18}+\frac{125}{570312}a^{17}-\frac{95}{190104}a^{16}+\frac{41}{63368}a^{15}-\frac{40}{71289}a^{14}+\frac{3}{15842}a^{13}+\frac{997}{570312}a^{12}-\frac{149}{570312}a^{11}-\frac{5355}{63368}a^{10}+\frac{283457}{570312}a^{9}+\frac{127145}{570312}a^{8}+\frac{15617}{190104}a^{7}+\frac{3608}{7921}a^{6}-\frac{110603}{285156}a^{5}+\frac{78341}{190104}a^{4}-\frac{73645}{190104}a^{3}+\frac{26327}{63368}a^{2}-\frac{153805}{570312}a-\frac{114001}{285156}$, $\frac{1}{2851560}a^{19}-\frac{1}{1425780}a^{18}-\frac{569}{712890}a^{17}+\frac{7}{79210}a^{16}-\frac{1259}{2851560}a^{15}-\frac{647}{356445}a^{14}+\frac{5437}{2851560}a^{13}-\frac{169}{79210}a^{12}+\frac{346}{356445}a^{11}+\frac{66997}{1425780}a^{10}-\frac{54997}{158420}a^{9}+\frac{20234}{356445}a^{8}+\frac{334997}{950520}a^{7}-\frac{88691}{1425780}a^{6}-\frac{1380371}{2851560}a^{5}-\frac{74881}{237630}a^{4}-\frac{70213}{237630}a^{3}+\frac{441181}{1425780}a^{2}-\frac{1268719}{2851560}a-\frac{186581}{1425780}$, $\frac{1}{22812480}a^{20}-\frac{1}{1140624}a^{18}-\frac{775}{1140624}a^{17}+\frac{3461}{4562496}a^{16}+\frac{1943}{11406240}a^{15}-\frac{2261}{1520832}a^{14}-\frac{3415}{2281248}a^{13}+\frac{53}{190104}a^{12}+\frac{1105}{2281248}a^{11}-\frac{849949}{11406240}a^{10}+\frac{12089}{126736}a^{9}+\frac{2175791}{4562496}a^{8}-\frac{16412}{71289}a^{7}+\frac{540109}{4562496}a^{6}+\frac{2047133}{11406240}a^{5}+\frac{136823}{380208}a^{4}+\frac{169997}{2281248}a^{3}+\frac{304427}{1520832}a^{2}-\frac{410261}{1140624}a+\frac{2708729}{5703120}$, $\frac{1}{45624960}a^{21}+\frac{1}{11406240}a^{19}+\frac{1}{3802080}a^{18}+\frac{28001}{45624960}a^{17}+\frac{9107}{22812480}a^{16}-\frac{5219}{5069440}a^{15}-\frac{26947}{22812480}a^{14}+\frac{2261}{950520}a^{13}+\frac{14099}{7604160}a^{12}+\frac{3509}{7604160}a^{11}-\frac{345167}{3802080}a^{10}-\frac{2149789}{45624960}a^{9}-\frac{11091}{633680}a^{8}+\frac{3992609}{45624960}a^{7}-\frac{5767771}{22812480}a^{6}+\frac{3698779}{11406240}a^{5}+\frac{1835341}{22812480}a^{4}+\frac{2216229}{5069440}a^{3}-\frac{3012703}{11406240}a^{2}-\frac{149429}{760416}a+\frac{220733}{712890}$, $\frac{1}{547499520}a^{22}-\frac{1}{109499904}a^{21}+\frac{1}{273749760}a^{20}+\frac{1}{22812480}a^{19}+\frac{157}{547499520}a^{18}+\frac{40297}{547499520}a^{17}+\frac{484613}{547499520}a^{16}-\frac{541259}{547499520}a^{15}-\frac{101471}{136874880}a^{14}-\frac{111721}{273749760}a^{13}-\frac{364471}{136874880}a^{12}+\frac{618521}{273749760}a^{11}-\frac{12814801}{109499904}a^{10}-\frac{37657013}{182499840}a^{9}-\frac{16227599}{182499840}a^{8}-\frac{17216399}{109499904}a^{7}-\frac{7660721}{68437440}a^{6}-\frac{105195931}{273749760}a^{5}+\frac{176814707}{547499520}a^{4}+\frac{17133391}{547499520}a^{3}+\frac{8280923}{30416640}a^{2}-\frac{15940673}{45624960}a+\frac{14366723}{68437440}$, $\frac{1}{53\cdots 80}a^{23}+\frac{27\cdots 25}{10\cdots 16}a^{22}-\frac{46\cdots 83}{67\cdots 60}a^{21}-\frac{10\cdots 77}{14\cdots 80}a^{20}-\frac{75\cdots 07}{53\cdots 80}a^{19}-\frac{27\cdots 97}{10\cdots 16}a^{18}-\frac{46\cdots 93}{53\cdots 80}a^{17}+\frac{25\cdots 27}{53\cdots 80}a^{16}-\frac{59\cdots 17}{26\cdots 40}a^{15}-\frac{60\cdots 89}{26\cdots 40}a^{14}-\frac{35\cdots 67}{21\cdots 80}a^{13}+\frac{72\cdots 09}{26\cdots 40}a^{12}+\frac{10\cdots 11}{53\cdots 80}a^{11}+\frac{98\cdots 67}{59\cdots 20}a^{10}-\frac{52\cdots 81}{17\cdots 60}a^{9}+\frac{24\cdots 07}{53\cdots 80}a^{8}+\frac{61\cdots 89}{26\cdots 40}a^{7}+\frac{21\cdots 05}{53\cdots 08}a^{6}+\frac{15\cdots 43}{10\cdots 16}a^{5}+\frac{20\cdots 89}{53\cdots 80}a^{4}-\frac{90\cdots 81}{44\cdots 40}a^{3}+\frac{37\cdots 61}{93\cdots 80}a^{2}-\frac{31\cdots 09}{84\cdots 20}a+\frac{27\cdots 69}{11\cdots 60}$ 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:  No
Index:  Not computed
Inessential primes:  $2$

Class group and class number

Ideal class group:  $C_{2}\times C_{4}$, which has order $8$ (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:  $C_{4}\times C_{2}\times C_{2}$, which has order $16$ (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:   $\frac{42\cdots 97}{21\cdots 20}a^{23}+\frac{17\cdots 61}{26\cdots 40}a^{22}-\frac{16\cdots 03}{42\cdots 44}a^{21}-\frac{19\cdots 51}{11\cdots 40}a^{20}+\frac{16\cdots 05}{42\cdots 44}a^{19}+\frac{53\cdots 61}{10\cdots 60}a^{18}-\frac{41\cdots 71}{10\cdots 60}a^{17}-\frac{83\cdots 65}{21\cdots 72}a^{16}-\frac{30\cdots 83}{21\cdots 20}a^{15}+\frac{39\cdots 49}{21\cdots 72}a^{14}+\frac{26\cdots 17}{10\cdots 60}a^{13}+\frac{11\cdots 83}{10\cdots 60}a^{12}-\frac{25\cdots 67}{42\cdots 44}a^{11}-\frac{13\cdots 97}{44\cdots 40}a^{10}-\frac{18\cdots 83}{99\cdots 92}a^{9}-\frac{26\cdots 97}{53\cdots 80}a^{8}+\frac{29\cdots 33}{21\cdots 20}a^{7}+\frac{23\cdots 41}{21\cdots 72}a^{6}+\frac{66\cdots 37}{21\cdots 20}a^{5}+\frac{65\cdots 19}{21\cdots 72}a^{4}-\frac{26\cdots 61}{71\cdots 40}a^{3}-\frac{41\cdots 77}{35\cdots 20}a^{2}+\frac{19\cdots 71}{10\cdots 36}a+\frac{46\cdots 53}{89\cdots 80}$, $\frac{73\cdots 07}{14\cdots 80}a^{23}+\frac{99\cdots 29}{10\cdots 60}a^{22}+\frac{96\cdots 71}{85\cdots 88}a^{21}-\frac{13\cdots 23}{21\cdots 20}a^{20}-\frac{44\cdots 87}{31\cdots 44}a^{19}-\frac{16\cdots 83}{21\cdots 20}a^{18}+\frac{21\cdots 63}{21\cdots 20}a^{17}+\frac{46\cdots 33}{42\cdots 44}a^{16}-\frac{12\cdots 01}{42\cdots 40}a^{15}-\frac{36\cdots 75}{42\cdots 44}a^{14}-\frac{75\cdots 61}{21\cdots 20}a^{13}+\frac{42\cdots 21}{21\cdots 20}a^{12}+\frac{18\cdots 75}{85\cdots 88}a^{11}+\frac{32\cdots 47}{10\cdots 60}a^{10}-\frac{12\cdots 71}{39\cdots 68}a^{9}-\frac{20\cdots 81}{11\cdots 40}a^{8}-\frac{98\cdots 29}{42\cdots 40}a^{7}+\frac{93\cdots 89}{42\cdots 44}a^{6}+\frac{52\cdots 79}{42\cdots 40}a^{5}-\frac{51\cdots 77}{42\cdots 44}a^{4}-\frac{63\cdots 61}{42\cdots 40}a^{3}+\frac{73\cdots 79}{79\cdots 60}a^{2}+\frac{51\cdots 83}{71\cdots 24}a-\frac{45\cdots 97}{53\cdots 80}$, $\frac{67\cdots 61}{13\cdots 20}a^{23}-\frac{32\cdots 27}{13\cdots 20}a^{22}-\frac{18\cdots 69}{16\cdots 40}a^{21}+\frac{29\cdots 91}{11\cdots 60}a^{20}+\frac{18\cdots 33}{13\cdots 20}a^{19}+\frac{16\cdots 47}{26\cdots 04}a^{18}-\frac{29\cdots 77}{26\cdots 04}a^{17}-\frac{15\cdots 61}{13\cdots 20}a^{16}+\frac{32\cdots 19}{13\cdots 52}a^{15}+\frac{54\cdots 91}{67\cdots 60}a^{14}+\frac{69\cdots 51}{16\cdots 40}a^{13}-\frac{13\cdots 23}{67\cdots 60}a^{12}-\frac{91\cdots 33}{26\cdots 04}a^{11}-\frac{89\cdots 47}{49\cdots 60}a^{10}-\frac{57\cdots 07}{14\cdots 80}a^{9}+\frac{27\cdots 27}{13\cdots 20}a^{8}+\frac{12\cdots 33}{67\cdots 60}a^{7}+\frac{26\cdots 21}{67\cdots 60}a^{6}-\frac{32\cdots 61}{13\cdots 20}a^{5}-\frac{76\cdots 51}{13\cdots 20}a^{4}-\frac{24\cdots 61}{11\cdots 60}a^{3}+\frac{25\cdots 49}{62\cdots 20}a^{2}+\frac{84\cdots 13}{42\cdots 60}a-\frac{27\cdots 91}{93\cdots 80}$, $\frac{69\cdots 23}{44\cdots 40}a^{23}+\frac{10\cdots 03}{44\cdots 40}a^{22}+\frac{54\cdots 93}{22\cdots 20}a^{21}-\frac{53\cdots 33}{93\cdots 80}a^{20}-\frac{91\cdots 53}{14\cdots 80}a^{19}+\frac{35\cdots 01}{89\cdots 68}a^{18}+\frac{91\cdots 39}{14\cdots 80}a^{17}-\frac{11\cdots 89}{14\cdots 80}a^{16}-\frac{21\cdots 87}{11\cdots 60}a^{15}+\frac{31\cdots 23}{24\cdots 80}a^{14}+\frac{10\cdots 99}{11\cdots 60}a^{13}-\frac{46\cdots 19}{22\cdots 20}a^{12}+\frac{78\cdots 15}{89\cdots 68}a^{11}-\frac{12\cdots 93}{14\cdots 80}a^{10}-\frac{29\cdots 41}{44\cdots 40}a^{9}+\frac{11\cdots 09}{44\cdots 40}a^{8}+\frac{35\cdots 31}{18\cdots 60}a^{7}+\frac{10\cdots 93}{24\cdots 80}a^{6}+\frac{90\cdots 03}{89\cdots 68}a^{5}-\frac{41\cdots 57}{44\cdots 40}a^{4}+\frac{26\cdots 41}{74\cdots 40}a^{3}+\frac{20\cdots 39}{22\cdots 92}a^{2}-\frac{12\cdots 61}{56\cdots 80}a+\frac{20\cdots 57}{14\cdots 20}$, $\frac{11\cdots 57}{53\cdots 80}a^{23}-\frac{73\cdots 71}{53\cdots 80}a^{22}-\frac{25\cdots 49}{67\cdots 60}a^{21}+\frac{11\cdots 01}{44\cdots 40}a^{20}+\frac{22\cdots 89}{53\cdots 80}a^{19}+\frac{10\cdots 19}{53\cdots 80}a^{18}-\frac{38\cdots 09}{10\cdots 16}a^{17}-\frac{70\cdots 69}{53\cdots 80}a^{16}+\frac{49\cdots 63}{26\cdots 40}a^{15}+\frac{37\cdots 23}{26\cdots 40}a^{14}+\frac{80\cdots 47}{21\cdots 80}a^{13}-\frac{76\cdots 19}{26\cdots 40}a^{12}-\frac{12\cdots 17}{53\cdots 80}a^{11}-\frac{23\cdots 71}{35\cdots 72}a^{10}+\frac{15\cdots 49}{59\cdots 20}a^{9}+\frac{27\cdots 47}{10\cdots 16}a^{8}+\frac{96\cdots 89}{26\cdots 40}a^{7}-\frac{14\cdots 43}{53\cdots 08}a^{6}-\frac{14\cdots 57}{53\cdots 80}a^{5}-\frac{31\cdots 63}{53\cdots 80}a^{4}+\frac{11\cdots 99}{44\cdots 40}a^{3}+\frac{11\cdots 17}{56\cdots 80}a^{2}-\frac{71\cdots 01}{42\cdots 60}a+\frac{17\cdots 53}{11\cdots 60}$, $\frac{15\cdots 29}{17\cdots 60}a^{23}-\frac{85\cdots 67}{53\cdots 80}a^{22}+\frac{10\cdots 09}{67\cdots 60}a^{21}+\frac{12\cdots 51}{26\cdots 04}a^{20}-\frac{19\cdots 99}{11\cdots 24}a^{19}-\frac{10\cdots 37}{53\cdots 80}a^{18}+\frac{15\cdots 23}{53\cdots 80}a^{17}+\frac{80\cdots 87}{53\cdots 80}a^{16}+\frac{23\cdots 27}{53\cdots 08}a^{15}-\frac{39\cdots 41}{53\cdots 08}a^{14}-\frac{15\cdots 07}{16\cdots 40}a^{13}-\frac{10\cdots 47}{26\cdots 40}a^{12}+\frac{11\cdots 99}{53\cdots 80}a^{11}+\frac{11\cdots 31}{10\cdots 16}a^{10}+\frac{22\cdots 15}{35\cdots 72}a^{9}+\frac{47\cdots 07}{59\cdots 20}a^{8}-\frac{23\cdots 91}{26\cdots 40}a^{7}-\frac{89\cdots 79}{26\cdots 40}a^{6}-\frac{57\cdots 65}{10\cdots 16}a^{5}+\frac{21\cdots 33}{10\cdots 16}a^{4}+\frac{25\cdots 37}{13\cdots 20}a^{3}-\frac{38\cdots 53}{38\cdots 20}a^{2}-\frac{14\cdots 53}{93\cdots 80}a+\frac{12\cdots 35}{67\cdots 76}$, $\frac{24\cdots 71}{26\cdots 40}a^{23}-\frac{72\cdots 91}{89\cdots 80}a^{22}-\frac{10\cdots 27}{84\cdots 20}a^{21}+\frac{69\cdots 53}{67\cdots 60}a^{20}+\frac{78\cdots 15}{53\cdots 08}a^{19}+\frac{14\cdots 27}{89\cdots 80}a^{18}-\frac{40\cdots 63}{26\cdots 40}a^{17}-\frac{94\cdots 19}{26\cdots 40}a^{16}+\frac{26\cdots 67}{44\cdots 40}a^{15}+\frac{18\cdots 05}{29\cdots 56}a^{14}+\frac{26\cdots 59}{21\cdots 80}a^{13}-\frac{15\cdots 93}{13\cdots 20}a^{12}-\frac{27\cdots 07}{29\cdots 60}a^{11}-\frac{72\cdots 31}{26\cdots 40}a^{10}+\frac{16\cdots 93}{17\cdots 36}a^{9}+\frac{31\cdots 41}{26\cdots 40}a^{8}+\frac{24\cdots 77}{44\cdots 40}a^{7}-\frac{81\cdots 37}{13\cdots 20}a^{6}-\frac{93\cdots 03}{26\cdots 40}a^{5}-\frac{81\cdots 29}{53\cdots 08}a^{4}+\frac{49\cdots 49}{67\cdots 60}a^{3}+\frac{81\cdots 27}{46\cdots 40}a^{2}-\frac{72\cdots 57}{84\cdots 20}a+\frac{12\cdots 57}{16\cdots 40}$, $\frac{36\cdots 09}{53\cdots 08}a^{23}+\frac{39\cdots 11}{89\cdots 80}a^{22}+\frac{11\cdots 37}{13\cdots 20}a^{21}-\frac{33\cdots 39}{16\cdots 40}a^{20}-\frac{30\cdots 49}{26\cdots 40}a^{19}-\frac{62\cdots 11}{99\cdots 20}a^{18}+\frac{19\cdots 47}{26\cdots 40}a^{17}+\frac{18\cdots 31}{26\cdots 40}a^{16}-\frac{40\cdots 19}{37\cdots 20}a^{15}-\frac{24\cdots 41}{44\cdots 40}a^{14}-\frac{24\cdots 87}{67\cdots 60}a^{13}-\frac{28\cdots 89}{13\cdots 20}a^{12}+\frac{25\cdots 29}{29\cdots 60}a^{11}+\frac{15\cdots 71}{26\cdots 40}a^{10}+\frac{12\cdots 63}{89\cdots 80}a^{9}-\frac{21\cdots 13}{26\cdots 40}a^{8}-\frac{40\cdots 31}{74\cdots 40}a^{7}-\frac{77\cdots 09}{13\cdots 20}a^{6}+\frac{14\cdots 73}{53\cdots 08}a^{5}+\frac{67\cdots 73}{26\cdots 40}a^{4}+\frac{67\cdots 27}{13\cdots 20}a^{3}-\frac{13\cdots 49}{44\cdots 84}a^{2}-\frac{32\cdots 05}{67\cdots 76}a+\frac{14\cdots 67}{84\cdots 20}$, $\frac{30\cdots 83}{26\cdots 40}a^{23}+\frac{60\cdots 49}{53\cdots 08}a^{22}+\frac{21\cdots 39}{11\cdots 60}a^{21}-\frac{12\cdots 41}{67\cdots 60}a^{20}-\frac{55\cdots 71}{26\cdots 40}a^{19}+\frac{32\cdots 19}{26\cdots 40}a^{18}+\frac{74\cdots 19}{29\cdots 60}a^{17}+\frac{75\cdots 53}{89\cdots 80}a^{16}-\frac{39\cdots 21}{26\cdots 04}a^{15}-\frac{13\cdots 77}{13\cdots 20}a^{14}-\frac{38\cdots 71}{56\cdots 80}a^{13}+\frac{30\cdots 51}{49\cdots 60}a^{12}+\frac{82\cdots 99}{26\cdots 40}a^{11}+\frac{16\cdots 83}{26\cdots 40}a^{10}-\frac{12\cdots 31}{29\cdots 60}a^{9}-\frac{31\cdots 29}{53\cdots 08}a^{8}-\frac{39\cdots 43}{13\cdots 20}a^{7}+\frac{29\cdots 27}{49\cdots 60}a^{6}+\frac{47\cdots 03}{29\cdots 60}a^{5}+\frac{27\cdots 73}{29\cdots 60}a^{4}-\frac{16\cdots 03}{67\cdots 60}a^{3}-\frac{89\cdots 09}{56\cdots 80}a^{2}+\frac{15\cdots 11}{47\cdots 24}a-\frac{18\cdots 97}{16\cdots 40}$, $\frac{52\cdots 53}{53\cdots 80}a^{23}+\frac{19\cdots 59}{53\cdots 80}a^{22}+\frac{63\cdots 77}{33\cdots 80}a^{21}+\frac{70\cdots 71}{44\cdots 40}a^{20}-\frac{10\cdots 29}{53\cdots 80}a^{19}-\frac{18\cdots 95}{10\cdots 16}a^{18}+\frac{46\cdots 93}{53\cdots 80}a^{17}+\frac{92\cdots 49}{53\cdots 80}a^{16}+\frac{34\cdots 49}{26\cdots 40}a^{15}-\frac{27\cdots 83}{26\cdots 40}a^{14}-\frac{31\cdots 19}{33\cdots 80}a^{13}-\frac{50\cdots 13}{26\cdots 40}a^{12}+\frac{90\cdots 17}{53\cdots 80}a^{11}+\frac{24\cdots 71}{17\cdots 60}a^{10}+\frac{85\cdots 13}{17\cdots 60}a^{9}-\frac{53\cdots 31}{53\cdots 80}a^{8}-\frac{81\cdots 53}{53\cdots 08}a^{7}-\frac{12\cdots 01}{53\cdots 08}a^{6}+\frac{22\cdots 21}{53\cdots 80}a^{5}+\frac{28\cdots 43}{53\cdots 80}a^{4}+\frac{32\cdots 81}{14\cdots 80}a^{3}-\frac{66\cdots 83}{11\cdots 60}a^{2}-\frac{38\cdots 21}{16\cdots 40}a+\frac{11\cdots 63}{22\cdots 92}$, $\frac{96\cdots 01}{53\cdots 80}a^{23}+\frac{53\cdots 51}{17\cdots 60}a^{22}-\frac{44\cdots 61}{13\cdots 20}a^{21}-\frac{25\cdots 35}{26\cdots 04}a^{20}+\frac{18\cdots 97}{53\cdots 80}a^{19}+\frac{69\cdots 49}{17\cdots 60}a^{18}-\frac{38\cdots 09}{53\cdots 80}a^{17}-\frac{34\cdots 49}{10\cdots 16}a^{16}-\frac{88\cdots 37}{99\cdots 20}a^{15}+\frac{15\cdots 37}{99\cdots 20}a^{14}+\frac{13\cdots 17}{67\cdots 60}a^{13}+\frac{21\cdots 49}{26\cdots 40}a^{12}-\frac{18\cdots 31}{19\cdots 40}a^{11}-\frac{13\cdots 37}{53\cdots 80}a^{10}-\frac{82\cdots 83}{59\cdots 20}a^{9}-\frac{18\cdots 69}{10\cdots 16}a^{8}+\frac{34\cdots 37}{17\cdots 36}a^{7}+\frac{19\cdots 41}{26\cdots 40}a^{6}+\frac{63\cdots 03}{53\cdots 80}a^{5}-\frac{23\cdots 19}{53\cdots 80}a^{4}-\frac{28\cdots 07}{67\cdots 60}a^{3}+\frac{46\cdots 23}{22\cdots 20}a^{2}+\frac{11\cdots 83}{33\cdots 80}a-\frac{13\cdots 99}{33\cdots 80}$, $\frac{60\cdots 03}{53\cdots 80}a^{23}-\frac{61\cdots 51}{17\cdots 60}a^{22}-\frac{58\cdots 71}{84\cdots 20}a^{21}+\frac{92\cdots 21}{13\cdots 20}a^{20}-\frac{13\cdots 57}{53\cdots 80}a^{19}-\frac{10\cdots 01}{17\cdots 60}a^{18}+\frac{85\cdots 29}{53\cdots 80}a^{17}+\frac{22\cdots 97}{53\cdots 80}a^{16}+\frac{13\cdots 03}{99\cdots 20}a^{15}-\frac{23\cdots 57}{99\cdots 20}a^{14}-\frac{31\cdots 43}{33\cdots 80}a^{13}+\frac{18\cdots 51}{26\cdots 40}a^{12}+\frac{26\cdots 27}{17\cdots 60}a^{11}+\frac{25\cdots 13}{10\cdots 16}a^{10}+\frac{11\cdots 23}{59\cdots 20}a^{9}-\frac{30\cdots 71}{53\cdots 80}a^{8}-\frac{41\cdots 43}{17\cdots 36}a^{7}-\frac{34\cdots 93}{53\cdots 08}a^{6}+\frac{40\cdots 41}{53\cdots 80}a^{5}+\frac{76\cdots 19}{53\cdots 80}a^{4}-\frac{89\cdots 67}{26\cdots 04}a^{3}-\frac{47\cdots 73}{37\cdots 20}a^{2}+\frac{78\cdots 57}{16\cdots 40}a-\frac{14\cdots 23}{37\cdots 20}$, $\frac{49\cdots 97}{59\cdots 12}a^{23}+\frac{29\cdots 83}{26\cdots 40}a^{22}+\frac{99\cdots 57}{67\cdots 76}a^{21}-\frac{14\cdots 11}{67\cdots 60}a^{20}-\frac{15\cdots 59}{89\cdots 80}a^{19}+\frac{30\cdots 89}{26\cdots 40}a^{18}+\frac{13\cdots 85}{53\cdots 08}a^{17}-\frac{38\cdots 79}{26\cdots 40}a^{16}-\frac{26\cdots 23}{13\cdots 20}a^{15}-\frac{68\cdots 79}{13\cdots 20}a^{14}+\frac{11\cdots 81}{16\cdots 40}a^{13}+\frac{86\cdots 67}{13\cdots 20}a^{12}-\frac{55\cdots 71}{26\cdots 40}a^{11}-\frac{55\cdots 31}{26\cdots 40}a^{10}-\frac{41\cdots 11}{89\cdots 80}a^{9}+\frac{11\cdots 87}{89\cdots 80}a^{8}+\frac{16\cdots 43}{13\cdots 20}a^{7}+\frac{10\cdots 87}{13\cdots 20}a^{6}-\frac{75\cdots 07}{26\cdots 40}a^{5}-\frac{43\cdots 01}{26\cdots 40}a^{4}+\frac{32\cdots 83}{67\cdots 60}a^{3}+\frac{30\cdots 61}{31\cdots 60}a^{2}-\frac{24\cdots 69}{58\cdots 30}a+\frac{11\cdots 57}{33\cdots 88}$ 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:  \( 27022688166806175000000 \) (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:  \( 3 \) (assuming GRH)

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 27022688166806175000000 \cdot 8}{2\cdot\sqrt{35863302598527284385359953566442393477815766818821430206298828125}}\cr\approx \mathstrut & 0.875753873514481 \end{aligned}\] (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 - 191*x^22 + 152*x^21 + 20801*x^20 + 145552*x^19 - 1176554*x^18 - 16138302*x^17 + 15768809*x^16 + 1059872352*x^15 + 7747419046*x^14 + 3640653858*x^13 - 202726936711*x^12 - 1134655611298*x^11 - 2506437023344*x^10 + 18528375557180*x^9 + 138487686799265*x^8 - 8715047372200*x^7 - 866046602082015*x^6 - 4747675198206540*x^5 - 13439138362502051*x^4 + 97786901456032952*x^3 + 139064771037891016*x^2 - 927383945402843232*x + 858450394618182224) 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 - 191*x^22 + 152*x^21 + 20801*x^20 + 145552*x^19 - 1176554*x^18 - 16138302*x^17 + 15768809*x^16 + 1059872352*x^15 + 7747419046*x^14 + 3640653858*x^13 - 202726936711*x^12 - 1134655611298*x^11 - 2506437023344*x^10 + 18528375557180*x^9 + 138487686799265*x^8 - 8715047372200*x^7 - 866046602082015*x^6 - 4747675198206540*x^5 - 13439138362502051*x^4 + 97786901456032952*x^3 + 139064771037891016*x^2 - 927383945402843232*x + 858450394618182224, 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 - 191*x^22 + 152*x^21 + 20801*x^20 + 145552*x^19 - 1176554*x^18 - 16138302*x^17 + 15768809*x^16 + 1059872352*x^15 + 7747419046*x^14 + 3640653858*x^13 - 202726936711*x^12 - 1134655611298*x^11 - 2506437023344*x^10 + 18528375557180*x^9 + 138487686799265*x^8 - 8715047372200*x^7 - 866046602082015*x^6 - 4747675198206540*x^5 - 13439138362502051*x^4 + 97786901456032952*x^3 + 139064771037891016*x^2 - 927383945402843232*x + 858450394618182224); 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 - 191*x^22 + 152*x^21 + 20801*x^20 + 145552*x^19 - 1176554*x^18 - 16138302*x^17 + 15768809*x^16 + 1059872352*x^15 + 7747419046*x^14 + 3640653858*x^13 - 202726936711*x^12 - 1134655611298*x^11 - 2506437023344*x^10 + 18528375557180*x^9 + 138487686799265*x^8 - 8715047372200*x^7 - 866046602082015*x^6 - 4747675198206540*x^5 - 13439138362502051*x^4 + 97786901456032952*x^3 + 139064771037891016*x^2 - 927383945402843232*x + 858450394618182224); 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

6.2.4901737578125.2, 12.4.951590574034612536651611328125.4

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: 24.4.896582564963182109633998839161059836945394170470535755157470703125.6, 24.4.896582564963182109633998839161059836945394170470535755157470703125.5
Arithmetically equivalent sibling: 24.4.35863302598527284385359953566442393477815766818821430206298828125.1
Minimal sibling: 24.4.35863302598527284385359953566442393477815766818821430206298828125.1

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type ${\href{/padicField/2.4.0.1}{4} }^{5}{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ ${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ R $24$ ${\href{/padicField/11.10.0.1}{10} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ $24$ ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ ${\href{/padicField/23.4.0.1}{4} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ ${\href{/padicField/31.12.0.1}{12} }^{2}$ ${\href{/padicField/37.8.0.1}{8} }^{3}$ $20{,}\,{\href{/padicField/41.4.0.1}{4} }$ $24$ $24$ ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ ${\href{/padicField/59.3.0.1}{3} }^{8}$

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 5.1.4.3a1.4$x^{4} + 20$$4$$1$$3$$C_4$$$[\ ]_{4}$$
5.2.5.14a4.1$x^{10} + 20 x^{9} + 170 x^{8} + 800 x^{7} + 2290 x^{6} + 4184 x^{5} + 5100 x^{4} + 4320 x^{3} + 2440 x^{2} + 800 x + 117$$5$$2$$14$$F_{5}\times C_2$$$[\frac{7}{4}]_{4}^{2}$$
5.2.5.14a4.1$x^{10} + 20 x^{9} + 170 x^{8} + 800 x^{7} + 2290 x^{6} + 4184 x^{5} + 5100 x^{4} + 4320 x^{3} + 2440 x^{2} + 800 x + 117$$5$$2$$14$$F_{5}\times C_2$$$[\frac{7}{4}]_{4}^{2}$$
\(89\) 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)