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Properties

Label 24.4.153...449.1
Degree $24$
Signature $[4, 10]$
Discriminant $1.538\times 10^{33}$
Root discriminant \(24.14\)
Ramified prime $2083$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $\SL(2,5):C_2$ (as 24T576)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 - 8*x^22 - 2*x^21 + 70*x^20 - 33*x^19 - 156*x^18 + 43*x^17 + 433*x^16 - 998*x^15 + 433*x^14 + 1850*x^13 - 1996*x^12 - 1334*x^11 + 1462*x^10 + 5*x^9 - 1195*x^8 - 1071*x^7 - 747*x^6 - 160*x^5 + 26*x^4 - 66*x^3 - 57*x^2 - 14*x - 1)
 
gp: K = bnfinit(y^24 - 2*y^23 - 8*y^22 - 2*y^21 + 70*y^20 - 33*y^19 - 156*y^18 + 43*y^17 + 433*y^16 - 998*y^15 + 433*y^14 + 1850*y^13 - 1996*y^12 - 1334*y^11 + 1462*y^10 + 5*y^9 - 1195*y^8 - 1071*y^7 - 747*y^6 - 160*y^5 + 26*y^4 - 66*y^3 - 57*y^2 - 14*y - 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 2*x^23 - 8*x^22 - 2*x^21 + 70*x^20 - 33*x^19 - 156*x^18 + 43*x^17 + 433*x^16 - 998*x^15 + 433*x^14 + 1850*x^13 - 1996*x^12 - 1334*x^11 + 1462*x^10 + 5*x^9 - 1195*x^8 - 1071*x^7 - 747*x^6 - 160*x^5 + 26*x^4 - 66*x^3 - 57*x^2 - 14*x - 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 2*x^23 - 8*x^22 - 2*x^21 + 70*x^20 - 33*x^19 - 156*x^18 + 43*x^17 + 433*x^16 - 998*x^15 + 433*x^14 + 1850*x^13 - 1996*x^12 - 1334*x^11 + 1462*x^10 + 5*x^9 - 1195*x^8 - 1071*x^7 - 747*x^6 - 160*x^5 + 26*x^4 - 66*x^3 - 57*x^2 - 14*x - 1)
 

\( x^{24} - 2 x^{23} - 8 x^{22} - 2 x^{21} + 70 x^{20} - 33 x^{19} - 156 x^{18} + 43 x^{17} + 433 x^{16} + \cdots - 1 \) Copy content Toggle raw display

sage: K.defining_polynomial()
 
gp: K.pol
 
magma: DefiningPolynomial(K);
 
oscar: defining_polynomial(K)
 

Invariants

Degree:  $24$
sage: K.degree()
 
gp: poldegree(K.pol)
 
magma: Degree(K);
 
oscar: degree(K)
 
Signature:  $[4, 10]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(1537774657557415544813485393913449\) \(\medspace = 2083^{10}\) Copy content Toggle raw display
sage: K.disc()
 
gp: K.disc
 
magma: OK := Integers(K); Discriminant(OK);
 
oscar: OK = ring_of_integers(K); discriminant(OK)
 
Root discriminant:  \(24.14\)
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
magma: Abs(Discriminant(OK))^(1/Degree(K));
 
oscar: (1.0 * dK)^(1/degree(K))
 
Galois root discriminant:  $2083^{1/2}\approx 45.639894828976104$
Ramified primes:   \(2083\) Copy content Toggle raw display
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
magma: PrimeDivisors(Discriminant(OK));
 
oscar: prime_divisors(discriminant((OK)))
 
Discriminant root field:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $4$
sage: K.automorphisms()
 
magma: Automorphisms(K);
 
oscar: automorphisms(K)
 
This field is not Galois over $\Q$.
This is not a CM field.

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

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{84\!\cdots\!71}a^{23}+\frac{88\!\cdots\!18}{84\!\cdots\!71}a^{22}-\frac{18\!\cdots\!92}{84\!\cdots\!71}a^{21}-\frac{36\!\cdots\!69}{84\!\cdots\!71}a^{20}-\frac{11\!\cdots\!87}{84\!\cdots\!71}a^{19}+\frac{25\!\cdots\!61}{84\!\cdots\!71}a^{18}+\frac{34\!\cdots\!16}{84\!\cdots\!71}a^{17}+\frac{14\!\cdots\!41}{84\!\cdots\!71}a^{16}-\frac{37\!\cdots\!23}{84\!\cdots\!71}a^{15}+\frac{18\!\cdots\!34}{84\!\cdots\!71}a^{14}+\frac{15\!\cdots\!83}{84\!\cdots\!71}a^{13}+\frac{20\!\cdots\!62}{19\!\cdots\!97}a^{12}+\frac{20\!\cdots\!63}{84\!\cdots\!71}a^{11}-\frac{12\!\cdots\!10}{84\!\cdots\!71}a^{10}+\frac{31\!\cdots\!15}{84\!\cdots\!71}a^{9}+\frac{20\!\cdots\!90}{84\!\cdots\!71}a^{8}-\frac{25\!\cdots\!27}{84\!\cdots\!71}a^{7}+\frac{36\!\cdots\!66}{84\!\cdots\!71}a^{6}+\frac{12\!\cdots\!25}{84\!\cdots\!71}a^{5}-\frac{38\!\cdots\!27}{84\!\cdots\!71}a^{4}+\frac{38\!\cdots\!71}{84\!\cdots\!71}a^{3}-\frac{37\!\cdots\!61}{84\!\cdots\!71}a^{2}-\frac{32\!\cdots\!84}{84\!\cdots\!71}a-\frac{33\!\cdots\!48}{84\!\cdots\!71}$ Copy content Toggle raw display

sage: K.integral_basis()
 
gp: K.zk
 
magma: IntegralBasis(K);
 
oscar: basis(OK)
 

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

Class group and class number

Trivial group, which has order $1$ (assuming GRH)

sage: K.class_group().invariants()
 
gp: K.clgp
 
magma: ClassGroup(K);
 
oscar: class_group(K)
 

Unit group

sage: UK = K.unit_group()
 
magma: UK, fUK := UnitGroup(K);
 
oscar: UK, fUK = unit_group(OK)
 
Rank:  $13$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -1 \)  (order $2$) Copy content Toggle raw display
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
oscar: torsion_units_generator(OK)
 
Fundamental units:   $\frac{85\!\cdots\!52}{84\!\cdots\!71}a^{23}-\frac{22\!\cdots\!46}{84\!\cdots\!71}a^{22}-\frac{55\!\cdots\!75}{84\!\cdots\!71}a^{21}+\frac{18\!\cdots\!73}{84\!\cdots\!71}a^{20}+\frac{59\!\cdots\!53}{84\!\cdots\!71}a^{19}-\frac{65\!\cdots\!99}{84\!\cdots\!71}a^{18}-\frac{98\!\cdots\!43}{84\!\cdots\!71}a^{17}+\frac{10\!\cdots\!36}{84\!\cdots\!71}a^{16}+\frac{31\!\cdots\!66}{84\!\cdots\!71}a^{15}-\frac{10\!\cdots\!53}{84\!\cdots\!71}a^{14}+\frac{99\!\cdots\!55}{84\!\cdots\!71}a^{13}+\frac{24\!\cdots\!46}{19\!\cdots\!97}a^{12}-\frac{24\!\cdots\!13}{84\!\cdots\!71}a^{11}+\frac{26\!\cdots\!18}{84\!\cdots\!71}a^{10}+\frac{12\!\cdots\!23}{84\!\cdots\!71}a^{9}-\frac{79\!\cdots\!30}{84\!\cdots\!71}a^{8}-\frac{64\!\cdots\!30}{84\!\cdots\!71}a^{7}-\frac{46\!\cdots\!31}{84\!\cdots\!71}a^{6}-\frac{29\!\cdots\!08}{84\!\cdots\!71}a^{5}+\frac{86\!\cdots\!76}{84\!\cdots\!71}a^{4}-\frac{58\!\cdots\!67}{84\!\cdots\!71}a^{3}-\frac{61\!\cdots\!23}{84\!\cdots\!71}a^{2}-\frac{12\!\cdots\!00}{84\!\cdots\!71}a+\frac{86\!\cdots\!61}{84\!\cdots\!71}$, $\frac{11\!\cdots\!70}{84\!\cdots\!71}a^{23}-\frac{17\!\cdots\!94}{84\!\cdots\!71}a^{22}-\frac{10\!\cdots\!69}{84\!\cdots\!71}a^{21}-\frac{60\!\cdots\!45}{84\!\cdots\!71}a^{20}+\frac{84\!\cdots\!79}{84\!\cdots\!71}a^{19}+\frac{21\!\cdots\!58}{84\!\cdots\!71}a^{18}-\frac{23\!\cdots\!61}{84\!\cdots\!71}a^{17}-\frac{19\!\cdots\!40}{84\!\cdots\!71}a^{16}+\frac{59\!\cdots\!72}{84\!\cdots\!71}a^{15}-\frac{95\!\cdots\!84}{84\!\cdots\!71}a^{14}-\frac{25\!\cdots\!94}{84\!\cdots\!71}a^{13}+\frac{67\!\cdots\!51}{19\!\cdots\!97}a^{12}-\frac{15\!\cdots\!47}{84\!\cdots\!71}a^{11}-\frac{35\!\cdots\!24}{84\!\cdots\!71}a^{10}+\frac{20\!\cdots\!74}{84\!\cdots\!71}a^{9}+\frac{11\!\cdots\!57}{84\!\cdots\!71}a^{8}-\frac{22\!\cdots\!58}{84\!\cdots\!71}a^{7}-\frac{16\!\cdots\!70}{84\!\cdots\!71}a^{6}-\frac{10\!\cdots\!12}{84\!\cdots\!71}a^{5}-\frac{31\!\cdots\!81}{84\!\cdots\!71}a^{4}+\frac{20\!\cdots\!03}{84\!\cdots\!71}a^{3}-\frac{93\!\cdots\!90}{84\!\cdots\!71}a^{2}-\frac{12\!\cdots\!87}{84\!\cdots\!71}a-\frac{20\!\cdots\!29}{84\!\cdots\!71}$, $\frac{43\!\cdots\!14}{84\!\cdots\!71}a^{23}-\frac{11\!\cdots\!55}{84\!\cdots\!71}a^{22}-\frac{26\!\cdots\!90}{84\!\cdots\!71}a^{21}+\frac{12\!\cdots\!18}{84\!\cdots\!71}a^{20}+\frac{29\!\cdots\!04}{84\!\cdots\!71}a^{19}-\frac{36\!\cdots\!41}{84\!\cdots\!71}a^{18}-\frac{45\!\cdots\!54}{84\!\cdots\!71}a^{17}+\frac{58\!\cdots\!24}{84\!\cdots\!71}a^{16}+\frac{15\!\cdots\!63}{84\!\cdots\!71}a^{15}-\frac{55\!\cdots\!88}{84\!\cdots\!71}a^{14}+\frac{57\!\cdots\!10}{84\!\cdots\!71}a^{13}+\frac{10\!\cdots\!57}{19\!\cdots\!97}a^{12}-\frac{12\!\cdots\!50}{84\!\cdots\!71}a^{11}+\frac{28\!\cdots\!61}{84\!\cdots\!71}a^{10}+\frac{63\!\cdots\!91}{84\!\cdots\!71}a^{9}-\frac{47\!\cdots\!62}{84\!\cdots\!71}a^{8}-\frac{28\!\cdots\!60}{84\!\cdots\!71}a^{7}-\frac{19\!\cdots\!59}{84\!\cdots\!71}a^{6}-\frac{11\!\cdots\!97}{84\!\cdots\!71}a^{5}+\frac{59\!\cdots\!49}{84\!\cdots\!71}a^{4}-\frac{88\!\cdots\!08}{84\!\cdots\!71}a^{3}-\frac{29\!\cdots\!85}{84\!\cdots\!71}a^{2}-\frac{30\!\cdots\!14}{84\!\cdots\!71}a+\frac{19\!\cdots\!90}{84\!\cdots\!71}$, $\frac{18\!\cdots\!07}{84\!\cdots\!71}a^{23}-\frac{70\!\cdots\!08}{84\!\cdots\!71}a^{22}-\frac{56\!\cdots\!90}{84\!\cdots\!71}a^{21}+\frac{18\!\cdots\!75}{84\!\cdots\!71}a^{20}+\frac{12\!\cdots\!07}{84\!\cdots\!71}a^{19}-\frac{29\!\cdots\!68}{84\!\cdots\!71}a^{18}-\frac{23\!\cdots\!43}{84\!\cdots\!71}a^{17}+\frac{45\!\cdots\!97}{84\!\cdots\!71}a^{16}+\frac{39\!\cdots\!77}{84\!\cdots\!71}a^{15}-\frac{30\!\cdots\!79}{84\!\cdots\!71}a^{14}+\frac{50\!\cdots\!57}{84\!\cdots\!71}a^{13}-\frac{16\!\cdots\!88}{19\!\cdots\!97}a^{12}-\frac{75\!\cdots\!36}{84\!\cdots\!71}a^{11}+\frac{70\!\cdots\!99}{84\!\cdots\!71}a^{10}+\frac{13\!\cdots\!00}{84\!\cdots\!71}a^{9}-\frac{45\!\cdots\!67}{84\!\cdots\!71}a^{8}+\frac{79\!\cdots\!98}{84\!\cdots\!71}a^{7}+\frac{43\!\cdots\!25}{84\!\cdots\!71}a^{6}+\frac{60\!\cdots\!79}{84\!\cdots\!71}a^{5}+\frac{96\!\cdots\!20}{84\!\cdots\!71}a^{4}-\frac{20\!\cdots\!51}{84\!\cdots\!71}a^{3}-\frac{67\!\cdots\!90}{84\!\cdots\!71}a^{2}+\frac{12\!\cdots\!99}{84\!\cdots\!71}a+\frac{29\!\cdots\!53}{84\!\cdots\!71}$, $\frac{14\!\cdots\!41}{84\!\cdots\!71}a^{23}-\frac{36\!\cdots\!66}{84\!\cdots\!71}a^{22}-\frac{10\!\cdots\!18}{84\!\cdots\!71}a^{21}+\frac{18\!\cdots\!04}{84\!\cdots\!71}a^{20}+\frac{10\!\cdots\!00}{84\!\cdots\!71}a^{19}-\frac{96\!\cdots\!11}{84\!\cdots\!71}a^{18}-\frac{18\!\cdots\!46}{84\!\cdots\!71}a^{17}+\frac{15\!\cdots\!31}{84\!\cdots\!71}a^{16}+\frac{57\!\cdots\!48}{84\!\cdots\!71}a^{15}-\frac{17\!\cdots\!07}{84\!\cdots\!71}a^{14}+\frac{14\!\cdots\!27}{84\!\cdots\!71}a^{13}+\frac{48\!\cdots\!61}{19\!\cdots\!97}a^{12}-\frac{39\!\cdots\!23}{84\!\cdots\!71}a^{11}-\frac{19\!\cdots\!56}{84\!\cdots\!71}a^{10}+\frac{23\!\cdots\!65}{84\!\cdots\!71}a^{9}-\frac{10\!\cdots\!51}{84\!\cdots\!71}a^{8}-\frac{13\!\cdots\!12}{84\!\cdots\!71}a^{7}-\frac{94\!\cdots\!04}{84\!\cdots\!71}a^{6}-\frac{62\!\cdots\!07}{84\!\cdots\!71}a^{5}+\frac{80\!\cdots\!06}{84\!\cdots\!71}a^{4}+\frac{18\!\cdots\!31}{84\!\cdots\!71}a^{3}-\frac{10\!\cdots\!72}{84\!\cdots\!71}a^{2}-\frac{35\!\cdots\!43}{84\!\cdots\!71}a-\frac{17\!\cdots\!81}{84\!\cdots\!71}$, $\frac{24\!\cdots\!47}{84\!\cdots\!71}a^{23}-\frac{55\!\cdots\!97}{84\!\cdots\!71}a^{22}-\frac{18\!\cdots\!02}{84\!\cdots\!71}a^{21}-\frac{19\!\cdots\!18}{84\!\cdots\!71}a^{20}+\frac{17\!\cdots\!00}{84\!\cdots\!71}a^{19}-\frac{12\!\cdots\!82}{84\!\cdots\!71}a^{18}-\frac{35\!\cdots\!87}{84\!\cdots\!71}a^{17}+\frac{20\!\cdots\!13}{84\!\cdots\!71}a^{16}+\frac{10\!\cdots\!69}{84\!\cdots\!71}a^{15}-\frac{27\!\cdots\!45}{84\!\cdots\!71}a^{14}+\frac{17\!\cdots\!35}{84\!\cdots\!71}a^{13}+\frac{97\!\cdots\!96}{19\!\cdots\!97}a^{12}-\frac{60\!\cdots\!13}{84\!\cdots\!71}a^{11}-\frac{18\!\cdots\!94}{84\!\cdots\!71}a^{10}+\frac{41\!\cdots\!46}{84\!\cdots\!71}a^{9}-\frac{10\!\cdots\!74}{84\!\cdots\!71}a^{8}-\frac{27\!\cdots\!26}{84\!\cdots\!71}a^{7}-\frac{19\!\cdots\!78}{84\!\cdots\!71}a^{6}-\frac{13\!\cdots\!68}{84\!\cdots\!71}a^{5}-\frac{41\!\cdots\!26}{84\!\cdots\!71}a^{4}+\frac{79\!\cdots\!40}{84\!\cdots\!71}a^{3}-\frac{19\!\cdots\!56}{84\!\cdots\!71}a^{2}-\frac{92\!\cdots\!97}{84\!\cdots\!71}a-\frac{10\!\cdots\!99}{84\!\cdots\!71}$, $\frac{32\!\cdots\!27}{84\!\cdots\!71}a^{23}-\frac{80\!\cdots\!22}{84\!\cdots\!71}a^{22}-\frac{21\!\cdots\!46}{84\!\cdots\!71}a^{21}+\frac{47\!\cdots\!76}{84\!\cdots\!71}a^{20}+\frac{22\!\cdots\!79}{84\!\cdots\!71}a^{19}-\frac{21\!\cdots\!30}{84\!\cdots\!71}a^{18}-\frac{39\!\cdots\!22}{84\!\cdots\!71}a^{17}+\frac{34\!\cdots\!99}{84\!\cdots\!71}a^{16}+\frac{12\!\cdots\!16}{84\!\cdots\!71}a^{15}-\frac{38\!\cdots\!48}{84\!\cdots\!71}a^{14}+\frac{33\!\cdots\!49}{84\!\cdots\!71}a^{13}+\frac{10\!\cdots\!42}{19\!\cdots\!97}a^{12}-\frac{86\!\cdots\!37}{84\!\cdots\!71}a^{11}+\frac{25\!\cdots\!77}{84\!\cdots\!71}a^{10}+\frac{47\!\cdots\!32}{84\!\cdots\!71}a^{9}-\frac{23\!\cdots\!70}{84\!\cdots\!71}a^{8}-\frac{27\!\cdots\!35}{84\!\cdots\!71}a^{7}-\frac{20\!\cdots\!61}{84\!\cdots\!71}a^{6}-\frac{12\!\cdots\!89}{84\!\cdots\!71}a^{5}+\frac{20\!\cdots\!72}{84\!\cdots\!71}a^{4}+\frac{74\!\cdots\!61}{84\!\cdots\!71}a^{3}-\frac{21\!\cdots\!84}{84\!\cdots\!71}a^{2}-\frac{80\!\cdots\!86}{84\!\cdots\!71}a-\frac{12\!\cdots\!85}{84\!\cdots\!71}$, $\frac{14\!\cdots\!58}{84\!\cdots\!71}a^{23}-\frac{36\!\cdots\!41}{84\!\cdots\!71}a^{22}-\frac{10\!\cdots\!35}{84\!\cdots\!71}a^{21}+\frac{19\!\cdots\!50}{84\!\cdots\!71}a^{20}+\frac{10\!\cdots\!84}{84\!\cdots\!71}a^{19}-\frac{97\!\cdots\!75}{84\!\cdots\!71}a^{18}-\frac{18\!\cdots\!52}{84\!\cdots\!71}a^{17}+\frac{15\!\cdots\!39}{84\!\cdots\!71}a^{16}+\frac{56\!\cdots\!35}{84\!\cdots\!71}a^{15}-\frac{17\!\cdots\!37}{84\!\cdots\!71}a^{14}+\frac{14\!\cdots\!21}{84\!\cdots\!71}a^{13}+\frac{47\!\cdots\!30}{19\!\cdots\!97}a^{12}-\frac{39\!\cdots\!40}{84\!\cdots\!71}a^{11}-\frac{82\!\cdots\!33}{84\!\cdots\!71}a^{10}+\frac{22\!\cdots\!53}{84\!\cdots\!71}a^{9}-\frac{11\!\cdots\!26}{84\!\cdots\!71}a^{8}-\frac{12\!\cdots\!56}{84\!\cdots\!71}a^{7}-\frac{94\!\cdots\!53}{84\!\cdots\!71}a^{6}-\frac{64\!\cdots\!42}{84\!\cdots\!71}a^{5}+\frac{81\!\cdots\!92}{84\!\cdots\!71}a^{4}+\frac{22\!\cdots\!23}{84\!\cdots\!71}a^{3}-\frac{10\!\cdots\!60}{84\!\cdots\!71}a^{2}-\frac{33\!\cdots\!71}{84\!\cdots\!71}a-\frac{31\!\cdots\!86}{84\!\cdots\!71}$, $\frac{32\!\cdots\!27}{84\!\cdots\!71}a^{23}-\frac{80\!\cdots\!22}{84\!\cdots\!71}a^{22}-\frac{21\!\cdots\!46}{84\!\cdots\!71}a^{21}+\frac{47\!\cdots\!76}{84\!\cdots\!71}a^{20}+\frac{22\!\cdots\!79}{84\!\cdots\!71}a^{19}-\frac{21\!\cdots\!30}{84\!\cdots\!71}a^{18}-\frac{39\!\cdots\!22}{84\!\cdots\!71}a^{17}+\frac{34\!\cdots\!99}{84\!\cdots\!71}a^{16}+\frac{12\!\cdots\!16}{84\!\cdots\!71}a^{15}-\frac{38\!\cdots\!48}{84\!\cdots\!71}a^{14}+\frac{33\!\cdots\!49}{84\!\cdots\!71}a^{13}+\frac{10\!\cdots\!42}{19\!\cdots\!97}a^{12}-\frac{86\!\cdots\!37}{84\!\cdots\!71}a^{11}+\frac{25\!\cdots\!77}{84\!\cdots\!71}a^{10}+\frac{47\!\cdots\!32}{84\!\cdots\!71}a^{9}-\frac{23\!\cdots\!70}{84\!\cdots\!71}a^{8}-\frac{27\!\cdots\!35}{84\!\cdots\!71}a^{7}-\frac{20\!\cdots\!61}{84\!\cdots\!71}a^{6}-\frac{12\!\cdots\!89}{84\!\cdots\!71}a^{5}+\frac{20\!\cdots\!72}{84\!\cdots\!71}a^{4}+\frac{74\!\cdots\!61}{84\!\cdots\!71}a^{3}-\frac{21\!\cdots\!84}{84\!\cdots\!71}a^{2}-\frac{80\!\cdots\!86}{84\!\cdots\!71}a-\frac{43\!\cdots\!14}{84\!\cdots\!71}$, $\frac{46\!\cdots\!19}{84\!\cdots\!71}a^{23}-\frac{10\!\cdots\!01}{84\!\cdots\!71}a^{22}-\frac{35\!\cdots\!29}{84\!\cdots\!71}a^{21}-\frac{23\!\cdots\!87}{84\!\cdots\!71}a^{20}+\frac{32\!\cdots\!88}{84\!\cdots\!71}a^{19}-\frac{21\!\cdots\!56}{84\!\cdots\!71}a^{18}-\frac{69\!\cdots\!93}{84\!\cdots\!71}a^{17}+\frac{33\!\cdots\!91}{84\!\cdots\!71}a^{16}+\frac{19\!\cdots\!64}{84\!\cdots\!71}a^{15}-\frac{49\!\cdots\!25}{84\!\cdots\!71}a^{14}+\frac{28\!\cdots\!51}{84\!\cdots\!71}a^{13}+\frac{19\!\cdots\!67}{19\!\cdots\!97}a^{12}-\frac{10\!\cdots\!67}{84\!\cdots\!71}a^{11}-\frac{42\!\cdots\!88}{84\!\cdots\!71}a^{10}+\frac{80\!\cdots\!68}{84\!\cdots\!71}a^{9}-\frac{16\!\cdots\!94}{84\!\cdots\!71}a^{8}-\frac{53\!\cdots\!02}{84\!\cdots\!71}a^{7}-\frac{37\!\cdots\!07}{84\!\cdots\!71}a^{6}-\frac{27\!\cdots\!23}{84\!\cdots\!71}a^{5}-\frac{12\!\cdots\!42}{84\!\cdots\!71}a^{4}+\frac{24\!\cdots\!35}{84\!\cdots\!71}a^{3}-\frac{31\!\cdots\!29}{84\!\cdots\!71}a^{2}-\frac{15\!\cdots\!10}{84\!\cdots\!71}a-\frac{17\!\cdots\!21}{84\!\cdots\!71}$, $\frac{32\!\cdots\!47}{84\!\cdots\!71}a^{23}-\frac{62\!\cdots\!87}{84\!\cdots\!71}a^{22}-\frac{27\!\cdots\!38}{84\!\cdots\!71}a^{21}-\frac{79\!\cdots\!27}{84\!\cdots\!71}a^{20}+\frac{23\!\cdots\!11}{84\!\cdots\!71}a^{19}-\frac{88\!\cdots\!82}{84\!\cdots\!71}a^{18}-\frac{54\!\cdots\!86}{84\!\cdots\!71}a^{17}+\frac{12\!\cdots\!77}{84\!\cdots\!71}a^{16}+\frac{14\!\cdots\!93}{84\!\cdots\!71}a^{15}-\frac{31\!\cdots\!25}{84\!\cdots\!71}a^{14}+\frac{98\!\cdots\!68}{84\!\cdots\!71}a^{13}+\frac{15\!\cdots\!60}{19\!\cdots\!97}a^{12}-\frac{64\!\cdots\!12}{84\!\cdots\!71}a^{11}-\frac{54\!\cdots\!65}{84\!\cdots\!71}a^{10}+\frac{54\!\cdots\!93}{84\!\cdots\!71}a^{9}+\frac{38\!\cdots\!48}{84\!\cdots\!71}a^{8}-\frac{44\!\cdots\!05}{84\!\cdots\!71}a^{7}-\frac{35\!\cdots\!04}{84\!\cdots\!71}a^{6}-\frac{25\!\cdots\!62}{84\!\cdots\!71}a^{5}-\frac{50\!\cdots\!80}{84\!\cdots\!71}a^{4}+\frac{20\!\cdots\!32}{84\!\cdots\!71}a^{3}-\frac{23\!\cdots\!00}{84\!\cdots\!71}a^{2}-\frac{19\!\cdots\!64}{84\!\cdots\!71}a-\frac{34\!\cdots\!93}{84\!\cdots\!71}$, $\frac{24\!\cdots\!76}{84\!\cdots\!71}a^{23}-\frac{39\!\cdots\!93}{84\!\cdots\!71}a^{22}-\frac{21\!\cdots\!80}{84\!\cdots\!71}a^{21}-\frac{11\!\cdots\!88}{84\!\cdots\!71}a^{20}+\frac{17\!\cdots\!23}{84\!\cdots\!71}a^{19}-\frac{13\!\cdots\!41}{84\!\cdots\!71}a^{18}-\frac{43\!\cdots\!36}{84\!\cdots\!71}a^{17}-\frac{29\!\cdots\!35}{84\!\cdots\!71}a^{16}+\frac{11\!\cdots\!23}{84\!\cdots\!71}a^{15}-\frac{20\!\cdots\!45}{84\!\cdots\!71}a^{14}-\frac{56\!\cdots\!91}{84\!\cdots\!71}a^{13}+\frac{12\!\cdots\!63}{19\!\cdots\!97}a^{12}-\frac{32\!\cdots\!53}{84\!\cdots\!71}a^{11}-\frac{58\!\cdots\!18}{84\!\cdots\!71}a^{10}+\frac{31\!\cdots\!96}{84\!\cdots\!71}a^{9}+\frac{19\!\cdots\!77}{84\!\cdots\!71}a^{8}-\frac{36\!\cdots\!39}{84\!\cdots\!71}a^{7}-\frac{36\!\cdots\!65}{84\!\cdots\!71}a^{6}-\frac{23\!\cdots\!89}{84\!\cdots\!71}a^{5}-\frac{82\!\cdots\!38}{84\!\cdots\!71}a^{4}+\frac{15\!\cdots\!66}{84\!\cdots\!71}a^{3}-\frac{11\!\cdots\!32}{84\!\cdots\!71}a^{2}-\frac{22\!\cdots\!46}{84\!\cdots\!71}a-\frac{53\!\cdots\!50}{84\!\cdots\!71}$, $\frac{77\!\cdots\!74}{84\!\cdots\!71}a^{23}-\frac{16\!\cdots\!46}{84\!\cdots\!71}a^{22}-\frac{59\!\cdots\!43}{84\!\cdots\!71}a^{21}-\frac{73\!\cdots\!71}{84\!\cdots\!71}a^{20}+\frac{53\!\cdots\!60}{84\!\cdots\!71}a^{19}-\frac{33\!\cdots\!24}{84\!\cdots\!71}a^{18}-\frac{11\!\cdots\!65}{84\!\cdots\!71}a^{17}+\frac{47\!\cdots\!91}{84\!\cdots\!71}a^{16}+\frac{32\!\cdots\!35}{84\!\cdots\!71}a^{15}-\frac{81\!\cdots\!69}{84\!\cdots\!71}a^{14}+\frac{45\!\cdots\!09}{84\!\cdots\!71}a^{13}+\frac{31\!\cdots\!91}{19\!\cdots\!97}a^{12}-\frac{17\!\cdots\!78}{84\!\cdots\!71}a^{11}-\frac{77\!\cdots\!28}{84\!\cdots\!71}a^{10}+\frac{11\!\cdots\!11}{84\!\cdots\!71}a^{9}-\frac{13\!\cdots\!21}{84\!\cdots\!71}a^{8}-\frac{88\!\cdots\!70}{84\!\cdots\!71}a^{7}-\frac{72\!\cdots\!40}{84\!\cdots\!71}a^{6}-\frac{47\!\cdots\!20}{84\!\cdots\!71}a^{5}-\frac{56\!\cdots\!43}{84\!\cdots\!71}a^{4}+\frac{28\!\cdots\!97}{84\!\cdots\!71}a^{3}-\frac{52\!\cdots\!93}{84\!\cdots\!71}a^{2}-\frac{37\!\cdots\!33}{84\!\cdots\!71}a-\frac{62\!\cdots\!15}{84\!\cdots\!71}$ Copy content Toggle raw display (assuming GRH)
sage: UK.fundamental_units()
 
gp: K.fu
 
magma: [K|fUK(g): g in Generators(UK)];
 
oscar: [K(fUK(a)) for a in gens(UK)]
 
Regulator:  \( 10781013.84232622 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 10781013.84232622 \cdot 1}{2\cdot\sqrt{1537774657557415544813485393913449}}\cr\approx \mathstrut & 0.210912308154547 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - 2*x^23 - 8*x^22 - 2*x^21 + 70*x^20 - 33*x^19 - 156*x^18 + 43*x^17 + 433*x^16 - 998*x^15 + 433*x^14 + 1850*x^13 - 1996*x^12 - 1334*x^11 + 1462*x^10 + 5*x^9 - 1195*x^8 - 1071*x^7 - 747*x^6 - 160*x^5 + 26*x^4 - 66*x^3 - 57*x^2 - 14*x - 1)
 
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))))
 
# self-contained Pari/GP code snippet to compute the analytic class number formula
 
K = bnfinit(x^24 - 2*x^23 - 8*x^22 - 2*x^21 + 70*x^20 - 33*x^19 - 156*x^18 + 43*x^17 + 433*x^16 - 998*x^15 + 433*x^14 + 1850*x^13 - 1996*x^12 - 1334*x^11 + 1462*x^10 + 5*x^9 - 1195*x^8 - 1071*x^7 - 747*x^6 - 160*x^5 + 26*x^4 - 66*x^3 - 57*x^2 - 14*x - 1, 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)))]
 
/* self-contained Magma code snippet to compute the analytic class number formula */
 
Qx<x> := PolynomialRing(QQ); K<a> := NumberField(x^24 - 2*x^23 - 8*x^22 - 2*x^21 + 70*x^20 - 33*x^19 - 156*x^18 + 43*x^17 + 433*x^16 - 998*x^15 + 433*x^14 + 1850*x^13 - 1996*x^12 - 1334*x^11 + 1462*x^10 + 5*x^9 - 1195*x^8 - 1071*x^7 - 747*x^6 - 160*x^5 + 26*x^4 - 66*x^3 - 57*x^2 - 14*x - 1);
 
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)));
 
# self-contained Oscar code snippet to compute the analytic class number formula
 
Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 2*x^23 - 8*x^22 - 2*x^21 + 70*x^20 - 33*x^19 - 156*x^18 + 43*x^17 + 433*x^16 - 998*x^15 + 433*x^14 + 1850*x^13 - 1996*x^12 - 1334*x^11 + 1462*x^10 + 5*x^9 - 1195*x^8 - 1071*x^7 - 747*x^6 - 160*x^5 + 26*x^4 - 66*x^3 - 57*x^2 - 14*x - 1);
 
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

$\SL(2,5):C_2$ (as 24T576):

sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
magma: G = GaloisGroup(K);
 
oscar: G, Gtx = galois_group(K); G, transitive_group_identification(G)
 
A non-solvable group of order 240
The 18 conjugacy class representatives for $\SL(2,5):C_2$
Character table for $\SL(2,5):C_2$

Intermediate fields

6.2.4338889.1, 12.4.18825957754321.1

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

sage: K.subfields()[1:-1]
 
gp: L = nfsubfields(K); L[2..length(b)]
 
magma: L := Subfields(K); L[2..#L];
 
oscar: subfields(K)[2:end-1]
 

Sibling fields

Degree 40 siblings: data not computed
Arithmetically equvalently sibling: 24.4.1537774657557415544813485393913449.2
Minimal sibling: This field is its own minimal sibling

Frobenius cycle types

$p$ $2$ $3$ $5$ $7$ $11$ $13$ $17$ $19$ $23$ $29$ $31$ $37$ $41$ $43$ $47$ $53$ $59$
Cycle type $20{,}\,{\href{/padicField/2.4.0.1}{4} }$ $20{,}\,{\href{/padicField/3.4.0.1}{4} }$ $20{,}\,{\href{/padicField/5.4.0.1}{4} }$ $20{,}\,{\href{/padicField/7.4.0.1}{4} }$ ${\href{/padicField/11.12.0.1}{12} }^{2}$ ${\href{/padicField/13.6.0.1}{6} }^{4}$ ${\href{/padicField/17.5.0.1}{5} }^{4}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ ${\href{/padicField/19.4.0.1}{4} }^{6}$ $20{,}\,{\href{/padicField/23.4.0.1}{4} }$ ${\href{/padicField/29.3.0.1}{3} }^{8}$ ${\href{/padicField/31.4.0.1}{4} }^{6}$ ${\href{/padicField/37.6.0.1}{6} }^{4}$ ${\href{/padicField/41.4.0.1}{4} }^{6}$ ${\href{/padicField/43.5.0.1}{5} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ ${\href{/padicField/47.5.0.1}{5} }^{4}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ ${\href{/padicField/53.4.0.1}{4} }^{6}$ ${\href{/padicField/59.5.0.1}{5} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$

Cycle lengths which are repeated in a cycle type are indicated by exponents.

# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Sage:
 
p = 7; [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
\\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7$ in Pari:
 
p = 7; pfac = idealprimedec(K, p); vector(length(pfac), j, [pfac[j][3], pfac[j][4]])
 
// to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ for $p=7 in Magma:
 
p := 7; [<pr[2], Valuation(Norm(pr[1]), p)> : pr in Factorization(p*Integers(K))];
 
# to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$ 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
\(2083\) Copy content Toggle raw display $\Q_{2083}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{2083}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{2083}$$x$$1$$1$$0$Trivial$[\ ]$
$\Q_{2083}$$x$$1$$1$$0$Trivial$[\ ]$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$
Deg $2$$2$$1$$1$$C_2$$[\ ]_{2}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
1.2083.2t1.a.a$1$ $ 2083 $ \(\Q(\sqrt{-2083}) \) $C_2$ (as 2T1) $1$ $-1$
2.2083.120.a.a$2$ $ 2083 $ 24.4.1537774657557415544813485393913449.1 $\SL(2,5):C_2$ (as 24T576) $0$ $0$
2.2083.120.a.b$2$ $ 2083 $ 24.4.1537774657557415544813485393913449.1 $\SL(2,5):C_2$ (as 24T576) $0$ $0$
2.2083.120.a.c$2$ $ 2083 $ 24.4.1537774657557415544813485393913449.1 $\SL(2,5):C_2$ (as 24T576) $0$ $0$
2.2083.120.a.d$2$ $ 2083 $ 24.4.1537774657557415544813485393913449.1 $\SL(2,5):C_2$ (as 24T576) $0$ $0$
* 3.2083.12t76.a.a$3$ $ 2083 $ 10.0.39214470002250643.1 $A_5\times C_2$ (as 10T11) $1$ $1$
* 3.2083.12t76.a.b$3$ $ 2083 $ 10.0.39214470002250643.1 $A_5\times C_2$ (as 10T11) $1$ $1$
3.4338889.12t33.a.a$3$ $ 2083^{2}$ 5.1.4338889.1 $A_5$ (as 5T4) $1$ $-1$
3.4338889.12t33.a.b$3$ $ 2083^{2}$ 5.1.4338889.1 $A_5$ (as 5T4) $1$ $-1$
4.4338889.10t11.a.a$4$ $ 2083^{2}$ 10.0.39214470002250643.1 $A_5\times C_2$ (as 10T11) $1$ $0$
4.4338889.5t4.a.a$4$ $ 2083^{2}$ 5.1.4338889.1 $A_5$ (as 5T4) $1$ $0$
4.4338889.40t188.a.a$4$ $ 2083^{2}$ 24.4.1537774657557415544813485393913449.1 $\SL(2,5):C_2$ (as 24T576) $0$ $0$
4.4338889.40t188.a.b$4$ $ 2083^{2}$ 24.4.1537774657557415544813485393913449.1 $\SL(2,5):C_2$ (as 24T576) $0$ $0$
5.9037905787.12t75.a.a$5$ $ 2083^{3}$ 10.0.39214470002250643.1 $A_5\times C_2$ (as 10T11) $1$ $-1$
* 5.4338889.6t12.a.a$5$ $ 2083^{2}$ 5.1.4338889.1 $A_5$ (as 5T4) $1$ $1$
* 6.9037905787.24t576.a.a$6$ $ 2083^{3}$ 24.4.1537774657557415544813485393913449.1 $\SL(2,5):C_2$ (as 24T576) $0$ $0$
* 6.9037905787.24t576.a.b$6$ $ 2083^{3}$ 24.4.1537774657557415544813485393913449.1 $\SL(2,5):C_2$ (as 24T576) $0$ $0$

Data is given for all irreducible representations of the Galois group for the Galois closure of this field. Those marked with * are summands in the permutation representation coming from this field. Representations which appear with multiplicity greater than one are indicated by exponents on the *.