Properties

Label 24.0.368...000.1
Degree $24$
Signature $[0, 12]$
Discriminant $3.689\times 10^{26}$
Root discriminant \(12.79\)
Ramified primes $2,5,7$
Class number $1$ (GRH)
Class group trivial (GRH)
Galois group $S_3\times C_{12}$ (as 24T65)

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Normalized defining polynomial

sage: x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^23 + 21*x^22 - 54*x^21 + 107*x^20 - 178*x^19 + 265*x^18 - 376*x^17 + 537*x^16 - 744*x^15 + 966*x^14 - 1140*x^13 + 1221*x^12 - 1228*x^11 + 1176*x^10 - 1074*x^9 + 914*x^8 - 696*x^7 + 471*x^6 - 280*x^5 + 140*x^4 - 58*x^3 + 21*x^2 - 6*x + 1)
 
gp: K = bnfinit(y^24 - 6*y^23 + 21*y^22 - 54*y^21 + 107*y^20 - 178*y^19 + 265*y^18 - 376*y^17 + 537*y^16 - 744*y^15 + 966*y^14 - 1140*y^13 + 1221*y^12 - 1228*y^11 + 1176*y^10 - 1074*y^9 + 914*y^8 - 696*y^7 + 471*y^6 - 280*y^5 + 140*y^4 - 58*y^3 + 21*y^2 - 6*y + 1, 1)
 
magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(x^24 - 6*x^23 + 21*x^22 - 54*x^21 + 107*x^20 - 178*x^19 + 265*x^18 - 376*x^17 + 537*x^16 - 744*x^15 + 966*x^14 - 1140*x^13 + 1221*x^12 - 1228*x^11 + 1176*x^10 - 1074*x^9 + 914*x^8 - 696*x^7 + 471*x^6 - 280*x^5 + 140*x^4 - 58*x^3 + 21*x^2 - 6*x + 1);
 
oscar: Qx, x = PolynomialRing(QQ); K, a = NumberField(x^24 - 6*x^23 + 21*x^22 - 54*x^21 + 107*x^20 - 178*x^19 + 265*x^18 - 376*x^17 + 537*x^16 - 744*x^15 + 966*x^14 - 1140*x^13 + 1221*x^12 - 1228*x^11 + 1176*x^10 - 1074*x^9 + 914*x^8 - 696*x^7 + 471*x^6 - 280*x^5 + 140*x^4 - 58*x^3 + 21*x^2 - 6*x + 1)
 

\( x^{24} - 6 x^{23} + 21 x^{22} - 54 x^{21} + 107 x^{20} - 178 x^{19} + 265 x^{18} - 376 x^{17} + 537 x^{16} - 744 x^{15} + 966 x^{14} - 1140 x^{13} + 1221 x^{12} - 1228 x^{11} + 1176 x^{10} + \cdots + 1 \) 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:  $[0, 12]$
sage: K.signature()
 
gp: K.sign
 
magma: Signature(K);
 
oscar: signature(K)
 
Discriminant:   \(368947264000000000000000000\) \(\medspace = 2^{24}\cdot 5^{18}\cdot 7^{8}\) 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:  \(12.79\)
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:  $2\cdot 5^{3/4}7^{2/3}\approx 24.471252165227245$
Ramified primes:   \(2\), \(5\), \(7\) 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:  \(\Q\)
$\card{ \Aut(K/\Q) }$:  $12$
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 \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{191679830229179}a^{23}+\frac{8422786950576}{191679830229179}a^{22}-\frac{22694145349783}{191679830229179}a^{21}-\frac{6161595013249}{191679830229179}a^{20}-\frac{73723363546300}{191679830229179}a^{19}+\frac{55063250317610}{191679830229179}a^{18}+\frac{33976018607284}{191679830229179}a^{17}-\frac{85199078093611}{191679830229179}a^{16}+\frac{33893792053320}{191679830229179}a^{15}+\frac{31808053967254}{191679830229179}a^{14}+\frac{54734987496610}{191679830229179}a^{13}+\frac{5199015835275}{191679830229179}a^{12}+\frac{59348984355966}{191679830229179}a^{11}-\frac{32238076066833}{191679830229179}a^{10}-\frac{79659112112203}{191679830229179}a^{9}+\frac{42917911640006}{191679830229179}a^{8}+\frac{15963632334858}{191679830229179}a^{7}-\frac{40481349346422}{191679830229179}a^{6}+\frac{21920184363788}{191679830229179}a^{5}-\frac{54317148824536}{191679830229179}a^{4}-\frac{64837609145563}{191679830229179}a^{3}-\frac{21523508422520}{191679830229179}a^{2}-\frac{69505981625860}{191679830229179}a+\frac{95071079886986}{191679830229179}$ 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:  $11$
sage: UK.rank()
 
gp: K.fu
 
magma: UnitRank(K);
 
oscar: rank(UK)
 
Torsion generator:   \( -\frac{246938601708342}{191679830229179} a^{23} + \frac{1390625883620859}{191679830229179} a^{22} - \frac{4690132406743003}{191679830229179} a^{21} + \frac{11693310212992961}{191679830229179} a^{20} - \frac{22407405942837677}{191679830229179} a^{19} + \frac{36436954373691654}{191679830229179} a^{18} - \frac{53417981694349363}{191679830229179} a^{17} + \frac{75464163619737171}{191679830229179} a^{16} - \frac{108114681538802938}{191679830229179} a^{15} + \frac{148486908248637245}{191679830229179} a^{14} - \frac{190461567438894078}{191679830229179} a^{13} + \frac{220408936121823028}{191679830229179} a^{12} - \frac{232075014107646222}{191679830229179} a^{11} + \frac{231190167248330130}{191679830229179} a^{10} - \frac{219003463204885278}{191679830229179} a^{9} + \frac{198179686228003322}{191679830229179} a^{8} - \frac{165442350017954611}{191679830229179} a^{7} + \frac{122617740183232305}{191679830229179} a^{6} - \frac{80797338352179268}{191679830229179} a^{5} + \frac{46132174266051938}{191679830229179} a^{4} - \frac{21969283656399143}{191679830229179} a^{3} + \frac{8687105857380701}{191679830229179} a^{2} - \frac{3018529100074687}{191679830229179} a + \frac{731375154086694}{191679830229179} \)  (order $20$) 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:   $\frac{313763220963504}{191679830229179}a^{23}-\frac{16\!\cdots\!54}{191679830229179}a^{22}+\frac{55\!\cdots\!58}{191679830229179}a^{21}-\frac{13\!\cdots\!65}{191679830229179}a^{20}+\frac{24\!\cdots\!43}{191679830229179}a^{19}-\frac{39\!\cdots\!72}{191679830229179}a^{18}+\frac{57\!\cdots\!31}{191679830229179}a^{17}-\frac{80\!\cdots\!44}{191679830229179}a^{16}+\frac{11\!\cdots\!37}{191679830229179}a^{15}-\frac{15\!\cdots\!99}{191679830229179}a^{14}+\frac{19\!\cdots\!06}{191679830229179}a^{13}-\frac{22\!\cdots\!86}{191679830229179}a^{12}+\frac{23\!\cdots\!09}{191679830229179}a^{11}-\frac{22\!\cdots\!92}{191679830229179}a^{10}+\frac{21\!\cdots\!46}{191679830229179}a^{9}-\frac{19\!\cdots\!40}{191679830229179}a^{8}+\frac{15\!\cdots\!09}{191679830229179}a^{7}-\frac{11\!\cdots\!31}{191679830229179}a^{6}+\frac{71\!\cdots\!32}{191679830229179}a^{5}-\frac{38\!\cdots\!54}{191679830229179}a^{4}+\frac{16\!\cdots\!61}{191679830229179}a^{3}-\frac{58\!\cdots\!81}{191679830229179}a^{2}+\frac{18\!\cdots\!08}{191679830229179}a-\frac{237902161820265}{191679830229179}$, $\frac{79182690531977}{191679830229179}a^{23}-\frac{523203456226157}{191679830229179}a^{22}+\frac{18\!\cdots\!47}{191679830229179}a^{21}-\frac{49\!\cdots\!74}{191679830229179}a^{20}+\frac{99\!\cdots\!40}{191679830229179}a^{19}-\frac{16\!\cdots\!12}{191679830229179}a^{18}+\frac{24\!\cdots\!05}{191679830229179}a^{17}-\frac{34\!\cdots\!03}{191679830229179}a^{16}+\frac{49\!\cdots\!07}{191679830229179}a^{15}-\frac{68\!\cdots\!23}{191679830229179}a^{14}+\frac{89\!\cdots\!58}{191679830229179}a^{13}-\frac{10\!\cdots\!57}{191679830229179}a^{12}+\frac{11\!\cdots\!19}{191679830229179}a^{11}-\frac{11\!\cdots\!70}{191679830229179}a^{10}+\frac{10\!\cdots\!90}{191679830229179}a^{9}-\frac{97\!\cdots\!96}{191679830229179}a^{8}+\frac{82\!\cdots\!98}{191679830229179}a^{7}-\frac{62\!\cdots\!24}{191679830229179}a^{6}+\frac{40\!\cdots\!88}{191679830229179}a^{5}-\frac{23\!\cdots\!09}{191679830229179}a^{4}+\frac{11\!\cdots\!98}{191679830229179}a^{3}-\frac{38\!\cdots\!41}{191679830229179}a^{2}+\frac{13\!\cdots\!44}{191679830229179}a-\frac{276077805680338}{191679830229179}$, $\frac{88654107573095}{191679830229179}a^{23}-\frac{286968977528091}{191679830229179}a^{22}+\frac{591938549466787}{191679830229179}a^{21}-\frac{721133030624724}{191679830229179}a^{20}-\frac{210776520387930}{191679830229179}a^{19}+\frac{18\!\cdots\!84}{191679830229179}a^{18}-\frac{41\!\cdots\!98}{191679830229179}a^{17}+\frac{61\!\cdots\!36}{191679830229179}a^{16}-\frac{76\!\cdots\!95}{191679830229179}a^{15}+\frac{13\!\cdots\!34}{191679830229179}a^{14}-\frac{21\!\cdots\!74}{191679830229179}a^{13}+\frac{33\!\cdots\!95}{191679830229179}a^{12}-\frac{42\!\cdots\!96}{191679830229179}a^{11}+\frac{43\!\cdots\!41}{191679830229179}a^{10}-\frac{44\!\cdots\!99}{191679830229179}a^{9}+\frac{42\!\cdots\!28}{191679830229179}a^{8}-\frac{41\!\cdots\!40}{191679830229179}a^{7}+\frac{36\!\cdots\!07}{191679830229179}a^{6}-\frac{25\!\cdots\!83}{191679830229179}a^{5}+\frac{16\!\cdots\!21}{191679830229179}a^{4}-\frac{86\!\cdots\!60}{191679830229179}a^{3}+\frac{29\!\cdots\!53}{191679830229179}a^{2}-\frac{928205194416882}{191679830229179}a+\frac{334917563525729}{191679830229179}$, $\frac{467578773838715}{191679830229179}a^{23}-\frac{24\!\cdots\!29}{191679830229179}a^{22}+\frac{81\!\cdots\!68}{191679830229179}a^{21}-\frac{19\!\cdots\!30}{191679830229179}a^{20}+\frac{36\!\cdots\!51}{191679830229179}a^{19}-\frac{58\!\cdots\!25}{191679830229179}a^{18}+\frac{84\!\cdots\!21}{191679830229179}a^{17}-\frac{11\!\cdots\!73}{191679830229179}a^{16}+\frac{17\!\cdots\!28}{191679830229179}a^{15}-\frac{23\!\cdots\!17}{191679830229179}a^{14}+\frac{29\!\cdots\!28}{191679830229179}a^{13}-\frac{33\!\cdots\!08}{191679830229179}a^{12}+\frac{34\!\cdots\!18}{191679830229179}a^{11}-\frac{34\!\cdots\!37}{191679830229179}a^{10}+\frac{32\!\cdots\!91}{191679830229179}a^{9}-\frac{28\!\cdots\!79}{191679830229179}a^{8}+\frac{23\!\cdots\!31}{191679830229179}a^{7}-\frac{17\!\cdots\!98}{191679830229179}a^{6}+\frac{11\!\cdots\!34}{191679830229179}a^{5}-\frac{61\!\cdots\!54}{191679830229179}a^{4}+\frac{27\!\cdots\!97}{191679830229179}a^{3}-\frac{10\!\cdots\!47}{191679830229179}a^{2}+\frac{36\!\cdots\!20}{191679830229179}a-\frac{560501170853498}{191679830229179}$, $\frac{137178468631118}{191679830229179}a^{23}-\frac{763704119341484}{191679830229179}a^{22}+\frac{25\!\cdots\!90}{191679830229179}a^{21}-\frac{62\!\cdots\!77}{191679830229179}a^{20}+\frac{11\!\cdots\!86}{191679830229179}a^{19}-\frac{18\!\cdots\!17}{191679830229179}a^{18}+\frac{27\!\cdots\!12}{191679830229179}a^{17}-\frac{38\!\cdots\!63}{191679830229179}a^{16}+\frac{55\!\cdots\!14}{191679830229179}a^{15}-\frac{75\!\cdots\!13}{191679830229179}a^{14}+\frac{95\!\cdots\!20}{191679830229179}a^{13}-\frac{10\!\cdots\!09}{191679830229179}a^{12}+\frac{11\!\cdots\!55}{191679830229179}a^{11}-\frac{11\!\cdots\!92}{191679830229179}a^{10}+\frac{10\!\cdots\!36}{191679830229179}a^{9}-\frac{93\!\cdots\!80}{191679830229179}a^{8}+\frac{76\!\cdots\!21}{191679830229179}a^{7}-\frac{54\!\cdots\!83}{191679830229179}a^{6}+\frac{34\!\cdots\!97}{191679830229179}a^{5}-\frac{18\!\cdots\!64}{191679830229179}a^{4}+\frac{80\!\cdots\!27}{191679830229179}a^{3}-\frac{26\!\cdots\!44}{191679830229179}a^{2}+\frac{883446793738667}{191679830229179}a-\frac{136879939765985}{191679830229179}$, $\frac{134917902018512}{191679830229179}a^{23}-\frac{595406819560583}{191679830229179}a^{22}+\frac{17\!\cdots\!12}{191679830229179}a^{21}-\frac{36\!\cdots\!89}{191679830229179}a^{20}+\frac{58\!\cdots\!21}{191679830229179}a^{19}-\frac{82\!\cdots\!75}{191679830229179}a^{18}+\frac{10\!\cdots\!93}{191679830229179}a^{17}-\frac{15\!\cdots\!14}{191679830229179}a^{16}+\frac{22\!\cdots\!27}{191679830229179}a^{15}-\frac{28\!\cdots\!94}{191679830229179}a^{14}+\frac{33\!\cdots\!75}{191679830229179}a^{13}-\frac{31\!\cdots\!47}{191679830229179}a^{12}+\frac{27\!\cdots\!48}{191679830229179}a^{11}-\frac{25\!\cdots\!69}{191679830229179}a^{10}+\frac{21\!\cdots\!73}{191679830229179}a^{9}-\frac{16\!\cdots\!25}{191679830229179}a^{8}+\frac{95\!\cdots\!47}{191679830229179}a^{7}-\frac{20\!\cdots\!07}{191679830229179}a^{6}-\frac{582505257031979}{191679830229179}a^{5}+\frac{24\!\cdots\!49}{191679830229179}a^{4}-\frac{25\!\cdots\!80}{191679830229179}a^{3}+\frac{838088717432422}{191679830229179}a^{2}-\frac{185808902177567}{191679830229179}a+\frac{332839733891005}{191679830229179}$, $\frac{103763337899096}{191679830229179}a^{23}-\frac{597060135753884}{191679830229179}a^{22}+\frac{20\!\cdots\!14}{191679830229179}a^{21}-\frac{54\!\cdots\!84}{191679830229179}a^{20}+\frac{10\!\cdots\!61}{191679830229179}a^{19}-\frac{18\!\cdots\!22}{191679830229179}a^{18}+\frac{27\!\cdots\!49}{191679830229179}a^{17}-\frac{38\!\cdots\!19}{191679830229179}a^{16}+\frac{54\!\cdots\!05}{191679830229179}a^{15}-\frac{75\!\cdots\!74}{191679830229179}a^{14}+\frac{98\!\cdots\!86}{191679830229179}a^{13}-\frac{11\!\cdots\!29}{191679830229179}a^{12}+\frac{12\!\cdots\!03}{191679830229179}a^{11}-\frac{12\!\cdots\!43}{191679830229179}a^{10}+\frac{11\!\cdots\!32}{191679830229179}a^{9}-\frac{10\!\cdots\!74}{191679830229179}a^{8}+\frac{91\!\cdots\!11}{191679830229179}a^{7}-\frac{69\!\cdots\!60}{191679830229179}a^{6}+\frac{45\!\cdots\!84}{191679830229179}a^{5}-\frac{25\!\cdots\!55}{191679830229179}a^{4}+\frac{11\!\cdots\!65}{191679830229179}a^{3}-\frac{40\!\cdots\!70}{191679830229179}a^{2}+\frac{11\!\cdots\!25}{191679830229179}a-\frac{145015384528938}{191679830229179}$, $\frac{124476743978622}{191679830229179}a^{23}-\frac{651947171615861}{191679830229179}a^{22}+\frac{20\!\cdots\!67}{191679830229179}a^{21}-\frac{47\!\cdots\!96}{191679830229179}a^{20}+\frac{82\!\cdots\!13}{191679830229179}a^{19}-\frac{12\!\cdots\!40}{191679830229179}a^{18}+\frac{17\!\cdots\!53}{191679830229179}a^{17}-\frac{23\!\cdots\!37}{191679830229179}a^{16}+\frac{34\!\cdots\!31}{191679830229179}a^{15}-\frac{46\!\cdots\!99}{191679830229179}a^{14}+\frac{56\!\cdots\!06}{191679830229179}a^{13}-\frac{60\!\cdots\!23}{191679830229179}a^{12}+\frac{58\!\cdots\!70}{191679830229179}a^{11}-\frac{55\!\cdots\!68}{191679830229179}a^{10}+\frac{52\!\cdots\!12}{191679830229179}a^{9}-\frac{45\!\cdots\!62}{191679830229179}a^{8}+\frac{35\!\cdots\!75}{191679830229179}a^{7}-\frac{21\!\cdots\!37}{191679830229179}a^{6}+\frac{12\!\cdots\!58}{191679830229179}a^{5}-\frac{67\!\cdots\!60}{191679830229179}a^{4}+\frac{30\!\cdots\!89}{191679830229179}a^{3}-\frac{15\!\cdots\!93}{191679830229179}a^{2}+\frac{10\!\cdots\!10}{191679830229179}a-\frac{201626230726472}{191679830229179}$, $\frac{338441737459678}{191679830229179}a^{23}-\frac{19\!\cdots\!12}{191679830229179}a^{22}+\frac{66\!\cdots\!13}{191679830229179}a^{21}-\frac{16\!\cdots\!40}{191679830229179}a^{20}+\frac{32\!\cdots\!71}{191679830229179}a^{19}-\frac{52\!\cdots\!11}{191679830229179}a^{18}+\frac{77\!\cdots\!57}{191679830229179}a^{17}-\frac{10\!\cdots\!27}{191679830229179}a^{16}+\frac{15\!\cdots\!79}{191679830229179}a^{15}-\frac{21\!\cdots\!88}{191679830229179}a^{14}+\frac{27\!\cdots\!98}{191679830229179}a^{13}-\frac{31\!\cdots\!07}{191679830229179}a^{12}+\frac{33\!\cdots\!75}{191679830229179}a^{11}-\frac{33\!\cdots\!97}{191679830229179}a^{10}+\frac{31\!\cdots\!24}{191679830229179}a^{9}-\frac{28\!\cdots\!41}{191679830229179}a^{8}+\frac{23\!\cdots\!44}{191679830229179}a^{7}-\frac{17\!\cdots\!03}{191679830229179}a^{6}+\frac{11\!\cdots\!69}{191679830229179}a^{5}-\frac{66\!\cdots\!92}{191679830229179}a^{4}+\frac{30\!\cdots\!18}{191679830229179}a^{3}-\frac{11\!\cdots\!58}{191679830229179}a^{2}+\frac{41\!\cdots\!70}{191679830229179}a-\frac{884858325660133}{191679830229179}$, $\frac{8537259204596}{191679830229179}a^{23}-\frac{14338870339241}{191679830229179}a^{22}+\frac{20945316892571}{191679830229179}a^{21}-\frac{1041265056030}{191679830229179}a^{20}-\frac{92554557882445}{191679830229179}a^{19}+\frac{119115536955379}{191679830229179}a^{18}-\frac{206032764670443}{191679830229179}a^{17}+\frac{244156048158941}{191679830229179}a^{16}-\frac{374200090640677}{191679830229179}a^{15}+\frac{10\!\cdots\!25}{191679830229179}a^{14}-\frac{12\!\cdots\!69}{191679830229179}a^{13}+\frac{18\!\cdots\!17}{191679830229179}a^{12}-\frac{17\!\cdots\!17}{191679830229179}a^{11}+\frac{18\!\cdots\!44}{191679830229179}a^{10}-\frac{27\!\cdots\!51}{191679830229179}a^{9}+\frac{28\!\cdots\!69}{191679830229179}a^{8}-\frac{32\!\cdots\!97}{191679830229179}a^{7}+\frac{24\!\cdots\!29}{191679830229179}a^{6}-\frac{18\!\cdots\!01}{191679830229179}a^{5}+\frac{20\!\cdots\!77}{191679830229179}a^{4}-\frac{14\!\cdots\!11}{191679830229179}a^{3}+\frac{12\!\cdots\!23}{191679830229179}a^{2}-\frac{615263025961386}{191679830229179}a+\frac{340458458469721}{191679830229179}$, $\frac{805942816283302}{191679830229179}a^{23}-\frac{44\!\cdots\!07}{191679830229179}a^{22}+\frac{14\!\cdots\!77}{191679830229179}a^{21}-\frac{35\!\cdots\!35}{191679830229179}a^{20}+\frac{67\!\cdots\!87}{191679830229179}a^{19}-\frac{10\!\cdots\!61}{191679830229179}a^{18}+\frac{15\!\cdots\!33}{191679830229179}a^{17}-\frac{21\!\cdots\!61}{191679830229179}a^{16}+\frac{31\!\cdots\!21}{191679830229179}a^{15}-\frac{43\!\cdots\!34}{191679830229179}a^{14}+\frac{54\!\cdots\!88}{191679830229179}a^{13}-\frac{62\!\cdots\!97}{191679830229179}a^{12}+\frac{64\!\cdots\!37}{191679830229179}a^{11}-\frac{63\!\cdots\!09}{191679830229179}a^{10}+\frac{59\!\cdots\!98}{191679830229179}a^{9}-\frac{53\!\cdots\!50}{191679830229179}a^{8}+\frac{43\!\cdots\!42}{191679830229179}a^{7}-\frac{31\!\cdots\!34}{191679830229179}a^{6}+\frac{20\!\cdots\!25}{191679830229179}a^{5}-\frac{11\!\cdots\!45}{191679830229179}a^{4}+\frac{48\!\cdots\!58}{191679830229179}a^{3}-\frac{17\!\cdots\!25}{191679830229179}a^{2}+\frac{59\!\cdots\!92}{191679830229179}a-\frac{10\!\cdots\!08}{191679830229179}$ 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:  \( 15456.373586053287 \) (assuming GRH)
sage: K.regulator()
 
gp: K.reg
 
magma: Regulator(K);
 
oscar: regulator(K)
 

Class number formula

\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 15456.373586053287 \cdot 1}{20\cdot\sqrt{368947264000000000000000000}}\cr\approx \mathstrut & 0.152318931258119 \end{aligned}\] (assuming GRH)

# self-contained SageMath code snippet to compute the analytic class number formula
 
x = polygen(QQ); K.<a> = NumberField(x^24 - 6*x^23 + 21*x^22 - 54*x^21 + 107*x^20 - 178*x^19 + 265*x^18 - 376*x^17 + 537*x^16 - 744*x^15 + 966*x^14 - 1140*x^13 + 1221*x^12 - 1228*x^11 + 1176*x^10 - 1074*x^9 + 914*x^8 - 696*x^7 + 471*x^6 - 280*x^5 + 140*x^4 - 58*x^3 + 21*x^2 - 6*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 - 6*x^23 + 21*x^22 - 54*x^21 + 107*x^20 - 178*x^19 + 265*x^18 - 376*x^17 + 537*x^16 - 744*x^15 + 966*x^14 - 1140*x^13 + 1221*x^12 - 1228*x^11 + 1176*x^10 - 1074*x^9 + 914*x^8 - 696*x^7 + 471*x^6 - 280*x^5 + 140*x^4 - 58*x^3 + 21*x^2 - 6*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 - 6*x^23 + 21*x^22 - 54*x^21 + 107*x^20 - 178*x^19 + 265*x^18 - 376*x^17 + 537*x^16 - 744*x^15 + 966*x^14 - 1140*x^13 + 1221*x^12 - 1228*x^11 + 1176*x^10 - 1074*x^9 + 914*x^8 - 696*x^7 + 471*x^6 - 280*x^5 + 140*x^4 - 58*x^3 + 21*x^2 - 6*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 - 6*x^23 + 21*x^22 - 54*x^21 + 107*x^20 - 178*x^19 + 265*x^18 - 376*x^17 + 537*x^16 - 744*x^15 + 966*x^14 - 1140*x^13 + 1221*x^12 - 1228*x^11 + 1176*x^10 - 1074*x^9 + 914*x^8 - 696*x^7 + 471*x^6 - 280*x^5 + 140*x^4 - 58*x^3 + 21*x^2 - 6*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

$S_3\times C_{12}$ (as 24T65):

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 solvable group of order 72
The 36 conjugacy class representatives for $S_3\times C_{12}$
Character table for $S_3\times C_{12}$ is not computed

Intermediate fields

\(\Q(\sqrt{5}) \), \(\Q(\sqrt{-5}) \), \(\Q(\sqrt{-1}) \), \(\Q(\zeta_{5})\), \(\Q(\zeta_{20})^+\), \(\Q(i, \sqrt{5})\), 6.0.392000.1, \(\Q(\zeta_{20})\), 12.0.153664000000.1

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 36 siblings: data not computed
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 R ${\href{/padicField/3.12.0.1}{12} }{,}\,{\href{/padicField/3.4.0.1}{4} }^{3}$ R R ${\href{/padicField/11.6.0.1}{6} }^{4}$ ${\href{/padicField/13.4.0.1}{4} }^{6}$ ${\href{/padicField/17.12.0.1}{12} }^{2}$ ${\href{/padicField/19.6.0.1}{6} }^{4}$ ${\href{/padicField/23.12.0.1}{12} }{,}\,{\href{/padicField/23.4.0.1}{4} }^{3}$ ${\href{/padicField/29.6.0.1}{6} }^{4}$ ${\href{/padicField/31.6.0.1}{6} }^{4}$ ${\href{/padicField/37.12.0.1}{12} }^{2}$ ${\href{/padicField/41.3.0.1}{3} }^{8}$ ${\href{/padicField/43.12.0.1}{12} }^{2}$ ${\href{/padicField/47.12.0.1}{12} }^{2}$ ${\href{/padicField/53.12.0.1}{12} }^{2}$ ${\href{/padicField/59.6.0.1}{6} }^{4}$

In the table, R denotes a ramified prime. 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
\(2\) Copy content Toggle raw display Deg $24$$2$$12$$24$
\(5\) Copy content Toggle raw display 5.12.9.2$x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
5.12.9.2$x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$$4$$3$$9$$C_{12}$$[\ ]_{4}^{3}$
\(7\) Copy content Toggle raw display 7.12.0.1$x^{12} + 2 x^{8} + 5 x^{7} + 3 x^{6} + 2 x^{5} + 4 x^{4} + 5 x^{2} + 3$$1$$12$$0$$C_{12}$$[\ ]^{12}$
7.12.8.3$x^{12} + 245 x^{6} - 1372 x^{3} + 7203$$3$$4$$8$$C_{12}$$[\ ]_{3}^{4}$

Artin representations

Label Dimension Conductor Artin stem field $G$ Ind $\chi(c)$
* 1.1.1t1.a.a$1$ $1$ \(\Q\) $C_1$ $1$ $1$
* 1.4.2t1.a.a$1$ $ 2^{2}$ \(\Q(\sqrt{-1}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.20.2t1.a.a$1$ $ 2^{2} \cdot 5 $ \(\Q(\sqrt{-5}) \) $C_2$ (as 2T1) $1$ $-1$
* 1.5.2t1.a.a$1$ $ 5 $ \(\Q(\sqrt{5}) \) $C_2$ (as 2T1) $1$ $1$
1.140.6t1.b.a$1$ $ 2^{2} \cdot 5 \cdot 7 $ 6.0.19208000.1 $C_6$ (as 6T1) $0$ $-1$
1.7.3t1.a.a$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
1.140.6t1.b.b$1$ $ 2^{2} \cdot 5 \cdot 7 $ 6.0.19208000.1 $C_6$ (as 6T1) $0$ $-1$
1.28.6t1.a.a$1$ $ 2^{2} \cdot 7 $ 6.0.153664.1 $C_6$ (as 6T1) $0$ $-1$
1.35.6t1.b.a$1$ $ 5 \cdot 7 $ 6.6.300125.1 $C_6$ (as 6T1) $0$ $1$
1.35.6t1.b.b$1$ $ 5 \cdot 7 $ 6.6.300125.1 $C_6$ (as 6T1) $0$ $1$
1.28.6t1.a.b$1$ $ 2^{2} \cdot 7 $ 6.0.153664.1 $C_6$ (as 6T1) $0$ $-1$
1.7.3t1.a.b$1$ $ 7 $ \(\Q(\zeta_{7})^+\) $C_3$ (as 3T1) $0$ $1$
* 1.5.4t1.a.a$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
* 1.20.4t1.a.a$1$ $ 2^{2} \cdot 5 $ \(\Q(\zeta_{20})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.20.4t1.a.b$1$ $ 2^{2} \cdot 5 $ \(\Q(\zeta_{20})^+\) $C_4$ (as 4T1) $0$ $1$
* 1.5.4t1.a.b$1$ $ 5 $ \(\Q(\zeta_{5})\) $C_4$ (as 4T1) $0$ $-1$
1.140.12t1.a.a$1$ $ 2^{2} \cdot 5 \cdot 7 $ 12.12.46118408000000000.1 $C_{12}$ (as 12T1) $0$ $1$
1.35.12t1.a.a$1$ $ 5 \cdot 7 $ 12.0.11259376953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.140.12t1.a.b$1$ $ 2^{2} \cdot 5 \cdot 7 $ 12.12.46118408000000000.1 $C_{12}$ (as 12T1) $0$ $1$
1.35.12t1.a.b$1$ $ 5 \cdot 7 $ 12.0.11259376953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.35.12t1.a.c$1$ $ 5 \cdot 7 $ 12.0.11259376953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.35.12t1.a.d$1$ $ 5 \cdot 7 $ 12.0.11259376953125.1 $C_{12}$ (as 12T1) $0$ $-1$
1.140.12t1.a.c$1$ $ 2^{2} \cdot 5 \cdot 7 $ 12.12.46118408000000000.1 $C_{12}$ (as 12T1) $0$ $1$
1.140.12t1.a.d$1$ $ 2^{2} \cdot 5 \cdot 7 $ 12.12.46118408000000000.1 $C_{12}$ (as 12T1) $0$ $1$
2.980.3t2.a.a$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ 3.1.980.1 $S_3$ (as 3T2) $1$ $0$
2.980.6t3.d.a$2$ $ 2^{2} \cdot 5 \cdot 7^{2}$ 6.2.4802000.1 $D_{6}$ (as 6T3) $1$ $0$
* 2.140.12t18.a.a$2$ $ 2^{2} \cdot 5 \cdot 7 $ 12.0.153664000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.140.6t5.a.a$2$ $ 2^{2} \cdot 5 \cdot 7 $ 6.0.392000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
* 2.140.12t18.a.b$2$ $ 2^{2} \cdot 5 \cdot 7 $ 12.0.153664000000.1 $C_6\times S_3$ (as 12T18) $0$ $0$
* 2.140.6t5.a.b$2$ $ 2^{2} \cdot 5 \cdot 7 $ 6.0.392000.1 $S_3\times C_3$ (as 6T5) $0$ $0$
2.4900.12t11.a.a$2$ $ 2^{2} \cdot 5^{2} \cdot 7^{2}$ 12.4.46118408000000000.2 $S_3 \times C_4$ (as 12T11) $0$ $0$
2.4900.12t11.a.b$2$ $ 2^{2} \cdot 5^{2} \cdot 7^{2}$ 12.4.46118408000000000.2 $S_3 \times C_4$ (as 12T11) $0$ $0$
* 2.700.24t65.a.a$2$ $ 2^{2} \cdot 5^{2} \cdot 7 $ 24.0.368947264000000000000000000.1 $S_3\times C_{12}$ (as 24T65) $0$ $0$
* 2.700.24t65.a.b$2$ $ 2^{2} \cdot 5^{2} \cdot 7 $ 24.0.368947264000000000000000000.1 $S_3\times C_{12}$ (as 24T65) $0$ $0$
* 2.700.24t65.a.c$2$ $ 2^{2} \cdot 5^{2} \cdot 7 $ 24.0.368947264000000000000000000.1 $S_3\times C_{12}$ (as 24T65) $0$ $0$
* 2.700.24t65.a.d$2$ $ 2^{2} \cdot 5^{2} \cdot 7 $ 24.0.368947264000000000000000000.1 $S_3\times C_{12}$ (as 24T65) $0$ $0$

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 *.