Normalized defining polynomial
\( x^{24} - 6 x^{23} + 21 x^{22} - 54 x^{21} + 107 x^{20} - 178 x^{19} + 265 x^{18} - 376 x^{17} + 537 x^{16} - 744 x^{15} + 966 x^{14} - 1140 x^{13} + 1221 x^{12} - 1228 x^{11} + 1176 x^{10} + \cdots + 1 \)
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: | \(368947264000000000000000000\) \(\medspace = 2^{24}\cdot 5^{18}\cdot 7^{8}\) | 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: | \(2\), \(5\), \(7\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $\frac{1}{191679830229179}a^{23}+\frac{8422786950576}{191679830229179}a^{22}-\frac{22694145349783}{191679830229179}a^{21}-\frac{6161595013249}{191679830229179}a^{20}-\frac{73723363546300}{191679830229179}a^{19}+\frac{55063250317610}{191679830229179}a^{18}+\frac{33976018607284}{191679830229179}a^{17}-\frac{85199078093611}{191679830229179}a^{16}+\frac{33893792053320}{191679830229179}a^{15}+\frac{31808053967254}{191679830229179}a^{14}+\frac{54734987496610}{191679830229179}a^{13}+\frac{5199015835275}{191679830229179}a^{12}+\frac{59348984355966}{191679830229179}a^{11}-\frac{32238076066833}{191679830229179}a^{10}-\frac{79659112112203}{191679830229179}a^{9}+\frac{42917911640006}{191679830229179}a^{8}+\frac{15963632334858}{191679830229179}a^{7}-\frac{40481349346422}{191679830229179}a^{6}+\frac{21920184363788}{191679830229179}a^{5}-\frac{54317148824536}{191679830229179}a^{4}-\frac{64837609145563}{191679830229179}a^{3}-\frac{21523508422520}{191679830229179}a^{2}-\frac{69505981625860}{191679830229179}a+\frac{95071079886986}{191679830229179}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
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 $20$) | 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: | $\frac{313763220963504}{191679830229179}a^{23}-\frac{16\!\cdots\!54}{191679830229179}a^{22}+\frac{55\!\cdots\!58}{191679830229179}a^{21}-\frac{13\!\cdots\!65}{191679830229179}a^{20}+\frac{24\!\cdots\!43}{191679830229179}a^{19}-\frac{39\!\cdots\!72}{191679830229179}a^{18}+\frac{57\!\cdots\!31}{191679830229179}a^{17}-\frac{80\!\cdots\!44}{191679830229179}a^{16}+\frac{11\!\cdots\!37}{191679830229179}a^{15}-\frac{15\!\cdots\!99}{191679830229179}a^{14}+\frac{19\!\cdots\!06}{191679830229179}a^{13}-\frac{22\!\cdots\!86}{191679830229179}a^{12}+\frac{23\!\cdots\!09}{191679830229179}a^{11}-\frac{22\!\cdots\!92}{191679830229179}a^{10}+\frac{21\!\cdots\!46}{191679830229179}a^{9}-\frac{19\!\cdots\!40}{191679830229179}a^{8}+\frac{15\!\cdots\!09}{191679830229179}a^{7}-\frac{11\!\cdots\!31}{191679830229179}a^{6}+\frac{71\!\cdots\!32}{191679830229179}a^{5}-\frac{38\!\cdots\!54}{191679830229179}a^{4}+\frac{16\!\cdots\!61}{191679830229179}a^{3}-\frac{58\!\cdots\!81}{191679830229179}a^{2}+\frac{18\!\cdots\!08}{191679830229179}a-\frac{237902161820265}{191679830229179}$, $\frac{79182690531977}{191679830229179}a^{23}-\frac{523203456226157}{191679830229179}a^{22}+\frac{18\!\cdots\!47}{191679830229179}a^{21}-\frac{49\!\cdots\!74}{191679830229179}a^{20}+\frac{99\!\cdots\!40}{191679830229179}a^{19}-\frac{16\!\cdots\!12}{191679830229179}a^{18}+\frac{24\!\cdots\!05}{191679830229179}a^{17}-\frac{34\!\cdots\!03}{191679830229179}a^{16}+\frac{49\!\cdots\!07}{191679830229179}a^{15}-\frac{68\!\cdots\!23}{191679830229179}a^{14}+\frac{89\!\cdots\!58}{191679830229179}a^{13}-\frac{10\!\cdots\!57}{191679830229179}a^{12}+\frac{11\!\cdots\!19}{191679830229179}a^{11}-\frac{11\!\cdots\!70}{191679830229179}a^{10}+\frac{10\!\cdots\!90}{191679830229179}a^{9}-\frac{97\!\cdots\!96}{191679830229179}a^{8}+\frac{82\!\cdots\!98}{191679830229179}a^{7}-\frac{62\!\cdots\!24}{191679830229179}a^{6}+\frac{40\!\cdots\!88}{191679830229179}a^{5}-\frac{23\!\cdots\!09}{191679830229179}a^{4}+\frac{11\!\cdots\!98}{191679830229179}a^{3}-\frac{38\!\cdots\!41}{191679830229179}a^{2}+\frac{13\!\cdots\!44}{191679830229179}a-\frac{276077805680338}{191679830229179}$, $\frac{88654107573095}{191679830229179}a^{23}-\frac{286968977528091}{191679830229179}a^{22}+\frac{591938549466787}{191679830229179}a^{21}-\frac{721133030624724}{191679830229179}a^{20}-\frac{210776520387930}{191679830229179}a^{19}+\frac{18\!\cdots\!84}{191679830229179}a^{18}-\frac{41\!\cdots\!98}{191679830229179}a^{17}+\frac{61\!\cdots\!36}{191679830229179}a^{16}-\frac{76\!\cdots\!95}{191679830229179}a^{15}+\frac{13\!\cdots\!34}{191679830229179}a^{14}-\frac{21\!\cdots\!74}{191679830229179}a^{13}+\frac{33\!\cdots\!95}{191679830229179}a^{12}-\frac{42\!\cdots\!96}{191679830229179}a^{11}+\frac{43\!\cdots\!41}{191679830229179}a^{10}-\frac{44\!\cdots\!99}{191679830229179}a^{9}+\frac{42\!\cdots\!28}{191679830229179}a^{8}-\frac{41\!\cdots\!40}{191679830229179}a^{7}+\frac{36\!\cdots\!07}{191679830229179}a^{6}-\frac{25\!\cdots\!83}{191679830229179}a^{5}+\frac{16\!\cdots\!21}{191679830229179}a^{4}-\frac{86\!\cdots\!60}{191679830229179}a^{3}+\frac{29\!\cdots\!53}{191679830229179}a^{2}-\frac{928205194416882}{191679830229179}a+\frac{334917563525729}{191679830229179}$, $\frac{467578773838715}{191679830229179}a^{23}-\frac{24\!\cdots\!29}{191679830229179}a^{22}+\frac{81\!\cdots\!68}{191679830229179}a^{21}-\frac{19\!\cdots\!30}{191679830229179}a^{20}+\frac{36\!\cdots\!51}{191679830229179}a^{19}-\frac{58\!\cdots\!25}{191679830229179}a^{18}+\frac{84\!\cdots\!21}{191679830229179}a^{17}-\frac{11\!\cdots\!73}{191679830229179}a^{16}+\frac{17\!\cdots\!28}{191679830229179}a^{15}-\frac{23\!\cdots\!17}{191679830229179}a^{14}+\frac{29\!\cdots\!28}{191679830229179}a^{13}-\frac{33\!\cdots\!08}{191679830229179}a^{12}+\frac{34\!\cdots\!18}{191679830229179}a^{11}-\frac{34\!\cdots\!37}{191679830229179}a^{10}+\frac{32\!\cdots\!91}{191679830229179}a^{9}-\frac{28\!\cdots\!79}{191679830229179}a^{8}+\frac{23\!\cdots\!31}{191679830229179}a^{7}-\frac{17\!\cdots\!98}{191679830229179}a^{6}+\frac{11\!\cdots\!34}{191679830229179}a^{5}-\frac{61\!\cdots\!54}{191679830229179}a^{4}+\frac{27\!\cdots\!97}{191679830229179}a^{3}-\frac{10\!\cdots\!47}{191679830229179}a^{2}+\frac{36\!\cdots\!20}{191679830229179}a-\frac{560501170853498}{191679830229179}$, $\frac{137178468631118}{191679830229179}a^{23}-\frac{763704119341484}{191679830229179}a^{22}+\frac{25\!\cdots\!90}{191679830229179}a^{21}-\frac{62\!\cdots\!77}{191679830229179}a^{20}+\frac{11\!\cdots\!86}{191679830229179}a^{19}-\frac{18\!\cdots\!17}{191679830229179}a^{18}+\frac{27\!\cdots\!12}{191679830229179}a^{17}-\frac{38\!\cdots\!63}{191679830229179}a^{16}+\frac{55\!\cdots\!14}{191679830229179}a^{15}-\frac{75\!\cdots\!13}{191679830229179}a^{14}+\frac{95\!\cdots\!20}{191679830229179}a^{13}-\frac{10\!\cdots\!09}{191679830229179}a^{12}+\frac{11\!\cdots\!55}{191679830229179}a^{11}-\frac{11\!\cdots\!92}{191679830229179}a^{10}+\frac{10\!\cdots\!36}{191679830229179}a^{9}-\frac{93\!\cdots\!80}{191679830229179}a^{8}+\frac{76\!\cdots\!21}{191679830229179}a^{7}-\frac{54\!\cdots\!83}{191679830229179}a^{6}+\frac{34\!\cdots\!97}{191679830229179}a^{5}-\frac{18\!\cdots\!64}{191679830229179}a^{4}+\frac{80\!\cdots\!27}{191679830229179}a^{3}-\frac{26\!\cdots\!44}{191679830229179}a^{2}+\frac{883446793738667}{191679830229179}a-\frac{136879939765985}{191679830229179}$, $\frac{134917902018512}{191679830229179}a^{23}-\frac{595406819560583}{191679830229179}a^{22}+\frac{17\!\cdots\!12}{191679830229179}a^{21}-\frac{36\!\cdots\!89}{191679830229179}a^{20}+\frac{58\!\cdots\!21}{191679830229179}a^{19}-\frac{82\!\cdots\!75}{191679830229179}a^{18}+\frac{10\!\cdots\!93}{191679830229179}a^{17}-\frac{15\!\cdots\!14}{191679830229179}a^{16}+\frac{22\!\cdots\!27}{191679830229179}a^{15}-\frac{28\!\cdots\!94}{191679830229179}a^{14}+\frac{33\!\cdots\!75}{191679830229179}a^{13}-\frac{31\!\cdots\!47}{191679830229179}a^{12}+\frac{27\!\cdots\!48}{191679830229179}a^{11}-\frac{25\!\cdots\!69}{191679830229179}a^{10}+\frac{21\!\cdots\!73}{191679830229179}a^{9}-\frac{16\!\cdots\!25}{191679830229179}a^{8}+\frac{95\!\cdots\!47}{191679830229179}a^{7}-\frac{20\!\cdots\!07}{191679830229179}a^{6}-\frac{582505257031979}{191679830229179}a^{5}+\frac{24\!\cdots\!49}{191679830229179}a^{4}-\frac{25\!\cdots\!80}{191679830229179}a^{3}+\frac{838088717432422}{191679830229179}a^{2}-\frac{185808902177567}{191679830229179}a+\frac{332839733891005}{191679830229179}$, $\frac{103763337899096}{191679830229179}a^{23}-\frac{597060135753884}{191679830229179}a^{22}+\frac{20\!\cdots\!14}{191679830229179}a^{21}-\frac{54\!\cdots\!84}{191679830229179}a^{20}+\frac{10\!\cdots\!61}{191679830229179}a^{19}-\frac{18\!\cdots\!22}{191679830229179}a^{18}+\frac{27\!\cdots\!49}{191679830229179}a^{17}-\frac{38\!\cdots\!19}{191679830229179}a^{16}+\frac{54\!\cdots\!05}{191679830229179}a^{15}-\frac{75\!\cdots\!74}{191679830229179}a^{14}+\frac{98\!\cdots\!86}{191679830229179}a^{13}-\frac{11\!\cdots\!29}{191679830229179}a^{12}+\frac{12\!\cdots\!03}{191679830229179}a^{11}-\frac{12\!\cdots\!43}{191679830229179}a^{10}+\frac{11\!\cdots\!32}{191679830229179}a^{9}-\frac{10\!\cdots\!74}{191679830229179}a^{8}+\frac{91\!\cdots\!11}{191679830229179}a^{7}-\frac{69\!\cdots\!60}{191679830229179}a^{6}+\frac{45\!\cdots\!84}{191679830229179}a^{5}-\frac{25\!\cdots\!55}{191679830229179}a^{4}+\frac{11\!\cdots\!65}{191679830229179}a^{3}-\frac{40\!\cdots\!70}{191679830229179}a^{2}+\frac{11\!\cdots\!25}{191679830229179}a-\frac{145015384528938}{191679830229179}$, $\frac{124476743978622}{191679830229179}a^{23}-\frac{651947171615861}{191679830229179}a^{22}+\frac{20\!\cdots\!67}{191679830229179}a^{21}-\frac{47\!\cdots\!96}{191679830229179}a^{20}+\frac{82\!\cdots\!13}{191679830229179}a^{19}-\frac{12\!\cdots\!40}{191679830229179}a^{18}+\frac{17\!\cdots\!53}{191679830229179}a^{17}-\frac{23\!\cdots\!37}{191679830229179}a^{16}+\frac{34\!\cdots\!31}{191679830229179}a^{15}-\frac{46\!\cdots\!99}{191679830229179}a^{14}+\frac{56\!\cdots\!06}{191679830229179}a^{13}-\frac{60\!\cdots\!23}{191679830229179}a^{12}+\frac{58\!\cdots\!70}{191679830229179}a^{11}-\frac{55\!\cdots\!68}{191679830229179}a^{10}+\frac{52\!\cdots\!12}{191679830229179}a^{9}-\frac{45\!\cdots\!62}{191679830229179}a^{8}+\frac{35\!\cdots\!75}{191679830229179}a^{7}-\frac{21\!\cdots\!37}{191679830229179}a^{6}+\frac{12\!\cdots\!58}{191679830229179}a^{5}-\frac{67\!\cdots\!60}{191679830229179}a^{4}+\frac{30\!\cdots\!89}{191679830229179}a^{3}-\frac{15\!\cdots\!93}{191679830229179}a^{2}+\frac{10\!\cdots\!10}{191679830229179}a-\frac{201626230726472}{191679830229179}$, $\frac{338441737459678}{191679830229179}a^{23}-\frac{19\!\cdots\!12}{191679830229179}a^{22}+\frac{66\!\cdots\!13}{191679830229179}a^{21}-\frac{16\!\cdots\!40}{191679830229179}a^{20}+\frac{32\!\cdots\!71}{191679830229179}a^{19}-\frac{52\!\cdots\!11}{191679830229179}a^{18}+\frac{77\!\cdots\!57}{191679830229179}a^{17}-\frac{10\!\cdots\!27}{191679830229179}a^{16}+\frac{15\!\cdots\!79}{191679830229179}a^{15}-\frac{21\!\cdots\!88}{191679830229179}a^{14}+\frac{27\!\cdots\!98}{191679830229179}a^{13}-\frac{31\!\cdots\!07}{191679830229179}a^{12}+\frac{33\!\cdots\!75}{191679830229179}a^{11}-\frac{33\!\cdots\!97}{191679830229179}a^{10}+\frac{31\!\cdots\!24}{191679830229179}a^{9}-\frac{28\!\cdots\!41}{191679830229179}a^{8}+\frac{23\!\cdots\!44}{191679830229179}a^{7}-\frac{17\!\cdots\!03}{191679830229179}a^{6}+\frac{11\!\cdots\!69}{191679830229179}a^{5}-\frac{66\!\cdots\!92}{191679830229179}a^{4}+\frac{30\!\cdots\!18}{191679830229179}a^{3}-\frac{11\!\cdots\!58}{191679830229179}a^{2}+\frac{41\!\cdots\!70}{191679830229179}a-\frac{884858325660133}{191679830229179}$, $\frac{8537259204596}{191679830229179}a^{23}-\frac{14338870339241}{191679830229179}a^{22}+\frac{20945316892571}{191679830229179}a^{21}-\frac{1041265056030}{191679830229179}a^{20}-\frac{92554557882445}{191679830229179}a^{19}+\frac{119115536955379}{191679830229179}a^{18}-\frac{206032764670443}{191679830229179}a^{17}+\frac{244156048158941}{191679830229179}a^{16}-\frac{374200090640677}{191679830229179}a^{15}+\frac{10\!\cdots\!25}{191679830229179}a^{14}-\frac{12\!\cdots\!69}{191679830229179}a^{13}+\frac{18\!\cdots\!17}{191679830229179}a^{12}-\frac{17\!\cdots\!17}{191679830229179}a^{11}+\frac{18\!\cdots\!44}{191679830229179}a^{10}-\frac{27\!\cdots\!51}{191679830229179}a^{9}+\frac{28\!\cdots\!69}{191679830229179}a^{8}-\frac{32\!\cdots\!97}{191679830229179}a^{7}+\frac{24\!\cdots\!29}{191679830229179}a^{6}-\frac{18\!\cdots\!01}{191679830229179}a^{5}+\frac{20\!\cdots\!77}{191679830229179}a^{4}-\frac{14\!\cdots\!11}{191679830229179}a^{3}+\frac{12\!\cdots\!23}{191679830229179}a^{2}-\frac{615263025961386}{191679830229179}a+\frac{340458458469721}{191679830229179}$, $\frac{805942816283302}{191679830229179}a^{23}-\frac{44\!\cdots\!07}{191679830229179}a^{22}+\frac{14\!\cdots\!77}{191679830229179}a^{21}-\frac{35\!\cdots\!35}{191679830229179}a^{20}+\frac{67\!\cdots\!87}{191679830229179}a^{19}-\frac{10\!\cdots\!61}{191679830229179}a^{18}+\frac{15\!\cdots\!33}{191679830229179}a^{17}-\frac{21\!\cdots\!61}{191679830229179}a^{16}+\frac{31\!\cdots\!21}{191679830229179}a^{15}-\frac{43\!\cdots\!34}{191679830229179}a^{14}+\frac{54\!\cdots\!88}{191679830229179}a^{13}-\frac{62\!\cdots\!97}{191679830229179}a^{12}+\frac{64\!\cdots\!37}{191679830229179}a^{11}-\frac{63\!\cdots\!09}{191679830229179}a^{10}+\frac{59\!\cdots\!98}{191679830229179}a^{9}-\frac{53\!\cdots\!50}{191679830229179}a^{8}+\frac{43\!\cdots\!42}{191679830229179}a^{7}-\frac{31\!\cdots\!34}{191679830229179}a^{6}+\frac{20\!\cdots\!25}{191679830229179}a^{5}-\frac{11\!\cdots\!45}{191679830229179}a^{4}+\frac{48\!\cdots\!58}{191679830229179}a^{3}-\frac{17\!\cdots\!25}{191679830229179}a^{2}+\frac{59\!\cdots\!92}{191679830229179}a-\frac{10\!\cdots\!08}{191679830229179}$ (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)
Galois group
$S_3\times C_{12}$ (as 24T65):
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.
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.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | Deg $24$ | $2$ | $12$ | $24$ | |||
\(5\) | 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 | $x^{12} + 12 x^{10} + 12 x^{9} + 69 x^{8} + 108 x^{7} + 42 x^{6} - 396 x^{5} + 840 x^{4} + 252 x^{3} + 1476 x^{2} + 684 x + 1601$ | $4$ | $3$ | $9$ | $C_{12}$ | $[\ ]_{4}^{3}$ | |
\(7\) | 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) | $0$ | $0$ |