Normalized defining polynomial
\( x^{24} - 8 x^{23} + 102 x^{22} - 612 x^{21} + 4179 x^{20} - 19236 x^{19} + 89532 x^{18} + \cdots + 58491904 \)
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: |
\(45709683595991970654497806956957696000000000000\)
\(\medspace = 2^{24}\cdot 3^{12}\cdot 5^{12}\cdot 7^{12}\cdot 79^{8}\)
| sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
| Root discriminant: | \(87.94\) | 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: | $2\cdot 3^{1/2}5^{1/2}7^{1/2}79^{1/2}\approx 182.1537811850196$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\), \(7\), \(79\)
| sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
| Discriminant root field: | \(\Q\) | ||
| $\card{ \Aut(K/\Q) }$: | $8$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
| |
| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{2048}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{7}+\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{8}a^{8}-\frac{1}{4}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{10}-\frac{1}{16}a^{9}-\frac{1}{8}a^{7}-\frac{1}{16}a^{6}+\frac{1}{16}a^{5}-\frac{3}{8}a^{3}-\frac{1}{2}a$, $\frac{1}{16}a^{11}-\frac{1}{16}a^{9}+\frac{1}{16}a^{7}-\frac{3}{16}a^{5}-\frac{1}{4}a^{4}-\frac{3}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{16}a^{12}-\frac{1}{16}a^{9}-\frac{1}{16}a^{8}-\frac{1}{8}a^{7}-\frac{3}{16}a^{5}-\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{32}a^{13}-\frac{1}{32}a^{12}-\frac{1}{32}a^{11}-\frac{1}{32}a^{10}+\frac{1}{32}a^{9}+\frac{1}{32}a^{8}+\frac{1}{32}a^{7}-\frac{3}{32}a^{6}-\frac{1}{16}a^{5}-\frac{1}{2}a^{3}+\frac{1}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{64}a^{14}-\frac{1}{32}a^{10}-\frac{1}{16}a^{9}-\frac{3}{64}a^{6}-\frac{3}{16}a^{5}-\frac{1}{2}a^{3}-\frac{7}{16}a^{2}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{128}a^{15}+\frac{1}{64}a^{11}+\frac{9}{128}a^{7}-\frac{1}{4}a^{4}+\frac{5}{32}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a$, $\frac{1}{128}a^{16}+\frac{1}{64}a^{12}-\frac{7}{128}a^{8}+\frac{1}{32}a^{4}$, $\frac{1}{128}a^{17}-\frac{1}{64}a^{13}-\frac{1}{32}a^{12}-\frac{1}{32}a^{11}-\frac{1}{32}a^{10}-\frac{3}{128}a^{9}+\frac{1}{32}a^{8}-\frac{3}{32}a^{7}-\frac{3}{32}a^{6}-\frac{7}{32}a^{5}-\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{1}{8}a^{2}-\frac{1}{2}a$, $\frac{1}{256}a^{18}-\frac{1}{256}a^{17}-\frac{1}{128}a^{14}-\frac{1}{128}a^{13}-\frac{1}{32}a^{12}-\frac{7}{256}a^{10}+\frac{7}{256}a^{9}+\frac{1}{32}a^{8}-\frac{1}{8}a^{7}-\frac{1}{32}a^{6}+\frac{15}{64}a^{5}-\frac{3}{8}a^{3}-\frac{7}{16}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{512}a^{19}+\frac{1}{512}a^{17}-\frac{1}{256}a^{16}+\frac{1}{256}a^{13}+\frac{3}{128}a^{12}+\frac{5}{512}a^{11}-\frac{1}{32}a^{10}+\frac{9}{512}a^{9}-\frac{1}{256}a^{8}+\frac{1}{256}a^{7}-\frac{3}{32}a^{6}-\frac{19}{128}a^{5}+\frac{15}{64}a^{4}-\frac{1}{64}a^{3}-\frac{1}{8}a^{2}+\frac{1}{8}a-\frac{1}{2}$, $\frac{1}{2048}a^{20}-\frac{1}{1024}a^{19}-\frac{1}{2048}a^{18}-\frac{3}{1024}a^{17}+\frac{1}{512}a^{16}+\frac{1}{512}a^{15}+\frac{7}{1024}a^{14}-\frac{3}{512}a^{13}+\frac{13}{2048}a^{12}-\frac{17}{1024}a^{11}+\frac{55}{2048}a^{10}+\frac{21}{1024}a^{9}+\frac{19}{1024}a^{8}-\frac{1}{16}a^{7}+\frac{15}{512}a^{6}-\frac{3}{256}a^{5}-\frac{23}{256}a^{4}-\frac{19}{64}a^{3}-\frac{3}{8}a^{2}-\frac{3}{8}a-\frac{1}{4}$, $\frac{1}{2048}a^{21}-\frac{1}{2048}a^{19}+\frac{1}{512}a^{17}+\frac{1}{512}a^{16}+\frac{3}{1024}a^{15}+\frac{13}{2048}a^{13}-\frac{3}{256}a^{12}-\frac{25}{2048}a^{11}+\frac{1}{64}a^{10}+\frac{51}{1024}a^{9}+\frac{1}{512}a^{8}-\frac{19}{512}a^{7}-\frac{5}{64}a^{6}+\frac{1}{256}a^{5}+\frac{1}{128}a^{4}-\frac{17}{64}a^{3}-\frac{7}{16}a^{2}-\frac{3}{8}a-\frac{1}{2}$, $\frac{1}{31\!\cdots\!96}a^{22}-\frac{11\!\cdots\!75}{15\!\cdots\!48}a^{21}-\frac{15\!\cdots\!13}{15\!\cdots\!48}a^{20}+\frac{45\!\cdots\!13}{19\!\cdots\!56}a^{19}+\frac{23\!\cdots\!65}{31\!\cdots\!96}a^{18}-\frac{48\!\cdots\!03}{15\!\cdots\!48}a^{17}+\frac{20\!\cdots\!53}{15\!\cdots\!48}a^{16}-\frac{69\!\cdots\!17}{39\!\cdots\!12}a^{15}+\frac{13\!\cdots\!19}{31\!\cdots\!96}a^{14}+\frac{12\!\cdots\!07}{15\!\cdots\!48}a^{13}+\frac{39\!\cdots\!17}{15\!\cdots\!48}a^{12}-\frac{25\!\cdots\!39}{24\!\cdots\!32}a^{11}+\frac{71\!\cdots\!59}{31\!\cdots\!96}a^{10}-\frac{73\!\cdots\!45}{15\!\cdots\!48}a^{9}-\frac{52\!\cdots\!17}{15\!\cdots\!48}a^{8}+\frac{34\!\cdots\!55}{39\!\cdots\!12}a^{7}+\frac{92\!\cdots\!91}{78\!\cdots\!24}a^{6}-\frac{93\!\cdots\!73}{39\!\cdots\!12}a^{5}+\frac{10\!\cdots\!39}{39\!\cdots\!12}a^{4}+\frac{25\!\cdots\!35}{57\!\cdots\!84}a^{3}-\frac{17\!\cdots\!97}{60\!\cdots\!08}a^{2}+\frac{38\!\cdots\!43}{12\!\cdots\!16}a+\frac{11\!\cdots\!43}{25\!\cdots\!72}$, $\frac{1}{44\!\cdots\!88}a^{23}+\frac{59\!\cdots\!93}{44\!\cdots\!88}a^{22}+\frac{47\!\cdots\!91}{22\!\cdots\!44}a^{21}-\frac{18\!\cdots\!77}{11\!\cdots\!72}a^{20}-\frac{99\!\cdots\!85}{44\!\cdots\!88}a^{19}+\frac{14\!\cdots\!71}{44\!\cdots\!88}a^{18}-\frac{18\!\cdots\!67}{11\!\cdots\!72}a^{17}+\frac{17\!\cdots\!07}{22\!\cdots\!44}a^{16}-\frac{10\!\cdots\!57}{44\!\cdots\!88}a^{15}+\frac{16\!\cdots\!31}{44\!\cdots\!88}a^{14}-\frac{17\!\cdots\!89}{22\!\cdots\!44}a^{13}-\frac{16\!\cdots\!17}{55\!\cdots\!36}a^{12}+\frac{93\!\cdots\!37}{44\!\cdots\!88}a^{11}-\frac{32\!\cdots\!47}{44\!\cdots\!88}a^{10}-\frac{15\!\cdots\!93}{27\!\cdots\!68}a^{9}+\frac{16\!\cdots\!35}{22\!\cdots\!44}a^{8}+\frac{21\!\cdots\!39}{11\!\cdots\!72}a^{7}+\frac{32\!\cdots\!41}{11\!\cdots\!72}a^{6}+\frac{16\!\cdots\!19}{13\!\cdots\!84}a^{5}-\frac{60\!\cdots\!69}{55\!\cdots\!36}a^{4}+\frac{46\!\cdots\!13}{13\!\cdots\!84}a^{3}-\frac{11\!\cdots\!11}{34\!\cdots\!96}a^{2}-\frac{14\!\cdots\!63}{17\!\cdots\!48}a+\frac{19\!\cdots\!89}{36\!\cdots\!16}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
$C_{2}\times C_{66572}$, which has order $133144$ (assuming GRH)
Unit group
| Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
| Torsion generator: |
\( -\frac{459411304400839888851}{17789599986227678503618405888} a^{23} + \frac{3763570167072784863913}{17789599986227678503618405888} a^{22} - \frac{46373891063689651132841}{17789599986227678503618405888} a^{21} + \frac{281682785030315058618989}{17789599986227678503618405888} a^{20} - \frac{932229259100234072227001}{8894799993113839251809202944} a^{19} + \frac{4303017318268350835872627}{8894799993113839251809202944} a^{18} - \frac{9700286202505765435474601}{4447399996556919625904601472} a^{17} + \frac{69255460343875853684581493}{8894799993113839251809202944} a^{16} - \frac{459045176726514790915831727}{17789599986227678503618405888} a^{15} + \frac{1292953926912676857447553641}{17789599986227678503618405888} a^{14} - \frac{3306857445918411568661139765}{17789599986227678503618405888} a^{13} + \frac{7512096432276593808669438709}{17789599986227678503618405888} a^{12} - \frac{59441082271093680366071865}{69490624946201869154759398} a^{11} + \frac{3435144018061755826062023117}{2223699998278459812952300736} a^{10} - \frac{22260024691537357712551032459}{8894799993113839251809202944} a^{9} + \frac{15521677245184072848993528781}{4447399996556919625904601472} a^{8} - \frac{19463329707555338752556281015}{4447399996556919625904601472} a^{7} + \frac{2540592952704045732681555407}{555924999569614953238075184} a^{6} - \frac{8871817588602631277458911821}{2223699998278459812952300736} a^{5} + \frac{217155206137257352045133623}{69490624946201869154759398} a^{4} - \frac{2588015391105294889128541707}{1111849999139229906476150368} a^{3} + \frac{426497101055702891661414857}{277962499784807476619037592} a^{2} - \frac{287870944438655549045472957}{138981249892403738309518796} a + \frac{87774549512710888105963}{145377876456489266014141} \)
(order $4$)
| 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: |
$\frac{23\!\cdots\!93}{66\!\cdots\!16}a^{23}-\frac{65\!\cdots\!85}{26\!\cdots\!64}a^{22}+\frac{88\!\cdots\!31}{26\!\cdots\!64}a^{21}-\frac{47\!\cdots\!07}{26\!\cdots\!64}a^{20}+\frac{19\!\cdots\!09}{15\!\cdots\!92}a^{19}-\frac{17\!\cdots\!19}{33\!\cdots\!08}a^{18}+\frac{16\!\cdots\!97}{66\!\cdots\!16}a^{17}-\frac{10\!\cdots\!41}{13\!\cdots\!32}a^{16}+\frac{21\!\cdots\!67}{78\!\cdots\!96}a^{15}-\frac{18\!\cdots\!49}{26\!\cdots\!64}a^{14}+\frac{48\!\cdots\!59}{26\!\cdots\!64}a^{13}-\frac{10\!\cdots\!75}{26\!\cdots\!64}a^{12}+\frac{19\!\cdots\!61}{26\!\cdots\!64}a^{11}-\frac{16\!\cdots\!33}{13\!\cdots\!32}a^{10}+\frac{24\!\cdots\!97}{13\!\cdots\!32}a^{9}-\frac{77\!\cdots\!47}{33\!\cdots\!08}a^{8}+\frac{16\!\cdots\!09}{66\!\cdots\!16}a^{7}-\frac{60\!\cdots\!79}{33\!\cdots\!08}a^{6}+\frac{41\!\cdots\!11}{33\!\cdots\!08}a^{5}-\frac{49\!\cdots\!95}{16\!\cdots\!04}a^{4}-\frac{54\!\cdots\!55}{41\!\cdots\!76}a^{3}-\frac{17\!\cdots\!19}{10\!\cdots\!44}a^{2}+\frac{20\!\cdots\!07}{25\!\cdots\!86}a+\frac{52\!\cdots\!07}{25\!\cdots\!86}$, $\frac{57\!\cdots\!27}{93\!\cdots\!12}a^{23}-\frac{65\!\cdots\!15}{14\!\cdots\!92}a^{22}+\frac{85\!\cdots\!97}{14\!\cdots\!92}a^{21}-\frac{46\!\cdots\!37}{14\!\cdots\!92}a^{20}+\frac{18\!\cdots\!31}{88\!\cdots\!76}a^{19}-\frac{84\!\cdots\!47}{93\!\cdots\!12}a^{18}+\frac{15\!\cdots\!33}{37\!\cdots\!48}a^{17}-\frac{99\!\cdots\!87}{74\!\cdots\!96}a^{16}+\frac{19\!\cdots\!87}{44\!\cdots\!88}a^{15}-\frac{16\!\cdots\!07}{14\!\cdots\!92}a^{14}+\frac{40\!\cdots\!41}{14\!\cdots\!92}a^{13}-\frac{84\!\cdots\!29}{14\!\cdots\!92}a^{12}+\frac{15\!\cdots\!67}{14\!\cdots\!92}a^{11}-\frac{12\!\cdots\!71}{74\!\cdots\!96}a^{10}+\frac{18\!\cdots\!19}{74\!\cdots\!96}a^{9}-\frac{53\!\cdots\!75}{18\!\cdots\!24}a^{8}+\frac{11\!\cdots\!19}{37\!\cdots\!48}a^{7}-\frac{35\!\cdots\!65}{18\!\cdots\!24}a^{6}+\frac{10\!\cdots\!73}{18\!\cdots\!24}a^{5}+\frac{57\!\cdots\!83}{93\!\cdots\!12}a^{4}-\frac{10\!\cdots\!09}{23\!\cdots\!28}a^{3}+\frac{13\!\cdots\!89}{11\!\cdots\!64}a^{2}-\frac{33\!\cdots\!33}{29\!\cdots\!16}a+\frac{99\!\cdots\!96}{30\!\cdots\!61}$, $\frac{10\!\cdots\!71}{55\!\cdots\!36}a^{23}-\frac{66\!\cdots\!51}{44\!\cdots\!88}a^{22}+\frac{84\!\cdots\!43}{44\!\cdots\!88}a^{21}-\frac{50\!\cdots\!75}{44\!\cdots\!88}a^{20}+\frac{20\!\cdots\!69}{25\!\cdots\!64}a^{19}-\frac{79\!\cdots\!31}{22\!\cdots\!44}a^{18}+\frac{91\!\cdots\!93}{55\!\cdots\!36}a^{17}-\frac{13\!\cdots\!23}{22\!\cdots\!44}a^{16}+\frac{25\!\cdots\!33}{12\!\cdots\!32}a^{15}-\frac{25\!\cdots\!47}{44\!\cdots\!88}a^{14}+\frac{65\!\cdots\!35}{44\!\cdots\!88}a^{13}-\frac{14\!\cdots\!19}{44\!\cdots\!88}a^{12}+\frac{30\!\cdots\!57}{44\!\cdots\!88}a^{11}-\frac{14\!\cdots\!39}{11\!\cdots\!72}a^{10}+\frac{46\!\cdots\!39}{22\!\cdots\!44}a^{9}-\frac{33\!\cdots\!47}{11\!\cdots\!72}a^{8}+\frac{43\!\cdots\!45}{11\!\cdots\!72}a^{7}-\frac{59\!\cdots\!51}{13\!\cdots\!84}a^{6}+\frac{22\!\cdots\!85}{55\!\cdots\!36}a^{5}-\frac{23\!\cdots\!77}{68\!\cdots\!92}a^{4}+\frac{17\!\cdots\!93}{68\!\cdots\!92}a^{3}-\frac{77\!\cdots\!41}{34\!\cdots\!96}a^{2}+\frac{22\!\cdots\!33}{86\!\cdots\!24}a-\frac{18\!\cdots\!63}{18\!\cdots\!58}$, $\frac{23\!\cdots\!93}{66\!\cdots\!16}a^{23}-\frac{65\!\cdots\!85}{26\!\cdots\!64}a^{22}+\frac{88\!\cdots\!31}{26\!\cdots\!64}a^{21}-\frac{47\!\cdots\!07}{26\!\cdots\!64}a^{20}+\frac{19\!\cdots\!09}{15\!\cdots\!92}a^{19}-\frac{17\!\cdots\!19}{33\!\cdots\!08}a^{18}+\frac{16\!\cdots\!97}{66\!\cdots\!16}a^{17}-\frac{10\!\cdots\!41}{13\!\cdots\!32}a^{16}+\frac{21\!\cdots\!67}{78\!\cdots\!96}a^{15}-\frac{18\!\cdots\!49}{26\!\cdots\!64}a^{14}+\frac{48\!\cdots\!59}{26\!\cdots\!64}a^{13}-\frac{10\!\cdots\!75}{26\!\cdots\!64}a^{12}+\frac{19\!\cdots\!61}{26\!\cdots\!64}a^{11}-\frac{16\!\cdots\!33}{13\!\cdots\!32}a^{10}+\frac{24\!\cdots\!97}{13\!\cdots\!32}a^{9}-\frac{77\!\cdots\!47}{33\!\cdots\!08}a^{8}+\frac{16\!\cdots\!09}{66\!\cdots\!16}a^{7}-\frac{60\!\cdots\!79}{33\!\cdots\!08}a^{6}+\frac{41\!\cdots\!11}{33\!\cdots\!08}a^{5}-\frac{49\!\cdots\!95}{16\!\cdots\!04}a^{4}-\frac{54\!\cdots\!55}{41\!\cdots\!76}a^{3}-\frac{17\!\cdots\!19}{10\!\cdots\!44}a^{2}+\frac{20\!\cdots\!07}{25\!\cdots\!86}a+\frac{10\!\cdots\!79}{25\!\cdots\!86}$, $\frac{13\!\cdots\!85}{22\!\cdots\!44}a^{23}-\frac{23\!\cdots\!41}{44\!\cdots\!88}a^{22}+\frac{27\!\cdots\!15}{44\!\cdots\!88}a^{21}-\frac{17\!\cdots\!47}{44\!\cdots\!88}a^{20}+\frac{64\!\cdots\!55}{25\!\cdots\!64}a^{19}-\frac{12\!\cdots\!25}{11\!\cdots\!72}a^{18}+\frac{27\!\cdots\!75}{55\!\cdots\!36}a^{17}-\frac{38\!\cdots\!21}{22\!\cdots\!44}a^{16}+\frac{17\!\cdots\!55}{32\!\cdots\!08}a^{15}-\frac{64\!\cdots\!93}{44\!\cdots\!88}a^{14}+\frac{15\!\cdots\!63}{44\!\cdots\!88}a^{13}-\frac{32\!\cdots\!23}{44\!\cdots\!88}a^{12}+\frac{60\!\cdots\!03}{44\!\cdots\!88}a^{11}-\frac{48\!\cdots\!55}{22\!\cdots\!44}a^{10}+\frac{71\!\cdots\!55}{22\!\cdots\!44}a^{9}-\frac{20\!\cdots\!43}{55\!\cdots\!36}a^{8}+\frac{42\!\cdots\!27}{11\!\cdots\!72}a^{7}-\frac{19\!\cdots\!97}{55\!\cdots\!36}a^{6}+\frac{98\!\cdots\!61}{55\!\cdots\!36}a^{5}-\frac{41\!\cdots\!35}{27\!\cdots\!68}a^{4}+\frac{29\!\cdots\!71}{86\!\cdots\!24}a^{3}-\frac{49\!\cdots\!31}{34\!\cdots\!96}a^{2}+\frac{68\!\cdots\!54}{21\!\cdots\!31}a-\frac{30\!\cdots\!32}{90\!\cdots\!29}$, $\frac{24\!\cdots\!31}{26\!\cdots\!64}a^{23}-\frac{22\!\cdots\!41}{26\!\cdots\!64}a^{22}+\frac{27\!\cdots\!45}{26\!\cdots\!64}a^{21}-\frac{17\!\cdots\!09}{26\!\cdots\!64}a^{20}+\frac{14\!\cdots\!55}{33\!\cdots\!08}a^{19}-\frac{27\!\cdots\!83}{13\!\cdots\!32}a^{18}+\frac{12\!\cdots\!81}{13\!\cdots\!32}a^{17}-\frac{45\!\cdots\!95}{13\!\cdots\!32}a^{16}+\frac{29\!\cdots\!51}{26\!\cdots\!64}a^{15}-\frac{83\!\cdots\!77}{26\!\cdots\!64}a^{14}+\frac{21\!\cdots\!37}{26\!\cdots\!64}a^{13}-\frac{46\!\cdots\!57}{26\!\cdots\!64}a^{12}+\frac{45\!\cdots\!35}{13\!\cdots\!32}a^{11}-\frac{49\!\cdots\!15}{83\!\cdots\!52}a^{10}+\frac{58\!\cdots\!25}{66\!\cdots\!16}a^{9}-\frac{94\!\cdots\!45}{83\!\cdots\!52}a^{8}+\frac{39\!\cdots\!23}{33\!\cdots\!08}a^{7}-\frac{15\!\cdots\!09}{16\!\cdots\!04}a^{6}+\frac{23\!\cdots\!63}{41\!\cdots\!76}a^{5}-\frac{91\!\cdots\!59}{16\!\cdots\!04}a^{4}+\frac{14\!\cdots\!13}{41\!\cdots\!76}a^{3}-\frac{15\!\cdots\!11}{20\!\cdots\!88}a^{2}+\frac{59\!\cdots\!72}{12\!\cdots\!93}a+\frac{13\!\cdots\!21}{12\!\cdots\!93}$, $\frac{15\!\cdots\!29}{34\!\cdots\!88}a^{23}-\frac{10\!\cdots\!61}{27\!\cdots\!04}a^{22}+\frac{12\!\cdots\!87}{27\!\cdots\!04}a^{21}-\frac{76\!\cdots\!69}{27\!\cdots\!04}a^{20}+\frac{52\!\cdots\!49}{27\!\cdots\!04}a^{19}-\frac{11\!\cdots\!01}{13\!\cdots\!52}a^{18}+\frac{13\!\cdots\!61}{34\!\cdots\!88}a^{17}-\frac{18\!\cdots\!37}{13\!\cdots\!52}a^{16}+\frac{62\!\cdots\!41}{13\!\cdots\!52}a^{15}-\frac{33\!\cdots\!37}{27\!\cdots\!04}a^{14}+\frac{86\!\cdots\!31}{27\!\cdots\!04}a^{13}-\frac{18\!\cdots\!41}{27\!\cdots\!04}a^{12}+\frac{36\!\cdots\!49}{27\!\cdots\!04}a^{11}-\frac{15\!\cdots\!37}{69\!\cdots\!76}a^{10}+\frac{45\!\cdots\!27}{13\!\cdots\!52}a^{9}-\frac{28\!\cdots\!03}{69\!\cdots\!76}a^{8}+\frac{29\!\cdots\!81}{69\!\cdots\!76}a^{7}-\frac{26\!\cdots\!27}{86\!\cdots\!72}a^{6}+\frac{54\!\cdots\!29}{34\!\cdots\!88}a^{5}+\frac{75\!\cdots\!73}{86\!\cdots\!72}a^{4}-\frac{13\!\cdots\!52}{26\!\cdots\!21}a^{3}-\frac{53\!\cdots\!95}{21\!\cdots\!68}a^{2}+\frac{10\!\cdots\!51}{53\!\cdots\!42}a-\frac{62\!\cdots\!68}{26\!\cdots\!21}$, $\frac{37\!\cdots\!19}{22\!\cdots\!44}a^{23}-\frac{37\!\cdots\!15}{22\!\cdots\!44}a^{22}+\frac{42\!\cdots\!31}{22\!\cdots\!44}a^{21}-\frac{28\!\cdots\!59}{22\!\cdots\!44}a^{20}+\frac{90\!\cdots\!31}{11\!\cdots\!72}a^{19}-\frac{44\!\cdots\!57}{11\!\cdots\!72}a^{18}+\frac{19\!\cdots\!35}{11\!\cdots\!72}a^{17}-\frac{73\!\cdots\!61}{11\!\cdots\!72}a^{16}+\frac{48\!\cdots\!83}{22\!\cdots\!44}a^{15}-\frac{13\!\cdots\!51}{22\!\cdots\!44}a^{14}+\frac{34\!\cdots\!67}{22\!\cdots\!44}a^{13}-\frac{46\!\cdots\!55}{12\!\cdots\!32}a^{12}+\frac{39\!\cdots\!97}{55\!\cdots\!36}a^{11}-\frac{35\!\cdots\!63}{27\!\cdots\!68}a^{10}+\frac{11\!\cdots\!41}{55\!\cdots\!36}a^{9}-\frac{38\!\cdots\!93}{13\!\cdots\!84}a^{8}+\frac{92\!\cdots\!45}{27\!\cdots\!68}a^{7}-\frac{12\!\cdots\!19}{34\!\cdots\!96}a^{6}+\frac{20\!\cdots\!21}{68\!\cdots\!92}a^{5}-\frac{31\!\cdots\!49}{13\!\cdots\!84}a^{4}+\frac{14\!\cdots\!73}{68\!\cdots\!92}a^{3}-\frac{69\!\cdots\!57}{34\!\cdots\!96}a^{2}+\frac{15\!\cdots\!86}{12\!\cdots\!43}a-\frac{11\!\cdots\!69}{18\!\cdots\!58}$, $\frac{24\!\cdots\!23}{26\!\cdots\!64}a^{23}-\frac{47\!\cdots\!23}{26\!\cdots\!64}a^{22}+\frac{45\!\cdots\!61}{26\!\cdots\!64}a^{21}-\frac{40\!\cdots\!13}{26\!\cdots\!64}a^{20}+\frac{30\!\cdots\!09}{33\!\cdots\!08}a^{19}-\frac{17\!\cdots\!07}{33\!\cdots\!08}a^{18}+\frac{30\!\cdots\!23}{13\!\cdots\!32}a^{17}-\frac{12\!\cdots\!65}{13\!\cdots\!32}a^{16}+\frac{84\!\cdots\!27}{26\!\cdots\!64}a^{15}-\frac{25\!\cdots\!95}{26\!\cdots\!64}a^{14}+\frac{67\!\cdots\!33}{26\!\cdots\!64}a^{13}-\frac{15\!\cdots\!65}{26\!\cdots\!64}a^{12}+\frac{16\!\cdots\!75}{13\!\cdots\!32}a^{11}-\frac{30\!\cdots\!55}{13\!\cdots\!32}a^{10}+\frac{12\!\cdots\!51}{33\!\cdots\!08}a^{9}-\frac{36\!\cdots\!15}{66\!\cdots\!16}a^{8}+\frac{22\!\cdots\!17}{33\!\cdots\!08}a^{7}-\frac{24\!\cdots\!57}{33\!\cdots\!08}a^{6}+\frac{11\!\cdots\!29}{16\!\cdots\!04}a^{5}-\frac{40\!\cdots\!07}{83\!\cdots\!52}a^{4}+\frac{89\!\cdots\!59}{20\!\cdots\!88}a^{3}-\frac{94\!\cdots\!11}{20\!\cdots\!88}a^{2}+\frac{14\!\cdots\!87}{51\!\cdots\!72}a-\frac{46\!\cdots\!35}{25\!\cdots\!86}$, $\frac{29\!\cdots\!73}{11\!\cdots\!72}a^{23}-\frac{65\!\cdots\!95}{44\!\cdots\!88}a^{22}+\frac{10\!\cdots\!33}{44\!\cdots\!88}a^{21}-\frac{27\!\cdots\!81}{25\!\cdots\!64}a^{20}+\frac{37\!\cdots\!59}{44\!\cdots\!88}a^{19}-\frac{34\!\cdots\!31}{11\!\cdots\!72}a^{18}+\frac{18\!\cdots\!67}{11\!\cdots\!72}a^{17}-\frac{10\!\cdots\!95}{22\!\cdots\!44}a^{16}+\frac{40\!\cdots\!53}{22\!\cdots\!44}a^{15}-\frac{18\!\cdots\!03}{44\!\cdots\!88}a^{14}+\frac{54\!\cdots\!37}{44\!\cdots\!88}a^{13}-\frac{10\!\cdots\!09}{44\!\cdots\!88}a^{12}+\frac{13\!\cdots\!07}{25\!\cdots\!64}a^{11}-\frac{18\!\cdots\!17}{22\!\cdots\!44}a^{10}+\frac{31\!\cdots\!07}{22\!\cdots\!44}a^{9}-\frac{49\!\cdots\!89}{27\!\cdots\!68}a^{8}+\frac{25\!\cdots\!95}{11\!\cdots\!72}a^{7}-\frac{10\!\cdots\!87}{55\!\cdots\!36}a^{6}+\frac{76\!\cdots\!69}{32\!\cdots\!08}a^{5}-\frac{23\!\cdots\!03}{27\!\cdots\!68}a^{4}+\frac{65\!\cdots\!77}{34\!\cdots\!96}a^{3}-\frac{29\!\cdots\!89}{34\!\cdots\!96}a^{2}+\frac{43\!\cdots\!27}{43\!\cdots\!62}a-\frac{21\!\cdots\!33}{90\!\cdots\!29}$, $\frac{88\!\cdots\!37}{44\!\cdots\!88}a^{23}-\frac{34\!\cdots\!39}{22\!\cdots\!44}a^{22}+\frac{21\!\cdots\!47}{11\!\cdots\!72}a^{21}-\frac{25\!\cdots\!81}{22\!\cdots\!44}a^{20}+\frac{33\!\cdots\!43}{44\!\cdots\!88}a^{19}-\frac{37\!\cdots\!27}{11\!\cdots\!72}a^{18}+\frac{33\!\cdots\!23}{22\!\cdots\!44}a^{17}-\frac{28\!\cdots\!19}{55\!\cdots\!36}a^{16}+\frac{73\!\cdots\!03}{44\!\cdots\!88}a^{15}-\frac{99\!\cdots\!67}{22\!\cdots\!44}a^{14}+\frac{61\!\cdots\!19}{55\!\cdots\!36}a^{13}-\frac{53\!\cdots\!89}{22\!\cdots\!44}a^{12}+\frac{20\!\cdots\!13}{44\!\cdots\!88}a^{11}-\frac{44\!\cdots\!75}{55\!\cdots\!36}a^{10}+\frac{26\!\cdots\!23}{22\!\cdots\!44}a^{9}-\frac{17\!\cdots\!51}{11\!\cdots\!72}a^{8}+\frac{20\!\cdots\!33}{11\!\cdots\!72}a^{7}-\frac{24\!\cdots\!61}{13\!\cdots\!84}a^{6}+\frac{73\!\cdots\!23}{55\!\cdots\!36}a^{5}-\frac{26\!\cdots\!69}{27\!\cdots\!68}a^{4}+\frac{56\!\cdots\!43}{68\!\cdots\!92}a^{3}-\frac{15\!\cdots\!55}{34\!\cdots\!96}a^{2}+\frac{25\!\cdots\!21}{86\!\cdots\!24}a-\frac{20\!\cdots\!88}{90\!\cdots\!29}$
| 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: | \( 1125992180218.7246 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 1125992180218.7246 \cdot 133144}{4\cdot\sqrt{45709683595991970654497806956957696000000000000}}\cr\approx \mathstrut & 663.668639092578 \end{aligned}\] (assuming GRH)
Galois group
$C_2^2\times D_6$ (as 24T30):
| A solvable group of order 48 |
| The 24 conjugacy class representatives for $C_2^2\times D_6$ |
| Character table for $C_2^2\times D_6$ |
Intermediate fields
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 24 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 | R | R | R | ${\href{/padicField/11.2.0.1}{2} }^{12}$ | ${\href{/padicField/13.6.0.1}{6} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{12}$ | ${\href{/padicField/19.2.0.1}{2} }^{12}$ | ${\href{/padicField/23.2.0.1}{2} }^{12}$ | ${\href{/padicField/29.2.0.1}{2} }^{12}$ | ${\href{/padicField/31.2.0.1}{2} }^{12}$ | ${\href{/padicField/37.2.0.1}{2} }^{12}$ | ${\href{/padicField/41.2.0.1}{2} }^{12}$ | ${\href{/padicField/43.2.0.1}{2} }^{12}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{8}{,}\,{\href{/padicField/53.1.0.1}{1} }^{8}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
| 2.2.2.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $[2]$ | |
|
\(3\)
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
| 3.12.6.2 | $x^{12} + 22 x^{10} + 177 x^{8} + 4 x^{7} + 644 x^{6} - 100 x^{5} + 876 x^{4} - 224 x^{3} + 1076 x^{2} + 344 x + 112$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(5\)
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
| 5.12.6.1 | $x^{12} + 120 x^{11} + 6032 x^{10} + 163208 x^{9} + 2529528 x^{8} + 21853448 x^{7} + 92223962 x^{6} + 138649448 x^{5} + 223472880 x^{4} + 401794296 x^{3} + 295909124 x^{2} + 118616440 x + 126881009$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(7\)
| 7.12.6.1 | $x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ |
| 7.12.6.1 | $x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
|
\(79\)
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.2.0.1 | $x^{2} + 78 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
| 79.4.2.1 | $x^{4} + 156 x^{3} + 6248 x^{2} + 12792 x + 486412$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |