Normalized defining polynomial
\( x^{24} + 42 x^{22} + 717 x^{20} + 6498 x^{18} + 34350 x^{16} + 110088 x^{14} + 216407 x^{12} + 257262 x^{10} + \cdots + 1 \)
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: | \(52792842355679189725978706186699601\) \(\medspace = 3^{36}\cdot 7^{12}\cdot 71^{4}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
| |
Root discriminant: | \(27.98\) | 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: | $3^{3/2}7^{1/2}71^{1/2}\approx 115.84040745784694$ | ||
Ramified primes: | \(3\), \(7\), \(71\) | 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{11}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{21\!\cdots\!26}a^{22}+\frac{35\!\cdots\!95}{21\!\cdots\!26}a^{20}-\frac{10\!\cdots\!77}{21\!\cdots\!26}a^{18}-\frac{32\!\cdots\!14}{10\!\cdots\!63}a^{16}-\frac{11\!\cdots\!52}{10\!\cdots\!63}a^{14}+\frac{72\!\cdots\!65}{10\!\cdots\!63}a^{12}-\frac{1}{2}a^{11}+\frac{41\!\cdots\!35}{10\!\cdots\!63}a^{10}-\frac{22\!\cdots\!26}{10\!\cdots\!63}a^{8}-\frac{1}{2}a^{7}-\frac{23\!\cdots\!47}{10\!\cdots\!63}a^{6}+\frac{39\!\cdots\!45}{10\!\cdots\!63}a^{4}-\frac{46\!\cdots\!05}{10\!\cdots\!63}a^{2}-\frac{1}{2}a-\frac{63\!\cdots\!29}{21\!\cdots\!26}$, $\frac{1}{21\!\cdots\!26}a^{23}+\frac{35\!\cdots\!95}{21\!\cdots\!26}a^{21}-\frac{10\!\cdots\!77}{21\!\cdots\!26}a^{19}-\frac{32\!\cdots\!14}{10\!\cdots\!63}a^{17}-\frac{11\!\cdots\!52}{10\!\cdots\!63}a^{15}+\frac{72\!\cdots\!65}{10\!\cdots\!63}a^{13}+\frac{41\!\cdots\!35}{10\!\cdots\!63}a^{11}-\frac{1}{2}a^{10}-\frac{22\!\cdots\!26}{10\!\cdots\!63}a^{9}-\frac{1}{2}a^{8}-\frac{23\!\cdots\!47}{10\!\cdots\!63}a^{7}-\frac{1}{2}a^{6}+\frac{39\!\cdots\!45}{10\!\cdots\!63}a^{5}-\frac{46\!\cdots\!05}{10\!\cdots\!63}a^{3}-\frac{1}{2}a^{2}+\frac{21\!\cdots\!67}{10\!\cdots\!63}a-\frac{1}{2}$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
$C_{5}$, which has order $5$ (assuming GRH)
Unit group
Rank: | $11$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
| |
Torsion generator: | \( \frac{35126590884879177}{1065173996492403463} a^{23} - \frac{32746920036131}{9772238499930307} a^{22} + \frac{1485480723094494431}{1065173996492403463} a^{21} - \frac{1384716754796059}{9772238499930307} a^{20} + \frac{51188870436250334755}{2130347992984806926} a^{19} - \frac{23862979590568251}{9772238499930307} a^{18} + \frac{469613327104460219481}{2130347992984806926} a^{17} - \frac{219117187070216119}{9772238499930307} a^{16} + \frac{1260854610858410396360}{1065173996492403463} a^{15} - \frac{1179422445993268052}{9772238499930307} a^{14} + \frac{8236431858755742405413}{2130347992984806926} a^{13} - \frac{7745851076014649295}{19544476999860614} a^{12} + \frac{8264584747522266795726}{1065173996492403463} a^{11} - \frac{15710147554858830857}{19544476999860614} a^{10} + \frac{10021235560883379344505}{1065173996492403463} a^{9} - \frac{19409001519627789083}{19544476999860614} a^{8} + \frac{13917471475244104259857}{2130347992984806926} a^{7} - \frac{13893958778888757251}{19544476999860614} a^{6} + \frac{2458399647644588765979}{1065173996492403463} a^{5} - \frac{2570649248547882730}{9772238499930307} a^{4} + \frac{341821424285860659032}{1065173996492403463} a^{3} - \frac{371528891039801659}{9772238499930307} a^{2} + \frac{29112002454062396145}{2130347992984806926} a - \frac{16790809112147057}{19544476999860614} \) (order $18$) | 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: | $\frac{79\!\cdots\!17}{21\!\cdots\!26}a^{23}+\frac{49\!\cdots\!02}{10\!\cdots\!63}a^{22}+\frac{33\!\cdots\!35}{21\!\cdots\!26}a^{21}+\frac{41\!\cdots\!75}{21\!\cdots\!26}a^{20}+\frac{28\!\cdots\!75}{10\!\cdots\!63}a^{19}+\frac{69\!\cdots\!13}{21\!\cdots\!26}a^{18}+\frac{51\!\cdots\!17}{21\!\cdots\!26}a^{17}+\frac{62\!\cdots\!69}{21\!\cdots\!26}a^{16}+\frac{13\!\cdots\!11}{10\!\cdots\!63}a^{15}+\frac{16\!\cdots\!17}{10\!\cdots\!63}a^{14}+\frac{85\!\cdots\!09}{21\!\cdots\!26}a^{13}+\frac{10\!\cdots\!43}{21\!\cdots\!26}a^{12}+\frac{82\!\cdots\!04}{10\!\cdots\!63}a^{11}+\frac{19\!\cdots\!49}{21\!\cdots\!26}a^{10}+\frac{18\!\cdots\!91}{21\!\cdots\!26}a^{9}+\frac{21\!\cdots\!25}{21\!\cdots\!26}a^{8}+\frac{11\!\cdots\!89}{21\!\cdots\!26}a^{7}+\frac{67\!\cdots\!85}{10\!\cdots\!63}a^{6}+\frac{30\!\cdots\!25}{21\!\cdots\!26}a^{5}+\frac{39\!\cdots\!91}{21\!\cdots\!26}a^{4}+\frac{29\!\cdots\!62}{10\!\cdots\!63}a^{3}+\frac{32\!\cdots\!75}{21\!\cdots\!26}a^{2}-\frac{18\!\cdots\!29}{21\!\cdots\!26}a+\frac{11\!\cdots\!92}{10\!\cdots\!63}$, $\frac{16\!\cdots\!37}{10\!\cdots\!63}a^{23}+\frac{13\!\cdots\!97}{10\!\cdots\!63}a^{22}+\frac{69\!\cdots\!88}{10\!\cdots\!63}a^{21}+\frac{55\!\cdots\!00}{10\!\cdots\!63}a^{20}+\frac{23\!\cdots\!55}{21\!\cdots\!26}a^{19}+\frac{94\!\cdots\!66}{10\!\cdots\!63}a^{18}+\frac{10\!\cdots\!05}{10\!\cdots\!63}a^{17}+\frac{17\!\cdots\!89}{21\!\cdots\!26}a^{16}+\frac{55\!\cdots\!42}{10\!\cdots\!63}a^{15}+\frac{45\!\cdots\!52}{10\!\cdots\!63}a^{14}+\frac{34\!\cdots\!45}{21\!\cdots\!26}a^{13}+\frac{29\!\cdots\!05}{21\!\cdots\!26}a^{12}+\frac{33\!\cdots\!57}{10\!\cdots\!63}a^{11}+\frac{58\!\cdots\!01}{21\!\cdots\!26}a^{10}+\frac{76\!\cdots\!11}{21\!\cdots\!26}a^{9}+\frac{34\!\cdots\!71}{10\!\cdots\!63}a^{8}+\frac{24\!\cdots\!76}{10\!\cdots\!63}a^{7}+\frac{47\!\cdots\!81}{21\!\cdots\!26}a^{6}+\frac{75\!\cdots\!19}{10\!\cdots\!63}a^{5}+\frac{76\!\cdots\!62}{10\!\cdots\!63}a^{4}+\frac{69\!\cdots\!60}{10\!\cdots\!63}a^{3}+\frac{68\!\cdots\!66}{10\!\cdots\!63}a^{2}+\frac{16\!\cdots\!63}{21\!\cdots\!26}a+\frac{44\!\cdots\!67}{10\!\cdots\!63}$, $\frac{14\!\cdots\!63}{21\!\cdots\!26}a^{23}+\frac{17709897201907}{97\!\cdots\!07}a^{22}+\frac{31\!\cdots\!79}{10\!\cdots\!63}a^{21}+\frac{761240155900490}{97\!\cdots\!07}a^{20}+\frac{10\!\cdots\!03}{21\!\cdots\!26}a^{19}+\frac{13\!\cdots\!42}{97\!\cdots\!07}a^{18}+\frac{98\!\cdots\!87}{21\!\cdots\!26}a^{17}+\frac{25\!\cdots\!37}{19\!\cdots\!14}a^{16}+\frac{26\!\cdots\!03}{10\!\cdots\!63}a^{15}+\frac{70\!\cdots\!09}{97\!\cdots\!07}a^{14}+\frac{17\!\cdots\!03}{21\!\cdots\!26}a^{13}+\frac{48\!\cdots\!99}{19\!\cdots\!14}a^{12}+\frac{35\!\cdots\!69}{21\!\cdots\!26}a^{11}+\frac{10\!\cdots\!05}{19\!\cdots\!14}a^{10}+\frac{21\!\cdots\!86}{10\!\cdots\!63}a^{9}+\frac{13\!\cdots\!37}{19\!\cdots\!14}a^{8}+\frac{14\!\cdots\!03}{10\!\cdots\!63}a^{7}+\frac{94\!\cdots\!19}{19\!\cdots\!14}a^{6}+\frac{10\!\cdots\!57}{21\!\cdots\!26}a^{5}+\frac{17\!\cdots\!62}{97\!\cdots\!07}a^{4}+\frac{75\!\cdots\!63}{10\!\cdots\!63}a^{3}+\frac{48\!\cdots\!17}{19\!\cdots\!14}a^{2}+\frac{73\!\cdots\!11}{21\!\cdots\!26}a+\frac{19\!\cdots\!11}{19\!\cdots\!14}$, $\frac{11\!\cdots\!26}{10\!\cdots\!63}a^{23}+\frac{25\!\cdots\!53}{10\!\cdots\!63}a^{22}+\frac{48\!\cdots\!66}{10\!\cdots\!63}a^{21}+\frac{10\!\cdots\!64}{10\!\cdots\!63}a^{20}+\frac{86\!\cdots\!53}{10\!\cdots\!63}a^{19}+\frac{18\!\cdots\!96}{10\!\cdots\!63}a^{18}+\frac{16\!\cdots\!91}{21\!\cdots\!26}a^{17}+\frac{32\!\cdots\!87}{21\!\cdots\!26}a^{16}+\frac{92\!\cdots\!25}{21\!\cdots\!26}a^{15}+\frac{84\!\cdots\!34}{10\!\cdots\!63}a^{14}+\frac{31\!\cdots\!71}{21\!\cdots\!26}a^{13}+\frac{53\!\cdots\!89}{21\!\cdots\!26}a^{12}+\frac{32\!\cdots\!49}{10\!\cdots\!63}a^{11}+\frac{10\!\cdots\!89}{21\!\cdots\!26}a^{10}+\frac{38\!\cdots\!28}{10\!\cdots\!63}a^{9}+\frac{11\!\cdots\!29}{21\!\cdots\!26}a^{8}+\frac{48\!\cdots\!87}{21\!\cdots\!26}a^{7}+\frac{73\!\cdots\!83}{21\!\cdots\!26}a^{6}+\frac{57\!\cdots\!27}{10\!\cdots\!63}a^{5}+\frac{21\!\cdots\!65}{21\!\cdots\!26}a^{4}-\frac{51\!\cdots\!51}{21\!\cdots\!26}a^{3}+\frac{76\!\cdots\!48}{10\!\cdots\!63}a^{2}-\frac{10\!\cdots\!83}{21\!\cdots\!26}a+\frac{63\!\cdots\!90}{10\!\cdots\!63}$, $\frac{66\!\cdots\!40}{10\!\cdots\!63}a^{23}+\frac{13\!\cdots\!44}{10\!\cdots\!63}a^{22}+\frac{27\!\cdots\!20}{10\!\cdots\!63}a^{21}+\frac{56\!\cdots\!89}{10\!\cdots\!63}a^{20}+\frac{47\!\cdots\!63}{10\!\cdots\!63}a^{19}+\frac{97\!\cdots\!63}{10\!\cdots\!63}a^{18}+\frac{84\!\cdots\!27}{21\!\cdots\!26}a^{17}+\frac{17\!\cdots\!13}{21\!\cdots\!26}a^{16}+\frac{22\!\cdots\!19}{10\!\cdots\!63}a^{15}+\frac{48\!\cdots\!46}{10\!\cdots\!63}a^{14}+\frac{69\!\cdots\!01}{10\!\cdots\!63}a^{13}+\frac{15\!\cdots\!16}{10\!\cdots\!63}a^{12}+\frac{13\!\cdots\!56}{10\!\cdots\!63}a^{11}+\frac{31\!\cdots\!89}{10\!\cdots\!63}a^{10}+\frac{29\!\cdots\!27}{21\!\cdots\!26}a^{9}+\frac{77\!\cdots\!61}{21\!\cdots\!26}a^{8}+\frac{17\!\cdots\!07}{21\!\cdots\!26}a^{7}+\frac{26\!\cdots\!82}{10\!\cdots\!63}a^{6}+\frac{24\!\cdots\!28}{10\!\cdots\!63}a^{5}+\frac{85\!\cdots\!08}{10\!\cdots\!63}a^{4}+\frac{93\!\cdots\!55}{10\!\cdots\!63}a^{3}+\frac{81\!\cdots\!88}{10\!\cdots\!63}a^{2}-\frac{13\!\cdots\!66}{10\!\cdots\!63}a+\frac{11\!\cdots\!93}{21\!\cdots\!26}$, $\frac{86\!\cdots\!55}{10\!\cdots\!63}a^{23}-\frac{377561537676535}{97\!\cdots\!07}a^{22}+\frac{71\!\cdots\!29}{21\!\cdots\!26}a^{21}-\frac{15\!\cdots\!13}{97\!\cdots\!07}a^{20}+\frac{60\!\cdots\!76}{10\!\cdots\!63}a^{19}-\frac{51\!\cdots\!13}{19\!\cdots\!14}a^{18}+\frac{10\!\cdots\!49}{21\!\cdots\!26}a^{17}-\frac{45\!\cdots\!45}{19\!\cdots\!14}a^{16}+\frac{56\!\cdots\!95}{21\!\cdots\!26}a^{15}-\frac{11\!\cdots\!86}{97\!\cdots\!07}a^{14}+\frac{17\!\cdots\!77}{21\!\cdots\!26}a^{13}-\frac{67\!\cdots\!75}{19\!\cdots\!14}a^{12}+\frac{16\!\cdots\!63}{10\!\cdots\!63}a^{11}-\frac{12\!\cdots\!67}{19\!\cdots\!14}a^{10}+\frac{37\!\cdots\!51}{21\!\cdots\!26}a^{9}-\frac{12\!\cdots\!65}{19\!\cdots\!14}a^{8}+\frac{11\!\cdots\!95}{10\!\cdots\!63}a^{7}-\frac{71\!\cdots\!79}{19\!\cdots\!14}a^{6}+\frac{30\!\cdots\!46}{10\!\cdots\!63}a^{5}-\frac{92\!\cdots\!52}{97\!\cdots\!07}a^{4}+\frac{15\!\cdots\!35}{10\!\cdots\!63}a^{3}-\frac{46\!\cdots\!46}{97\!\cdots\!07}a^{2}-\frac{77\!\cdots\!38}{10\!\cdots\!63}a+\frac{19\!\cdots\!23}{97\!\cdots\!07}$, $\frac{89\!\cdots\!57}{10\!\cdots\!63}a^{23}-\frac{56\!\cdots\!11}{10\!\cdots\!63}a^{22}+\frac{37\!\cdots\!98}{10\!\cdots\!63}a^{21}-\frac{23\!\cdots\!51}{10\!\cdots\!63}a^{20}+\frac{12\!\cdots\!53}{21\!\cdots\!26}a^{19}-\frac{40\!\cdots\!21}{10\!\cdots\!63}a^{18}+\frac{11\!\cdots\!31}{21\!\cdots\!26}a^{17}-\frac{36\!\cdots\!54}{10\!\cdots\!63}a^{16}+\frac{60\!\cdots\!31}{21\!\cdots\!26}a^{15}-\frac{19\!\cdots\!56}{10\!\cdots\!63}a^{14}+\frac{18\!\cdots\!05}{21\!\cdots\!26}a^{13}-\frac{62\!\cdots\!88}{10\!\cdots\!63}a^{12}+\frac{36\!\cdots\!77}{21\!\cdots\!26}a^{11}-\frac{12\!\cdots\!44}{10\!\cdots\!63}a^{10}+\frac{41\!\cdots\!07}{21\!\cdots\!26}a^{9}-\frac{27\!\cdots\!51}{21\!\cdots\!26}a^{8}+\frac{13\!\cdots\!37}{10\!\cdots\!63}a^{7}-\frac{87\!\cdots\!76}{10\!\cdots\!63}a^{6}+\frac{40\!\cdots\!89}{10\!\cdots\!63}a^{5}-\frac{51\!\cdots\!93}{21\!\cdots\!26}a^{4}+\frac{62\!\cdots\!17}{21\!\cdots\!26}a^{3}-\frac{41\!\cdots\!41}{21\!\cdots\!26}a^{2}-\frac{14\!\cdots\!63}{21\!\cdots\!26}a-\frac{63\!\cdots\!44}{10\!\cdots\!63}$, $\frac{62\!\cdots\!29}{19\!\cdots\!14}a^{23}+\frac{14\!\cdots\!28}{10\!\cdots\!63}a^{22}+\frac{13\!\cdots\!36}{97\!\cdots\!07}a^{21}+\frac{12\!\cdots\!99}{21\!\cdots\!26}a^{20}+\frac{44\!\cdots\!69}{19\!\cdots\!14}a^{19}+\frac{23\!\cdots\!07}{21\!\cdots\!26}a^{18}+\frac{39\!\cdots\!43}{19\!\cdots\!14}a^{17}+\frac{22\!\cdots\!73}{21\!\cdots\!26}a^{16}+\frac{10\!\cdots\!15}{97\!\cdots\!07}a^{15}+\frac{12\!\cdots\!61}{21\!\cdots\!26}a^{14}+\frac{32\!\cdots\!29}{97\!\cdots\!07}a^{13}+\frac{44\!\cdots\!99}{21\!\cdots\!26}a^{12}+\frac{63\!\cdots\!67}{97\!\cdots\!07}a^{11}+\frac{96\!\cdots\!49}{21\!\cdots\!26}a^{10}+\frac{14\!\cdots\!69}{19\!\cdots\!14}a^{9}+\frac{62\!\cdots\!56}{10\!\cdots\!63}a^{8}+\frac{93\!\cdots\!95}{19\!\cdots\!14}a^{7}+\frac{89\!\cdots\!71}{21\!\cdots\!26}a^{6}+\frac{28\!\cdots\!73}{19\!\cdots\!14}a^{5}+\frac{14\!\cdots\!70}{10\!\cdots\!63}a^{4}+\frac{13\!\cdots\!50}{97\!\cdots\!07}a^{3}+\frac{13\!\cdots\!48}{10\!\cdots\!63}a^{2}+\frac{22\!\cdots\!70}{97\!\cdots\!07}a+\frac{24\!\cdots\!67}{21\!\cdots\!26}$, $\frac{25\!\cdots\!30}{10\!\cdots\!63}a^{22}+\frac{82\!\cdots\!27}{10\!\cdots\!63}a^{20}+\frac{87\!\cdots\!71}{10\!\cdots\!63}a^{18}+\frac{14\!\cdots\!89}{10\!\cdots\!63}a^{16}-\frac{36\!\cdots\!03}{10\!\cdots\!63}a^{14}-\frac{28\!\cdots\!20}{10\!\cdots\!63}a^{12}-\frac{91\!\cdots\!72}{10\!\cdots\!63}a^{10}-\frac{14\!\cdots\!22}{10\!\cdots\!63}a^{8}-\frac{11\!\cdots\!19}{10\!\cdots\!63}a^{6}-\frac{39\!\cdots\!09}{10\!\cdots\!63}a^{4}-\frac{31\!\cdots\!38}{10\!\cdots\!63}a^{2}+\frac{54\!\cdots\!57}{10\!\cdots\!63}$, $\frac{16\!\cdots\!70}{10\!\cdots\!63}a^{23}+\frac{440190612274949}{97\!\cdots\!07}a^{22}+\frac{13\!\cdots\!01}{21\!\cdots\!26}a^{21}+\frac{36\!\cdots\!93}{19\!\cdots\!14}a^{20}+\frac{23\!\cdots\!17}{21\!\cdots\!26}a^{19}+\frac{31\!\cdots\!54}{97\!\cdots\!07}a^{18}+\frac{21\!\cdots\!15}{21\!\cdots\!26}a^{17}+\frac{28\!\cdots\!80}{97\!\cdots\!07}a^{16}+\frac{11\!\cdots\!97}{21\!\cdots\!26}a^{15}+\frac{29\!\cdots\!35}{19\!\cdots\!14}a^{14}+\frac{36\!\cdots\!03}{21\!\cdots\!26}a^{13}+\frac{46\!\cdots\!16}{97\!\cdots\!07}a^{12}+\frac{72\!\cdots\!35}{21\!\cdots\!26}a^{11}+\frac{88\!\cdots\!24}{97\!\cdots\!07}a^{10}+\frac{87\!\cdots\!31}{21\!\cdots\!26}a^{9}+\frac{10\!\cdots\!62}{97\!\cdots\!07}a^{8}+\frac{60\!\cdots\!19}{21\!\cdots\!26}a^{7}+\frac{64\!\cdots\!36}{97\!\cdots\!07}a^{6}+\frac{10\!\cdots\!42}{10\!\cdots\!63}a^{5}+\frac{19\!\cdots\!62}{97\!\cdots\!07}a^{4}+\frac{14\!\cdots\!18}{10\!\cdots\!63}a^{3}+\frac{31\!\cdots\!07}{19\!\cdots\!14}a^{2}+\frac{61\!\cdots\!92}{10\!\cdots\!63}a-\frac{57\!\cdots\!03}{97\!\cdots\!07}$, $\frac{23\!\cdots\!19}{21\!\cdots\!26}a^{23}-\frac{93\!\cdots\!07}{10\!\cdots\!63}a^{22}+\frac{98\!\cdots\!83}{21\!\cdots\!26}a^{21}-\frac{39\!\cdots\!80}{10\!\cdots\!63}a^{20}+\frac{83\!\cdots\!89}{10\!\cdots\!63}a^{19}-\frac{13\!\cdots\!61}{21\!\cdots\!26}a^{18}+\frac{74\!\cdots\!97}{10\!\cdots\!63}a^{17}-\frac{11\!\cdots\!49}{21\!\cdots\!26}a^{16}+\frac{38\!\cdots\!77}{10\!\cdots\!63}a^{15}-\frac{61\!\cdots\!03}{21\!\cdots\!26}a^{14}+\frac{24\!\cdots\!83}{21\!\cdots\!26}a^{13}-\frac{96\!\cdots\!49}{10\!\cdots\!63}a^{12}+\frac{23\!\cdots\!24}{10\!\cdots\!63}a^{11}-\frac{36\!\cdots\!85}{21\!\cdots\!26}a^{10}+\frac{52\!\cdots\!81}{21\!\cdots\!26}a^{9}-\frac{41\!\cdots\!75}{21\!\cdots\!26}a^{8}+\frac{33\!\cdots\!61}{21\!\cdots\!26}a^{7}-\frac{26\!\cdots\!75}{21\!\cdots\!26}a^{6}+\frac{10\!\cdots\!55}{21\!\cdots\!26}a^{5}-\frac{79\!\cdots\!79}{21\!\cdots\!26}a^{4}+\frac{50\!\cdots\!05}{10\!\cdots\!63}a^{3}-\frac{71\!\cdots\!61}{21\!\cdots\!26}a^{2}+\frac{10\!\cdots\!51}{21\!\cdots\!26}a-\frac{22\!\cdots\!89}{21\!\cdots\!26}$ (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: | \( 51307562.10637229 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{12}\cdot 51307562.10637229 \cdot 5}{18\cdot\sqrt{52792842355679189725978706186699601}}\cr\approx \mathstrut & 0.234827944458463 \end{aligned}\] (assuming GRH)
Galois group
$C_2^3\times A_4$ (as 24T135):
A solvable group of order 96 |
The 32 conjugacy class representatives for $C_2^3\times A_4$ |
Character table for $C_2^3\times A_4$ is not computed |
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 |
Degree 32 sibling: | 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 | ${\href{/padicField/2.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/5.6.0.1}{6} }^{4}$ | R | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | ${\href{/padicField/13.6.0.1}{6} }^{4}$ | ${\href{/padicField/17.2.0.1}{2} }^{12}$ | ${\href{/padicField/19.2.0.1}{2} }^{12}$ | ${\href{/padicField/23.6.0.1}{6} }^{4}$ | ${\href{/padicField/29.6.0.1}{6} }^{4}$ | ${\href{/padicField/31.6.0.1}{6} }^{4}$ | ${\href{/padicField/37.2.0.1}{2} }^{8}{,}\,{\href{/padicField/37.1.0.1}{1} }^{8}$ | ${\href{/padicField/41.6.0.1}{6} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }^{8}$ | ${\href{/padicField/47.6.0.1}{6} }^{4}$ | ${\href{/padicField/53.2.0.1}{2} }^{12}$ | ${\href{/padicField/59.6.0.1}{6} }^{4}$ |
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 |
---|---|---|---|---|---|---|---|
\(3\) | 3.12.18.82 | $x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277$ | $6$ | $2$ | $18$ | $C_6\times C_2$ | $[2]_{2}^{2}$ |
3.12.18.82 | $x^{12} + 24 x^{11} + 252 x^{10} + 1558 x^{9} + 6450 x^{8} + 19068 x^{7} + 41627 x^{6} + 68094 x^{5} + 83298 x^{4} + 74306 x^{3} + 45618 x^{2} + 17400 x + 3277$ | $6$ | $2$ | $18$ | $C_6\times C_2$ | $[2]_{2}^{2}$ | |
\(7\) | 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 | $x^{12} + 44 x^{10} + 10 x^{9} + 786 x^{8} + 22 x^{7} + 6899 x^{6} - 3434 x^{5} + 31050 x^{4} - 28440 x^{3} + 84557 x^{2} - 48082 x + 107648$ | $2$ | $6$ | $6$ | $C_6\times C_2$ | $[\ ]_{2}^{6}$ | |
\(71\) | 71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ |
71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
71.2.0.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $[\ ]^{2}$ | |
71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ | |
71.4.2.1 | $x^{4} + 138 x^{3} + 4917 x^{2} + 10764 x + 342127$ | $2$ | $2$ | $2$ | $C_2^2$ | $[\ ]_{2}^{2}$ |