Normalized defining polynomial
\( x^{24} - 120 x^{22} - 10 x^{21} + 4680 x^{20} - 6321 x^{19} - 96055 x^{18} + 164600 x^{17} + \cdots + 2006228467125 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $(4, 10)$ |
| |
| Discriminant: |
\(91810054652229848026521481130092527303208363056182861328125\)
\(\medspace = 5^{23}\cdot 89^{22}\)
|
| |
| Root discriminant: | \(286.28\) |
| |
| Galois root discriminant: | $5^{23/20}89^{19/20}\approx 452.62236684313837$ | ||
| Ramified primes: |
\(5\), \(89\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{5}a^{12}-\frac{2}{5}a^{11}-\frac{2}{5}a^{10}+\frac{1}{5}a^{9}-\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{5}-\frac{1}{5}a^{4}$, $\frac{1}{5}a^{13}-\frac{1}{5}a^{11}+\frac{2}{5}a^{10}+\frac{2}{5}a^{9}-\frac{1}{5}a^{8}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{4}$, $\frac{1}{5}a^{14}-\frac{1}{5}a^{4}$, $\frac{1}{5}a^{15}-\frac{1}{5}a^{5}$, $\frac{1}{5}a^{16}-\frac{1}{5}a^{6}$, $\frac{1}{5}a^{17}-\frac{1}{5}a^{7}$, $\frac{1}{5}a^{18}-\frac{1}{5}a^{8}$, $\frac{1}{15}a^{19}+\frac{1}{15}a^{18}-\frac{1}{15}a^{14}-\frac{1}{15}a^{13}-\frac{1}{15}a^{12}+\frac{1}{5}a^{11}+\frac{1}{15}a^{9}+\frac{2}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{3}a^{5}-\frac{2}{5}a^{4}+\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{15}a^{20}-\frac{1}{15}a^{18}-\frac{1}{15}a^{15}+\frac{1}{15}a^{12}+\frac{1}{5}a^{11}+\frac{7}{15}a^{10}-\frac{4}{15}a^{9}+\frac{2}{5}a^{8}-\frac{2}{5}a^{7}+\frac{2}{15}a^{6}-\frac{2}{15}a^{5}-\frac{1}{15}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{615}a^{21}-\frac{4}{615}a^{20}+\frac{1}{615}a^{19}+\frac{14}{205}a^{18}+\frac{16}{205}a^{17}-\frac{28}{615}a^{16}+\frac{19}{615}a^{15}+\frac{11}{123}a^{14}+\frac{4}{123}a^{13}+\frac{2}{41}a^{12}+\frac{94}{615}a^{11}+\frac{124}{615}a^{10}-\frac{67}{205}a^{9}-\frac{74}{205}a^{8}+\frac{62}{615}a^{7}-\frac{52}{615}a^{6}-\frac{274}{615}a^{5}-\frac{4}{41}a^{4}-\frac{23}{123}a^{3}-\frac{55}{123}a^{2}+\frac{18}{41}a-\frac{15}{41}$, $\frac{1}{3075}a^{22}-\frac{1}{205}a^{20}+\frac{1}{615}a^{19}+\frac{7}{123}a^{18}+\frac{4}{75}a^{17}+\frac{2}{205}a^{16}-\frac{23}{615}a^{15}+\frac{7}{615}a^{14}-\frac{19}{615}a^{13}+\frac{3}{1025}a^{12}-\frac{146}{615}a^{11}-\frac{64}{615}a^{10}+\frac{8}{615}a^{9}+\frac{253}{615}a^{8}+\frac{1426}{3075}a^{7}+\frac{25}{123}a^{6}-\frac{20}{123}a^{5}-\frac{71}{615}a^{4}+\frac{181}{615}a^{3}-\frac{25}{123}a^{2}-\frac{56}{123}a-\frac{12}{41}$, $\frac{1}{13\cdots 75}a^{23}+\frac{12\cdots 31}{13\cdots 75}a^{22}+\frac{50\cdots 51}{87\cdots 85}a^{21}-\frac{63\cdots 76}{17\cdots 57}a^{20}-\frac{40\cdots 67}{26\cdots 55}a^{19}+\frac{74\cdots 29}{13\cdots 75}a^{18}-\frac{32\cdots 76}{13\cdots 75}a^{17}-\frac{85\cdots 96}{26\cdots 55}a^{16}+\frac{53\cdots 24}{26\cdots 55}a^{15}+\frac{16\cdots 21}{87\cdots 85}a^{14}-\frac{81\cdots 32}{43\cdots 25}a^{13}+\frac{24\cdots 09}{13\cdots 75}a^{12}-\frac{10\cdots 83}{26\cdots 55}a^{11}+\frac{33\cdots 04}{26\cdots 55}a^{10}-\frac{66\cdots 48}{26\cdots 55}a^{9}+\frac{59\cdots 61}{13\cdots 75}a^{8}-\frac{16\cdots 38}{43\cdots 25}a^{7}+\frac{12\cdots 91}{26\cdots 55}a^{6}+\frac{11\cdots 44}{87\cdots 85}a^{5}+\frac{15\cdots 87}{26\cdots 55}a^{4}+\frac{30\cdots 87}{87\cdots 85}a^{3}+\frac{23\cdots 80}{52\cdots 71}a^{2}+\frac{14\cdots 41}{17\cdots 57}a-\frac{56\cdots 87}{17\cdots 57}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{4}$, which has order $16$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{4}$, which has order $16$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{35\cdots 32}{76\cdots 75}a^{23}+\frac{64\cdots 62}{76\cdots 75}a^{22}-\frac{24\cdots 78}{45\cdots 65}a^{21}-\frac{47\cdots 79}{45\cdots 65}a^{20}+\frac{90\cdots 96}{45\cdots 65}a^{19}+\frac{51\cdots 43}{76\cdots 75}a^{18}-\frac{32\cdots 02}{76\cdots 75}a^{17}-\frac{11\cdots 52}{45\cdots 65}a^{16}+\frac{84\cdots 93}{45\cdots 65}a^{15}-\frac{62\cdots 59}{91\cdots 53}a^{14}-\frac{42\cdots 16}{22\cdots 25}a^{13}-\frac{51\cdots 42}{76\cdots 75}a^{12}+\frac{43\cdots 43}{91\cdots 53}a^{11}-\frac{13\cdots 82}{91\cdots 53}a^{10}-\frac{21\cdots 94}{15\cdots 55}a^{9}+\frac{55\cdots 62}{76\cdots 75}a^{8}+\frac{12\cdots 21}{22\cdots 25}a^{7}+\frac{29\cdots 35}{91\cdots 53}a^{6}-\frac{22\cdots 94}{45\cdots 65}a^{5}+\frac{32\cdots 72}{15\cdots 55}a^{4}+\frac{48\cdots 96}{45\cdots 65}a^{3}-\frac{41\cdots 79}{91\cdots 53}a^{2}-\frac{23\cdots 26}{30\cdots 51}a-\frac{15\cdots 21}{30\cdots 51}$, $\frac{34\cdots 44}{65\cdots 25}a^{23}+\frac{10\cdots 48}{65\cdots 25}a^{22}+\frac{25\cdots 66}{39\cdots 15}a^{21}-\frac{11\cdots 13}{13\cdots 05}a^{20}-\frac{99\cdots 58}{39\cdots 15}a^{19}+\frac{22\cdots 94}{65\cdots 25}a^{18}+\frac{32\cdots 67}{65\cdots 25}a^{17}-\frac{29\cdots 53}{39\cdots 15}a^{16}-\frac{20\cdots 27}{13\cdots 05}a^{15}+\frac{52\cdots 83}{13\cdots 05}a^{14}-\frac{41\cdots 48}{19\cdots 75}a^{13}+\frac{12\cdots 77}{65\cdots 25}a^{12}-\frac{26\cdots 19}{79\cdots 23}a^{11}+\frac{33\cdots 46}{39\cdots 15}a^{10}+\frac{14\cdots 87}{13\cdots 05}a^{9}-\frac{29\cdots 99}{65\cdots 25}a^{8}-\frac{36\cdots 51}{19\cdots 75}a^{7}-\frac{20\cdots 78}{39\cdots 15}a^{6}+\frac{99\cdots 79}{39\cdots 15}a^{5}-\frac{27\cdots 52}{39\cdots 15}a^{4}-\frac{18\cdots 37}{13\cdots 05}a^{3}+\frac{27\cdots 19}{79\cdots 23}a^{2}-\frac{17\cdots 81}{26\cdots 41}a+\frac{54\cdots 04}{26\cdots 41}$, $\frac{17\cdots 84}{22\cdots 25}a^{23}+\frac{68\cdots 37}{22\cdots 25}a^{22}-\frac{35\cdots 86}{45\cdots 65}a^{21}-\frac{99\cdots 51}{30\cdots 51}a^{20}+\frac{83\cdots 04}{45\cdots 65}a^{19}+\frac{14\cdots 81}{22\cdots 25}a^{18}-\frac{31\cdots 22}{22\cdots 25}a^{17}-\frac{11\cdots 36}{45\cdots 65}a^{16}-\frac{13\cdots 08}{15\cdots 55}a^{15}+\frac{14\cdots 36}{30\cdots 51}a^{14}-\frac{23\cdots 14}{22\cdots 25}a^{13}-\frac{63\cdots 89}{76\cdots 75}a^{12}+\frac{17\cdots 34}{45\cdots 65}a^{11}-\frac{51\cdots 88}{45\cdots 65}a^{10}-\frac{79\cdots 82}{45\cdots 65}a^{9}+\frac{19\cdots 38}{76\cdots 75}a^{8}-\frac{18\cdots 98}{22\cdots 25}a^{7}+\frac{16\cdots 33}{45\cdots 65}a^{6}-\frac{34\cdots 46}{15\cdots 55}a^{5}-\frac{53\cdots 52}{45\cdots 65}a^{4}+\frac{16\cdots 89}{15\cdots 55}a^{3}-\frac{14\cdots 90}{91\cdots 53}a^{2}-\frac{61\cdots 46}{30\cdots 51}a-\frac{56\cdots 82}{30\cdots 51}$, $\frac{29\cdots 42}{19\cdots 75}a^{23}-\frac{45\cdots 76}{19\cdots 75}a^{22}+\frac{13\cdots 94}{79\cdots 23}a^{21}+\frac{11\cdots 77}{39\cdots 15}a^{20}-\frac{87\cdots 39}{13\cdots 05}a^{19}-\frac{27\cdots 28}{19\cdots 75}a^{18}+\frac{28\cdots 01}{19\cdots 75}a^{17}+\frac{39\cdots 97}{13\cdots 05}a^{16}-\frac{21\cdots 19}{39\cdots 15}a^{15}+\frac{50\cdots 68}{39\cdots 15}a^{14}+\frac{47\cdots 32}{19\cdots 75}a^{13}+\frac{82\cdots 56}{19\cdots 75}a^{12}-\frac{67\cdots 08}{79\cdots 23}a^{11}+\frac{26\cdots 82}{13\cdots 05}a^{10}+\frac{16\cdots 49}{39\cdots 15}a^{9}-\frac{91\cdots 07}{19\cdots 75}a^{8}-\frac{31\cdots 56}{19\cdots 75}a^{7}-\frac{79\cdots 42}{13\cdots 05}a^{6}+\frac{75\cdots 06}{39\cdots 15}a^{5}-\frac{20\cdots 54}{79\cdots 23}a^{4}-\frac{89\cdots 71}{39\cdots 15}a^{3}+\frac{26\cdots 59}{26\cdots 41}a^{2}+\frac{45\cdots 49}{26\cdots 41}a+\frac{29\cdots 41}{26\cdots 41}$, $\frac{14\cdots 23}{76\cdots 75}a^{23}-\frac{26\cdots 68}{15\cdots 55}a^{22}+\frac{20\cdots 03}{91\cdots 53}a^{21}+\frac{19\cdots 99}{91\cdots 53}a^{20}-\frac{77\cdots 46}{91\cdots 53}a^{19}+\frac{11\cdots 64}{22\cdots 25}a^{18}+\frac{27\cdots 37}{15\cdots 55}a^{17}-\frac{73\cdots 11}{45\cdots 65}a^{16}-\frac{35\cdots 59}{45\cdots 65}a^{15}+\frac{56\cdots 08}{15\cdots 55}a^{14}+\frac{13\cdots 49}{22\cdots 25}a^{13}+\frac{88\cdots 91}{45\cdots 65}a^{12}-\frac{99\cdots 87}{45\cdots 65}a^{11}+\frac{12\cdots 07}{15\cdots 55}a^{10}+\frac{23\cdots 89}{45\cdots 65}a^{9}-\frac{63\cdots 23}{76\cdots 75}a^{8}-\frac{82\cdots 93}{45\cdots 65}a^{7}-\frac{39\cdots 99}{37\cdots 55}a^{6}+\frac{54\cdots 56}{15\cdots 55}a^{5}-\frac{54\cdots 83}{45\cdots 65}a^{4}-\frac{35\cdots 77}{91\cdots 53}a^{3}+\frac{20\cdots 47}{91\cdots 53}a^{2}-\frac{18\cdots 47}{91\cdots 53}a-\frac{15\cdots 34}{30\cdots 51}$, $\frac{13\cdots 38}{43\cdots 25}a^{23}+\frac{71\cdots 36}{13\cdots 75}a^{22}-\frac{18\cdots 10}{52\cdots 71}a^{21}-\frac{58\cdots 77}{87\cdots 85}a^{20}+\frac{33\cdots 58}{26\cdots 55}a^{19}+\frac{19\cdots 42}{43\cdots 25}a^{18}-\frac{36\cdots 31}{13\cdots 75}a^{17}-\frac{46\cdots 57}{26\cdots 55}a^{16}+\frac{31\cdots 38}{26\cdots 55}a^{15}-\frac{38\cdots 17}{87\cdots 85}a^{14}-\frac{15\cdots 24}{13\cdots 75}a^{13}-\frac{57\cdots 11}{13\cdots 75}a^{12}+\frac{80\cdots 88}{26\cdots 55}a^{11}-\frac{24\cdots 16}{26\cdots 55}a^{10}-\frac{81\cdots 29}{87\cdots 85}a^{9}+\frac{62\cdots 54}{13\cdots 75}a^{8}+\frac{47\cdots 81}{13\cdots 75}a^{7}+\frac{18\cdots 08}{87\cdots 85}a^{6}-\frac{28\cdots 87}{87\cdots 85}a^{5}+\frac{24\cdots 33}{17\cdots 57}a^{4}+\frac{60\cdots 07}{87\cdots 85}a^{3}-\frac{15\cdots 74}{52\cdots 71}a^{2}-\frac{88\cdots 16}{17\cdots 57}a-\frac{57\cdots 78}{17\cdots 57}$, $\frac{11\cdots 57}{13\cdots 75}a^{23}-\frac{73\cdots 66}{26\cdots 55}a^{22}-\frac{24\cdots 37}{26\cdots 55}a^{21}+\frac{15\cdots 80}{52\cdots 71}a^{20}+\frac{82\cdots 13}{26\cdots 55}a^{19}-\frac{66\cdots 99}{43\cdots 25}a^{18}-\frac{98\cdots 73}{26\cdots 55}a^{17}+\frac{21\cdots 78}{87\cdots 85}a^{16}-\frac{26\cdots 52}{87\cdots 85}a^{15}-\frac{34\cdots 58}{87\cdots 85}a^{14}-\frac{16\cdots 37}{13\cdots 75}a^{13}-\frac{81\cdots 12}{87\cdots 85}a^{12}+\frac{37\cdots 38}{26\cdots 55}a^{11}-\frac{20\cdots 57}{26\cdots 55}a^{10}+\frac{47\cdots 26}{87\cdots 85}a^{9}+\frac{48\cdots 02}{13\cdots 75}a^{8}-\frac{21\cdots 28}{52\cdots 71}a^{7}+\frac{28\cdots 67}{52\cdots 71}a^{6}-\frac{34\cdots 99}{87\cdots 85}a^{5}+\frac{45\cdots 43}{26\cdots 55}a^{4}-\frac{22\cdots 40}{52\cdots 71}a^{3}+\frac{18\cdots 74}{52\cdots 71}a^{2}+\frac{15\cdots 24}{52\cdots 71}a+\frac{80\cdots 79}{17\cdots 57}$, $\frac{13\cdots 31}{26\cdots 55}a^{23}+\frac{17\cdots 33}{13\cdots 75}a^{22}-\frac{98\cdots 27}{17\cdots 57}a^{21}-\frac{13\cdots 74}{87\cdots 85}a^{20}+\frac{50\cdots 56}{26\cdots 55}a^{19}+\frac{48\cdots 48}{26\cdots 55}a^{18}-\frac{55\cdots 63}{13\cdots 75}a^{17}-\frac{76\cdots 56}{26\cdots 55}a^{16}+\frac{29\cdots 06}{26\cdots 55}a^{15}-\frac{68\cdots 56}{87\cdots 85}a^{14}-\frac{10\cdots 67}{42\cdots 77}a^{13}-\frac{13\cdots 48}{13\cdots 75}a^{12}+\frac{96\cdots 72}{26\cdots 55}a^{11}-\frac{80\cdots 81}{52\cdots 71}a^{10}-\frac{42\cdots 72}{26\cdots 55}a^{9}-\frac{18\cdots 57}{26\cdots 55}a^{8}+\frac{10\cdots 26}{43\cdots 25}a^{7}+\frac{82\cdots 34}{26\cdots 55}a^{6}-\frac{26\cdots 38}{87\cdots 85}a^{5}+\frac{69\cdots 44}{26\cdots 55}a^{4}+\frac{12\cdots 51}{87\cdots 85}a^{3}-\frac{16\cdots 69}{52\cdots 71}a^{2}-\frac{10\cdots 50}{17\cdots 57}a-\frac{16\cdots 77}{17\cdots 57}$, $\frac{76\cdots 62}{13\cdots 75}a^{23}+\frac{13\cdots 71}{26\cdots 55}a^{22}+\frac{19\cdots 79}{26\cdots 55}a^{21}-\frac{83\cdots 10}{17\cdots 57}a^{20}-\frac{61\cdots 60}{17\cdots 57}a^{19}+\frac{65\cdots 12}{13\cdots 75}a^{18}+\frac{22\cdots 91}{26\cdots 55}a^{17}-\frac{11\cdots 17}{87\cdots 85}a^{16}-\frac{85\cdots 99}{87\cdots 85}a^{15}+\frac{31\cdots 31}{26\cdots 55}a^{14}+\frac{10\cdots 77}{13\cdots 75}a^{13}+\frac{23\cdots 52}{26\cdots 55}a^{12}-\frac{37\cdots 69}{26\cdots 55}a^{11}+\frac{39\cdots 69}{26\cdots 55}a^{10}+\frac{57\cdots 99}{26\cdots 55}a^{9}-\frac{22\cdots 84}{43\cdots 25}a^{8}-\frac{19\cdots 43}{87\cdots 85}a^{7}-\frac{66\cdots 63}{87\cdots 85}a^{6}+\frac{35\cdots 76}{87\cdots 85}a^{5}+\frac{62\cdots 12}{52\cdots 71}a^{4}-\frac{16\cdots 28}{52\cdots 71}a^{3}+\frac{53\cdots 98}{17\cdots 57}a^{2}+\frac{19\cdots 98}{52\cdots 71}a+\frac{71\cdots 98}{17\cdots 57}$, $\frac{21\cdots 10}{12\cdots 31}a^{23}-\frac{32\cdots 01}{17\cdots 57}a^{22}-\frac{68\cdots 69}{26\cdots 55}a^{21}+\frac{19\cdots 93}{87\cdots 85}a^{20}+\frac{13\cdots 84}{87\cdots 85}a^{19}-\frac{25\cdots 13}{26\cdots 55}a^{18}-\frac{35\cdots 89}{87\cdots 85}a^{17}+\frac{10\cdots 51}{42\cdots 77}a^{16}+\frac{16\cdots 37}{26\cdots 55}a^{15}-\frac{17\cdots 42}{87\cdots 85}a^{14}-\frac{14\cdots 93}{26\cdots 55}a^{13}+\frac{25\cdots 82}{26\cdots 55}a^{12}+\frac{99\cdots 41}{87\cdots 85}a^{11}-\frac{16\cdots 83}{87\cdots 85}a^{10}-\frac{21\cdots 81}{26\cdots 55}a^{9}+\frac{18\cdots 71}{26\cdots 55}a^{8}+\frac{59\cdots 49}{52\cdots 71}a^{7}-\frac{13\cdots 19}{26\cdots 55}a^{6}-\frac{30\cdots 39}{87\cdots 85}a^{5}+\frac{15\cdots 49}{52\cdots 71}a^{4}+\frac{46\cdots 73}{17\cdots 57}a^{3}-\frac{56\cdots 31}{17\cdots 57}a^{2}-\frac{74\cdots 11}{17\cdots 57}a-\frac{70\cdots 09}{17\cdots 57}$, $\frac{24\cdots 36}{76\cdots 75}a^{23}-\frac{19\cdots 21}{22\cdots 25}a^{22}-\frac{19\cdots 56}{45\cdots 65}a^{21}+\frac{44\cdots 44}{45\cdots 65}a^{20}+\frac{32\cdots 62}{15\cdots 55}a^{19}-\frac{42\cdots 06}{76\cdots 75}a^{18}-\frac{11\cdots 44}{22\cdots 25}a^{17}+\frac{64\cdots 16}{45\cdots 65}a^{16}+\frac{29\cdots 58}{45\cdots 65}a^{15}-\frac{58\cdots 43}{45\cdots 65}a^{14}-\frac{42\cdots 96}{76\cdots 75}a^{13}-\frac{78\cdots 54}{22\cdots 25}a^{12}+\frac{46\cdots 42}{45\cdots 65}a^{11}-\frac{22\cdots 74}{15\cdots 55}a^{10}-\frac{19\cdots 11}{15\cdots 55}a^{9}+\frac{11\cdots 03}{22\cdots 25}a^{8}+\frac{30\cdots 19}{22\cdots 25}a^{7}-\frac{65\cdots 87}{45\cdots 65}a^{6}-\frac{90\cdots 93}{30\cdots 51}a^{5}+\frac{78\cdots 03}{91\cdots 53}a^{4}+\frac{10\cdots 64}{45\cdots 65}a^{3}-\frac{82\cdots 66}{30\cdots 51}a^{2}-\frac{80\cdots 28}{30\cdots 51}a-\frac{10\cdots 46}{30\cdots 51}$, $\frac{95\cdots 91}{43\cdots 25}a^{23}+\frac{52\cdots 84}{13\cdots 75}a^{22}-\frac{66\cdots 19}{26\cdots 55}a^{21}-\frac{42\cdots 32}{87\cdots 85}a^{20}+\frac{24\cdots 96}{26\cdots 55}a^{19}+\frac{14\cdots 99}{43\cdots 25}a^{18}-\frac{26\cdots 94}{13\cdots 75}a^{17}-\frac{34\cdots 78}{26\cdots 55}a^{16}+\frac{22\cdots 92}{26\cdots 55}a^{15}-\frac{28\cdots 08}{87\cdots 85}a^{14}-\frac{11\cdots 38}{13\cdots 75}a^{13}-\frac{41\cdots 44}{13\cdots 75}a^{12}+\frac{58\cdots 26}{26\cdots 55}a^{11}-\frac{18\cdots 87}{26\cdots 55}a^{10}-\frac{59\cdots 78}{87\cdots 85}a^{9}+\frac{45\cdots 23}{13\cdots 75}a^{8}+\frac{34\cdots 29}{13\cdots 75}a^{7}+\frac{13\cdots 72}{87\cdots 85}a^{6}-\frac{20\cdots 38}{87\cdots 85}a^{5}+\frac{87\cdots 24}{87\cdots 85}a^{4}+\frac{43\cdots 23}{87\cdots 85}a^{3}-\frac{11\cdots 14}{52\cdots 71}a^{2}-\frac{64\cdots 15}{17\cdots 57}a-\frac{41\cdots 17}{17\cdots 57}$, $\frac{15\cdots 42}{26\cdots 55}a^{23}-\frac{89\cdots 17}{43\cdots 25}a^{22}-\frac{34\cdots 11}{52\cdots 71}a^{21}+\frac{57\cdots 31}{26\cdots 55}a^{20}+\frac{55\cdots 79}{26\cdots 55}a^{19}-\frac{97\cdots 83}{87\cdots 85}a^{18}-\frac{92\cdots 53}{43\cdots 25}a^{17}+\frac{46\cdots 29}{26\cdots 55}a^{16}-\frac{18\cdots 80}{52\cdots 71}a^{15}-\frac{17\cdots 58}{87\cdots 85}a^{14}+\frac{38\cdots 31}{87\cdots 85}a^{13}-\frac{92\cdots 84}{13\cdots 75}a^{12}+\frac{52\cdots 35}{52\cdots 71}a^{11}-\frac{53\cdots 34}{87\cdots 85}a^{10}+\frac{37\cdots 58}{87\cdots 85}a^{9}+\frac{16\cdots 05}{52\cdots 71}a^{8}-\frac{23\cdots 77}{43\cdots 25}a^{7}+\frac{10\cdots 16}{26\cdots 55}a^{6}-\frac{22\cdots 03}{87\cdots 85}a^{5}+\frac{32\cdots 41}{26\cdots 55}a^{4}-\frac{20\cdots 16}{63\cdots 55}a^{3}+\frac{13\cdots 19}{52\cdots 71}a^{2}+\frac{16\cdots 45}{52\cdots 71}a+\frac{72\cdots 48}{17\cdots 57}$
|
| |
| Regulator: | \( 68610484458411210000 \) (assuming GRH) |
| |
| Unit signature rank: | \( 4 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 68610484458411210000 \cdot 16}{2\cdot\sqrt{91810054652229848026521481130092527303208363056182861328125}}\cr\approx \mathstrut & 2.77941896256117 \end{aligned}\] (assuming GRH)
Galois group
$\GL(2,5)$ (as 24T1353):
| A non-solvable group of order 480 |
| The 24 conjugacy class representatives for $\GL(2,5)$ |
| Character table for $\GL(2,5)$ |
Intermediate fields
| 6.2.196069503125.1, 12.4.1522544918455380058642578125.3 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $24$ | ${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | R | $24$ | ${\href{/padicField/11.10.0.1}{10} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | $24$ | ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{6}$ | $24$ | ${\href{/padicField/29.4.0.1}{4} }^{6}$ | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | $24$ | ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | $24$ | ${\href{/padicField/59.5.0.1}{5} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.1.5.5a1.1 | $x^{5} + 5 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ | |
| 5.1.5.5a1.1 | $x^{5} + 5 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ | |
| 5.1.5.5a1.1 | $x^{5} + 5 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ | |
| 5.1.5.5a1.1 | $x^{5} + 5 x + 5$ | $5$ | $1$ | $5$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ | |
|
\(89\)
| 89.1.4.3a1.1 | $x^{4} + 89$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 89.1.20.19a1.3 | $x^{20} + 801$ | $20$ | $1$ | $19$ | 20T6 | $$[\ ]_{20}^{2}$$ |