Normalized defining polynomial
\( x^{24} - 8 x^{23} + 64 x^{22} - 902 x^{21} + 6316 x^{20} - 136934 x^{19} + 609372 x^{18} + \cdots - 21\!\cdots\!68 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $(4, 10)$ |
| |
| Discriminant: |
\(35863302598527284385359953566442393477815766818821430206298828125\)
\(\medspace = 5^{31}\cdot 89^{22}\)
|
| |
| Root discriminant: | \(489.53\) |
| |
| Galois root discriminant: | $5^{31/20}89^{19/20}\approx 861.6363513918435$ | ||
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{7}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{17}-\frac{1}{4}a^{14}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{8}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}+\frac{1}{8}a^{5}+\frac{1}{4}a^{4}-\frac{1}{8}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{16}a^{18}+\frac{1}{8}a^{15}-\frac{1}{4}a^{14}+\frac{1}{8}a^{13}-\frac{1}{4}a^{12}-\frac{1}{8}a^{11}-\frac{1}{16}a^{10}-\frac{1}{2}a^{7}-\frac{3}{16}a^{6}+\frac{1}{8}a^{5}+\frac{7}{16}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{32}a^{19}+\frac{1}{16}a^{16}-\frac{1}{8}a^{15}-\frac{3}{16}a^{14}-\frac{1}{8}a^{13}-\frac{1}{16}a^{12}+\frac{7}{32}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}+\frac{13}{32}a^{7}+\frac{1}{16}a^{6}+\frac{7}{32}a^{5}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{192}a^{20}+\frac{1}{96}a^{19}-\frac{1}{48}a^{18}+\frac{1}{96}a^{17}+\frac{5}{96}a^{15}+\frac{23}{96}a^{13}+\frac{35}{192}a^{12}+\frac{5}{32}a^{11}-\frac{1}{48}a^{10}-\frac{1}{4}a^{9}-\frac{35}{192}a^{8}-\frac{13}{48}a^{7}+\frac{55}{192}a^{6}+\frac{19}{96}a^{5}+\frac{13}{48}a^{4}+\frac{1}{24}a^{3}-\frac{1}{3}a$, $\frac{1}{384}a^{21}+\frac{1}{96}a^{19}+\frac{5}{192}a^{18}-\frac{1}{96}a^{17}+\frac{17}{192}a^{16}-\frac{17}{96}a^{15}-\frac{13}{192}a^{14}+\frac{29}{128}a^{13}-\frac{1}{6}a^{12}+\frac{5}{96}a^{11}+\frac{7}{48}a^{10}+\frac{61}{384}a^{9}+\frac{3}{64}a^{8}+\frac{41}{128}a^{7}+\frac{3}{8}a^{6}+\frac{13}{32}a^{5}+\frac{1}{4}a^{4}+\frac{5}{24}a^{3}-\frac{1}{6}a^{2}-\frac{1}{6}a$, $\frac{1}{472320}a^{22}-\frac{1}{7380}a^{21}-\frac{3}{6560}a^{20}+\frac{265}{47232}a^{19}+\frac{187}{23616}a^{18}+\frac{311}{78720}a^{17}-\frac{4367}{39360}a^{16}-\frac{38393}{236160}a^{15}-\frac{2693}{94464}a^{14}-\frac{1373}{11808}a^{13}-\frac{4733}{29520}a^{12}+\frac{801}{6560}a^{11}-\frac{9089}{157440}a^{10}-\frac{7579}{47232}a^{9}-\frac{13949}{94464}a^{8}-\frac{2519}{11808}a^{7}-\frac{5545}{11808}a^{6}-\frac{2425}{5904}a^{5}-\frac{2917}{5904}a^{4}+\frac{169}{738}a^{3}-\frac{979}{2460}a^{2}+\frac{349}{3690}a-\frac{119}{615}$, $\frac{1}{52\cdots 80}a^{23}-\frac{13\cdots 87}{32\cdots 80}a^{22}+\frac{10\cdots 17}{10\cdots 60}a^{21}-\frac{67\cdots 71}{26\cdots 40}a^{20}+\frac{28\cdots 91}{26\cdots 04}a^{19}+\frac{54\cdots 93}{32\cdots 40}a^{18}+\frac{34\cdots 83}{14\cdots 80}a^{17}+\frac{63\cdots 63}{63\cdots 40}a^{16}-\frac{11\cdots 57}{52\cdots 80}a^{15}+\frac{13\cdots 91}{13\cdots 52}a^{14}-\frac{23\cdots 71}{65\cdots 60}a^{13}+\frac{89\cdots 01}{24\cdots 80}a^{12}+\frac{40\cdots 39}{17\cdots 60}a^{11}-\frac{50\cdots 67}{26\cdots 40}a^{10}+\frac{74\cdots 67}{10\cdots 16}a^{9}-\frac{21\cdots 81}{13\cdots 52}a^{8}+\frac{23\cdots 95}{65\cdots 76}a^{7}-\frac{57\cdots 23}{16\cdots 44}a^{6}-\frac{19\cdots 85}{65\cdots 76}a^{5}+\frac{41\cdots 97}{16\cdots 44}a^{4}+\frac{11\cdots 41}{27\cdots 40}a^{3}+\frac{92\cdots 17}{40\cdots 60}a^{2}+\frac{26\cdots 59}{68\cdots 60}a+\frac{68\cdots 41}{56\cdots 05}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ (assuming GRH) |
| |
| Narrow class group: | $C_{4}$, which has order $4$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{45\cdots 93}{39\cdots 80}a^{23}+\frac{67\cdots 43}{65\cdots 80}a^{22}-\frac{81\cdots 29}{97\cdots 20}a^{21}+\frac{45\cdots 19}{40\cdots 80}a^{20}-\frac{81\cdots 07}{97\cdots 32}a^{19}+\frac{10\cdots 37}{65\cdots 80}a^{18}-\frac{83\cdots 53}{97\cdots 20}a^{17}+\frac{21\cdots 77}{19\cdots 40}a^{16}-\frac{59\cdots 57}{13\cdots 60}a^{15}+\frac{85\cdots 61}{13\cdots 76}a^{14}-\frac{15\cdots 67}{48\cdots 60}a^{13}+\frac{67\cdots 03}{19\cdots 40}a^{12}-\frac{16\cdots 39}{13\cdots 60}a^{11}+\frac{20\cdots 77}{16\cdots 20}a^{10}-\frac{91\cdots 99}{19\cdots 16}a^{9}+\frac{32\cdots 45}{97\cdots 32}a^{8}-\frac{11\cdots 77}{97\cdots 32}a^{7}+\frac{24\cdots 27}{39\cdots 28}a^{6}-\frac{11\cdots 25}{81\cdots 36}a^{5}+\frac{39\cdots 57}{97\cdots 32}a^{4}+\frac{21\cdots 89}{24\cdots 80}a^{3}+\frac{22\cdots 09}{60\cdots 70}a^{2}+\frac{34\cdots 52}{10\cdots 45}a+\frac{27\cdots 67}{10\cdots 45}$, $\frac{20\cdots 41}{65\cdots 80}a^{23}-\frac{74\cdots 33}{19\cdots 40}a^{22}+\frac{79\cdots 87}{39\cdots 80}a^{21}-\frac{10\cdots 89}{32\cdots 40}a^{20}+\frac{18\cdots 85}{65\cdots 88}a^{19}-\frac{84\cdots 47}{19\cdots 40}a^{18}+\frac{16\cdots 23}{48\cdots 60}a^{17}-\frac{44\cdots 59}{19\cdots 40}a^{16}+\frac{35\cdots 01}{19\cdots 40}a^{15}-\frac{26\cdots 63}{19\cdots 64}a^{14}+\frac{48\cdots 37}{39\cdots 80}a^{13}-\frac{87\cdots 61}{12\cdots 40}a^{12}+\frac{96\cdots 87}{19\cdots 40}a^{11}-\frac{41\cdots 83}{19\cdots 40}a^{10}+\frac{14\cdots 17}{78\cdots 56}a^{9}-\frac{47\cdots 97}{97\cdots 32}a^{8}+\frac{30\cdots 49}{78\cdots 56}a^{7}-\frac{49\cdots 67}{48\cdots 16}a^{6}+\frac{37\cdots 25}{97\cdots 32}a^{5}+\frac{17\cdots 03}{48\cdots 16}a^{4}+\frac{18\cdots 14}{30\cdots 35}a^{3}+\frac{34\cdots 37}{60\cdots 70}a^{2}+\frac{81\cdots 26}{10\cdots 45}a+\frac{31\cdots 51}{10\cdots 45}$, $\frac{22\cdots 61}{23\cdots 36}a^{23}+\frac{10\cdots 75}{96\cdots 64}a^{22}-\frac{18\cdots 43}{28\cdots 92}a^{21}+\frac{35\cdots 57}{38\cdots 56}a^{20}-\frac{44\cdots 83}{57\cdots 84}a^{19}+\frac{51\cdots 65}{38\cdots 56}a^{18}-\frac{50\cdots 43}{57\cdots 84}a^{17}+\frac{85\cdots 21}{11\cdots 68}a^{16}-\frac{35\cdots 93}{76\cdots 12}a^{15}+\frac{10\cdots 27}{24\cdots 16}a^{14}-\frac{47\cdots 01}{14\cdots 96}a^{13}+\frac{16\cdots 25}{72\cdots 48}a^{12}-\frac{10\cdots 51}{76\cdots 12}a^{11}+\frac{27\cdots 61}{38\cdots 56}a^{10}-\frac{11\cdots 79}{23\cdots 36}a^{9}+\frac{49\cdots 05}{28\cdots 92}a^{8}-\frac{31\cdots 29}{28\cdots 92}a^{7}+\frac{84\cdots 87}{28\cdots 92}a^{6}-\frac{15\cdots 09}{12\cdots 08}a^{5}-\frac{18\cdots 99}{10\cdots 33}a^{4}-\frac{62\cdots 21}{18\cdots 12}a^{3}-\frac{39\cdots 17}{18\cdots 12}a^{2}-\frac{11\cdots 79}{36\cdots 11}a-\frac{18\cdots 03}{15\cdots 51}$, $\frac{43\cdots 51}{15\cdots 20}a^{23}+\frac{49\cdots 57}{19\cdots 40}a^{22}-\frac{51\cdots 13}{39\cdots 80}a^{21}+\frac{49\cdots 63}{26\cdots 20}a^{20}-\frac{12\cdots 65}{78\cdots 56}a^{19}+\frac{27\cdots 17}{78\cdots 60}a^{18}-\frac{20\cdots 27}{13\cdots 60}a^{17}+\frac{42\cdots 13}{26\cdots 20}a^{16}-\frac{79\cdots 57}{15\cdots 20}a^{15}+\frac{40\cdots 81}{39\cdots 28}a^{14}-\frac{71\cdots 71}{13\cdots 60}a^{13}+\frac{17\cdots 49}{32\cdots 40}a^{12}-\frac{17\cdots 99}{15\cdots 20}a^{11}+\frac{99\cdots 93}{78\cdots 60}a^{10}-\frac{58\cdots 95}{10\cdots 08}a^{9}+\frac{30\cdots 91}{81\cdots 36}a^{8}-\frac{15\cdots 25}{26\cdots 52}a^{7}+\frac{15\cdots 97}{13\cdots 76}a^{6}+\frac{99\cdots 41}{19\cdots 64}a^{5}+\frac{25\cdots 70}{20\cdots 09}a^{4}+\frac{22\cdots 68}{10\cdots 45}a^{3}+\frac{22\cdots 96}{30\cdots 35}a^{2}+\frac{74\cdots 76}{10\cdots 45}a+\frac{25\cdots 31}{10\cdots 45}$, $\frac{33\cdots 07}{23\cdots 36}a^{23}+\frac{15\cdots 25}{19\cdots 28}a^{22}-\frac{16\cdots 73}{28\cdots 92}a^{21}+\frac{39\cdots 07}{38\cdots 56}a^{20}-\frac{33\cdots 95}{57\cdots 84}a^{19}+\frac{65\cdots 75}{38\cdots 56}a^{18}-\frac{21\cdots 59}{57\cdots 84}a^{17}+\frac{11\cdots 75}{11\cdots 68}a^{16}-\frac{10\cdots 75}{76\cdots 12}a^{15}+\frac{12\cdots 17}{19\cdots 28}a^{14}-\frac{20\cdots 45}{14\cdots 96}a^{13}+\frac{10\cdots 55}{36\cdots 24}a^{12}-\frac{17\cdots 89}{76\cdots 12}a^{11}+\frac{40\cdots 05}{38\cdots 56}a^{10}-\frac{32\cdots 21}{23\cdots 36}a^{9}+\frac{13\cdots 61}{57\cdots 84}a^{8}-\frac{62\cdots 15}{28\cdots 92}a^{7}+\frac{10\cdots 69}{28\cdots 92}a^{6}+\frac{36\cdots 83}{30\cdots 02}a^{5}+\frac{26\cdots 35}{21\cdots 66}a^{4}+\frac{17\cdots 69}{18\cdots 12}a^{3}+\frac{28\cdots 43}{18\cdots 12}a^{2}+\frac{36\cdots 65}{36\cdots 11}a+\frac{28\cdots 23}{15\cdots 51}$, $\frac{24\cdots 25}{15\cdots 12}a^{23}-\frac{74\cdots 67}{65\cdots 88}a^{22}+\frac{63\cdots 05}{78\cdots 56}a^{21}-\frac{33\cdots 57}{26\cdots 52}a^{20}+\frac{32\cdots 97}{39\cdots 28}a^{19}-\frac{52\cdots 43}{26\cdots 52}a^{18}+\frac{29\cdots 57}{39\cdots 28}a^{17}-\frac{91\cdots 87}{78\cdots 56}a^{16}+\frac{18\cdots 33}{52\cdots 04}a^{15}-\frac{94\cdots 59}{13\cdots 76}a^{14}+\frac{21\cdots 23}{78\cdots 56}a^{13}-\frac{70\cdots 65}{19\cdots 64}a^{12}+\frac{46\cdots 87}{52\cdots 04}a^{11}-\frac{32\cdots 57}{26\cdots 52}a^{10}+\frac{55\cdots 13}{15\cdots 12}a^{9}-\frac{10\cdots 27}{39\cdots 28}a^{8}+\frac{59\cdots 79}{78\cdots 56}a^{7}-\frac{44\cdots 89}{97\cdots 32}a^{6}+\frac{28\cdots 79}{65\cdots 88}a^{5}-\frac{37\cdots 55}{97\cdots 32}a^{4}-\frac{36\cdots 43}{48\cdots 16}a^{3}-\frac{70\cdots 49}{12\cdots 54}a^{2}+\frac{71\cdots 88}{20\cdots 09}a+\frac{64\cdots 65}{20\cdots 09}$, $\frac{14\cdots 29}{26\cdots 40}a^{23}-\frac{19\cdots 39}{65\cdots 60}a^{22}+\frac{46\cdots 43}{21\cdots 20}a^{21}-\frac{11\cdots 27}{26\cdots 04}a^{20}+\frac{28\cdots 41}{13\cdots 52}a^{19}-\frac{10\cdots 83}{16\cdots 20}a^{18}+\frac{10\cdots 89}{72\cdots 40}a^{17}-\frac{49\cdots 37}{13\cdots 20}a^{16}+\frac{34\cdots 59}{52\cdots 08}a^{15}-\frac{30\cdots 21}{13\cdots 52}a^{14}+\frac{38\cdots 37}{65\cdots 60}a^{13}-\frac{33\cdots 83}{30\cdots 60}a^{12}+\frac{12\cdots 79}{87\cdots 80}a^{11}-\frac{10\cdots 25}{26\cdots 04}a^{10}+\frac{38\cdots 87}{52\cdots 08}a^{9}-\frac{95\cdots 91}{13\cdots 52}a^{8}+\frac{26\cdots 07}{13\cdots 52}a^{7}-\frac{71\cdots 71}{65\cdots 76}a^{6}+\frac{70\cdots 03}{39\cdots 84}a^{5}-\frac{19\cdots 45}{16\cdots 44}a^{4}-\frac{69\cdots 33}{68\cdots 60}a^{3}+\frac{15\cdots 59}{51\cdots 45}a^{2}+\frac{73\cdots 59}{34\cdots 30}a+\frac{12\cdots 23}{11\cdots 01}$, $\frac{33\cdots 39}{26\cdots 40}a^{23}-\frac{40\cdots 91}{13\cdots 20}a^{22}-\frac{75\cdots 77}{21\cdots 20}a^{21}+\frac{82\cdots 89}{13\cdots 20}a^{20}+\frac{43\cdots 97}{32\cdots 88}a^{19}+\frac{22\cdots 73}{16\cdots 20}a^{18}+\frac{45\cdots 66}{56\cdots 05}a^{17}+\frac{13\cdots 83}{13\cdots 20}a^{16}+\frac{15\cdots 03}{26\cdots 40}a^{15}+\frac{18\cdots 79}{26\cdots 04}a^{14}+\frac{21\cdots 73}{65\cdots 60}a^{13}+\frac{95\cdots 39}{30\cdots 60}a^{12}+\frac{16\cdots 79}{87\cdots 80}a^{11}+\frac{90\cdots 59}{65\cdots 60}a^{10}+\frac{36\cdots 27}{52\cdots 08}a^{9}+\frac{88\cdots 99}{26\cdots 04}a^{8}+\frac{18\cdots 07}{13\cdots 52}a^{7}+\frac{29\cdots 07}{65\cdots 76}a^{6}+\frac{69\cdots 21}{32\cdots 88}a^{5}+\frac{85\cdots 37}{20\cdots 18}a^{4}+\frac{36\cdots 69}{34\cdots 30}a^{3}+\frac{50\cdots 07}{20\cdots 80}a^{2}+\frac{84\cdots 49}{34\cdots 30}a+\frac{48\cdots 67}{56\cdots 05}$, $\frac{34\cdots 05}{52\cdots 08}a^{23}+\frac{74\cdots 39}{13\cdots 52}a^{22}-\frac{14\cdots 61}{43\cdots 84}a^{21}+\frac{14\cdots 15}{26\cdots 04}a^{20}-\frac{53\cdots 65}{13\cdots 52}a^{19}+\frac{26\cdots 63}{32\cdots 84}a^{18}-\frac{61\cdots 97}{14\cdots 28}a^{17}+\frac{11\cdots 89}{26\cdots 04}a^{16}-\frac{11\cdots 79}{52\cdots 08}a^{15}+\frac{35\cdots 05}{13\cdots 52}a^{14}-\frac{21\cdots 31}{13\cdots 52}a^{13}+\frac{82\cdots 53}{60\cdots 72}a^{12}-\frac{10\cdots 79}{17\cdots 36}a^{11}+\frac{11\cdots 49}{26\cdots 04}a^{10}-\frac{12\cdots 95}{52\cdots 08}a^{9}+\frac{11\cdots 91}{13\cdots 52}a^{8}-\frac{67\cdots 21}{13\cdots 52}a^{7}+\frac{10\cdots 87}{65\cdots 76}a^{6}-\frac{65\cdots 43}{16\cdots 44}a^{5}-\frac{15\cdots 73}{40\cdots 36}a^{4}-\frac{24\cdots 29}{27\cdots 24}a^{3}-\frac{13\cdots 83}{20\cdots 18}a^{2}-\frac{71\cdots 67}{68\cdots 06}a-\frac{50\cdots 19}{11\cdots 01}$, $\frac{30\cdots 43}{26\cdots 40}a^{23}-\frac{11\cdots 63}{16\cdots 40}a^{22}+\frac{10\cdots 33}{21\cdots 20}a^{21}-\frac{12\cdots 77}{13\cdots 20}a^{20}+\frac{69\cdots 01}{13\cdots 52}a^{19}-\frac{22\cdots 81}{16\cdots 20}a^{18}+\frac{30\cdots 17}{72\cdots 40}a^{17}-\frac{10\cdots 27}{13\cdots 20}a^{16}+\frac{55\cdots 21}{26\cdots 40}a^{15}-\frac{16\cdots 97}{32\cdots 88}a^{14}+\frac{11\cdots 59}{65\cdots 60}a^{13}-\frac{28\cdots 11}{12\cdots 40}a^{12}+\frac{44\cdots 49}{87\cdots 80}a^{11}-\frac{10\cdots 29}{13\cdots 20}a^{10}+\frac{12\cdots 37}{52\cdots 08}a^{9}-\frac{50\cdots 63}{32\cdots 88}a^{8}+\frac{73\cdots 73}{13\cdots 52}a^{7}-\frac{80\cdots 19}{32\cdots 88}a^{6}+\frac{18\cdots 25}{81\cdots 72}a^{5}+\frac{56\cdots 43}{16\cdots 44}a^{4}-\frac{20\cdots 57}{13\cdots 20}a^{3}+\frac{73\cdots 33}{20\cdots 80}a^{2}+\frac{28\cdots 79}{34\cdots 30}a+\frac{21\cdots 79}{56\cdots 05}$, $\frac{80\cdots 83}{26\cdots 40}a^{23}-\frac{36\cdots 79}{13\cdots 20}a^{22}+\frac{34\cdots 23}{21\cdots 20}a^{21}-\frac{36\cdots 97}{13\cdots 20}a^{20}+\frac{13\cdots 13}{65\cdots 76}a^{19}-\frac{64\cdots 81}{16\cdots 20}a^{18}+\frac{79\cdots 81}{36\cdots 20}a^{17}-\frac{27\cdots 27}{13\cdots 20}a^{16}+\frac{32\cdots 01}{26\cdots 40}a^{15}-\frac{32\cdots 17}{26\cdots 04}a^{14}+\frac{59\cdots 49}{65\cdots 60}a^{13}-\frac{72\cdots 11}{12\cdots 40}a^{12}+\frac{31\cdots 89}{87\cdots 80}a^{11}-\frac{29\cdots 43}{16\cdots 40}a^{10}+\frac{77\cdots 93}{52\cdots 08}a^{9}-\frac{75\cdots 69}{26\cdots 04}a^{8}+\frac{47\cdots 63}{13\cdots 52}a^{7}-\frac{13\cdots 99}{40\cdots 36}a^{6}+\frac{13\cdots 55}{32\cdots 88}a^{5}+\frac{11\cdots 17}{16\cdots 44}a^{4}+\frac{19\cdots 23}{13\cdots 20}a^{3}+\frac{31\cdots 42}{51\cdots 45}a^{2}+\frac{28\cdots 89}{34\cdots 30}a+\frac{18\cdots 69}{56\cdots 05}$, $\frac{59\cdots 99}{26\cdots 40}a^{23}-\frac{12\cdots 83}{13\cdots 20}a^{22}+\frac{79\cdots 61}{21\cdots 20}a^{21}-\frac{41\cdots 19}{13\cdots 20}a^{20}+\frac{11\cdots 27}{32\cdots 88}a^{19}-\frac{20\cdots 07}{16\cdots 20}a^{18}+\frac{79\cdots 81}{18\cdots 60}a^{17}-\frac{46\cdots 09}{13\cdots 20}a^{16}+\frac{11\cdots 67}{26\cdots 40}a^{15}-\frac{53\cdots 37}{26\cdots 04}a^{14}+\frac{12\cdots 03}{65\cdots 60}a^{13}-\frac{12\cdots 27}{12\cdots 40}a^{12}+\frac{97\cdots 83}{87\cdots 80}a^{11}-\frac{40\cdots 09}{65\cdots 60}a^{10}+\frac{22\cdots 19}{52\cdots 08}a^{9}-\frac{46\cdots 61}{26\cdots 04}a^{8}+\frac{95\cdots 73}{13\cdots 52}a^{7}-\frac{46\cdots 51}{32\cdots 88}a^{6}+\frac{72\cdots 89}{32\cdots 88}a^{5}+\frac{81\cdots 09}{16\cdots 44}a^{4}-\frac{26\cdots 77}{68\cdots 60}a^{3}+\frac{27\cdots 83}{10\cdots 90}a^{2}+\frac{19\cdots 23}{34\cdots 30}a+\frac{15\cdots 73}{56\cdots 05}$, $\frac{14\cdots 73}{13\cdots 20}a^{23}-\frac{41\cdots 43}{32\cdots 80}a^{22}+\frac{11\cdots 49}{13\cdots 20}a^{21}+\frac{63\cdots 27}{65\cdots 60}a^{20}+\frac{10\cdots 57}{65\cdots 76}a^{19}+\frac{67\cdots 71}{80\cdots 60}a^{18}+\frac{52\cdots 03}{36\cdots 20}a^{17}+\frac{24\cdots 57}{65\cdots 60}a^{16}+\frac{45\cdots 69}{13\cdots 20}a^{15}+\frac{18\cdots 73}{65\cdots 76}a^{14}+\frac{10\cdots 23}{16\cdots 40}a^{13}+\frac{32\cdots 63}{30\cdots 60}a^{12}+\frac{31\cdots 01}{43\cdots 40}a^{11}+\frac{16\cdots 89}{65\cdots 60}a^{10}+\frac{27\cdots 61}{26\cdots 04}a^{9}+\frac{31\cdots 31}{65\cdots 76}a^{8}+\frac{10\cdots 73}{32\cdots 88}a^{7}+\frac{88\cdots 77}{32\cdots 88}a^{6}+\frac{52\cdots 47}{32\cdots 88}a^{5}+\frac{85\cdots 07}{40\cdots 36}a^{4}+\frac{25\cdots 61}{34\cdots 30}a^{3}+\frac{21\cdots 17}{10\cdots 90}a^{2}+\frac{76\cdots 17}{34\cdots 30}a+\frac{43\cdots 27}{56\cdots 05}$
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| Regulator: | \( 792553891707177200000000 \) (assuming GRH) |
| |
| Unit signature rank: | \( 4 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 792553891707177200000000 \cdot 4}{2\cdot\sqrt{35863302598527284385359953566442393477815766818821430206298828125}}\cr\approx \mathstrut & 12.8425813217969 \end{aligned}\] (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
| 6.2.4901737578125.1, 12.4.951590574034612536651611328125.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{5}{,}\,{\href{/padicField/2.1.0.1}{1} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{8}$ | $24$ | ${\href{/padicField/17.8.0.1}{8} }^{3}$ | ${\href{/padicField/19.4.0.1}{4} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | $24$ | ${\href{/padicField/47.8.0.1}{8} }^{3}$ | $24$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.4.3a1.3 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.4.5.28a1.1 | $x^{20} + 20 x^{18} + 20 x^{17} + 170 x^{16} + 320 x^{15} + 960 x^{14} + 2080 x^{13} + 4215 x^{12} + 7680 x^{11} + 12884 x^{10} + 18580 x^{9} + 24570 x^{8} + 28320 x^{7} + 28000 x^{6} + 23184 x^{5} + 16100 x^{4} + 8960 x^{3} + 3760 x^{2} + 1040 x + 157$ | $5$ | $4$ | $28$ | 20T20 | $not computed$ | |
|
\(89\)
| 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 | $$[\ ]_{20}^{2}$$ |