Normalized defining polynomial
\( x^{24} - 8 x^{23} + 64 x^{22} + 433 x^{21} - 12374 x^{20} + 111376 x^{19} - 796828 x^{18} + \cdots + 52\!\cdots\!56 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $(4, 10)$ |
| |
| Discriminant: |
\(896582564963182109633998839161059836945394170470535755157470703125\)
\(\medspace = 5^{33}\cdot 89^{22}\)
|
| |
| Root discriminant: | \(559.79\) |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3}a^{15}-\frac{1}{3}a^{14}+\frac{1}{3}a^{13}+\frac{1}{3}a^{12}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{6}a^{16}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}-\frac{1}{3}a^{9}-\frac{1}{3}a^{8}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}+\frac{1}{3}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{2}-\frac{1}{6}a-\frac{1}{3}$, $\frac{1}{6}a^{17}+\frac{1}{3}a^{14}-\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{3}a^{10}-\frac{1}{3}a^{9}+\frac{1}{3}a^{8}-\frac{1}{3}a^{7}+\frac{1}{3}a^{6}-\frac{1}{3}a^{5}-\frac{1}{3}a^{3}-\frac{1}{6}a^{2}-\frac{1}{3}a$, $\frac{1}{6}a^{18}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}-\frac{1}{3}a^{11}+\frac{1}{3}a^{10}-\frac{1}{3}a^{9}-\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{6}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a-\frac{1}{3}$, $\frac{1}{12}a^{19}-\frac{1}{12}a^{18}-\frac{1}{12}a^{17}-\frac{1}{6}a^{15}+\frac{1}{6}a^{14}+\frac{1}{6}a^{13}-\frac{1}{2}a^{12}-\frac{1}{6}a^{11}-\frac{1}{3}a^{10}-\frac{1}{6}a^{8}+\frac{1}{3}a^{7}-\frac{1}{6}a^{6}+\frac{1}{3}a^{5}-\frac{5}{12}a^{4}+\frac{1}{12}a^{3}+\frac{1}{12}a^{2}$, $\frac{1}{24}a^{20}-\frac{1}{24}a^{19}+\frac{1}{24}a^{18}-\frac{1}{12}a^{16}+\frac{1}{12}a^{15}+\frac{1}{12}a^{14}-\frac{1}{12}a^{13}-\frac{1}{4}a^{12}+\frac{1}{6}a^{11}+\frac{1}{6}a^{10}+\frac{1}{4}a^{9}+\frac{1}{6}a^{8}+\frac{1}{4}a^{7}-\frac{1}{2}a^{6}+\frac{1}{8}a^{5}+\frac{5}{24}a^{4}+\frac{11}{24}a^{3}+\frac{1}{3}a^{2}-\frac{1}{6}a+\frac{1}{3}$, $\frac{1}{528}a^{21}+\frac{5}{528}a^{20}+\frac{1}{48}a^{19}+\frac{5}{264}a^{18}+\frac{7}{88}a^{17}+\frac{19}{264}a^{16}+\frac{7}{264}a^{15}+\frac{53}{264}a^{14}-\frac{17}{264}a^{13}+\frac{8}{33}a^{12}+\frac{61}{132}a^{11}+\frac{7}{264}a^{10}+\frac{4}{33}a^{9}-\frac{53}{264}a^{8}+\frac{16}{33}a^{7}+\frac{65}{176}a^{6}+\frac{175}{528}a^{5}-\frac{21}{176}a^{4}-\frac{1}{264}a^{3}+\frac{19}{66}a^{2}+\frac{3}{11}a+\frac{4}{33}$, $\frac{1}{5280}a^{22}+\frac{1}{5280}a^{21}+\frac{79}{5280}a^{20}-\frac{7}{176}a^{19}-\frac{35}{528}a^{18}-\frac{197}{2640}a^{17}-\frac{157}{2640}a^{16}-\frac{21}{880}a^{15}+\frac{95}{528}a^{14}+\frac{5}{12}a^{13}+\frac{373}{1320}a^{12}-\frac{1009}{2640}a^{11}+\frac{37}{220}a^{10}-\frac{59}{176}a^{9}-\frac{9}{44}a^{8}+\frac{193}{1760}a^{7}+\frac{89}{480}a^{6}+\frac{127}{1760}a^{5}-\frac{151}{528}a^{4}+\frac{1}{33}a^{3}-\frac{1}{220}a^{2}-\frac{109}{330}a+\frac{47}{165}$, $\frac{1}{11\cdots 80}a^{23}+\frac{52\cdots 69}{11\cdots 80}a^{22}-\frac{92\cdots 73}{11\cdots 80}a^{21}+\frac{93\cdots 51}{57\cdots 40}a^{20}-\frac{46\cdots 33}{11\cdots 28}a^{19}-\frac{25\cdots 37}{57\cdots 40}a^{18}-\frac{77\cdots 51}{19\cdots 80}a^{17}-\frac{51\cdots 93}{19\cdots 80}a^{16}-\frac{16\cdots 69}{57\cdots 40}a^{15}+\frac{10\cdots 97}{28\cdots 32}a^{14}-\frac{62\cdots 17}{28\cdots 20}a^{13}-\frac{12\cdots 51}{51\cdots 40}a^{12}+\frac{39\cdots 01}{17\cdots 95}a^{11}-\frac{26\cdots 73}{57\cdots 40}a^{10}-\frac{32\cdots 11}{71\cdots 58}a^{9}+\frac{16\cdots 13}{38\cdots 60}a^{8}-\frac{95\cdots 29}{11\cdots 80}a^{7}-\frac{16\cdots 29}{38\cdots 60}a^{6}+\frac{53\cdots 89}{57\cdots 40}a^{5}-\frac{22\cdots 67}{57\cdots 64}a^{4}+\frac{61\cdots 57}{23\cdots 60}a^{3}-\frac{12\cdots 93}{35\cdots 90}a^{2}+\frac{13\cdots 21}{35\cdots 90}a+\frac{42\cdots 03}{16\cdots 45}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}\times C_{4}$, which has order $8$ (assuming GRH) |
| |
| Narrow class group: | $C_{4}\times C_{2}$, which has order $8$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{20\cdots 77}{12\cdots 40}a^{23}+\frac{37\cdots 97}{21\cdots 40}a^{22}-\frac{85\cdots 97}{64\cdots 20}a^{21}-\frac{41\cdots 49}{12\cdots 40}a^{20}+\frac{14\cdots 91}{64\cdots 32}a^{19}-\frac{48\cdots 77}{21\cdots 40}a^{18}+\frac{48\cdots 64}{26\cdots 05}a^{17}-\frac{71\cdots 43}{16\cdots 30}a^{16}-\frac{26\cdots 31}{53\cdots 10}a^{15}+\frac{37\cdots 63}{42\cdots 88}a^{14}-\frac{94\cdots 47}{10\cdots 20}a^{13}+\frac{12\cdots 79}{21\cdots 40}a^{12}-\frac{75\cdots 33}{21\cdots 40}a^{11}+\frac{37\cdots 97}{21\cdots 40}a^{10}-\frac{11\cdots 69}{12\cdots 64}a^{9}+\frac{73\cdots 97}{12\cdots 40}a^{8}-\frac{64\cdots 27}{21\cdots 40}a^{7}+\frac{11\cdots 37}{64\cdots 20}a^{6}-\frac{95\cdots 11}{12\cdots 40}a^{5}+\frac{35\cdots 25}{12\cdots 64}a^{4}-\frac{17\cdots 17}{19\cdots 40}a^{3}+\frac{17\cdots 39}{16\cdots 30}a^{2}-\frac{66\cdots 83}{26\cdots 05}a-\frac{15\cdots 78}{26\cdots 05}$, $\frac{33\cdots 79}{12\cdots 40}a^{23}+\frac{30\cdots 57}{64\cdots 20}a^{22}+\frac{75\cdots 13}{32\cdots 60}a^{21}-\frac{24\cdots 03}{12\cdots 40}a^{20}+\frac{14\cdots 15}{64\cdots 32}a^{19}-\frac{62\cdots 13}{16\cdots 30}a^{18}-\frac{14\cdots 91}{80\cdots 15}a^{17}+\frac{22\cdots 94}{26\cdots 05}a^{16}-\frac{18\cdots 96}{26\cdots 05}a^{15}+\frac{16\cdots 01}{42\cdots 88}a^{14}-\frac{76\cdots 87}{32\cdots 60}a^{13}+\frac{12\cdots 49}{64\cdots 20}a^{12}-\frac{22\cdots 43}{58\cdots 20}a^{11}+\frac{29\cdots 69}{21\cdots 40}a^{10}-\frac{61\cdots 23}{42\cdots 88}a^{9}+\frac{20\cdots 39}{12\cdots 40}a^{8}+\frac{94\cdots 93}{58\cdots 20}a^{7}+\frac{12\cdots 59}{10\cdots 20}a^{6}+\frac{46\cdots 03}{12\cdots 40}a^{5}-\frac{22\cdots 97}{39\cdots 08}a^{4}-\frac{27\cdots 21}{48\cdots 10}a^{3}-\frac{40\cdots 73}{97\cdots 20}a^{2}-\frac{84\cdots 37}{53\cdots 10}a-\frac{32\cdots 46}{26\cdots 05}$, $\frac{37\cdots 43}{15\cdots 48}a^{23}-\frac{78\cdots 17}{19\cdots 06}a^{22}+\frac{22\cdots 49}{79\cdots 24}a^{21}-\frac{12\cdots 65}{52\cdots 16}a^{20}-\frac{79\cdots 87}{79\cdots 24}a^{19}+\frac{38\cdots 77}{79\cdots 24}a^{18}-\frac{54\cdots 73}{19\cdots 06}a^{17}+\frac{56\cdots 34}{33\cdots 51}a^{16}+\frac{37\cdots 83}{66\cdots 02}a^{15}-\frac{10\cdots 49}{79\cdots 24}a^{14}+\frac{21\cdots 37}{19\cdots 06}a^{13}-\frac{70\cdots 13}{79\cdots 24}a^{12}+\frac{79\cdots 59}{26\cdots 08}a^{11}-\frac{13\cdots 95}{72\cdots 84}a^{10}+\frac{67\cdots 83}{79\cdots 24}a^{9}-\frac{24\cdots 67}{52\cdots 16}a^{8}+\frac{20\cdots 91}{39\cdots 12}a^{7}-\frac{11\cdots 33}{79\cdots 24}a^{6}+\frac{12\cdots 35}{15\cdots 48}a^{5}-\frac{27\cdots 52}{99\cdots 53}a^{4}-\frac{13\cdots 57}{72\cdots 84}a^{3}-\frac{48\cdots 73}{66\cdots 02}a^{2}-\frac{23\cdots 30}{33\cdots 51}a-\frac{64\cdots 82}{99\cdots 53}$, $\frac{59\cdots 65}{76\cdots 52}a^{23}+\frac{24\cdots 87}{11\cdots 80}a^{22}-\frac{44\cdots 63}{11\cdots 80}a^{21}-\frac{51\cdots 31}{95\cdots 40}a^{20}+\frac{78\cdots 33}{11\cdots 28}a^{19}-\frac{19\cdots 05}{38\cdots 76}a^{18}+\frac{67\cdots 67}{19\cdots 80}a^{17}+\frac{90\cdots 91}{57\cdots 40}a^{16}-\frac{34\cdots 17}{19\cdots 80}a^{15}+\frac{52\cdots 19}{19\cdots 88}a^{14}-\frac{10\cdots 85}{57\cdots 64}a^{13}+\frac{53\cdots 47}{57\cdots 40}a^{12}-\frac{17\cdots 09}{28\cdots 20}a^{11}+\frac{94\cdots 23}{57\cdots 40}a^{10}-\frac{37\cdots 35}{19\cdots 88}a^{9}+\frac{57\cdots 77}{76\cdots 52}a^{8}-\frac{59\cdots 27}{11\cdots 80}a^{7}+\frac{32\cdots 03}{11\cdots 80}a^{6}-\frac{10\cdots 27}{28\cdots 20}a^{5}+\frac{21\cdots 65}{47\cdots 72}a^{4}+\frac{35\cdots 63}{47\cdots 72}a^{3}+\frac{14\cdots 67}{71\cdots 80}a^{2}+\frac{32\cdots 71}{35\cdots 90}a+\frac{14\cdots 87}{17\cdots 95}$, $\frac{10\cdots 11}{11\cdots 80}a^{23}-\frac{32\cdots 01}{38\cdots 60}a^{22}+\frac{15\cdots 59}{22\cdots 56}a^{21}+\frac{94\cdots 21}{28\cdots 20}a^{20}-\frac{12\cdots 73}{10\cdots 48}a^{19}+\frac{67\cdots 23}{57\cdots 40}a^{18}-\frac{50\cdots 89}{57\cdots 40}a^{17}+\frac{14\cdots 17}{11\cdots 28}a^{16}+\frac{17\cdots 27}{57\cdots 40}a^{15}-\frac{91\cdots 67}{19\cdots 88}a^{14}+\frac{12\cdots 23}{28\cdots 20}a^{13}-\frac{47\cdots 11}{17\cdots 80}a^{12}+\frac{31\cdots 85}{19\cdots 88}a^{11}-\frac{44\cdots 31}{57\cdots 40}a^{10}+\frac{24\cdots 79}{57\cdots 64}a^{9}-\frac{29\cdots 51}{11\cdots 80}a^{8}+\frac{15\cdots 03}{11\cdots 80}a^{7}-\frac{18\cdots 39}{22\cdots 56}a^{6}+\frac{29\cdots 33}{95\cdots 40}a^{5}-\frac{64\cdots 69}{57\cdots 64}a^{4}+\frac{70\cdots 77}{21\cdots 60}a^{3}-\frac{93\cdots 33}{71\cdots 80}a^{2}+\frac{10\cdots 21}{35\cdots 79}a+\frac{80\cdots 11}{17\cdots 95}$, $\frac{28\cdots 99}{57\cdots 40}a^{23}+\frac{36\cdots 19}{17\cdots 80}a^{22}-\frac{10\cdots 69}{38\cdots 76}a^{21}-\frac{18\cdots 93}{43\cdots 20}a^{20}+\frac{17\cdots 83}{57\cdots 64}a^{19}-\frac{27\cdots 09}{95\cdots 40}a^{18}+\frac{14\cdots 07}{95\cdots 40}a^{17}+\frac{22\cdots 93}{19\cdots 88}a^{16}-\frac{25\cdots 43}{28\cdots 20}a^{15}+\frac{46\cdots 81}{28\cdots 32}a^{14}-\frac{36\cdots 09}{47\cdots 20}a^{13}+\frac{14\cdots 67}{28\cdots 20}a^{12}-\frac{76\cdots 79}{28\cdots 32}a^{11}+\frac{66\cdots 13}{95\cdots 40}a^{10}-\frac{29\cdots 95}{28\cdots 32}a^{9}+\frac{15\cdots 99}{57\cdots 40}a^{8}-\frac{17\cdots 07}{57\cdots 40}a^{7}+\frac{11\cdots 67}{11\cdots 28}a^{6}-\frac{27\cdots 91}{14\cdots 60}a^{5}+\frac{47\cdots 83}{28\cdots 32}a^{4}+\frac{27\cdots 96}{59\cdots 65}a^{3}+\frac{19\cdots 23}{23\cdots 60}a^{2}+\frac{17\cdots 81}{35\cdots 79}a+\frac{11\cdots 77}{17\cdots 95}$, $\frac{23\cdots 39}{85\cdots 60}a^{23}+\frac{10\cdots 31}{12\cdots 40}a^{22}-\frac{59\cdots 89}{16\cdots 30}a^{21}-\frac{11\cdots 83}{85\cdots 60}a^{20}+\frac{36\cdots 51}{12\cdots 64}a^{19}-\frac{24\cdots 91}{21\cdots 40}a^{18}+\frac{49\cdots 83}{64\cdots 20}a^{17}+\frac{34\cdots 68}{80\cdots 15}a^{16}-\frac{14\cdots 59}{16\cdots 30}a^{15}+\frac{52\cdots 23}{85\cdots 76}a^{14}-\frac{36\cdots 01}{64\cdots 20}a^{13}+\frac{10\cdots 29}{42\cdots 80}a^{12}-\frac{14\cdots 99}{12\cdots 40}a^{11}+\frac{10\cdots 41}{12\cdots 40}a^{10}-\frac{77\cdots 35}{25\cdots 28}a^{9}+\frac{24\cdots 79}{85\cdots 60}a^{8}-\frac{17\cdots 71}{12\cdots 40}a^{7}+\frac{73\cdots 59}{16\cdots 30}a^{6}-\frac{54\cdots 91}{25\cdots 80}a^{5}+\frac{62\cdots 29}{25\cdots 28}a^{4}-\frac{58\cdots 23}{64\cdots 20}a^{3}-\frac{22\cdots 63}{14\cdots 30}a^{2}+\frac{56\cdots 88}{80\cdots 15}a+\frac{69\cdots 53}{80\cdots 15}$, $\frac{90\cdots 77}{19\cdots 06}a^{23}+\frac{12\cdots 85}{39\cdots 12}a^{22}-\frac{16\cdots 97}{72\cdots 84}a^{21}-\frac{16\cdots 99}{66\cdots 02}a^{20}+\frac{21\cdots 75}{39\cdots 12}a^{19}-\frac{35\cdots 55}{79\cdots 24}a^{18}+\frac{11\cdots 17}{39\cdots 12}a^{17}+\frac{25\cdots 95}{66\cdots 02}a^{16}-\frac{11\cdots 33}{66\cdots 02}a^{15}+\frac{19\cdots 99}{99\cdots 53}a^{14}-\frac{31\cdots 59}{19\cdots 06}a^{13}+\frac{32\cdots 23}{39\cdots 12}a^{12}-\frac{30\cdots 65}{66\cdots 02}a^{11}+\frac{77\cdots 03}{39\cdots 12}a^{10}-\frac{46\cdots 17}{39\cdots 12}a^{9}+\frac{10\cdots 91}{13\cdots 04}a^{8}-\frac{39\cdots 92}{99\cdots 53}a^{7}+\frac{18\cdots 19}{79\cdots 24}a^{6}-\frac{12\cdots 25}{19\cdots 06}a^{5}+\frac{61\cdots 05}{39\cdots 12}a^{4}-\frac{36\cdots 25}{79\cdots 24}a^{3}-\frac{66\cdots 56}{33\cdots 51}a^{2}+\frac{54\cdots 06}{30\cdots 41}a+\frac{47\cdots 16}{90\cdots 23}$, $\frac{63\cdots 97}{11\cdots 80}a^{23}-\frac{80\cdots 37}{11\cdots 80}a^{22}+\frac{29\cdots 23}{38\cdots 60}a^{21}+\frac{44\cdots 76}{17\cdots 95}a^{20}-\frac{98\cdots 35}{10\cdots 48}a^{19}+\frac{62\cdots 61}{57\cdots 40}a^{18}-\frac{16\cdots 77}{19\cdots 80}a^{17}+\frac{40\cdots 09}{19\cdots 80}a^{16}+\frac{12\cdots 67}{51\cdots 40}a^{15}-\frac{74\cdots 93}{17\cdots 08}a^{14}+\frac{11\cdots 21}{28\cdots 20}a^{13}-\frac{47\cdots 99}{19\cdots 80}a^{12}+\frac{39\cdots 87}{28\cdots 20}a^{11}-\frac{35\cdots 61}{57\cdots 40}a^{10}+\frac{17\cdots 03}{57\cdots 64}a^{9}-\frac{74\cdots 89}{34\cdots 60}a^{8}+\frac{13\cdots 37}{11\cdots 80}a^{7}-\frac{79\cdots 29}{11\cdots 80}a^{6}+\frac{66\cdots 27}{23\cdots 60}a^{5}-\frac{36\cdots 51}{57\cdots 64}a^{4}+\frac{14\cdots 07}{71\cdots 80}a^{3}+\frac{15\cdots 43}{71\cdots 80}a^{2}-\frac{88\cdots 43}{35\cdots 90}a-\frac{45\cdots 34}{17\cdots 95}$, $\frac{16\cdots 41}{11\cdots 80}a^{23}-\frac{24\cdots 41}{11\cdots 80}a^{22}-\frac{41\cdots 43}{11\cdots 80}a^{21}+\frac{34\cdots 93}{28\cdots 20}a^{20}-\frac{45\cdots 63}{38\cdots 76}a^{19}-\frac{32\cdots 29}{19\cdots 80}a^{18}+\frac{14\cdots 77}{57\cdots 40}a^{17}-\frac{33\cdots 69}{57\cdots 40}a^{16}+\frac{20\cdots 61}{57\cdots 40}a^{15}-\frac{79\cdots 59}{57\cdots 64}a^{14}+\frac{47\cdots 51}{95\cdots 40}a^{13}+\frac{32\cdots 19}{57\cdots 40}a^{12}-\frac{16\cdots 09}{25\cdots 20}a^{11}+\frac{11\cdots 27}{57\cdots 40}a^{10}+\frac{63\cdots 57}{17\cdots 08}a^{9}-\frac{14\cdots 47}{38\cdots 60}a^{8}-\frac{42\cdots 59}{11\cdots 80}a^{7}+\frac{21\cdots 01}{38\cdots 60}a^{6}-\frac{81\cdots 31}{95\cdots 40}a^{5}+\frac{86\cdots 67}{19\cdots 88}a^{4}+\frac{33\cdots 81}{11\cdots 30}a^{3}+\frac{50\cdots 13}{23\cdots 60}a^{2}+\frac{14\cdots 67}{11\cdots 30}a+\frac{18\cdots 18}{17\cdots 95}$, $\frac{87\cdots 89}{11\cdots 80}a^{23}+\frac{31\cdots 87}{38\cdots 60}a^{22}+\frac{17\cdots 03}{11\cdots 80}a^{21}+\frac{50\cdots 39}{57\cdots 40}a^{20}-\frac{58\cdots 67}{11\cdots 28}a^{19}+\frac{14\cdots 67}{57\cdots 40}a^{18}+\frac{21\cdots 43}{57\cdots 40}a^{17}-\frac{86\cdots 77}{19\cdots 80}a^{16}+\frac{15\cdots 79}{57\cdots 40}a^{15}-\frac{87\cdots 21}{28\cdots 32}a^{14}+\frac{29\cdots 17}{28\cdots 20}a^{13}-\frac{28\cdots 69}{57\cdots 40}a^{12}+\frac{29\cdots 97}{14\cdots 60}a^{11}+\frac{40\cdots 21}{19\cdots 80}a^{10}+\frac{35\cdots 31}{28\cdots 32}a^{9}-\frac{41\cdots 29}{11\cdots 80}a^{8}+\frac{11\cdots 33}{38\cdots 60}a^{7}+\frac{75\cdots 17}{11\cdots 80}a^{6}-\frac{31\cdots 79}{57\cdots 40}a^{5}+\frac{52\cdots 31}{95\cdots 44}a^{4}-\frac{33\cdots 07}{14\cdots 60}a^{3}+\frac{80\cdots 19}{17\cdots 95}a^{2}-\frac{53\cdots 51}{35\cdots 90}a-\frac{35\cdots 28}{17\cdots 95}$, $\frac{20\cdots 23}{57\cdots 40}a^{23}-\frac{23\cdots 79}{25\cdots 20}a^{22}+\frac{43\cdots 93}{28\cdots 20}a^{21}-\frac{10\cdots 79}{57\cdots 40}a^{20}+\frac{10\cdots 31}{57\cdots 64}a^{19}-\frac{64\cdots 51}{43\cdots 20}a^{18}+\frac{59\cdots 42}{59\cdots 65}a^{17}-\frac{83\cdots 11}{14\cdots 60}a^{16}+\frac{14\cdots 63}{47\cdots 20}a^{15}-\frac{79\cdots 89}{57\cdots 64}a^{14}+\frac{25\cdots 01}{35\cdots 90}a^{13}-\frac{99\cdots 73}{28\cdots 20}a^{12}+\frac{56\cdots 81}{28\cdots 20}a^{11}-\frac{25\cdots 09}{28\cdots 20}a^{10}+\frac{55\cdots 35}{19\cdots 88}a^{9}-\frac{12\cdots 21}{19\cdots 80}a^{8}-\frac{90\cdots 11}{28\cdots 20}a^{7}+\frac{49\cdots 67}{28\cdots 20}a^{6}-\frac{56\cdots 81}{57\cdots 40}a^{5}+\frac{47\cdots 23}{14\cdots 16}a^{4}+\frac{26\cdots 27}{14\cdots 60}a^{3}+\frac{70\cdots 89}{11\cdots 30}a^{2}+\frac{44\cdots 11}{59\cdots 65}a+\frac{41\cdots 71}{59\cdots 65}$, $\frac{45\cdots 63}{11\cdots 80}a^{23}-\frac{14\cdots 59}{76\cdots 52}a^{22}+\frac{15\cdots 49}{11\cdots 80}a^{21}+\frac{14\cdots 89}{57\cdots 40}a^{20}-\frac{16\cdots 77}{38\cdots 76}a^{19}+\frac{15\cdots 89}{57\cdots 40}a^{18}-\frac{63\cdots 81}{38\cdots 76}a^{17}-\frac{48\cdots 33}{57\cdots 40}a^{16}+\frac{77\cdots 29}{57\cdots 40}a^{15}-\frac{39\cdots 49}{28\cdots 32}a^{14}+\frac{29\cdots 49}{28\cdots 20}a^{13}-\frac{16\cdots 77}{38\cdots 76}a^{12}+\frac{32\cdots 33}{11\cdots 30}a^{11}-\frac{50\cdots 47}{57\cdots 40}a^{10}+\frac{50\cdots 25}{71\cdots 58}a^{9}-\frac{58\cdots 23}{11\cdots 80}a^{8}+\frac{43\cdots 61}{22\cdots 56}a^{7}-\frac{54\cdots 43}{38\cdots 60}a^{6}+\frac{79\cdots 11}{57\cdots 40}a^{5}-\frac{26\cdots 65}{57\cdots 64}a^{4}+\frac{26\cdots 13}{11\cdots 30}a^{3}+\frac{13\cdots 99}{47\cdots 72}a^{2}+\frac{30\cdots 32}{59\cdots 65}a+\frac{46\cdots 47}{17\cdots 95}$
|
| |
| Regulator: | \( 268297290075719700000000 \) (assuming GRH) |
| |
| Unit signature rank: | \( 4 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 268297290075719700000000 \cdot 8}{2\cdot\sqrt{896582564963182109633998839161059836945394170470535755157470703125}}\cr\approx \mathstrut & 1.73900086909836 \end{aligned}\] (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 | ${\href{/padicField/2.4.0.1}{4} }^{5}{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{8}$ | $24$ | ${\href{/padicField/17.8.0.1}{8} }^{3}$ | ${\href{/padicField/19.4.0.1}{4} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | $24$ | ${\href{/padicField/47.8.0.1}{8} }^{3}$ | $24$ | ${\href{/padicField/59.5.0.1}{5} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.4.3a1.2 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.2.10.30a6.5 | $x^{20} + 40 x^{19} + 740 x^{18} + 8400 x^{17} + 65460 x^{16} + 371328 x^{15} + 1586895 x^{14} + 5218990 x^{13} + 13387420 x^{12} + 27012280 x^{11} + 43108740 x^{10} + 54631720 x^{9} + 55176000 x^{8} + 44559680 x^{7} + 28824080 x^{6} + 14877236 x^{5} + 6032495 x^{4} + 1855885 x^{3} + 406090 x^{2} + 56055 x + 3664$ | $10$ | $2$ | $30$ | 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 | $$[\ ]_{20}^{2}$$ |