Normalized defining polynomial
\( x^{24} - 4 x^{23} - 99 x^{22} + 431 x^{21} - 14784 x^{20} - 3090 x^{19} + 940100 x^{18} + \cdots + 15\!\cdots\!16 \)
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{15}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{4}a^{16}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}+\frac{1}{4}a^{6}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{14}-\frac{1}{4}a^{13}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}-\frac{1}{2}a$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{10}+\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}+\frac{1}{4}a^{5}-\frac{1}{4}a^{4}+\frac{1}{4}a^{2}$, $\frac{1}{12}a^{19}+\frac{1}{12}a^{18}+\frac{1}{12}a^{17}-\frac{1}{12}a^{15}-\frac{1}{4}a^{14}-\frac{1}{12}a^{13}+\frac{1}{12}a^{12}-\frac{1}{4}a^{11}-\frac{1}{12}a^{10}-\frac{1}{6}a^{9}+\frac{1}{6}a^{7}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}+\frac{1}{3}a^{4}+\frac{1}{3}a^{3}+\frac{1}{3}a^{2}+\frac{1}{6}a$, $\frac{1}{120}a^{20}-\frac{1}{8}a^{18}+\frac{1}{24}a^{17}-\frac{1}{12}a^{16}+\frac{1}{12}a^{15}-\frac{1}{12}a^{14}+\frac{1}{6}a^{13}+\frac{1}{24}a^{12}-\frac{5}{24}a^{11}+\frac{1}{15}a^{10}-\frac{5}{24}a^{9}-\frac{1}{3}a^{8}+\frac{1}{12}a^{7}-\frac{3}{8}a^{6}-\frac{7}{24}a^{5}-\frac{3}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{6}a^{2}+\frac{1}{3}a-\frac{1}{5}$, $\frac{1}{240}a^{21}+\frac{1}{48}a^{19}-\frac{1}{48}a^{18}-\frac{1}{12}a^{17}+\frac{1}{24}a^{16}-\frac{1}{6}a^{14}-\frac{3}{16}a^{13}+\frac{11}{48}a^{12}-\frac{13}{60}a^{11}-\frac{3}{16}a^{10}-\frac{1}{12}a^{9}+\frac{1}{6}a^{8}-\frac{19}{48}a^{7}-\frac{19}{48}a^{6}-\frac{5}{16}a^{5}+\frac{19}{48}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{13}{30}a$, $\frac{1}{11040}a^{22}+\frac{1}{1104}a^{21}+\frac{13}{11040}a^{20}-\frac{7}{2208}a^{19}+\frac{43}{368}a^{18}+\frac{31}{1104}a^{17}-\frac{4}{69}a^{16}+\frac{3}{184}a^{15}-\frac{35}{736}a^{14}-\frac{187}{2208}a^{13}-\frac{1301}{5520}a^{12}-\frac{1}{32}a^{11}-\frac{313}{5520}a^{10}+\frac{143}{552}a^{9}+\frac{67}{736}a^{8}+\frac{197}{736}a^{7}+\frac{683}{2208}a^{6}-\frac{619}{2208}a^{5}+\frac{59}{184}a^{4}+\frac{97}{276}a^{3}-\frac{313}{1380}a^{2}+\frac{55}{138}a-\frac{57}{115}$, $\frac{1}{58\cdots 60}a^{23}-\frac{92\cdots 61}{29\cdots 80}a^{22}+\frac{74\cdots 19}{52\cdots 60}a^{21}-\frac{13\cdots 85}{38\cdots 44}a^{20}+\frac{27\cdots 07}{58\cdots 84}a^{19}-\frac{77\cdots 25}{19\cdots 72}a^{18}-\frac{10\cdots 35}{14\cdots 04}a^{17}-\frac{25\cdots 13}{29\cdots 08}a^{16}-\frac{68\cdots 57}{11\cdots 32}a^{15}-\frac{21\cdots 91}{11\cdots 32}a^{14}-\frac{85\cdots 01}{29\cdots 80}a^{13}-\frac{12\cdots 09}{64\cdots 40}a^{12}-\frac{62\cdots 99}{29\cdots 80}a^{11}+\frac{19\cdots 95}{63\cdots 48}a^{10}-\frac{54\cdots 21}{38\cdots 44}a^{9}+\frac{24\cdots 47}{11\cdots 32}a^{8}+\frac{12\cdots 65}{42\cdots 16}a^{7}-\frac{18\cdots 69}{50\cdots 84}a^{6}+\frac{51\cdots 09}{29\cdots 08}a^{5}-\frac{46\cdots 13}{15\cdots 32}a^{4}-\frac{97\cdots 29}{36\cdots 60}a^{3}+\frac{19\cdots 73}{36\cdots 60}a^{2}-\frac{33\cdots 33}{18\cdots 80}a-\frac{75\cdots 34}{27\cdots 93}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
| Ideal class group: | $C_{2}\times C_{4}\times C_{4}$, which has order $32$ (assuming GRH) |
| |
| Narrow class group: | $C_{4}\times C_{4}\times C_{2}\times C_{2}$, which has order $64$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{21\cdots 91}{11\cdots 60}a^{23}-\frac{16\cdots 69}{16\cdots 40}a^{22}-\frac{62\cdots 63}{33\cdots 80}a^{21}+\frac{35\cdots 13}{33\cdots 80}a^{20}-\frac{93\cdots 79}{33\cdots 08}a^{19}+\frac{29\cdots 07}{11\cdots 36}a^{18}+\frac{79\cdots 67}{41\cdots 26}a^{17}-\frac{37\cdots 07}{55\cdots 68}a^{16}+\frac{93\cdots 19}{67\cdots 16}a^{15}+\frac{16\cdots 69}{67\cdots 16}a^{14}-\frac{89\cdots 73}{16\cdots 40}a^{13}+\frac{11\cdots 71}{33\cdots 80}a^{12}-\frac{48\cdots 87}{16\cdots 40}a^{11}+\frac{68\cdots 49}{20\cdots 30}a^{10}+\frac{31\cdots 89}{67\cdots 16}a^{9}+\frac{32\cdots 35}{67\cdots 16}a^{8}+\frac{10\cdots 07}{67\cdots 16}a^{7}-\frac{19\cdots 39}{67\cdots 16}a^{6}+\frac{78\cdots 69}{16\cdots 04}a^{5}+\frac{89\cdots 45}{16\cdots 04}a^{4}-\frac{46\cdots 63}{13\cdots 20}a^{3}+\frac{67\cdots 72}{34\cdots 05}a^{2}+\frac{42\cdots 21}{10\cdots 15}a-\frac{15\cdots 47}{34\cdots 05}$, $\frac{14\cdots 91}{16\cdots 40}a^{23}+\frac{59\cdots 09}{13\cdots 42}a^{22}+\frac{25\cdots 63}{16\cdots 40}a^{21}-\frac{42\cdots 33}{55\cdots 80}a^{20}+\frac{11\cdots 41}{16\cdots 04}a^{19}+\frac{38\cdots 55}{16\cdots 04}a^{18}-\frac{30\cdots 41}{16\cdots 04}a^{17}+\frac{10\cdots 25}{27\cdots 84}a^{16}-\frac{20\cdots 01}{33\cdots 08}a^{15}-\frac{10\cdots 17}{33\cdots 08}a^{14}+\frac{19\cdots 17}{27\cdots 40}a^{13}+\frac{21\cdots 47}{33\cdots 08}a^{12}-\frac{12\cdots 83}{83\cdots 20}a^{11}-\frac{77\cdots 21}{83\cdots 20}a^{10}-\frac{40\cdots 57}{33\cdots 08}a^{9}+\frac{31\cdots 97}{33\cdots 08}a^{8}+\frac{14\cdots 67}{33\cdots 08}a^{7}+\frac{16\cdots 07}{33\cdots 08}a^{6}+\frac{73\cdots 19}{16\cdots 04}a^{5}-\frac{19\cdots 15}{13\cdots 42}a^{4}+\frac{41\cdots 47}{83\cdots 20}a^{3}+\frac{16\cdots 12}{20\cdots 63}a^{2}-\frac{35\cdots 39}{20\cdots 30}a+\frac{91\cdots 37}{34\cdots 05}$, $\frac{33\cdots 33}{22\cdots 20}a^{23}+\frac{89\cdots 89}{33\cdots 08}a^{22}+\frac{11\cdots 57}{67\cdots 60}a^{21}-\frac{19\cdots 21}{67\cdots 60}a^{20}+\frac{69\cdots 49}{33\cdots 08}a^{19}+\frac{11\cdots 23}{22\cdots 72}a^{18}-\frac{48\cdots 71}{33\cdots 08}a^{17}+\frac{10\cdots 85}{33\cdots 08}a^{16}-\frac{46\cdots 79}{44\cdots 44}a^{15}-\frac{24\cdots 67}{44\cdots 44}a^{14}+\frac{24\cdots 01}{83\cdots 20}a^{13}+\frac{12\cdots 95}{13\cdots 32}a^{12}+\frac{13\cdots 73}{55\cdots 80}a^{11}+\frac{22\cdots 87}{41\cdots 60}a^{10}-\frac{40\cdots 83}{13\cdots 32}a^{9}-\frac{70\cdots 13}{44\cdots 44}a^{8}-\frac{68\cdots 85}{44\cdots 44}a^{7}-\frac{26\cdots 05}{13\cdots 32}a^{6}-\frac{26\cdots 93}{67\cdots 16}a^{5}-\frac{30\cdots 89}{55\cdots 68}a^{4}+\frac{10\cdots 93}{20\cdots 30}a^{3}+\frac{83\cdots 17}{83\cdots 52}a^{2}-\frac{68\cdots 69}{20\cdots 30}a+\frac{10\cdots 47}{34\cdots 05}$, $\frac{35\cdots 19}{67\cdots 60}a^{23}+\frac{45\cdots 93}{11\cdots 60}a^{22}+\frac{84\cdots 67}{22\cdots 20}a^{21}-\frac{83\cdots 37}{22\cdots 20}a^{20}+\frac{62\cdots 89}{67\cdots 16}a^{19}-\frac{21\cdots 31}{67\cdots 16}a^{18}-\frac{64\cdots 49}{16\cdots 04}a^{17}+\frac{95\cdots 59}{33\cdots 08}a^{16}-\frac{22\cdots 51}{44\cdots 44}a^{15}+\frac{95\cdots 93}{13\cdots 32}a^{14}+\frac{32\cdots 59}{33\cdots 80}a^{13}-\frac{28\cdots 41}{67\cdots 60}a^{12}+\frac{33\cdots 49}{33\cdots 80}a^{11}-\frac{99\cdots 07}{27\cdots 40}a^{10}+\frac{52\cdots 25}{13\cdots 32}a^{9}-\frac{10\cdots 83}{44\cdots 44}a^{8}-\frac{52\cdots 09}{13\cdots 32}a^{7}+\frac{73\cdots 67}{44\cdots 44}a^{6}-\frac{33\cdots 25}{33\cdots 08}a^{5}+\frac{43\cdots 05}{33\cdots 08}a^{4}-\frac{12\cdots 93}{13\cdots 20}a^{3}-\frac{12\cdots 83}{10\cdots 15}a^{2}+\frac{19\cdots 29}{10\cdots 15}a-\frac{73\cdots 53}{34\cdots 05}$, $\frac{32\cdots 23}{80\cdots 40}a^{23}+\frac{19\cdots 63}{21\cdots 40}a^{22}-\frac{15\cdots 83}{32\cdots 36}a^{21}-\frac{67\cdots 87}{64\cdots 20}a^{20}-\frac{21\cdots 23}{43\cdots 48}a^{19}-\frac{37\cdots 97}{10\cdots 12}a^{18}+\frac{21\cdots 53}{64\cdots 72}a^{17}+\frac{50\cdots 61}{32\cdots 36}a^{16}+\frac{84\cdots 93}{32\cdots 36}a^{15}+\frac{31\cdots 79}{12\cdots 44}a^{14}-\frac{49\cdots 91}{21\cdots 40}a^{13}-\frac{49\cdots 51}{80\cdots 40}a^{12}-\frac{97\cdots 01}{12\cdots 44}a^{11}-\frac{60\cdots 59}{16\cdots 80}a^{10}+\frac{24\cdots 05}{53\cdots 56}a^{9}+\frac{10\cdots 97}{12\cdots 44}a^{8}+\frac{70\cdots 93}{12\cdots 44}a^{7}+\frac{22\cdots 53}{12\cdots 44}a^{6}+\frac{57\cdots 89}{43\cdots 48}a^{5}+\frac{98\cdots 55}{64\cdots 72}a^{4}+\frac{88\cdots 13}{13\cdots 90}a^{3}+\frac{74\cdots 53}{80\cdots 40}a^{2}-\frac{18\cdots 59}{80\cdots 34}a+\frac{15\cdots 73}{67\cdots 95}$, $\frac{16\cdots 15}{32\cdots 36}a^{23}-\frac{41\cdots 29}{12\cdots 44}a^{22}-\frac{71\cdots 45}{16\cdots 68}a^{21}+\frac{41\cdots 19}{12\cdots 44}a^{20}-\frac{36\cdots 81}{43\cdots 48}a^{19}+\frac{54\cdots 21}{32\cdots 36}a^{18}+\frac{29\cdots 43}{64\cdots 72}a^{17}-\frac{25\cdots 47}{10\cdots 12}a^{16}+\frac{23\cdots 93}{53\cdots 56}a^{15}+\frac{18\cdots 49}{12\cdots 44}a^{14}-\frac{15\cdots 03}{12\cdots 44}a^{13}+\frac{33\cdots 23}{10\cdots 12}a^{12}-\frac{11\cdots 47}{12\cdots 44}a^{11}+\frac{58\cdots 25}{32\cdots 36}a^{10}+\frac{51\cdots 59}{10\cdots 12}a^{9}+\frac{98\cdots 77}{43\cdots 48}a^{8}+\frac{57\cdots 59}{12\cdots 44}a^{7}-\frac{14\cdots 01}{12\cdots 44}a^{6}+\frac{25\cdots 31}{43\cdots 48}a^{5}+\frac{73\cdots 73}{64\cdots 72}a^{4}-\frac{10\cdots 69}{16\cdots 68}a^{3}+\frac{46\cdots 51}{16\cdots 68}a^{2}-\frac{10\cdots 61}{26\cdots 78}a+\frac{47\cdots 45}{13\cdots 39}$, $\frac{88\cdots 77}{36\cdots 60}a^{23}-\frac{14\cdots 07}{14\cdots 40}a^{22}+\frac{11\cdots 73}{65\cdots 20}a^{21}+\frac{20\cdots 27}{48\cdots 80}a^{20}+\frac{11\cdots 19}{29\cdots 92}a^{19}+\frac{15\cdots 37}{48\cdots 68}a^{18}+\frac{18\cdots 25}{14\cdots 04}a^{17}+\frac{18\cdots 25}{36\cdots 76}a^{16}-\frac{27\cdots 31}{18\cdots 38}a^{15}-\frac{52\cdots 05}{29\cdots 08}a^{14}-\frac{12\cdots 99}{14\cdots 40}a^{13}-\frac{51\cdots 57}{80\cdots 80}a^{12}-\frac{70\cdots 97}{14\cdots 40}a^{11}+\frac{10\cdots 13}{31\cdots 40}a^{10}+\frac{42\cdots 55}{24\cdots 84}a^{9}+\frac{61\cdots 25}{29\cdots 08}a^{8}+\frac{35\cdots 27}{32\cdots 12}a^{7}+\frac{34\cdots 65}{12\cdots 96}a^{6}+\frac{31\cdots 13}{29\cdots 08}a^{5}-\frac{63\cdots 88}{47\cdots 51}a^{4}+\frac{30\cdots 73}{36\cdots 60}a^{3}+\frac{98\cdots 33}{90\cdots 90}a^{2}-\frac{94\cdots 77}{45\cdots 45}a+\frac{54\cdots 61}{13\cdots 65}$, $\frac{36\cdots 55}{82\cdots 79}a^{23}-\frac{47\cdots 69}{26\cdots 28}a^{22}-\frac{36\cdots 71}{65\cdots 20}a^{21}+\frac{12\cdots 37}{43\cdots 80}a^{20}-\frac{15\cdots 77}{29\cdots 92}a^{19}-\frac{54\cdots 27}{43\cdots 88}a^{18}+\frac{78\cdots 11}{13\cdots 64}a^{17}-\frac{12\cdots 13}{82\cdots 79}a^{16}+\frac{12\cdots 71}{65\cdots 32}a^{15}+\frac{49\cdots 97}{26\cdots 28}a^{14}-\frac{48\cdots 29}{26\cdots 28}a^{13}-\frac{20\cdots 43}{14\cdots 96}a^{12}-\frac{20\cdots 51}{13\cdots 40}a^{11}-\frac{11\cdots 41}{65\cdots 20}a^{10}+\frac{96\cdots 71}{54\cdots 86}a^{9}+\frac{51\cdots 11}{26\cdots 28}a^{8}-\frac{10\cdots 69}{97\cdots 64}a^{7}+\frac{31\cdots 89}{26\cdots 28}a^{6}+\frac{33\cdots 43}{26\cdots 28}a^{5}-\frac{13\cdots 91}{34\cdots 28}a^{4}+\frac{16\cdots 97}{65\cdots 32}a^{3}+\frac{14\cdots 10}{82\cdots 79}a^{2}-\frac{18\cdots 86}{41\cdots 95}a+\frac{12\cdots 31}{13\cdots 65}$, $\frac{11\cdots 51}{72\cdots 20}a^{23}+\frac{15\cdots 21}{14\cdots 40}a^{22}+\frac{22\cdots 05}{13\cdots 64}a^{21}-\frac{14\cdots 95}{96\cdots 36}a^{20}+\frac{23\cdots 43}{97\cdots 64}a^{19}-\frac{26\cdots 31}{12\cdots 92}a^{18}-\frac{30\cdots 47}{14\cdots 04}a^{17}+\frac{80\cdots 59}{72\cdots 52}a^{16}-\frac{14\cdots 47}{14\cdots 04}a^{15}-\frac{11\cdots 23}{29\cdots 08}a^{14}+\frac{10\cdots 79}{14\cdots 40}a^{13}-\frac{12\cdots 09}{80\cdots 80}a^{12}+\frac{34\cdots 99}{29\cdots 08}a^{11}+\frac{20\cdots 87}{72\cdots 52}a^{10}-\frac{33\cdots 37}{48\cdots 68}a^{9}+\frac{32\cdots 43}{29\cdots 08}a^{8}-\frac{24\cdots 25}{32\cdots 12}a^{7}-\frac{11\cdots 05}{29\cdots 08}a^{6}+\frac{13\cdots 75}{29\cdots 08}a^{5}-\frac{51\cdots 83}{76\cdots 16}a^{4}-\frac{49\cdots 27}{90\cdots 90}a^{3}+\frac{30\cdots 41}{90\cdots 90}a^{2}-\frac{81\cdots 94}{39\cdots 03}a-\frac{27\cdots 53}{27\cdots 93}$, $\frac{92\cdots 32}{45\cdots 45}a^{23}+\frac{41\cdots 37}{14\cdots 40}a^{22}-\frac{16\cdots 45}{65\cdots 32}a^{21}-\frac{15\cdots 81}{48\cdots 80}a^{20}-\frac{75\cdots 11}{29\cdots 92}a^{19}-\frac{61\cdots 19}{12\cdots 92}a^{18}+\frac{56\cdots 11}{14\cdots 04}a^{17}+\frac{20\cdots 21}{72\cdots 52}a^{16}+\frac{11\cdots 09}{72\cdots 52}a^{15}+\frac{86\cdots 47}{29\cdots 08}a^{14}+\frac{14\cdots 87}{14\cdots 40}a^{13}-\frac{75\cdots 53}{13\cdots 80}a^{12}-\frac{16\cdots 09}{29\cdots 08}a^{11}-\frac{22\cdots 71}{36\cdots 60}a^{10}-\frac{32\cdots 69}{24\cdots 84}a^{9}+\frac{18\cdots 97}{29\cdots 08}a^{8}+\frac{18\cdots 89}{32\cdots 12}a^{7}+\frac{11\cdots 93}{29\cdots 08}a^{6}+\frac{26\cdots 79}{29\cdots 08}a^{5}+\frac{20\cdots 43}{76\cdots 16}a^{4}+\frac{47\cdots 11}{36\cdots 60}a^{3}+\frac{31\cdots 67}{90\cdots 90}a^{2}-\frac{82\cdots 17}{18\cdots 38}a+\frac{61\cdots 77}{13\cdots 65}$, $\frac{22\cdots 67}{86\cdots 96}a^{23}+\frac{11\cdots 61}{64\cdots 72}a^{22}+\frac{20\cdots 97}{86\cdots 96}a^{21}-\frac{25\cdots 07}{12\cdots 40}a^{20}+\frac{26\cdots 57}{64\cdots 72}a^{19}-\frac{13\cdots 61}{12\cdots 44}a^{18}-\frac{17\cdots 71}{64\cdots 72}a^{17}+\frac{30\cdots 17}{21\cdots 24}a^{16}-\frac{52\cdots 51}{25\cdots 88}a^{15}+\frac{14\cdots 67}{86\cdots 96}a^{14}+\frac{11\cdots 23}{13\cdots 39}a^{13}-\frac{13\cdots 57}{86\cdots 96}a^{12}+\frac{86\cdots 15}{21\cdots 24}a^{11}-\frac{33\cdots 67}{26\cdots 80}a^{10}-\frac{17\cdots 93}{25\cdots 88}a^{9}+\frac{59\cdots 87}{25\cdots 88}a^{8}-\frac{15\cdots 47}{86\cdots 96}a^{7}+\frac{69\cdots 07}{86\cdots 96}a^{6}-\frac{11\cdots 79}{12\cdots 44}a^{5}-\frac{83\cdots 11}{16\cdots 68}a^{4}+\frac{19\cdots 35}{32\cdots 36}a^{3}-\frac{69\cdots 33}{16\cdots 68}a^{2}+\frac{60\cdots 91}{40\cdots 17}a+\frac{56\cdots 99}{67\cdots 95}$, $\frac{41\cdots 27}{36\cdots 60}a^{23}-\frac{36\cdots 53}{36\cdots 60}a^{22}-\frac{83\cdots 19}{65\cdots 20}a^{21}+\frac{83\cdots 59}{12\cdots 20}a^{20}-\frac{22\cdots 39}{14\cdots 96}a^{19}-\frac{31\cdots 81}{48\cdots 68}a^{18}+\frac{72\cdots 63}{72\cdots 52}a^{17}+\frac{10\cdots 63}{72\cdots 52}a^{16}+\frac{58\cdots 87}{72\cdots 52}a^{15}+\frac{19\cdots 93}{36\cdots 76}a^{14}-\frac{95\cdots 23}{72\cdots 20}a^{13}-\frac{25\cdots 19}{26\cdots 60}a^{12}-\frac{18\cdots 39}{90\cdots 90}a^{11}-\frac{23\cdots 91}{31\cdots 40}a^{10}+\frac{14\cdots 27}{12\cdots 92}a^{9}+\frac{52\cdots 01}{36\cdots 76}a^{8}+\frac{75\cdots 67}{53\cdots 52}a^{7}+\frac{23\cdots 85}{63\cdots 48}a^{6}+\frac{12\cdots 13}{14\cdots 04}a^{5}+\frac{40\cdots 59}{76\cdots 16}a^{4}+\frac{86\cdots 91}{18\cdots 80}a^{3}-\frac{30\cdots 77}{90\cdots 90}a^{2}+\frac{15\cdots 37}{90\cdots 90}a+\frac{11\cdots 33}{13\cdots 65}$, $\frac{36\cdots 48}{45\cdots 45}a^{23}+\frac{86\cdots 19}{72\cdots 20}a^{22}+\frac{14\cdots 93}{16\cdots 80}a^{21}-\frac{31\cdots 67}{24\cdots 40}a^{20}+\frac{51\cdots 81}{48\cdots 32}a^{19}+\frac{91\cdots 64}{30\cdots 23}a^{18}-\frac{56\cdots 49}{72\cdots 52}a^{17}-\frac{60\cdots 71}{36\cdots 76}a^{16}-\frac{18\cdots 39}{36\cdots 76}a^{15}-\frac{43\cdots 19}{14\cdots 04}a^{14}+\frac{11\cdots 01}{72\cdots 20}a^{13}+\frac{10\cdots 01}{16\cdots 35}a^{12}+\frac{92\cdots 41}{72\cdots 20}a^{11}+\frac{12\cdots 37}{45\cdots 45}a^{10}-\frac{23\cdots 99}{12\cdots 92}a^{9}-\frac{13\cdots 89}{14\cdots 04}a^{8}-\frac{11\cdots 09}{16\cdots 56}a^{7}-\frac{10\cdots 01}{14\cdots 04}a^{6}+\frac{47\cdots 45}{14\cdots 04}a^{5}-\frac{11\cdots 59}{38\cdots 08}a^{4}-\frac{16\cdots 61}{90\cdots 90}a^{3}+\frac{79\cdots 61}{18\cdots 80}a^{2}-\frac{10\cdots 51}{90\cdots 90}a-\frac{59\cdots 27}{13\cdots 65}$
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| Regulator: | \( 1641545903294094000000000 \) (assuming GRH) |
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| Unit signature rank: | \( 3 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 1641545903294094000000000 \cdot 32}{2\cdot\sqrt{896582564963182109633998839161059836945394170470535755157470703125}}\cr\approx \mathstrut & 42.5595018374973 \end{aligned}\] (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
| 6.2.4901737578125.2, 12.4.951590574034612536651611328125.4 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{5}{,}\,{\href{/padicField/2.1.0.1}{1} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | R | $24$ | ${\href{/padicField/11.10.0.1}{10} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | $24$ | ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{3}$ | $20{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $24$ | $24$ | ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{8}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.4.3a1.4 | $x^{4} + 20$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.1.10.15a1.10 | $x^{10} + 15 x^{6} + 10$ | $10$ | $1$ | $15$ | $F_{5}\times C_2$ | $$[\frac{7}{4}]_{4}^{2}$$ | |
| 5.1.10.15a1.10 | $x^{10} + 15 x^{6} + 10$ | $10$ | $1$ | $15$ | $F_{5}\times C_2$ | $$[\frac{7}{4}]_{4}^{2}$$ | |
|
\(89\)
| 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 | $$[\ ]_{20}^{2}$$ |