Normalized defining polynomial
\( x^{24} - 9790 x^{20} - 699540 x^{18} - 14942655 x^{16} + 343628555 x^{14} + 25447488760 x^{12} + \cdots + 18205372928000 \)
Invariants
| Degree: | $24$ |
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| Signature: | $(4, 10)$ |
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| Discriminant: |
\(35863302598527284385359953566442393477815766818821430206298828125\)
\(\medspace = 5^{31}\cdot 89^{22}\)
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| Root discriminant: | \(489.53\) |
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| Galois root discriminant: | $5^{31/20}89^{19/20}\approx 861.6363513918435$ | ||
| Ramified primes: |
\(5\), \(89\)
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| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{11}-\frac{1}{2}a$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{2}$, $\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{4}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{3}{8}a^{3}+\frac{1}{8}a^{2}-\frac{1}{4}a$, $\frac{1}{8}a^{14}-\frac{1}{8}a^{12}-\frac{1}{4}a^{11}-\frac{1}{8}a^{4}-\frac{1}{2}a^{3}-\frac{3}{8}a^{2}+\frac{1}{4}a$, $\frac{1}{16}a^{15}-\frac{1}{16}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{16}a^{5}+\frac{5}{16}a^{3}+\frac{1}{4}a$, $\frac{1}{32}a^{16}-\frac{1}{32}a^{15}-\frac{1}{32}a^{14}+\frac{1}{32}a^{13}-\frac{1}{8}a^{12}-\frac{1}{8}a^{11}-\frac{1}{8}a^{10}+\frac{1}{8}a^{9}-\frac{1}{32}a^{6}-\frac{15}{32}a^{5}+\frac{5}{32}a^{4}-\frac{5}{32}a^{3}+\frac{1}{8}a^{2}-\frac{3}{8}a$, $\frac{1}{32}a^{17}-\frac{1}{32}a^{13}-\frac{1}{8}a^{12}-\frac{1}{4}a^{11}-\frac{1}{8}a^{9}-\frac{1}{32}a^{7}+\frac{1}{8}a^{5}-\frac{1}{2}a^{4}-\frac{11}{32}a^{3}-\frac{3}{8}a^{2}-\frac{3}{8}a$, $\frac{1}{192}a^{18}-\frac{1}{96}a^{16}+\frac{5}{192}a^{14}+\frac{1}{24}a^{12}+\frac{1}{48}a^{10}+\frac{5}{64}a^{8}-\frac{5}{96}a^{6}-\frac{1}{2}a^{5}+\frac{29}{64}a^{4}-\frac{1}{2}a^{3}-\frac{19}{48}a^{2}-\frac{1}{2}a+\frac{1}{3}$, $\frac{1}{384}a^{19}-\frac{1}{384}a^{18}+\frac{1}{96}a^{17}+\frac{1}{192}a^{16}-\frac{7}{384}a^{15}-\frac{5}{384}a^{14}-\frac{5}{192}a^{13}+\frac{5}{48}a^{12}+\frac{13}{96}a^{11}+\frac{23}{96}a^{10}-\frac{19}{128}a^{9}+\frac{27}{128}a^{8}-\frac{1}{24}a^{7}-\frac{43}{192}a^{6}-\frac{55}{128}a^{5}-\frac{29}{128}a^{4}-\frac{41}{192}a^{3}+\frac{7}{96}a^{2}+\frac{11}{48}a+\frac{1}{3}$, $\frac{1}{1401216}a^{20}+\frac{5}{5248}a^{18}-\frac{1}{64}a^{17}+\frac{227}{15744}a^{16}-\frac{1}{32}a^{15}-\frac{121}{5248}a^{14}-\frac{1}{64}a^{13}+\frac{403}{3936}a^{12}+\frac{1}{8}a^{11}-\frac{53}{15744}a^{10}+\frac{3}{16}a^{9}-\frac{2219}{15744}a^{8}-\frac{15}{64}a^{7}-\frac{2707}{15744}a^{6}-\frac{9}{32}a^{5}+\frac{3715}{15744}a^{4}-\frac{11}{64}a^{3}+\frac{589}{3936}a^{2}+\frac{7}{16}a-\frac{4}{123}$, $\frac{1}{2802432}a^{21}+\frac{5}{10496}a^{19}-\frac{1}{384}a^{18}+\frac{227}{31488}a^{17}+\frac{1}{192}a^{16}+\frac{207}{10496}a^{15}-\frac{5}{384}a^{14}+\frac{157}{7872}a^{13}+\frac{5}{48}a^{12}-\frac{3989}{31488}a^{11}+\frac{23}{96}a^{10}+\frac{1717}{31488}a^{9}+\frac{27}{128}a^{8}+\frac{5165}{31488}a^{7}-\frac{43}{192}a^{6}+\frac{10603}{31488}a^{5}+\frac{35}{128}a^{4}+\frac{3787}{7872}a^{3}-\frac{41}{96}a^{2}-\frac{385}{984}a+\frac{1}{3}$, $\frac{1}{48\cdots 20}a^{22}-\frac{1}{5604864}a^{21}+\frac{16\cdots 61}{96\cdots 84}a^{20}+\frac{67}{62976}a^{19}-\frac{28\cdots 59}{15\cdots 08}a^{18}-\frac{883}{62976}a^{17}+\frac{11\cdots 51}{10\cdots 56}a^{16}+\frac{773}{62976}a^{15}+\frac{18\cdots 55}{33\cdots 08}a^{14}+\frac{47}{1968}a^{13}+\frac{10\cdots 71}{10\cdots 56}a^{12}+\frac{127}{20992}a^{11}+\frac{16\cdots 19}{10\cdots 56}a^{10}+\frac{5417}{62976}a^{9}+\frac{32\cdots 07}{36\cdots 52}a^{8}-\frac{1831}{20992}a^{7}-\frac{15\cdots 53}{36\cdots 52}a^{6}+\frac{17195}{62976}a^{5}+\frac{45\cdots 31}{13\cdots 32}a^{4}+\frac{1325}{7872}a^{3}+\frac{10\cdots 93}{33\cdots 80}a^{2}-\frac{3}{1312}a+\frac{96\cdots 19}{42\cdots 26}$, $\frac{1}{16\cdots 00}a^{23}+\frac{42\cdots 59}{24\cdots 96}a^{21}+\frac{13\cdots 67}{25\cdots 40}a^{19}-\frac{1}{384}a^{18}+\frac{13\cdots 97}{15\cdots 40}a^{17}+\frac{1}{192}a^{16}-\frac{85\cdots 19}{36\cdots 60}a^{15}-\frac{5}{384}a^{14}+\frac{75\cdots 39}{36\cdots 60}a^{13}-\frac{1}{48}a^{12}+\frac{38\cdots 97}{90\cdots 40}a^{11}+\frac{23}{96}a^{10}-\frac{31\cdots 07}{18\cdots 80}a^{9}+\frac{27}{128}a^{8}+\frac{77\cdots 57}{37\cdots 60}a^{7}-\frac{43}{192}a^{6}-\frac{27\cdots 87}{12\cdots 20}a^{5}-\frac{29}{128}a^{4}+\frac{83\cdots 69}{56\cdots 00}a^{3}+\frac{19}{96}a^{2}-\frac{37\cdots 07}{45\cdots 72}a-\frac{1}{6}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$, $3$ |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ (assuming GRH) |
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| Narrow class group: | $C_{4}$, which has order $4$ (assuming GRH) |
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Unit group
| Rank: | $13$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{11\cdots 27}{12\cdots 80}a^{22}-\frac{15\cdots 95}{27\cdots 64}a^{20}+\frac{35\cdots 27}{38\cdots 52}a^{18}+\frac{19\cdots 75}{27\cdots 64}a^{16}+\frac{27\cdots 61}{13\cdots 32}a^{14}-\frac{55\cdots 59}{90\cdots 88}a^{12}-\frac{15\cdots 35}{90\cdots 88}a^{10}-\frac{10\cdots 41}{27\cdots 64}a^{8}-\frac{58\cdots 91}{27\cdots 64}a^{6}+\frac{47\cdots 81}{45\cdots 44}a^{4}+\frac{44\cdots 01}{56\cdots 80}a^{2}-\frac{51\cdots 07}{21\cdots 13}$, $\frac{23\cdots 29}{80\cdots 20}a^{22}-\frac{95\cdots 53}{48\cdots 92}a^{20}-\frac{22\cdots 25}{77\cdots 04}a^{18}-\frac{10\cdots 55}{54\cdots 28}a^{16}-\frac{41\cdots 07}{13\cdots 32}a^{14}+\frac{67\cdots 25}{54\cdots 28}a^{12}+\frac{12\cdots 31}{18\cdots 76}a^{10}+\frac{52\cdots 55}{54\cdots 28}a^{8}+\frac{69\cdots 09}{18\cdots 76}a^{6}-\frac{61\cdots 91}{13\cdots 32}a^{4}+\frac{41\cdots 69}{42\cdots 60}a^{2}+\frac{20\cdots 87}{17\cdots 31}$, $\frac{84\cdots 63}{16\cdots 68}a^{22}-\frac{78\cdots 59}{49\cdots 04}a^{20}+\frac{93\cdots 43}{18\cdots 12}a^{18}+\frac{20\cdots 03}{55\cdots 36}a^{16}+\frac{20\cdots 71}{23\cdots 64}a^{14}-\frac{84\cdots 99}{55\cdots 36}a^{12}-\frac{75\cdots 57}{55\cdots 36}a^{10}-\frac{16\cdots 53}{55\cdots 36}a^{8}-\frac{11\cdots 47}{55\cdots 36}a^{6}+\frac{18\cdots 45}{16\cdots 12}a^{4}+\frac{41\cdots 25}{84\cdots 56}a^{2}+\frac{13\cdots 43}{43\cdots 37}$, $\frac{74\cdots 69}{12\cdots 80}a^{22}+\frac{85\cdots 61}{27\cdots 64}a^{20}-\frac{31\cdots 71}{38\cdots 52}a^{18}-\frac{64\cdots 81}{27\cdots 64}a^{16}-\frac{89\cdots 97}{13\cdots 32}a^{14}+\frac{15\cdots 99}{90\cdots 88}a^{12}+\frac{22\cdots 43}{27\cdots 64}a^{10}+\frac{62\cdots 59}{90\cdots 88}a^{8}-\frac{45\cdots 95}{27\cdots 64}a^{6}-\frac{32\cdots 91}{13\cdots 32}a^{4}+\frac{31\cdots 07}{56\cdots 80}a^{2}+\frac{15\cdots 67}{70\cdots 71}$, $\frac{84\cdots 91}{41\cdots 92}a^{22}+\frac{16\cdots 17}{24\cdots 52}a^{20}+\frac{43\cdots 31}{23\cdots 64}a^{18}+\frac{20\cdots 77}{27\cdots 68}a^{16}-\frac{79\cdots 53}{23\cdots 64}a^{14}-\frac{11\cdots 65}{13\cdots 84}a^{12}-\frac{47\cdots 21}{27\cdots 68}a^{10}-\frac{32\cdots 79}{34\cdots 96}a^{8}+\frac{12\cdots 27}{27\cdots 68}a^{6}+\frac{14\cdots 41}{42\cdots 28}a^{4}-\frac{41\cdots 25}{42\cdots 28}a^{2}-\frac{27\cdots 47}{43\cdots 37}$, $\frac{32\cdots 93}{20\cdots 00}a^{23}-\frac{81\cdots 77}{30\cdots 20}a^{22}+\frac{13\cdots 69}{48\cdots 92}a^{21}-\frac{12\cdots 43}{60\cdots 24}a^{20}-\frac{40\cdots 27}{25\cdots 40}a^{19}+\frac{13\cdots 01}{48\cdots 44}a^{18}-\frac{21\cdots 19}{18\cdots 80}a^{17}+\frac{13\cdots 03}{67\cdots 16}a^{16}-\frac{48\cdots 63}{18\cdots 80}a^{15}+\frac{33\cdots 95}{67\cdots 16}a^{14}+\frac{71\cdots 39}{15\cdots 40}a^{13}-\frac{18\cdots 49}{22\cdots 72}a^{12}+\frac{75\cdots 01}{18\cdots 80}a^{11}-\frac{41\cdots 51}{55\cdots 92}a^{10}+\frac{54\cdots 99}{60\cdots 60}a^{9}-\frac{54\cdots 27}{33\cdots 08}a^{8}+\frac{12\cdots 19}{18\cdots 80}a^{7}-\frac{85\cdots 03}{67\cdots 16}a^{6}+\frac{57\cdots 23}{18\cdots 80}a^{5}-\frac{12\cdots 33}{22\cdots 72}a^{4}-\frac{26\cdots 13}{18\cdots 00}a^{3}+\frac{17\cdots 63}{70\cdots 10}a^{2}-\frac{65\cdots 13}{75\cdots 12}a+\frac{33\cdots 68}{21\cdots 13}$, $\frac{81\cdots 91}{53\cdots 00}a^{23}+\frac{63\cdots 57}{48\cdots 20}a^{22}-\frac{25\cdots 33}{32\cdots 28}a^{21}-\frac{23\cdots 23}{96\cdots 84}a^{20}+\frac{34\cdots 01}{17\cdots 60}a^{19}-\frac{20\cdots 63}{15\cdots 08}a^{18}+\frac{19\cdots 87}{12\cdots 20}a^{17}-\frac{64\cdots 49}{10\cdots 56}a^{16}+\frac{50\cdots 13}{15\cdots 40}a^{15}-\frac{77\cdots 49}{27\cdots 64}a^{14}-\frac{11\cdots 29}{12\cdots 20}a^{13}+\frac{21\cdots 85}{36\cdots 52}a^{12}-\frac{73\cdots 33}{12\cdots 20}a^{11}+\frac{62\cdots 49}{36\cdots 52}a^{10}-\frac{10\cdots 71}{12\cdots 20}a^{9}+\frac{14\cdots 93}{10\cdots 56}a^{8}+\frac{71\cdots 93}{12\cdots 20}a^{7}-\frac{12\cdots 67}{10\cdots 56}a^{6}+\frac{12\cdots 23}{37\cdots 60}a^{5}-\frac{59\cdots 55}{90\cdots 88}a^{4}-\frac{59\cdots 91}{75\cdots 00}a^{3}+\frac{44\cdots 63}{28\cdots 40}a^{2}-\frac{38\cdots 87}{75\cdots 12}a+\frac{21\cdots 08}{21\cdots 13}$, $\frac{33\cdots 45}{28\cdots 44}a^{22}-\frac{25\cdots 95}{86\cdots 32}a^{20}-\frac{99\cdots 55}{92\cdots 56}a^{18}-\frac{13\cdots 03}{24\cdots 72}a^{16}-\frac{25\cdots 39}{64\cdots 92}a^{14}+\frac{48\cdots 01}{96\cdots 88}a^{12}+\frac{17\cdots 17}{96\cdots 88}a^{10}+\frac{27\cdots 95}{19\cdots 76}a^{8}-\frac{18\cdots 43}{48\cdots 44}a^{6}-\frac{99\cdots 27}{19\cdots 76}a^{4}+\frac{65\cdots 87}{48\cdots 44}a^{2}+\frac{26\cdots 41}{30\cdots 59}$, $\frac{18\cdots 17}{80\cdots 00}a^{23}+\frac{17\cdots 91}{40\cdots 60}a^{22}-\frac{59\cdots 93}{60\cdots 24}a^{21}+\frac{24\cdots 67}{80\cdots 32}a^{20}-\frac{13\cdots 23}{64\cdots 60}a^{19}-\frac{40\cdots 67}{64\cdots 92}a^{18}-\frac{28\cdots 41}{45\cdots 20}a^{17}-\frac{48\cdots 69}{90\cdots 88}a^{16}+\frac{19\cdots 07}{18\cdots 80}a^{15}-\frac{16\cdots 47}{22\cdots 68}a^{14}+\frac{44\cdots 01}{60\cdots 60}a^{13}+\frac{43\cdots 09}{90\cdots 88}a^{12}+\frac{86\cdots 89}{75\cdots 20}a^{11}+\frac{13\cdots 45}{90\cdots 88}a^{10}+\frac{15\cdots 19}{22\cdots 60}a^{9}+\frac{45\cdots 41}{45\cdots 44}a^{8}-\frac{13\cdots 19}{45\cdots 20}a^{7}-\frac{37\cdots 15}{90\cdots 88}a^{6}-\frac{14\cdots 09}{60\cdots 60}a^{5}-\frac{28\cdots 17}{90\cdots 88}a^{4}+\frac{54\cdots 03}{75\cdots 00}a^{3}+\frac{10\cdots 97}{11\cdots 60}a^{2}+\frac{52\cdots 03}{11\cdots 68}a+\frac{42\cdots 29}{70\cdots 71}$, $\frac{15\cdots 01}{53\cdots 00}a^{23}+\frac{46\cdots 63}{48\cdots 20}a^{22}+\frac{73\cdots 65}{96\cdots 84}a^{21}+\frac{61\cdots 89}{32\cdots 28}a^{20}+\frac{12\cdots 43}{51\cdots 80}a^{19}-\frac{25\cdots 25}{15\cdots 08}a^{18}+\frac{15\cdots 27}{12\cdots 20}a^{17}-\frac{39\cdots 53}{36\cdots 52}a^{16}+\frac{27\cdots 81}{18\cdots 80}a^{15}-\frac{43\cdots 31}{27\cdots 64}a^{14}-\frac{11\cdots 79}{12\cdots 20}a^{13}+\frac{10\cdots 77}{10\cdots 56}a^{12}-\frac{14\cdots 99}{36\cdots 60}a^{11}+\frac{26\cdots 37}{10\cdots 56}a^{10}-\frac{21\cdots 33}{36\cdots 60}a^{9}-\frac{20\cdots 53}{10\cdots 56}a^{8}-\frac{10\cdots 81}{36\cdots 60}a^{7}-\frac{56\cdots 41}{10\cdots 56}a^{6}+\frac{21\cdots 59}{18\cdots 80}a^{5}+\frac{28\cdots 65}{27\cdots 64}a^{4}+\frac{90\cdots 99}{75\cdots 00}a^{3}+\frac{20\cdots 21}{84\cdots 20}a^{2}+\frac{69\cdots 27}{11\cdots 68}a+\frac{10\cdots 49}{70\cdots 71}$, $\frac{97\cdots 47}{32\cdots 40}a^{23}+\frac{27\cdots 89}{16\cdots 40}a^{22}-\frac{23\cdots 53}{32\cdots 28}a^{21}-\frac{10\cdots 97}{10\cdots 56}a^{20}-\frac{10\cdots 11}{34\cdots 12}a^{19}-\frac{65\cdots 39}{51\cdots 36}a^{18}-\frac{93\cdots 95}{72\cdots 52}a^{17}-\frac{44\cdots 43}{10\cdots 56}a^{16}-\frac{26\cdots 29}{60\cdots 96}a^{15}+\frac{36\cdots 81}{90\cdots 88}a^{14}+\frac{32\cdots 03}{24\cdots 84}a^{13}+\frac{55\cdots 69}{10\cdots 56}a^{12}+\frac{26\cdots 17}{72\cdots 52}a^{11}+\frac{10\cdots 93}{10\cdots 56}a^{10}+\frac{18\cdots 87}{72\cdots 52}a^{9}+\frac{52\cdots 79}{10\cdots 56}a^{8}-\frac{21\cdots 15}{24\cdots 84}a^{7}-\frac{27\cdots 61}{10\cdots 56}a^{6}-\frac{16\cdots 31}{18\cdots 88}a^{5}-\frac{46\cdots 77}{27\cdots 64}a^{4}+\frac{37\cdots 89}{15\cdots 40}a^{3}+\frac{89\cdots 43}{16\cdots 40}a^{2}+\frac{35\cdots 29}{22\cdots 36}a+\frac{71\cdots 66}{21\cdots 13}$, $\frac{20\cdots 01}{40\cdots 80}a^{23}-\frac{42\cdots 09}{24\cdots 60}a^{22}-\frac{37\cdots 49}{40\cdots 16}a^{21}+\frac{59\cdots 61}{54\cdots 28}a^{20}-\frac{65\cdots 45}{12\cdots 92}a^{19}+\frac{11\cdots 53}{77\cdots 04}a^{18}-\frac{35\cdots 29}{15\cdots 24}a^{17}+\frac{86\cdots 43}{54\cdots 28}a^{16}+\frac{59\cdots 49}{22\cdots 36}a^{15}-\frac{46\cdots 41}{27\cdots 64}a^{14}+\frac{55\cdots 83}{22\cdots 36}a^{13}-\frac{18\cdots 07}{54\cdots 28}a^{12}+\frac{49\cdots 57}{90\cdots 44}a^{11}+\frac{16\cdots 35}{54\cdots 28}a^{10}+\frac{28\cdots 55}{90\cdots 44}a^{9}+\frac{12\cdots 15}{18\cdots 76}a^{8}-\frac{59\cdots 73}{45\cdots 72}a^{7}-\frac{63\cdots 97}{18\cdots 76}a^{6}-\frac{12\cdots 11}{11\cdots 68}a^{5}-\frac{64\cdots 11}{27\cdots 64}a^{4}+\frac{13\cdots 23}{45\cdots 20}a^{3}+\frac{18\cdots 61}{33\cdots 80}a^{2}+\frac{14\cdots 71}{75\cdots 12}a+\frac{74\cdots 74}{21\cdots 13}$, $\frac{18\cdots 83}{66\cdots 60}a^{23}+\frac{72\cdots 11}{48\cdots 20}a^{22}-\frac{62\cdots 93}{65\cdots 72}a^{21}+\frac{44\cdots 23}{96\cdots 84}a^{20}+\frac{11\cdots 19}{49\cdots 16}a^{19}-\frac{21\cdots 37}{15\cdots 08}a^{18}+\frac{39\cdots 45}{14\cdots 48}a^{17}-\frac{55\cdots 29}{36\cdots 52}a^{16}+\frac{31\cdots 75}{24\cdots 08}a^{15}-\frac{20\cdots 45}{27\cdots 64}a^{14}+\frac{16\cdots 01}{49\cdots 16}a^{13}-\frac{21\cdots 59}{10\cdots 56}a^{12}+\frac{66\cdots 85}{14\cdots 48}a^{11}-\frac{27\cdots 67}{10\cdots 56}a^{10}+\frac{26\cdots 15}{14\cdots 48}a^{9}-\frac{10\cdots 05}{10\cdots 56}a^{8}-\frac{60\cdots 31}{49\cdots 16}a^{7}+\frac{25\cdots 45}{36\cdots 52}a^{6}-\frac{52\cdots 89}{74\cdots 24}a^{5}+\frac{36\cdots 21}{90\cdots 88}a^{4}+\frac{71\cdots 69}{30\cdots 60}a^{3}-\frac{22\cdots 71}{16\cdots 40}a^{2}+\frac{42\cdots 71}{29\cdots 79}a-\frac{17\cdots 17}{21\cdots 13}$
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| Regulator: | \( 1877883798506105000000000 \) (assuming GRH) |
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| Unit signature rank: | \( 4 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 1877883798506105000000000 \cdot 4}{2\cdot\sqrt{35863302598527284385359953566442393477815766818821430206298828125}}\cr\approx \mathstrut & 30.4293192520339 \end{aligned}\] (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 | ${\href{/padicField/2.4.0.1}{4} }^{5}{,}\,{\href{/padicField/2.1.0.1}{1} }^{4}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{8}$ | $24$ | ${\href{/padicField/17.8.0.1}{8} }^{3}$ | ${\href{/padicField/19.4.0.1}{4} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{3}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | $24$ | ${\href{/padicField/47.8.0.1}{8} }^{3}$ | $24$ | ${\href{/padicField/59.5.0.1}{5} }^{4}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.4.3a1.3 | $x^{4} + 15$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 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 | $$[\ ]_{20}^{2}$$ |