Normalized defining polynomial
\( x^{24} - 2 x^{23} - 51 x^{22} + 192 x^{21} + 466 x^{20} - 5962 x^{19} + 36309 x^{18} - 164988 x^{17} + \cdots - 601196449 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $[4, 10]$ |
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| Discriminant: |
\(61343664831565854732948944775316051495075225830078125\)
\(\medspace = 5^{23}\cdot 197^{16}\)
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| |
| Root discriminant: | \(158.30\) |
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| Galois root discriminant: | $5^{23/24}197^{4/5}\approx 320.19845723009803$ | ||
| Ramified primes: |
\(5\), \(197\)
|
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{18}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{19}-\frac{1}{2}a^{4}$, $\frac{1}{2}a^{20}-\frac{1}{2}a^{5}$, $\frac{1}{2}a^{21}-\frac{1}{2}a^{6}$, $\frac{1}{236}a^{22}-\frac{27}{118}a^{21}+\frac{9}{118}a^{20}+\frac{4}{59}a^{19}+\frac{5}{118}a^{18}-\frac{19}{118}a^{17}+\frac{39}{236}a^{16}-\frac{10}{59}a^{15}+\frac{31}{236}a^{14}-\frac{1}{118}a^{13}-\frac{8}{59}a^{12}-\frac{7}{118}a^{11}+\frac{23}{236}a^{10}+\frac{49}{118}a^{9}-\frac{15}{59}a^{8}-\frac{8}{59}a^{7}-\frac{103}{236}a^{6}-\frac{53}{118}a^{5}+\frac{103}{236}a^{4}+\frac{11}{59}a^{3}+\frac{77}{236}a^{2}-\frac{8}{59}a+\frac{37}{236}$, $\frac{1}{30\cdots 24}a^{23}+\frac{34\cdots 15}{30\cdots 24}a^{22}-\frac{78\cdots 71}{38\cdots 03}a^{21}+\frac{92\cdots 75}{77\cdots 06}a^{20}+\frac{71\cdots 13}{15\cdots 12}a^{19}+\frac{34\cdots 25}{38\cdots 03}a^{18}-\frac{40\cdots 23}{30\cdots 24}a^{17}-\frac{72\cdots 67}{30\cdots 24}a^{16}+\frac{69\cdots 05}{30\cdots 24}a^{15}-\frac{36\cdots 87}{30\cdots 24}a^{14}+\frac{21\cdots 71}{15\cdots 12}a^{13}+\frac{17\cdots 01}{15\cdots 12}a^{12}-\frac{62\cdots 21}{30\cdots 24}a^{11}-\frac{69\cdots 07}{30\cdots 24}a^{10}-\frac{65\cdots 97}{15\cdots 12}a^{9}-\frac{35\cdots 37}{15\cdots 12}a^{8}+\frac{39\cdots 39}{30\cdots 24}a^{7}+\frac{86\cdots 71}{30\cdots 24}a^{6}+\frac{16\cdots 13}{30\cdots 24}a^{5}+\frac{40\cdots 79}{30\cdots 24}a^{4}+\frac{12\cdots 71}{30\cdots 24}a^{3}-\frac{21\cdots 95}{52\cdots 36}a^{2}-\frac{13\cdots 51}{30\cdots 24}a-\frac{64\cdots 51}{30\cdots 24}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ (assuming GRH) |
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| Narrow class group: | $C_{4}$, which has order $4$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{33\cdots 79}{20\cdots 88}a^{23}-\frac{69\cdots 11}{20\cdots 88}a^{22}-\frac{20\cdots 87}{25\cdots 11}a^{21}+\frac{16\cdots 89}{50\cdots 22}a^{20}+\frac{53\cdots 65}{10\cdots 44}a^{19}-\frac{48\cdots 65}{50\cdots 22}a^{18}+\frac{12\cdots 43}{20\cdots 88}a^{17}-\frac{60\cdots 89}{20\cdots 88}a^{16}+\frac{23\cdots 91}{20\cdots 88}a^{15}-\frac{86\cdots 61}{20\cdots 88}a^{14}+\frac{13\cdots 13}{10\cdots 44}a^{13}-\frac{36\cdots 01}{10\cdots 44}a^{12}+\frac{17\cdots 53}{20\cdots 88}a^{11}-\frac{34\cdots 21}{20\cdots 88}a^{10}+\frac{26\cdots 13}{10\cdots 44}a^{9}-\frac{53\cdots 59}{10\cdots 44}a^{8}+\frac{13\cdots 89}{20\cdots 88}a^{7}+\frac{34\cdots 33}{20\cdots 88}a^{6}-\frac{51\cdots 49}{20\cdots 88}a^{5}-\frac{14\cdots 99}{20\cdots 88}a^{4}+\frac{13\cdots 61}{20\cdots 88}a^{3}+\frac{13\cdots 41}{20\cdots 88}a^{2}+\frac{96\cdots 23}{20\cdots 88}a-\frac{20\cdots 49}{20\cdots 88}$, $\frac{26\cdots 29}{20\cdots 88}a^{23}-\frac{49\cdots 23}{20\cdots 88}a^{22}-\frac{31\cdots 73}{50\cdots 22}a^{21}+\frac{12\cdots 09}{50\cdots 22}a^{20}+\frac{42\cdots 07}{10\cdots 44}a^{19}-\frac{37\cdots 45}{50\cdots 22}a^{18}+\frac{99\cdots 29}{20\cdots 88}a^{17}-\frac{46\cdots 21}{20\cdots 88}a^{16}+\frac{18\cdots 77}{20\cdots 88}a^{15}-\frac{68\cdots 97}{20\cdots 88}a^{14}+\frac{11\cdots 49}{10\cdots 44}a^{13}-\frac{29\cdots 51}{10\cdots 44}a^{12}+\frac{14\cdots 75}{20\cdots 88}a^{11}-\frac{29\cdots 85}{20\cdots 88}a^{10}+\frac{23\cdots 49}{10\cdots 44}a^{9}-\frac{47\cdots 61}{10\cdots 44}a^{8}+\frac{10\cdots 95}{20\cdots 88}a^{7}+\frac{28\cdots 25}{20\cdots 88}a^{6}-\frac{56\cdots 95}{20\cdots 88}a^{5}-\frac{86\cdots 11}{20\cdots 88}a^{4}+\frac{16\cdots 19}{20\cdots 88}a^{3}+\frac{26\cdots 81}{20\cdots 88}a^{2}-\frac{77\cdots 31}{20\cdots 88}a-\frac{12\cdots 25}{20\cdots 88}$, $\frac{56\cdots 31}{15\cdots 12}a^{23}+\frac{36\cdots 27}{77\cdots 06}a^{22}-\frac{13\cdots 67}{77\cdots 06}a^{21}+\frac{10\cdots 91}{77\cdots 06}a^{20}+\frac{16\cdots 21}{77\cdots 06}a^{19}-\frac{11\cdots 97}{77\cdots 06}a^{18}+\frac{12\cdots 25}{15\cdots 12}a^{17}-\frac{20\cdots 42}{65\cdots 17}a^{16}+\frac{18\cdots 59}{15\cdots 12}a^{15}-\frac{17\cdots 75}{38\cdots 03}a^{14}+\frac{91\cdots 47}{77\cdots 06}a^{13}-\frac{21\cdots 71}{77\cdots 06}a^{12}+\frac{10\cdots 33}{15\cdots 12}a^{11}-\frac{67\cdots 05}{77\cdots 06}a^{10}+\frac{10\cdots 35}{77\cdots 06}a^{9}-\frac{20\cdots 72}{38\cdots 03}a^{8}-\frac{10\cdots 53}{15\cdots 12}a^{7}+\frac{13\cdots 39}{38\cdots 03}a^{6}+\frac{81\cdots 13}{15\cdots 12}a^{5}-\frac{30\cdots 88}{38\cdots 03}a^{4}-\frac{23\cdots 35}{15\cdots 12}a^{3}-\frac{35\cdots 18}{38\cdots 03}a^{2}-\frac{25\cdots 83}{15\cdots 12}a+\frac{14\cdots 11}{38\cdots 03}$, $\frac{42\cdots 63}{30\cdots 24}a^{23}+\frac{10\cdots 49}{30\cdots 24}a^{22}+\frac{26\cdots 48}{38\cdots 03}a^{21}-\frac{39\cdots 83}{13\cdots 34}a^{20}-\frac{77\cdots 49}{15\cdots 12}a^{19}+\frac{33\cdots 86}{38\cdots 03}a^{18}-\frac{16\cdots 11}{30\cdots 24}a^{17}+\frac{77\cdots 63}{30\cdots 24}a^{16}-\frac{29\cdots 43}{30\cdots 24}a^{15}+\frac{11\cdots 47}{30\cdots 24}a^{14}-\frac{18\cdots 61}{15\cdots 12}a^{13}+\frac{47\cdots 87}{15\cdots 12}a^{12}-\frac{22\cdots 57}{30\cdots 24}a^{11}+\frac{44\cdots 79}{30\cdots 24}a^{10}-\frac{32\cdots 15}{15\cdots 12}a^{9}+\frac{65\cdots 49}{15\cdots 12}a^{8}-\frac{17\cdots 93}{30\cdots 24}a^{7}-\frac{62\cdots 35}{30\cdots 24}a^{6}+\frac{11\cdots 33}{30\cdots 24}a^{5}+\frac{25\cdots 73}{30\cdots 24}a^{4}-\frac{32\cdots 25}{30\cdots 24}a^{3}-\frac{37\cdots 51}{30\cdots 24}a^{2}+\frac{32\cdots 77}{30\cdots 24}a+\frac{44\cdots 51}{30\cdots 24}$, $\frac{14\cdots 31}{15\cdots 12}a^{23}+\frac{13\cdots 50}{38\cdots 03}a^{22}+\frac{32\cdots 13}{77\cdots 06}a^{21}-\frac{20\cdots 83}{77\cdots 06}a^{20}-\frac{41\cdots 37}{77\cdots 06}a^{19}+\frac{46\cdots 61}{77\cdots 06}a^{18}-\frac{67\cdots 73}{15\cdots 12}a^{17}+\frac{83\cdots 91}{38\cdots 03}a^{16}-\frac{13\cdots 73}{15\cdots 12}a^{15}+\frac{12\cdots 81}{38\cdots 03}a^{14}-\frac{73\cdots 21}{65\cdots 17}a^{13}+\frac{12\cdots 90}{38\cdots 03}a^{12}-\frac{12\cdots 85}{15\cdots 12}a^{11}+\frac{65\cdots 25}{38\cdots 03}a^{10}-\frac{22\cdots 65}{77\cdots 06}a^{9}+\frac{40\cdots 01}{77\cdots 06}a^{8}-\frac{12\cdots 27}{15\cdots 12}a^{7}-\frac{53\cdots 79}{77\cdots 06}a^{6}+\frac{61\cdots 57}{15\cdots 12}a^{5}+\frac{60\cdots 68}{38\cdots 03}a^{4}-\frac{18\cdots 01}{15\cdots 12}a^{3}+\frac{11\cdots 09}{38\cdots 03}a^{2}+\frac{83\cdots 73}{15\cdots 12}a-\frac{53\cdots 57}{77\cdots 06}$, $\frac{16\cdots 11}{77\cdots 06}a^{23}+\frac{18\cdots 09}{15\cdots 12}a^{22}+\frac{90\cdots 67}{77\cdots 06}a^{21}-\frac{57\cdots 17}{77\cdots 06}a^{20}-\frac{48\cdots 03}{77\cdots 06}a^{19}+\frac{62\cdots 48}{38\cdots 03}a^{18}-\frac{82\cdots 81}{77\cdots 06}a^{17}+\frac{84\cdots 25}{15\cdots 12}a^{16}-\frac{84\cdots 12}{38\cdots 03}a^{15}+\frac{12\cdots 21}{15\cdots 12}a^{14}-\frac{10\cdots 63}{38\cdots 03}a^{13}+\frac{57\cdots 59}{77\cdots 06}a^{12}-\frac{69\cdots 57}{38\cdots 03}a^{11}+\frac{60\cdots 35}{15\cdots 12}a^{10}-\frac{45\cdots 05}{77\cdots 06}a^{9}+\frac{85\cdots 51}{77\cdots 06}a^{8}-\frac{13\cdots 93}{77\cdots 06}a^{7}-\frac{77\cdots 11}{15\cdots 12}a^{6}+\frac{72\cdots 91}{77\cdots 06}a^{5}+\frac{43\cdots 03}{15\cdots 12}a^{4}-\frac{60\cdots 53}{38\cdots 03}a^{3}-\frac{80\cdots 91}{15\cdots 12}a^{2}-\frac{21\cdots 92}{38\cdots 03}a-\frac{28\cdots 99}{15\cdots 12}$, $\frac{10\cdots 91}{30\cdots 24}a^{23}+\frac{89\cdots 43}{30\cdots 24}a^{22}-\frac{78\cdots 90}{38\cdots 03}a^{21}+\frac{15\cdots 17}{77\cdots 06}a^{20}+\frac{61\cdots 49}{15\cdots 12}a^{19}-\frac{72\cdots 58}{38\cdots 03}a^{18}+\frac{19\cdots 67}{30\cdots 24}a^{17}-\frac{38\cdots 19}{30\cdots 24}a^{16}-\frac{31\cdots 65}{30\cdots 24}a^{15}+\frac{41\cdots 45}{30\cdots 24}a^{14}-\frac{15\cdots 87}{15\cdots 12}a^{13}+\frac{82\cdots 63}{15\cdots 12}a^{12}-\frac{52\cdots 27}{30\cdots 24}a^{11}+\frac{15\cdots 01}{30\cdots 24}a^{10}-\frac{19\cdots 87}{15\cdots 12}a^{9}+\frac{28\cdots 79}{15\cdots 12}a^{8}-\frac{11\cdots 63}{30\cdots 24}a^{7}+\frac{32\cdots 87}{30\cdots 24}a^{6}+\frac{25\cdots 79}{30\cdots 24}a^{5}-\frac{17\cdots 73}{30\cdots 24}a^{4}-\frac{12\cdots 59}{30\cdots 24}a^{3}+\frac{42\cdots 87}{30\cdots 24}a^{2}+\frac{22\cdots 99}{30\cdots 24}a-\frac{33\cdots 15}{30\cdots 24}$, $\frac{57\cdots 53}{15\cdots 12}a^{23}+\frac{32\cdots 72}{38\cdots 03}a^{22}+\frac{72\cdots 31}{38\cdots 03}a^{21}-\frac{60\cdots 15}{77\cdots 06}a^{20}-\frac{55\cdots 59}{38\cdots 03}a^{19}+\frac{17\cdots 97}{77\cdots 06}a^{18}-\frac{22\cdots 47}{15\cdots 12}a^{17}+\frac{25\cdots 52}{38\cdots 03}a^{16}-\frac{39\cdots 51}{15\cdots 12}a^{15}+\frac{36\cdots 83}{38\cdots 03}a^{14}-\frac{23\cdots 03}{77\cdots 06}a^{13}+\frac{61\cdots 31}{77\cdots 06}a^{12}-\frac{29\cdots 25}{15\cdots 12}a^{11}+\frac{29\cdots 03}{77\cdots 06}a^{10}-\frac{43\cdots 31}{77\cdots 06}a^{9}+\frac{88\cdots 79}{77\cdots 06}a^{8}-\frac{21\cdots 13}{15\cdots 12}a^{7}-\frac{41\cdots 73}{77\cdots 06}a^{6}+\frac{13\cdots 99}{15\cdots 12}a^{5}+\frac{16\cdots 11}{77\cdots 06}a^{4}-\frac{36\cdots 17}{15\cdots 12}a^{3}-\frac{22\cdots 79}{77\cdots 06}a^{2}-\frac{65\cdots 53}{15\cdots 12}a+\frac{39\cdots 22}{38\cdots 03}$, $\frac{21\cdots 66}{38\cdots 03}a^{23}-\frac{13\cdots 69}{77\cdots 06}a^{22}-\frac{10\cdots 47}{38\cdots 03}a^{21}+\frac{53\cdots 69}{38\cdots 03}a^{20}+\frac{67\cdots 93}{77\cdots 06}a^{19}-\frac{13\cdots 99}{38\cdots 03}a^{18}+\frac{18\cdots 91}{77\cdots 06}a^{17}-\frac{46\cdots 46}{38\cdots 03}a^{16}+\frac{37\cdots 45}{77\cdots 06}a^{15}-\frac{72\cdots 92}{38\cdots 03}a^{14}+\frac{48\cdots 87}{77\cdots 06}a^{13}-\frac{13\cdots 47}{77\cdots 06}a^{12}+\frac{17\cdots 17}{38\cdots 03}a^{11}-\frac{12\cdots 51}{13\cdots 34}a^{10}+\frac{63\cdots 11}{38\cdots 03}a^{9}-\frac{23\cdots 03}{77\cdots 06}a^{8}+\frac{27\cdots 46}{65\cdots 17}a^{7}+\frac{44\cdots 37}{77\cdots 06}a^{6}-\frac{16\cdots 41}{77\cdots 06}a^{5}-\frac{37\cdots 13}{77\cdots 06}a^{4}+\frac{69\cdots 29}{38\cdots 03}a^{3}+\frac{25\cdots 97}{77\cdots 06}a^{2}+\frac{99\cdots 07}{77\cdots 06}a-\frac{16\cdots 89}{77\cdots 06}$, $\frac{35\cdots 69}{77\cdots 06}a^{23}+\frac{63\cdots 62}{38\cdots 03}a^{22}-\frac{84\cdots 34}{38\cdots 03}a^{21}+\frac{14\cdots 55}{38\cdots 03}a^{20}+\frac{19\cdots 31}{77\cdots 06}a^{19}-\frac{15\cdots 75}{77\cdots 06}a^{18}+\frac{94\cdots 87}{77\cdots 06}a^{17}-\frac{39\cdots 63}{77\cdots 06}a^{16}+\frac{76\cdots 49}{38\cdots 03}a^{15}-\frac{28\cdots 68}{38\cdots 03}a^{14}+\frac{28\cdots 89}{13\cdots 34}a^{13}-\frac{42\cdots 11}{77\cdots 06}a^{12}+\frac{52\cdots 49}{38\cdots 03}a^{11}-\frac{18\cdots 29}{77\cdots 06}a^{10}+\frac{15\cdots 84}{38\cdots 03}a^{9}-\frac{41\cdots 70}{38\cdots 03}a^{8}+\frac{17\cdots 27}{77\cdots 06}a^{7}+\frac{33\cdots 95}{77\cdots 06}a^{6}+\frac{27\cdots 77}{77\cdots 06}a^{5}-\frac{10\cdots 93}{77\cdots 06}a^{4}-\frac{11\cdots 31}{77\cdots 06}a^{3}-\frac{26\cdots 09}{77\cdots 06}a^{2}+\frac{83\cdots 27}{38\cdots 03}a+\frac{50\cdots 71}{38\cdots 03}$, $\frac{62\cdots 89}{30\cdots 24}a^{23}+\frac{98\cdots 53}{30\cdots 24}a^{22}+\frac{81\cdots 57}{77\cdots 06}a^{21}-\frac{13\cdots 06}{38\cdots 03}a^{20}-\frac{17\cdots 15}{15\cdots 12}a^{19}+\frac{92\cdots 99}{77\cdots 06}a^{18}-\frac{35\cdots 07}{52\cdots 36}a^{17}+\frac{90\cdots 83}{30\cdots 24}a^{16}-\frac{33\cdots 61}{30\cdots 24}a^{15}+\frac{12\cdots 23}{30\cdots 24}a^{14}-\frac{18\cdots 43}{15\cdots 12}a^{13}+\frac{44\cdots 37}{15\cdots 12}a^{12}-\frac{19\cdots 47}{30\cdots 24}a^{11}+\frac{35\cdots 11}{30\cdots 24}a^{10}-\frac{20\cdots 53}{15\cdots 12}a^{9}+\frac{59\cdots 61}{15\cdots 12}a^{8}-\frac{12\cdots 27}{30\cdots 24}a^{7}-\frac{10\cdots 83}{30\cdots 24}a^{6}+\frac{60\cdots 07}{30\cdots 24}a^{5}+\frac{58\cdots 01}{30\cdots 24}a^{4}-\frac{23\cdots 75}{30\cdots 24}a^{3}-\frac{98\cdots 07}{30\cdots 24}a^{2}-\frac{30\cdots 53}{30\cdots 24}a+\frac{51\cdots 27}{30\cdots 24}$, $\frac{43\cdots 47}{30\cdots 24}a^{23}-\frac{70\cdots 83}{30\cdots 24}a^{22}+\frac{25\cdots 87}{38\cdots 03}a^{21}-\frac{13\cdots 49}{38\cdots 03}a^{20}-\frac{13\cdots 87}{15\cdots 12}a^{19}+\frac{41\cdots 89}{77\cdots 06}a^{18}-\frac{94\cdots 63}{30\cdots 24}a^{17}+\frac{35\cdots 23}{30\cdots 24}a^{16}-\frac{13\cdots 23}{30\cdots 24}a^{15}+\frac{49\cdots 11}{30\cdots 24}a^{14}-\frac{64\cdots 99}{15\cdots 12}a^{13}+\frac{15\cdots 21}{15\cdots 12}a^{12}-\frac{71\cdots 05}{30\cdots 24}a^{11}+\frac{89\cdots 07}{30\cdots 24}a^{10}-\frac{72\cdots 79}{15\cdots 12}a^{9}+\frac{30\cdots 17}{15\cdots 12}a^{8}+\frac{94\cdots 87}{30\cdots 24}a^{7}-\frac{38\cdots 43}{30\cdots 24}a^{6}-\frac{73\cdots 27}{30\cdots 24}a^{5}+\frac{68\cdots 77}{30\cdots 24}a^{4}+\frac{19\cdots 31}{30\cdots 24}a^{3}+\frac{18\cdots 57}{30\cdots 24}a^{2}+\frac{59\cdots 89}{30\cdots 24}a-\frac{10\cdots 93}{30\cdots 24}$, $\frac{17\cdots 15}{30\cdots 24}a^{23}-\frac{26\cdots 33}{30\cdots 24}a^{22}+\frac{99\cdots 16}{38\cdots 03}a^{21}-\frac{13\cdots 75}{77\cdots 06}a^{20}-\frac{50\cdots 71}{15\cdots 12}a^{19}+\frac{17\cdots 89}{77\cdots 06}a^{18}-\frac{39\cdots 35}{30\cdots 24}a^{17}+\frac{14\cdots 85}{30\cdots 24}a^{16}-\frac{56\cdots 59}{30\cdots 24}a^{15}+\frac{21\cdots 85}{30\cdots 24}a^{14}-\frac{28\cdots 13}{15\cdots 12}a^{13}+\frac{67\cdots 55}{15\cdots 12}a^{12}-\frac{32\cdots 97}{30\cdots 24}a^{11}+\frac{44\cdots 81}{30\cdots 24}a^{10}-\frac{37\cdots 03}{15\cdots 12}a^{9}+\frac{12\cdots 47}{15\cdots 12}a^{8}+\frac{31\cdots 11}{30\cdots 24}a^{7}-\frac{14\cdots 97}{30\cdots 24}a^{6}-\frac{25\cdots 99}{30\cdots 24}a^{5}+\frac{28\cdots 19}{30\cdots 24}a^{4}+\frac{66\cdots 99}{30\cdots 24}a^{3}+\frac{59\cdots 47}{30\cdots 24}a^{2}+\frac{15\cdots 05}{30\cdots 24}a-\frac{29\cdots 11}{30\cdots 24}$
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| Regulator: | \( 219783402990552930 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 219783402990552930 \cdot 2}{2\cdot\sqrt{61343664831565854732948944775316051495075225830078125}}\cr\approx \mathstrut & 1.36153465706632 \end{aligned}\] (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
| 6.2.4706682753125.1, 12.4.110764312692821648486328125.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{5}{,}\,{\href{/padicField/2.2.0.1}{2} }^{2}$ | ${\href{/padicField/3.8.0.1}{8} }^{3}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.3.0.1}{3} }^{8}$ | $24$ | ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.12.0.1}{12} }^{2}$ | $24$ | ${\href{/padicField/29.12.0.1}{12} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | ${\href{/padicField/41.3.0.1}{3} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | $24$ | ${\href{/padicField/53.8.0.1}{8} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $24$ | $24$ | $1$ | $23$ | |||
|
\(197\)
| 197.4.1.0a1.1 | $x^{4} + 16 x^{2} + 124 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 197.1.5.4a1.1 | $x^{5} + 197$ | $5$ | $1$ | $4$ | $F_5$ | $$[\ ]_{5}^{4}$$ | |
| 197.1.5.4a1.1 | $x^{5} + 197$ | $5$ | $1$ | $4$ | $F_5$ | $$[\ ]_{5}^{4}$$ | |
| 197.1.5.4a1.1 | $x^{5} + 197$ | $5$ | $1$ | $4$ | $F_5$ | $$[\ ]_{5}^{4}$$ | |
| 197.1.5.4a1.1 | $x^{5} + 197$ | $5$ | $1$ | $4$ | $F_5$ | $$[\ ]_{5}^{4}$$ |