Normalized defining polynomial
\( x^{24} - 4 x^{23} + 6 x^{22} - 4 x^{21} - 2174 x^{20} + 2465 x^{19} + 15515 x^{18} + 81490 x^{17} + \cdots + 338358763849 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $(4, 10)$ |
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| Discriminant: |
\(270761008401829353605241639483649123576469719409942626953125\)
\(\medspace = 5^{39}\cdot 29^{22}\)
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| |
| Root discriminant: | \(299.47\) |
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| Galois root discriminant: | $5^{39/20}29^{19/20}\approx 565.287791942645$ | ||
| Ramified primes: |
\(5\), \(29\)
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{29}a^{14}-\frac{4}{29}a^{13}+\frac{6}{29}a^{12}-\frac{4}{29}a^{11}+\frac{1}{29}a^{10}$, $\frac{1}{29}a^{15}-\frac{10}{29}a^{13}-\frac{9}{29}a^{12}+\frac{14}{29}a^{11}+\frac{4}{29}a^{10}$, $\frac{1}{29}a^{16}+\frac{9}{29}a^{13}-\frac{13}{29}a^{12}-\frac{7}{29}a^{11}+\frac{10}{29}a^{10}$, $\frac{1}{29}a^{17}-\frac{6}{29}a^{13}-\frac{3}{29}a^{12}-\frac{12}{29}a^{11}-\frac{9}{29}a^{10}$, $\frac{1}{29}a^{18}+\frac{2}{29}a^{13}-\frac{5}{29}a^{12}-\frac{4}{29}a^{11}+\frac{6}{29}a^{10}$, $\frac{1}{145}a^{19}+\frac{1}{145}a^{18}+\frac{1}{145}a^{17}+\frac{1}{145}a^{16}+\frac{1}{145}a^{15}+\frac{1}{145}a^{14}-\frac{64}{145}a^{13}-\frac{69}{145}a^{12}+\frac{1}{145}a^{11}-\frac{19}{145}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{145}a^{20}+\frac{5}{29}a^{13}+\frac{5}{29}a^{12}+\frac{2}{29}a^{11}+\frac{11}{29}a^{10}+\frac{1}{5}$, $\frac{1}{145}a^{21}-\frac{4}{29}a^{13}+\frac{1}{29}a^{12}+\frac{2}{29}a^{11}-\frac{5}{29}a^{10}+\frac{1}{5}a$, $\frac{1}{725}a^{22}-\frac{2}{725}a^{21}+\frac{1}{725}a^{20}-\frac{2}{145}a^{17}-\frac{1}{145}a^{16}-\frac{2}{145}a^{15}+\frac{2}{145}a^{14}+\frac{42}{145}a^{13}+\frac{49}{145}a^{12}-\frac{57}{145}a^{11}+\frac{27}{145}a^{10}+\frac{1}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{1}{5}a^{4}-\frac{1}{5}a^{3}-\frac{9}{25}a^{2}+\frac{8}{25}a+\frac{6}{25}$, $\frac{1}{51\cdots 75}a^{23}+\frac{32\cdots 62}{51\cdots 75}a^{22}+\frac{95\cdots 78}{51\cdots 75}a^{21}+\frac{68\cdots 62}{73\cdots 25}a^{20}+\frac{30\cdots 14}{10\cdots 15}a^{19}-\frac{19\cdots 19}{14\cdots 45}a^{18}-\frac{93\cdots 95}{20\cdots 63}a^{17}+\frac{26\cdots 43}{10\cdots 15}a^{16}-\frac{58\cdots 02}{10\cdots 15}a^{15}-\frac{12\cdots 96}{10\cdots 15}a^{14}+\frac{36\cdots 31}{10\cdots 15}a^{13}+\frac{27\cdots 79}{14\cdots 45}a^{12}+\frac{27\cdots 53}{10\cdots 15}a^{11}-\frac{42\cdots 54}{10\cdots 15}a^{10}+\frac{74\cdots 11}{35\cdots 35}a^{9}-\frac{16\cdots 47}{35\cdots 35}a^{8}-\frac{12\cdots 31}{35\cdots 35}a^{7}+\frac{57\cdots 13}{35\cdots 35}a^{6}+\frac{14\cdots 66}{50\cdots 05}a^{5}+\frac{71\cdots 76}{35\cdots 35}a^{4}+\frac{60\cdots 26}{17\cdots 75}a^{3}+\frac{14\cdots 12}{17\cdots 75}a^{2}-\frac{92\cdots 22}{17\cdots 75}a-\frac{62\cdots 14}{31\cdots 75}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{4}$, which has order $8$ (assuming GRH) |
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| Narrow class group: | $C_{4}\times C_{2}\times C_{2}$, which has order $16$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{20\cdots 82}{28\cdots 25}a^{23}-\frac{60\cdots 93}{28\cdots 25}a^{22}+\frac{52\cdots 01}{57\cdots 05}a^{21}-\frac{53\cdots 74}{28\cdots 25}a^{20}-\frac{86\cdots 28}{57\cdots 05}a^{19}+\frac{91\cdots 93}{57\cdots 05}a^{18}+\frac{78\cdots 38}{57\cdots 05}a^{17}+\frac{44\cdots 07}{57\cdots 05}a^{16}+\frac{65\cdots 89}{57\cdots 05}a^{15}+\frac{63\cdots 48}{57\cdots 05}a^{14}-\frac{48\cdots 03}{57\cdots 05}a^{13}+\frac{24\cdots 77}{57\cdots 05}a^{12}+\frac{10\cdots 01}{19\cdots 45}a^{11}+\frac{21\cdots 83}{19\cdots 45}a^{10}-\frac{75\cdots 89}{19\cdots 45}a^{9}-\frac{83\cdots 03}{39\cdots 29}a^{8}-\frac{83\cdots 92}{19\cdots 45}a^{7}-\frac{14\cdots 29}{39\cdots 29}a^{6}+\frac{87\cdots 59}{19\cdots 45}a^{5}+\frac{55\cdots 07}{39\cdots 29}a^{4}+\frac{94\cdots 22}{98\cdots 25}a^{3}-\frac{23\cdots 68}{98\cdots 25}a^{2}-\frac{48\cdots 92}{19\cdots 45}a+\frac{27\cdots 59}{17\cdots 25}$, $\frac{16\cdots 81}{10\cdots 25}a^{23}-\frac{60\cdots 08}{10\cdots 25}a^{22}+\frac{80\cdots 73}{10\cdots 25}a^{21}-\frac{54\cdots 96}{10\cdots 25}a^{20}-\frac{73\cdots 93}{20\cdots 45}a^{19}+\frac{10\cdots 44}{40\cdots 09}a^{18}+\frac{53\cdots 13}{20\cdots 45}a^{17}+\frac{29\cdots 51}{20\cdots 45}a^{16}+\frac{54\cdots 21}{20\cdots 45}a^{15}+\frac{27\cdots 52}{20\cdots 45}a^{14}-\frac{33\cdots 31}{20\cdots 45}a^{13}+\frac{26\cdots 86}{20\cdots 45}a^{12}+\frac{22\cdots 31}{20\cdots 45}a^{11}+\frac{13\cdots 56}{69\cdots 05}a^{10}-\frac{54\cdots 52}{69\cdots 05}a^{9}-\frac{50\cdots 81}{13\cdots 21}a^{8}-\frac{60\cdots 69}{69\cdots 05}a^{7}-\frac{23\cdots 23}{13\cdots 21}a^{6}-\frac{57\cdots 51}{69\cdots 05}a^{5}+\frac{82\cdots 73}{69\cdots 05}a^{4}+\frac{53\cdots 91}{34\cdots 25}a^{3}+\frac{18\cdots 52}{34\cdots 25}a^{2}-\frac{79\cdots 17}{34\cdots 25}a-\frac{74\cdots 69}{61\cdots 25}$, $\frac{12\cdots 03}{40\cdots 09}a^{23}-\frac{38\cdots 77}{20\cdots 45}a^{22}+\frac{11\cdots 34}{20\cdots 45}a^{21}-\frac{24\cdots 41}{20\cdots 45}a^{20}-\frac{13\cdots 89}{20\cdots 45}a^{19}+\frac{43\cdots 46}{20\cdots 45}a^{18}+\frac{57\cdots 64}{69\cdots 05}a^{17}+\frac{48\cdots 41}{20\cdots 45}a^{16}+\frac{93\cdots 26}{20\cdots 45}a^{15}-\frac{17\cdots 29}{20\cdots 45}a^{14}-\frac{35\cdots 09}{20\cdots 45}a^{13}+\frac{61\cdots 86}{20\cdots 45}a^{12}+\frac{28\cdots 51}{20\cdots 45}a^{11}-\frac{21\cdots 26}{69\cdots 05}a^{10}-\frac{11\cdots 26}{69\cdots 05}a^{9}-\frac{20\cdots 61}{69\cdots 05}a^{8}-\frac{52\cdots 26}{69\cdots 05}a^{7}-\frac{69\cdots 51}{69\cdots 05}a^{6}+\frac{13\cdots 49}{69\cdots 05}a^{5}-\frac{61\cdots 36}{69\cdots 05}a^{4}+\frac{25\cdots 79}{69\cdots 05}a^{3}+\frac{12\cdots 67}{69\cdots 05}a^{2}-\frac{91\cdots 97}{69\cdots 05}a+\frac{64\cdots 82}{12\cdots 45}$, $\frac{19\cdots 69}{20\cdots 45}a^{23}+\frac{36\cdots 64}{10\cdots 25}a^{22}-\frac{50\cdots 33}{10\cdots 25}a^{21}+\frac{38\cdots 54}{10\cdots 25}a^{20}+\frac{43\cdots 77}{20\cdots 45}a^{19}-\frac{35\cdots 53}{20\cdots 45}a^{18}-\frac{31\cdots 41}{20\cdots 45}a^{17}-\frac{17\cdots 02}{20\cdots 45}a^{16}-\frac{32\cdots 46}{20\cdots 45}a^{15}-\frac{27\cdots 65}{40\cdots 09}a^{14}+\frac{40\cdots 55}{40\cdots 09}a^{13}-\frac{15\cdots 42}{20\cdots 45}a^{12}-\frac{12\cdots 51}{20\cdots 45}a^{11}-\frac{14\cdots 91}{13\cdots 21}a^{10}+\frac{33\cdots 02}{69\cdots 05}a^{9}+\frac{14\cdots 37}{69\cdots 05}a^{8}+\frac{68\cdots 83}{13\cdots 21}a^{7}+\frac{64\cdots 62}{69\cdots 05}a^{6}+\frac{56\cdots 25}{13\cdots 21}a^{5}-\frac{54\cdots 91}{69\cdots 05}a^{4}-\frac{14\cdots 53}{13\cdots 21}a^{3}-\frac{11\cdots 11}{34\cdots 25}a^{2}+\frac{41\cdots 47}{34\cdots 25}a+\frac{40\cdots 06}{61\cdots 25}$, $\frac{97\cdots 89}{11\cdots 41}a^{23}-\frac{16\cdots 41}{28\cdots 25}a^{22}+\frac{53\cdots 32}{28\cdots 25}a^{21}-\frac{12\cdots 41}{28\cdots 25}a^{20}-\frac{20\cdots 54}{11\cdots 41}a^{19}+\frac{81\cdots 11}{11\cdots 41}a^{18}-\frac{10\cdots 18}{57\cdots 05}a^{17}+\frac{34\cdots 86}{57\cdots 05}a^{16}+\frac{67\cdots 47}{57\cdots 05}a^{15}-\frac{18\cdots 02}{57\cdots 05}a^{14}-\frac{17\cdots 62}{57\cdots 05}a^{13}+\frac{48\cdots 86}{57\cdots 05}a^{12}+\frac{60\cdots 78}{19\cdots 45}a^{11}-\frac{60\cdots 03}{19\cdots 45}a^{10}-\frac{89\cdots 06}{19\cdots 45}a^{9}-\frac{89\cdots 01}{19\cdots 45}a^{8}-\frac{26\cdots 93}{19\cdots 45}a^{7}-\frac{17\cdots 76}{19\cdots 45}a^{6}+\frac{16\cdots 72}{19\cdots 45}a^{5}-\frac{10\cdots 59}{19\cdots 45}a^{4}+\frac{17\cdots 61}{19\cdots 45}a^{3}-\frac{42\cdots 81}{98\cdots 25}a^{2}-\frac{53\cdots 53}{98\cdots 25}a+\frac{42\cdots 16}{17\cdots 25}$, $\frac{26\cdots 93}{10\cdots 25}a^{23}+\frac{84\cdots 52}{10\cdots 25}a^{22}-\frac{34\cdots 71}{20\cdots 45}a^{21}+\frac{17\cdots 01}{10\cdots 25}a^{20}+\frac{22\cdots 88}{40\cdots 09}a^{19}-\frac{43\cdots 99}{20\cdots 45}a^{18}-\frac{49\cdots 19}{20\cdots 45}a^{17}-\frac{89\cdots 51}{40\cdots 09}a^{16}-\frac{90\cdots 78}{20\cdots 45}a^{15}-\frac{94\cdots 54}{20\cdots 45}a^{14}+\frac{44\cdots 01}{40\cdots 09}a^{13}-\frac{90\cdots 19}{40\cdots 09}a^{12}-\frac{33\cdots 03}{20\cdots 45}a^{11}-\frac{56\cdots 21}{13\cdots 21}a^{10}+\frac{29\cdots 28}{69\cdots 05}a^{9}+\frac{28\cdots 67}{69\cdots 05}a^{8}+\frac{26\cdots 47}{13\cdots 21}a^{7}+\frac{40\cdots 52}{69\cdots 05}a^{6}+\frac{61\cdots 66}{69\cdots 05}a^{5}+\frac{69\cdots 52}{69\cdots 05}a^{4}+\frac{39\cdots 57}{34\cdots 25}a^{3}-\frac{22\cdots 63}{34\cdots 25}a^{2}-\frac{95\cdots 02}{69\cdots 05}a+\frac{35\cdots 49}{61\cdots 25}$, $\frac{60\cdots 17}{51\cdots 75}a^{23}-\frac{22\cdots 32}{20\cdots 63}a^{22}+\frac{14\cdots 29}{51\cdots 75}a^{21}-\frac{50\cdots 93}{73\cdots 25}a^{20}-\frac{26\cdots 22}{10\cdots 15}a^{19}+\frac{24\cdots 97}{14\cdots 45}a^{18}+\frac{13\cdots 83}{10\cdots 15}a^{17}-\frac{10\cdots 43}{20\cdots 63}a^{16}+\frac{12\cdots 19}{10\cdots 15}a^{15}-\frac{20\cdots 90}{20\cdots 63}a^{14}-\frac{20\cdots 91}{10\cdots 15}a^{13}+\frac{55\cdots 53}{29\cdots 09}a^{12}+\frac{29\cdots 64}{10\cdots 15}a^{11}-\frac{36\cdots 06}{10\cdots 15}a^{10}-\frac{48\cdots 67}{35\cdots 35}a^{9}+\frac{62\cdots 12}{35\cdots 35}a^{8}+\frac{45\cdots 96}{35\cdots 35}a^{7}+\frac{66\cdots 62}{35\cdots 35}a^{6}+\frac{10\cdots 66}{50\cdots 05}a^{5}-\frac{13\cdots 14}{35\cdots 35}a^{4}-\frac{22\cdots 63}{17\cdots 75}a^{3}+\frac{27\cdots 93}{35\cdots 35}a^{2}+\frac{29\cdots 39}{17\cdots 75}a-\frac{24\cdots 59}{31\cdots 75}$, $\frac{98\cdots 01}{10\cdots 75}a^{23}+\frac{19\cdots 89}{10\cdots 75}a^{22}-\frac{22\cdots 18}{21\cdots 35}a^{21}-\frac{37\cdots 79}{15\cdots 25}a^{20}+\frac{86\cdots 06}{42\cdots 87}a^{19}+\frac{56\cdots 76}{30\cdots 05}a^{18}-\frac{26\cdots 03}{21\cdots 35}a^{17}-\frac{43\cdots 44}{42\cdots 87}a^{16}-\frac{35\cdots 11}{21\cdots 35}a^{15}-\frac{74\cdots 33}{21\cdots 35}a^{14}+\frac{17\cdots 02}{42\cdots 87}a^{13}-\frac{39\cdots 95}{60\cdots 41}a^{12}-\frac{15\cdots 56}{21\cdots 35}a^{11}-\frac{93\cdots 81}{42\cdots 87}a^{10}+\frac{69\cdots 96}{72\cdots 15}a^{9}+\frac{16\cdots 29}{72\cdots 15}a^{8}+\frac{12\cdots 11}{14\cdots 03}a^{7}+\frac{16\cdots 99}{72\cdots 15}a^{6}+\frac{42\cdots 56}{10\cdots 45}a^{5}+\frac{37\cdots 84}{72\cdots 15}a^{4}+\frac{19\cdots 99}{36\cdots 75}a^{3}+\frac{45\cdots 09}{36\cdots 75}a^{2}-\frac{34\cdots 34}{14\cdots 03}a-\frac{16\cdots 47}{63\cdots 75}$, $\frac{41\cdots 12}{51\cdots 75}a^{23}+\frac{23\cdots 88}{51\cdots 75}a^{22}-\frac{25\cdots 49}{20\cdots 63}a^{21}+\frac{18\cdots 72}{73\cdots 25}a^{20}+\frac{17\cdots 94}{10\cdots 15}a^{19}-\frac{68\cdots 16}{14\cdots 45}a^{18}-\frac{54\cdots 02}{10\cdots 15}a^{17}-\frac{64\cdots 06}{10\cdots 15}a^{16}-\frac{11\cdots 38}{10\cdots 15}a^{15}+\frac{17\cdots 68}{10\cdots 15}a^{14}+\frac{60\cdots 79}{10\cdots 15}a^{13}-\frac{10\cdots 23}{14\cdots 45}a^{12}-\frac{41\cdots 33}{10\cdots 15}a^{11}-\frac{24\cdots 81}{10\cdots 15}a^{10}+\frac{16\cdots 53}{35\cdots 35}a^{9}+\frac{28\cdots 04}{35\cdots 35}a^{8}+\frac{75\cdots 76}{35\cdots 35}a^{7}+\frac{10\cdots 19}{35\cdots 35}a^{6}-\frac{56\cdots 50}{10\cdots 21}a^{5}+\frac{98\cdots 59}{35\cdots 35}a^{4}-\frac{18\cdots 57}{17\cdots 75}a^{3}-\frac{10\cdots 17}{17\cdots 75}a^{2}+\frac{13\cdots 92}{35\cdots 35}a-\frac{47\cdots 59}{31\cdots 75}$, $\frac{30\cdots 55}{20\cdots 63}a^{23}+\frac{32\cdots 73}{51\cdots 75}a^{22}-\frac{46\cdots 41}{51\cdots 75}a^{21}+\frac{24\cdots 79}{73\cdots 25}a^{20}+\frac{33\cdots 17}{10\cdots 15}a^{19}-\frac{64\cdots 04}{14\cdots 45}a^{18}-\frac{25\cdots 14}{10\cdots 15}a^{17}-\frac{11\cdots 71}{10\cdots 15}a^{16}-\frac{24\cdots 89}{10\cdots 15}a^{15}+\frac{33\cdots 08}{10\cdots 15}a^{14}+\frac{18\cdots 43}{10\cdots 15}a^{13}-\frac{18\cdots 28}{14\cdots 45}a^{12}-\frac{93\cdots 24}{10\cdots 15}a^{11}-\frac{10\cdots 77}{10\cdots 15}a^{10}+\frac{30\cdots 16}{35\cdots 35}a^{9}+\frac{10\cdots 86}{35\cdots 35}a^{8}+\frac{19\cdots 42}{35\cdots 35}a^{7}+\frac{28\cdots 76}{35\cdots 35}a^{6}-\frac{39\cdots 84}{50\cdots 05}a^{5}-\frac{20\cdots 12}{71\cdots 47}a^{4}-\frac{14\cdots 56}{71\cdots 47}a^{3}-\frac{66\cdots 17}{17\cdots 75}a^{2}+\frac{12\cdots 79}{17\cdots 75}a+\frac{50\cdots 32}{31\cdots 75}$, $\frac{12\cdots 26}{73\cdots 25}a^{23}+\frac{91\cdots 31}{73\cdots 25}a^{22}-\frac{39\cdots 69}{73\cdots 25}a^{21}+\frac{20\cdots 87}{10\cdots 75}a^{20}+\frac{42\cdots 51}{14\cdots 45}a^{19}-\frac{29\cdots 41}{21\cdots 35}a^{18}+\frac{36\cdots 29}{14\cdots 45}a^{17}-\frac{32\cdots 06}{14\cdots 45}a^{16}-\frac{26\cdots 29}{14\cdots 45}a^{15}+\frac{90\cdots 02}{14\cdots 45}a^{14}-\frac{12\cdots 01}{29\cdots 09}a^{13}-\frac{25\cdots 78}{21\cdots 35}a^{12}-\frac{88\cdots 94}{14\cdots 45}a^{11}+\frac{20\cdots 43}{29\cdots 09}a^{10}+\frac{31\cdots 17}{50\cdots 05}a^{9}+\frac{11\cdots 59}{10\cdots 21}a^{8}+\frac{31\cdots 26}{10\cdots 21}a^{7}+\frac{17\cdots 26}{10\cdots 21}a^{6}-\frac{35\cdots 94}{72\cdots 15}a^{5}+\frac{67\cdots 91}{50\cdots 05}a^{4}-\frac{35\cdots 16}{25\cdots 25}a^{3}+\frac{66\cdots 51}{25\cdots 25}a^{2}+\frac{92\cdots 86}{25\cdots 25}a-\frac{70\cdots 29}{44\cdots 25}$, $\frac{46\cdots 34}{51\cdots 75}a^{23}-\frac{11\cdots 64}{10\cdots 15}a^{22}+\frac{17\cdots 63}{51\cdots 75}a^{21}+\frac{76\cdots 24}{73\cdots 25}a^{20}-\frac{20\cdots 93}{10\cdots 15}a^{19}-\frac{46\cdots 48}{14\cdots 45}a^{18}+\frac{32\cdots 57}{10\cdots 15}a^{17}+\frac{69\cdots 71}{10\cdots 15}a^{16}+\frac{16\cdots 39}{10\cdots 15}a^{15}+\frac{49\cdots 01}{10\cdots 15}a^{14}+\frac{60\cdots 69}{10\cdots 15}a^{13}+\frac{15\cdots 68}{14\cdots 45}a^{12}+\frac{92\cdots 64}{10\cdots 15}a^{11}+\frac{35\cdots 89}{10\cdots 15}a^{10}+\frac{45\cdots 38}{71\cdots 47}a^{9}+\frac{31\cdots 08}{35\cdots 35}a^{8}+\frac{41\cdots 06}{35\cdots 35}a^{7}-\frac{31\cdots 82}{35\cdots 35}a^{6}-\frac{12\cdots 56}{50\cdots 05}a^{5}+\frac{20\cdots 31}{35\cdots 35}a^{4}-\frac{56\cdots 81}{17\cdots 75}a^{3}+\frac{17\cdots 51}{35\cdots 35}a^{2}+\frac{20\cdots 53}{17\cdots 75}a-\frac{18\cdots 83}{31\cdots 75}$, $\frac{22\cdots 22}{10\cdots 25}a^{23}+\frac{13\cdots 43}{20\cdots 45}a^{22}+\frac{15\cdots 51}{10\cdots 25}a^{21}-\frac{64\cdots 86}{34\cdots 25}a^{20}+\frac{19\cdots 17}{40\cdots 09}a^{19}+\frac{41\cdots 39}{20\cdots 45}a^{18}-\frac{20\cdots 90}{40\cdots 09}a^{17}-\frac{45\cdots 02}{20\cdots 45}a^{16}-\frac{73\cdots 76}{20\cdots 45}a^{15}-\frac{78\cdots 82}{20\cdots 45}a^{14}+\frac{63\cdots 89}{20\cdots 45}a^{13}-\frac{30\cdots 67}{20\cdots 45}a^{12}-\frac{34\cdots 36}{20\cdots 45}a^{11}-\frac{22\cdots 89}{69\cdots 05}a^{10}+\frac{93\cdots 24}{69\cdots 05}a^{9}+\frac{92\cdots 31}{13\cdots 21}a^{8}+\frac{80\cdots 56}{69\cdots 05}a^{7}+\frac{16\cdots 92}{13\cdots 21}a^{6}-\frac{94\cdots 31}{13\cdots 21}a^{5}-\frac{28\cdots 29}{69\cdots 05}a^{4}-\frac{18\cdots 57}{34\cdots 25}a^{3}+\frac{29\cdots 17}{69\cdots 05}a^{2}+\frac{60\cdots 26}{34\cdots 25}a-\frac{44\cdots 06}{61\cdots 25}$
|
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| Regulator: | \( 109965381349596890000 \) (assuming GRH) |
| |
| Unit signature rank: | \( 3 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 109965381349596890000 \cdot 8}{2\cdot\sqrt{270761008401829353605241639483649123576469719409942626953125}}\cr\approx \mathstrut & 1.29700479236038 \end{aligned}\] (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
| 6.2.1381408203125.1, 12.4.8024353662494678497314453125.5 |
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 | $24$ | ${\href{/padicField/3.8.0.1}{8} }^{3}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | $24$ | ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.1.0.1}{1} }^{4}$ | ${\href{/padicField/19.3.0.1}{3} }^{8}$ | $24$ | R | $20{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.8.0.1}{8} }^{3}$ | $20{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $24$ | ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.1.5.9a1.2 | $x^{5} + 25 x + 5$ | $5$ | $1$ | $9$ | $F_5$ | $$[\frac{9}{4}]_{4}$$ | |
| 5.1.5.9a1.2 | $x^{5} + 25 x + 5$ | $5$ | $1$ | $9$ | $F_5$ | $$[\frac{9}{4}]_{4}$$ | |
| 5.1.5.9a1.2 | $x^{5} + 25 x + 5$ | $5$ | $1$ | $9$ | $F_5$ | $$[\frac{9}{4}]_{4}$$ | |
| 5.1.5.9a1.2 | $x^{5} + 25 x + 5$ | $5$ | $1$ | $9$ | $F_5$ | $$[\frac{9}{4}]_{4}$$ | |
|
\(29\)
| 29.1.4.3a1.1 | $x^{4} + 29$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 29.1.20.19a1.3 | $x^{20} + 116$ | $20$ | $1$ | $19$ | $C_4\times D_5$ | $$[\ ]_{20}^{2}$$ |