Normalized defining polynomial
\( x^{24} - 2 x^{23} + 119 x^{22} - 123 x^{21} - 2284 x^{20} + 35021 x^{19} - 287932 x^{18} + \cdots + 4794247932259 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $(4, 10)$ |
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| Discriminant: |
\(6769025210045733840131040987091228089411742985248565673828125\)
\(\medspace = 5^{41}\cdot 29^{22}\)
|
| |
| Root discriminant: | \(342.46\) |
| |
| Galois root discriminant: | $5^{39/20}29^{19/20}\approx 565.287791942645$ | ||
| Ramified primes: |
\(5\), \(29\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5}a^{15}-\frac{1}{5}a^{10}+\frac{2}{5}$, $\frac{1}{5}a^{16}-\frac{1}{5}a^{11}+\frac{2}{5}a$, $\frac{1}{5}a^{17}-\frac{1}{5}a^{12}+\frac{2}{5}a^{2}$, $\frac{1}{5}a^{18}-\frac{1}{5}a^{13}+\frac{2}{5}a^{3}$, $\frac{1}{75}a^{19}-\frac{2}{75}a^{18}-\frac{1}{75}a^{17}+\frac{7}{75}a^{16}+\frac{2}{25}a^{15}-\frac{31}{75}a^{14}+\frac{22}{75}a^{13}+\frac{31}{75}a^{12}-\frac{22}{75}a^{11}+\frac{34}{75}a^{10}-\frac{1}{15}a^{9}-\frac{2}{15}a^{8}-\frac{4}{15}a^{7}+\frac{1}{3}a^{6}-\frac{4}{15}a^{5}-\frac{1}{25}a^{4}+\frac{1}{75}a^{3}+\frac{1}{25}a^{2}+\frac{19}{75}a-\frac{28}{75}$, $\frac{1}{75}a^{20}-\frac{1}{15}a^{18}+\frac{1}{15}a^{17}+\frac{1}{15}a^{16}-\frac{4}{75}a^{15}+\frac{7}{15}a^{14}-\frac{7}{15}a^{12}+\frac{1}{15}a^{11}-\frac{9}{25}a^{10}-\frac{4}{15}a^{9}+\frac{7}{15}a^{8}-\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{32}{75}a^{5}-\frac{1}{15}a^{4}+\frac{1}{15}a^{3}+\frac{1}{3}a^{2}-\frac{4}{15}a-\frac{26}{75}$, $\frac{1}{75}a^{21}-\frac{1}{15}a^{18}+\frac{1}{75}a^{16}+\frac{1}{15}a^{15}-\frac{1}{15}a^{14}+\frac{2}{15}a^{12}-\frac{32}{75}a^{11}-\frac{1}{5}a^{10}+\frac{2}{15}a^{9}+\frac{2}{15}a^{8}+\frac{1}{15}a^{7}+\frac{7}{75}a^{6}-\frac{2}{5}a^{5}-\frac{2}{15}a^{4}+\frac{2}{5}a^{3}-\frac{1}{15}a^{2}+\frac{3}{25}a-\frac{7}{15}$, $\frac{1}{525}a^{22}-\frac{2}{525}a^{21}-\frac{1}{525}a^{20}-\frac{2}{525}a^{19}-\frac{12}{175}a^{18}+\frac{23}{525}a^{17}+\frac{7}{75}a^{16}+\frac{37}{525}a^{15}+\frac{26}{75}a^{14}+\frac{121}{525}a^{13}+\frac{46}{525}a^{12}+\frac{14}{75}a^{11}+\frac{139}{525}a^{10}+\frac{44}{105}a^{9}+\frac{2}{15}a^{8}+\frac{9}{175}a^{7}+\frac{76}{525}a^{6}+\frac{11}{175}a^{5}-\frac{104}{525}a^{4}+\frac{218}{525}a^{3}-\frac{79}{175}a^{2}-\frac{22}{175}a+\frac{24}{175}$, $\frac{1}{13\cdots 25}a^{23}-\frac{11\cdots 91}{34\cdots 75}a^{22}-\frac{38\cdots 53}{13\cdots 25}a^{21}+\frac{66\cdots 33}{13\cdots 25}a^{20}-\frac{15\cdots 74}{20\cdots 25}a^{19}+\frac{34\cdots 82}{44\cdots 75}a^{18}-\frac{17\cdots 47}{19\cdots 75}a^{17}+\frac{42\cdots 32}{13\cdots 25}a^{16}-\frac{32\cdots 52}{63\cdots 25}a^{15}+\frac{52\cdots 19}{89\cdots 75}a^{14}-\frac{20\cdots 64}{44\cdots 75}a^{13}+\frac{74\cdots 49}{19\cdots 75}a^{12}-\frac{19\cdots 88}{44\cdots 75}a^{11}+\frac{26\cdots 03}{44\cdots 75}a^{10}-\frac{37\cdots 66}{76\cdots 55}a^{9}+\frac{55\cdots 54}{44\cdots 75}a^{8}-\frac{59\cdots 11}{44\cdots 75}a^{7}+\frac{10\cdots 93}{44\cdots 75}a^{6}-\frac{38\cdots 74}{13\cdots 25}a^{5}-\frac{55\cdots 08}{89\cdots 75}a^{4}-\frac{12\cdots 31}{13\cdots 25}a^{3}-\frac{26\cdots 97}{10\cdots 25}a^{2}+\frac{28\cdots 56}{44\cdots 75}a-\frac{15\cdots 14}{19\cdots 75}$
| 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) |
| |
| 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: |
$\frac{62\cdots 74}{52\cdots 25}a^{23}-\frac{34\cdots 68}{10\cdots 05}a^{22}+\frac{75\cdots 42}{52\cdots 25}a^{21}-\frac{17\cdots 41}{69\cdots 47}a^{20}-\frac{45\cdots 66}{17\cdots 75}a^{19}+\frac{45\cdots 82}{10\cdots 05}a^{18}-\frac{19\cdots 97}{52\cdots 25}a^{17}-\frac{20\cdots 79}{52\cdots 25}a^{16}+\frac{18\cdots 12}{52\cdots 25}a^{15}+\frac{46\cdots 18}{52\cdots 25}a^{14}+\frac{20\cdots 51}{52\cdots 25}a^{13}+\frac{27\cdots 59}{17\cdots 75}a^{12}+\frac{58\cdots 67}{52\cdots 25}a^{11}-\frac{15\cdots 82}{52\cdots 25}a^{10}-\frac{20\cdots 04}{10\cdots 05}a^{9}-\frac{12\cdots 44}{17\cdots 75}a^{8}-\frac{25\cdots 83}{14\cdots 15}a^{7}-\frac{99\cdots 96}{52\cdots 25}a^{6}+\frac{19\cdots 84}{34\cdots 35}a^{5}+\frac{23\cdots 47}{74\cdots 75}a^{4}+\frac{49\cdots 79}{74\cdots 75}a^{3}+\frac{36\cdots 46}{52\cdots 25}a^{2}+\frac{18\cdots 36}{52\cdots 25}a+\frac{33\cdots 69}{52\cdots 25}$, $\frac{14\cdots 63}{17\cdots 75}a^{23}-\frac{57\cdots 87}{17\cdots 75}a^{22}+\frac{18\cdots 82}{17\cdots 75}a^{21}-\frac{52\cdots 11}{17\cdots 75}a^{20}-\frac{26\cdots 29}{17\cdots 75}a^{19}+\frac{58\cdots 21}{17\cdots 75}a^{18}-\frac{53\cdots 38}{17\cdots 75}a^{17}-\frac{43\cdots 81}{17\cdots 75}a^{16}+\frac{37\cdots 90}{69\cdots 47}a^{15}+\frac{10\cdots 69}{17\cdots 75}a^{14}+\frac{37\cdots 46}{17\cdots 75}a^{13}+\frac{30\cdots 12}{34\cdots 35}a^{12}-\frac{16\cdots 26}{17\cdots 75}a^{11}-\frac{37\cdots 34}{17\cdots 75}a^{10}-\frac{40\cdots 14}{34\cdots 35}a^{9}-\frac{65\cdots 69}{17\cdots 75}a^{8}-\frac{20\cdots 67}{24\cdots 25}a^{7}-\frac{95\cdots 01}{17\cdots 75}a^{6}+\frac{73\cdots 33}{17\cdots 75}a^{5}+\frac{44\cdots 11}{24\cdots 25}a^{4}+\frac{72\cdots 09}{24\cdots 25}a^{3}+\frac{77\cdots 72}{34\cdots 35}a^{2}+\frac{12\cdots 42}{17\cdots 75}a+\frac{11\cdots 53}{17\cdots 75}$, $\frac{18\cdots 97}{14\cdots 15}a^{23}-\frac{32\cdots 82}{49\cdots 05}a^{22}+\frac{12\cdots 66}{74\cdots 75}a^{21}-\frac{17\cdots 66}{24\cdots 25}a^{20}-\frac{38\cdots 12}{74\cdots 75}a^{19}+\frac{11\cdots 33}{24\cdots 25}a^{18}-\frac{37\cdots 28}{74\cdots 75}a^{17}-\frac{19\cdots 48}{74\cdots 75}a^{16}+\frac{14\cdots 96}{14\cdots 15}a^{15}+\frac{15\cdots 79}{24\cdots 25}a^{14}+\frac{18\cdots 61}{74\cdots 75}a^{13}+\frac{74\cdots 23}{74\cdots 75}a^{12}-\frac{29\cdots 41}{24\cdots 25}a^{11}-\frac{20\cdots 82}{74\cdots 75}a^{10}-\frac{19\cdots 29}{14\cdots 15}a^{9}-\frac{20\cdots 27}{49\cdots 05}a^{8}-\frac{12\cdots 09}{14\cdots 15}a^{7}-\frac{19\cdots 53}{74\cdots 75}a^{6}+\frac{40\cdots 24}{74\cdots 75}a^{5}+\frac{13\cdots 66}{74\cdots 75}a^{4}+\frac{20\cdots 53}{74\cdots 75}a^{3}+\frac{15\cdots 74}{74\cdots 75}a^{2}+\frac{20\cdots 57}{24\cdots 25}a+\frac{31\cdots 53}{24\cdots 25}$, $\frac{25\cdots 14}{17\cdots 75}a^{23}-\frac{84\cdots 17}{52\cdots 25}a^{22}+\frac{29\cdots 41}{17\cdots 75}a^{21}-\frac{12\cdots 03}{10\cdots 05}a^{20}-\frac{20\cdots 48}{52\cdots 25}a^{19}+\frac{25\cdots 78}{52\cdots 25}a^{18}-\frac{63\cdots 58}{17\cdots 75}a^{17}-\frac{29\cdots 68}{52\cdots 25}a^{16}-\frac{46\cdots 96}{17\cdots 75}a^{15}+\frac{64\cdots 53}{52\cdots 25}a^{14}+\frac{13\cdots 95}{20\cdots 41}a^{13}+\frac{13\cdots 21}{52\cdots 25}a^{12}+\frac{39\cdots 11}{10\cdots 05}a^{11}-\frac{19\cdots 72}{52\cdots 25}a^{10}-\frac{30\cdots 48}{10\cdots 05}a^{9}-\frac{61\cdots 56}{52\cdots 25}a^{8}-\frac{23\cdots 72}{74\cdots 75}a^{7}-\frac{79\cdots 58}{17\cdots 75}a^{6}+\frac{56\cdots 98}{10\cdots 05}a^{5}+\frac{38\cdots 82}{74\cdots 75}a^{4}+\frac{64\cdots 63}{49\cdots 05}a^{3}+\frac{85\cdots 38}{52\cdots 25}a^{2}+\frac{10\cdots 79}{10\cdots 05}a+\frac{11\cdots 79}{52\cdots 25}$, $\frac{46\cdots 47}{10\cdots 25}a^{23}-\frac{48\cdots 36}{34\cdots 75}a^{22}+\frac{27\cdots 44}{48\cdots 25}a^{21}-\frac{11\cdots 64}{10\cdots 25}a^{20}-\frac{54\cdots 63}{58\cdots 35}a^{19}+\frac{58\cdots 99}{34\cdots 75}a^{18}-\frac{51\cdots 46}{34\cdots 75}a^{17}-\frac{49\cdots 22}{34\cdots 75}a^{16}+\frac{62\cdots 02}{34\cdots 75}a^{15}+\frac{13\cdots 72}{41\cdots 45}a^{14}+\frac{47\cdots 72}{34\cdots 75}a^{13}+\frac{56\cdots 76}{10\cdots 25}a^{12}+\frac{26\cdots 47}{10\cdots 25}a^{11}-\frac{57\cdots 77}{48\cdots 25}a^{10}-\frac{28\cdots 32}{41\cdots 45}a^{9}-\frac{25\cdots 36}{10\cdots 25}a^{8}-\frac{59\cdots 86}{10\cdots 25}a^{7}-\frac{57\cdots 07}{10\cdots 25}a^{6}+\frac{22\cdots 17}{10\cdots 25}a^{5}+\frac{46\cdots 73}{41\cdots 45}a^{4}+\frac{22\cdots 13}{10\cdots 25}a^{3}+\frac{30\cdots 89}{14\cdots 75}a^{2}+\frac{13\cdots 98}{14\cdots 75}a+\frac{15\cdots 49}{10\cdots 25}$, $\frac{18\cdots 99}{26\cdots 25}a^{23}+\frac{34\cdots 09}{68\cdots 75}a^{22}-\frac{13\cdots 24}{12\cdots 25}a^{21}+\frac{57\cdots 92}{89\cdots 75}a^{20}-\frac{46\cdots 66}{29\cdots 75}a^{19}-\frac{49\cdots 27}{26\cdots 25}a^{18}+\frac{83\cdots 52}{26\cdots 25}a^{17}+\frac{20\cdots 04}{26\cdots 25}a^{16}-\frac{26\cdots 24}{53\cdots 85}a^{15}-\frac{22\cdots 43}{89\cdots 75}a^{14}-\frac{11\cdots 08}{89\cdots 75}a^{13}-\frac{10\cdots 43}{26\cdots 25}a^{12}+\frac{16\cdots 16}{17\cdots 95}a^{11}+\frac{15\cdots 69}{12\cdots 25}a^{10}+\frac{10\cdots 29}{17\cdots 95}a^{9}+\frac{15\cdots 64}{89\cdots 75}a^{8}+\frac{88\cdots 37}{26\cdots 25}a^{7}-\frac{72\cdots 63}{26\cdots 25}a^{6}-\frac{21\cdots 66}{89\cdots 75}a^{5}-\frac{65\cdots 54}{89\cdots 75}a^{4}-\frac{97\cdots 34}{89\cdots 75}a^{3}-\frac{86\cdots 48}{97\cdots 25}a^{2}-\frac{28\cdots 74}{76\cdots 55}a-\frac{55\cdots 81}{89\cdots 75}$, $\frac{13\cdots 26}{19\cdots 75}a^{23}+\frac{22\cdots 31}{10\cdots 25}a^{22}-\frac{39\cdots 98}{44\cdots 75}a^{21}+\frac{82\cdots 18}{44\cdots 75}a^{20}+\frac{30\cdots 37}{20\cdots 25}a^{19}-\frac{12\cdots 64}{44\cdots 75}a^{18}+\frac{31\cdots 78}{13\cdots 25}a^{17}+\frac{14\cdots 61}{63\cdots 25}a^{16}-\frac{12\cdots 37}{44\cdots 75}a^{15}-\frac{20\cdots 58}{38\cdots 75}a^{14}-\frac{98\cdots 37}{44\cdots 75}a^{13}-\frac{11\cdots 61}{13\cdots 25}a^{12}-\frac{82\cdots 31}{19\cdots 75}a^{11}+\frac{82\cdots 69}{44\cdots 75}a^{10}+\frac{19\cdots 84}{17\cdots 95}a^{9}+\frac{25\cdots 21}{63\cdots 25}a^{8}+\frac{12\cdots 76}{13\cdots 25}a^{7}+\frac{12\cdots 17}{13\cdots 25}a^{6}-\frac{46\cdots 37}{13\cdots 25}a^{5}-\frac{16\cdots 16}{89\cdots 75}a^{4}-\frac{15\cdots 66}{44\cdots 75}a^{3}-\frac{11\cdots 27}{34\cdots 75}a^{2}-\frac{21\cdots 96}{13\cdots 25}a-\frac{37\cdots 79}{13\cdots 25}$, $\frac{61\cdots 99}{10\cdots 57}a^{23}-\frac{14\cdots 89}{68\cdots 75}a^{22}+\frac{18\cdots 97}{26\cdots 25}a^{21}-\frac{44\cdots 53}{26\cdots 25}a^{20}-\frac{58\cdots 66}{41\cdots 45}a^{19}+\frac{12\cdots 22}{53\cdots 85}a^{18}-\frac{17\cdots 72}{89\cdots 75}a^{17}-\frac{49\cdots 13}{26\cdots 25}a^{16}+\frac{11\cdots 37}{26\cdots 25}a^{15}+\frac{77\cdots 97}{17\cdots 95}a^{14}+\frac{20\cdots 09}{17\cdots 95}a^{13}+\frac{48\cdots 84}{89\cdots 75}a^{12}-\frac{56\cdots 13}{89\cdots 75}a^{11}-\frac{14\cdots 98}{89\cdots 75}a^{10}-\frac{39\cdots 86}{53\cdots 85}a^{9}-\frac{38\cdots 76}{17\cdots 95}a^{8}-\frac{13\cdots 17}{26\cdots 25}a^{7}-\frac{28\cdots 56}{26\cdots 25}a^{6}+\frac{12\cdots 72}{38\cdots 75}a^{5}+\frac{35\cdots 95}{35\cdots 19}a^{4}+\frac{75\cdots 82}{53\cdots 85}a^{3}+\frac{21\cdots 57}{20\cdots 25}a^{2}+\frac{11\cdots 63}{26\cdots 25}a+\frac{16\cdots 53}{26\cdots 25}$, $\frac{90\cdots 72}{13\cdots 25}a^{23}+\frac{26\cdots 41}{10\cdots 25}a^{22}-\frac{11\cdots 39}{13\cdots 25}a^{21}+\frac{10\cdots 78}{44\cdots 75}a^{20}+\frac{82\cdots 31}{68\cdots 75}a^{19}-\frac{34\cdots 37}{13\cdots 25}a^{18}+\frac{32\cdots 93}{13\cdots 25}a^{17}+\frac{26\cdots 91}{13\cdots 25}a^{16}-\frac{53\cdots 41}{13\cdots 25}a^{15}-\frac{12\cdots 54}{26\cdots 25}a^{14}-\frac{23\cdots 76}{13\cdots 25}a^{13}-\frac{31\cdots 02}{44\cdots 75}a^{12}+\frac{10\cdots 43}{13\cdots 25}a^{11}+\frac{22\cdots 32}{13\cdots 25}a^{10}+\frac{49\cdots 26}{53\cdots 85}a^{9}+\frac{13\cdots 62}{44\cdots 75}a^{8}+\frac{88\cdots 61}{13\cdots 25}a^{7}+\frac{56\cdots 77}{13\cdots 25}a^{6}-\frac{15\cdots 84}{44\cdots 75}a^{5}-\frac{38\cdots 17}{26\cdots 25}a^{4}-\frac{30\cdots 43}{13\cdots 25}a^{3}-\frac{17\cdots 01}{10\cdots 25}a^{2}-\frac{69\cdots 66}{13\cdots 25}a-\frac{61\cdots 29}{13\cdots 25}$, $\frac{10\cdots 37}{12\cdots 25}a^{23}+\frac{12\cdots 96}{41\cdots 75}a^{22}-\frac{12\cdots 64}{12\cdots 25}a^{21}+\frac{32\cdots 09}{12\cdots 25}a^{20}+\frac{38\cdots 69}{24\cdots 25}a^{19}-\frac{39\cdots 77}{12\cdots 25}a^{18}+\frac{36\cdots 48}{12\cdots 25}a^{17}+\frac{10\cdots 22}{41\cdots 75}a^{16}-\frac{53\cdots 66}{12\cdots 25}a^{15}-\frac{14\cdots 74}{24\cdots 25}a^{14}-\frac{95\cdots 57}{41\cdots 75}a^{13}-\frac{11\cdots 91}{12\cdots 25}a^{12}-\frac{17\cdots 07}{12\cdots 25}a^{11}+\frac{87\cdots 19}{41\cdots 75}a^{10}+\frac{39\cdots 73}{33\cdots 47}a^{9}+\frac{50\cdots 81}{12\cdots 25}a^{8}+\frac{11\cdots 46}{12\cdots 25}a^{7}+\frac{93\cdots 27}{12\cdots 25}a^{6}-\frac{51\cdots 77}{12\cdots 25}a^{5}-\frac{46\cdots 42}{24\cdots 25}a^{4}-\frac{41\cdots 78}{12\cdots 25}a^{3}-\frac{12\cdots 56}{41\cdots 75}a^{2}-\frac{52\cdots 47}{41\cdots 75}a-\frac{82\cdots 68}{41\cdots 75}$, $\frac{86\cdots 29}{52\cdots 25}a^{23}+\frac{28\cdots 81}{10\cdots 05}a^{22}-\frac{99\cdots 01}{52\cdots 25}a^{21}+\frac{62\cdots 28}{52\cdots 25}a^{20}+\frac{22\cdots 84}{52\cdots 25}a^{19}-\frac{30\cdots 07}{52\cdots 25}a^{18}+\frac{23\cdots 76}{52\cdots 25}a^{17}+\frac{10\cdots 74}{17\cdots 75}a^{16}-\frac{64\cdots 23}{52\cdots 25}a^{15}-\frac{71\cdots 59}{52\cdots 25}a^{14}-\frac{11\cdots 33}{17\cdots 75}a^{13}-\frac{13\cdots 46}{52\cdots 25}a^{12}-\frac{16\cdots 66}{52\cdots 25}a^{11}+\frac{15\cdots 27}{34\cdots 35}a^{10}+\frac{10\cdots 86}{34\cdots 35}a^{9}+\frac{61\cdots 02}{52\cdots 25}a^{8}+\frac{29\cdots 13}{99\cdots 21}a^{7}+\frac{20\cdots 73}{52\cdots 25}a^{6}-\frac{12\cdots 63}{17\cdots 75}a^{5}-\frac{39\cdots 91}{74\cdots 75}a^{4}-\frac{90\cdots 56}{74\cdots 75}a^{3}-\frac{23\cdots 21}{17\cdots 75}a^{2}-\frac{37\cdots 48}{52\cdots 25}a-\frac{14\cdots 61}{10\cdots 05}$, $\frac{90\cdots 46}{17\cdots 75}a^{23}-\frac{79\cdots 46}{52\cdots 25}a^{22}+\frac{10\cdots 02}{17\cdots 75}a^{21}-\frac{63\cdots 59}{52\cdots 25}a^{20}-\frac{18\cdots 52}{17\cdots 75}a^{19}+\frac{20\cdots 47}{10\cdots 05}a^{18}-\frac{58\cdots 78}{34\cdots 35}a^{17}-\frac{87\cdots 51}{52\cdots 25}a^{16}+\frac{19\cdots 57}{10\cdots 05}a^{15}+\frac{19\cdots 26}{52\cdots 25}a^{14}+\frac{28\cdots 74}{17\cdots 75}a^{13}+\frac{33\cdots 16}{52\cdots 25}a^{12}+\frac{37\cdots 93}{10\cdots 05}a^{11}-\frac{23\cdots 32}{17\cdots 75}a^{10}-\frac{16\cdots 89}{20\cdots 41}a^{9}-\frac{15\cdots 59}{52\cdots 25}a^{8}-\frac{17\cdots 37}{24\cdots 25}a^{7}-\frac{36\cdots 53}{52\cdots 25}a^{6}+\frac{13\cdots 82}{52\cdots 25}a^{5}+\frac{98\cdots 14}{74\cdots 75}a^{4}+\frac{19\cdots 63}{74\cdots 75}a^{3}+\frac{13\cdots 33}{52\cdots 25}a^{2}+\frac{12\cdots 06}{10\cdots 05}a+\frac{37\cdots 24}{17\cdots 75}$, $\frac{13\cdots 67}{42\cdots 75}a^{23}+\frac{11\cdots 52}{14\cdots 25}a^{22}-\frac{10\cdots 81}{28\cdots 45}a^{21}+\frac{24\cdots 66}{42\cdots 75}a^{20}+\frac{33\cdots 46}{42\cdots 75}a^{19}-\frac{16\cdots 68}{14\cdots 25}a^{18}+\frac{27\cdots 42}{28\cdots 45}a^{17}+\frac{14\cdots 84}{14\cdots 25}a^{16}-\frac{16\cdots 56}{14\cdots 25}a^{15}-\frac{10\cdots 41}{42\cdots 75}a^{14}-\frac{13\cdots 98}{14\cdots 25}a^{13}-\frac{16\cdots 46}{42\cdots 75}a^{12}-\frac{85\cdots 37}{42\cdots 75}a^{11}+\frac{11\cdots 94}{14\cdots 25}a^{10}+\frac{82\cdots 74}{17\cdots 07}a^{9}+\frac{72\cdots 36}{42\cdots 75}a^{8}+\frac{17\cdots 67}{42\cdots 75}a^{7}+\frac{34\cdots 33}{85\cdots 35}a^{6}-\frac{62\cdots 48}{42\cdots 75}a^{5}-\frac{32\cdots 43}{42\cdots 75}a^{4}-\frac{65\cdots 37}{42\cdots 75}a^{3}-\frac{66\cdots 88}{42\cdots 75}a^{2}-\frac{32\cdots 91}{42\cdots 75}a-\frac{57\cdots 74}{42\cdots 75}$
|
| |
| Regulator: | \( 164472864839769980000 \) (assuming GRH) |
| |
| Unit signature rank: | \( 3 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 164472864839769980000 \cdot 8}{2\cdot\sqrt{6769025210045733840131040987091228089411742985248565673828125}}\cr\approx \mathstrut & 0.387980455834985 \end{aligned}\] (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
| 6.2.1381408203125.4, 12.4.8024353662494678497314453125.6 |
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 | ${\href{/padicField/2.8.0.1}{8} }^{3}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{5}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | $24$ | ${\href{/padicField/19.3.0.1}{3} }^{8}$ | $24$ | R | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | $20{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.8.0.1}{8} }^{3}$ | $24$ | $24$ | ${\href{/padicField/59.2.0.1}{2} }^{10}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.1.10.19a2.3 | $x^{10} + 50 x^{2} + 5$ | $10$ | $1$ | $19$ | $F_5$ | $$[\frac{9}{4}]_{4}$$ | |
| 5.1.10.19a2.3 | $x^{10} + 50 x^{2} + 5$ | $10$ | $1$ | $19$ | $F_5$ | $$[\frac{9}{4}]_{4}$$ | |
|
\(29\)
| 29.1.4.3a1.3 | $x^{4} + 116$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 29.1.20.19a1.1 | $x^{20} + 29$ | $20$ | $1$ | $19$ | 20T6 | $$[\ ]_{20}^{2}$$ |