Normalized defining polynomial
\( x^{24} - 10 x^{23} + 10 x^{22} + 430 x^{21} - 2305 x^{20} + 1339 x^{19} - 64445 x^{18} + \cdots - 771099274575 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $(4, 10)$ |
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| Discriminant: |
\(270761008401829353605241639483649123576469719409942626953125\)
\(\medspace = 5^{39}\cdot 29^{22}\)
|
| |
| Root discriminant: | \(299.47\) |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{5}a^{15}-\frac{2}{5}a^{10}+\frac{2}{5}a^{5}$, $\frac{1}{5}a^{16}-\frac{2}{5}a^{11}+\frac{2}{5}a^{6}$, $\frac{1}{5}a^{17}-\frac{2}{5}a^{12}+\frac{2}{5}a^{7}$, $\frac{1}{15}a^{18}-\frac{1}{15}a^{17}-\frac{1}{15}a^{15}+\frac{1}{3}a^{14}+\frac{1}{5}a^{13}+\frac{7}{15}a^{12}+\frac{2}{15}a^{10}+\frac{7}{15}a^{8}-\frac{7}{15}a^{7}-\frac{1}{3}a^{6}-\frac{2}{15}a^{5}-\frac{1}{3}a$, $\frac{1}{15}a^{19}-\frac{1}{15}a^{17}-\frac{1}{15}a^{16}+\frac{1}{15}a^{15}-\frac{7}{15}a^{14}-\frac{1}{3}a^{13}+\frac{7}{15}a^{12}+\frac{2}{15}a^{11}-\frac{7}{15}a^{10}+\frac{7}{15}a^{9}+\frac{1}{5}a^{7}-\frac{7}{15}a^{6}+\frac{7}{15}a^{5}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{15}a^{20}+\frac{1}{15}a^{17}+\frac{1}{15}a^{16}+\frac{1}{15}a^{15}-\frac{1}{3}a^{13}+\frac{1}{5}a^{12}-\frac{7}{15}a^{11}+\frac{2}{5}a^{10}-\frac{1}{3}a^{8}+\frac{7}{15}a^{7}+\frac{2}{15}a^{6}+\frac{1}{15}a^{5}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{15}a^{21}-\frac{1}{15}a^{17}+\frac{1}{15}a^{16}+\frac{1}{15}a^{15}+\frac{1}{3}a^{14}+\frac{7}{15}a^{12}+\frac{2}{5}a^{11}-\frac{2}{15}a^{10}-\frac{1}{3}a^{9}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{2}{15}a^{5}-\frac{1}{3}a^{4}-\frac{1}{3}a^{3}-\frac{1}{3}a^{2}+\frac{1}{3}a$, $\frac{1}{975}a^{22}+\frac{6}{325}a^{21}+\frac{23}{975}a^{20}-\frac{29}{975}a^{19}+\frac{4}{195}a^{18}+\frac{83}{975}a^{17}+\frac{14}{975}a^{16}-\frac{76}{975}a^{15}-\frac{67}{975}a^{14}-\frac{37}{195}a^{13}+\frac{19}{75}a^{12}+\frac{196}{975}a^{11}+\frac{17}{325}a^{10}+\frac{127}{975}a^{9}+\frac{17}{39}a^{8}+\frac{5}{39}a^{7}+\frac{17}{39}a^{6}-\frac{7}{195}a^{5}-\frac{88}{195}a^{4}-\frac{1}{13}a^{3}+\frac{11}{39}a^{2}+\frac{6}{13}a-\frac{5}{13}$, $\frac{1}{96\cdots 25}a^{23}+\frac{33\cdots 16}{19\cdots 45}a^{22}-\frac{57\cdots 07}{32\cdots 75}a^{21}+\frac{28\cdots 97}{96\cdots 25}a^{20}+\frac{16\cdots 82}{96\cdots 25}a^{19}+\frac{17\cdots 32}{10\cdots 25}a^{18}+\frac{27\cdots 75}{29\cdots 53}a^{17}+\frac{57\cdots 37}{96\cdots 25}a^{16}+\frac{93\cdots 96}{96\cdots 25}a^{15}+\frac{69\cdots 82}{74\cdots 25}a^{14}-\frac{39\cdots 78}{96\cdots 25}a^{13}-\frac{19\cdots 53}{64\cdots 15}a^{12}-\frac{43\cdots 42}{96\cdots 25}a^{11}-\frac{66\cdots 61}{96\cdots 25}a^{10}-\frac{32\cdots 56}{96\cdots 25}a^{9}+\frac{77\cdots 31}{19\cdots 45}a^{8}-\frac{76\cdots 03}{64\cdots 15}a^{7}-\frac{58\cdots 63}{21\cdots 05}a^{6}+\frac{78\cdots 88}{19\cdots 45}a^{5}+\frac{67\cdots 98}{14\cdots 65}a^{4}+\frac{86\cdots 31}{38\cdots 89}a^{3}+\frac{99\cdots 07}{38\cdots 89}a^{2}-\frac{62\cdots 75}{12\cdots 63}a-\frac{15\cdots 72}{43\cdots 21}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | $C_{4}$, which has order $4$ (assuming GRH) |
| |
| Narrow class group: | $C_{4}\times C_{2}$, which has order $8$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{65\cdots 74}{40\cdots 75}a^{23}-\frac{23\cdots 72}{40\cdots 75}a^{22}+\frac{25\cdots 41}{13\cdots 25}a^{21}-\frac{31\cdots 64}{40\cdots 75}a^{20}-\frac{46\cdots 54}{81\cdots 15}a^{19}+\frac{57\cdots 11}{13\cdots 25}a^{18}+\frac{41\cdots 04}{40\cdots 75}a^{17}+\frac{78\cdots 44}{40\cdots 75}a^{16}+\frac{13\cdots 88}{40\cdots 75}a^{15}-\frac{59\cdots 51}{81\cdots 15}a^{14}-\frac{13\cdots 28}{40\cdots 75}a^{13}-\frac{18\cdots 18}{13\cdots 25}a^{12}+\frac{13\cdots 11}{40\cdots 75}a^{11}+\frac{10\cdots 12}{40\cdots 75}a^{10}+\frac{10\cdots 59}{81\cdots 15}a^{9}+\frac{10\cdots 09}{16\cdots 63}a^{8}+\frac{49\cdots 08}{27\cdots 05}a^{7}+\frac{12\cdots 57}{27\cdots 05}a^{6}+\frac{78\cdots 01}{81\cdots 15}a^{5}+\frac{17\cdots 60}{16\cdots 63}a^{4}+\frac{28\cdots 73}{16\cdots 63}a^{3}+\frac{22\cdots 29}{16\cdots 63}a^{2}-\frac{73\cdots 93}{54\cdots 21}a+\frac{29\cdots 49}{18\cdots 07}$, $\frac{61\cdots 90}{18\cdots 07}a^{23}-\frac{16\cdots 09}{45\cdots 75}a^{22}+\frac{21\cdots 03}{45\cdots 75}a^{21}+\frac{73\cdots 28}{45\cdots 75}a^{20}-\frac{43\cdots 14}{45\cdots 75}a^{19}+\frac{37\cdots 56}{90\cdots 35}a^{18}-\frac{78\cdots 22}{45\cdots 75}a^{17}-\frac{44\cdots 61}{45\cdots 75}a^{16}+\frac{53\cdots 39}{45\cdots 75}a^{15}+\frac{13\cdots 53}{45\cdots 75}a^{14}+\frac{84\cdots 64}{90\cdots 35}a^{13}-\frac{97\cdots 23}{45\cdots 75}a^{12}-\frac{18\cdots 74}{45\cdots 75}a^{11}-\frac{40\cdots 39}{45\cdots 75}a^{10}-\frac{19\cdots 43}{45\cdots 75}a^{9}-\frac{76\cdots 61}{90\cdots 35}a^{8}-\frac{18\cdots 78}{18\cdots 07}a^{7}-\frac{16\cdots 72}{90\cdots 35}a^{6}+\frac{74\cdots 03}{90\cdots 35}a^{5}+\frac{94\cdots 77}{90\cdots 35}a^{4}-\frac{92\cdots 13}{18\cdots 07}a^{3}+\frac{10\cdots 79}{18\cdots 07}a^{2}-\frac{41\cdots 87}{18\cdots 07}a+\frac{12\cdots 39}{18\cdots 07}$, $\frac{64\cdots 11}{13\cdots 25}a^{23}+\frac{10\cdots 92}{90\cdots 35}a^{22}-\frac{27\cdots 68}{45\cdots 75}a^{21}-\frac{53\cdots 17}{13\cdots 25}a^{20}+\frac{85\cdots 63}{13\cdots 25}a^{19}-\frac{31\cdots 43}{13\cdots 25}a^{18}+\frac{16\cdots 37}{90\cdots 35}a^{17}+\frac{21\cdots 08}{13\cdots 25}a^{16}-\frac{26\cdots 77}{45\cdots 75}a^{15}+\frac{41\cdots 69}{13\cdots 25}a^{14}+\frac{83\cdots 61}{45\cdots 75}a^{13}+\frac{14\cdots 16}{90\cdots 35}a^{12}+\frac{36\cdots 52}{13\cdots 25}a^{11}-\frac{14\cdots 29}{13\cdots 25}a^{10}+\frac{18\cdots 46}{13\cdots 25}a^{9}-\frac{32\cdots 41}{27\cdots 05}a^{8}+\frac{75\cdots 73}{90\cdots 35}a^{7}-\frac{15\cdots 13}{54\cdots 21}a^{6}-\frac{60\cdots 93}{27\cdots 05}a^{5}+\frac{11\cdots 76}{27\cdots 05}a^{4}-\frac{21\cdots 05}{18\cdots 07}a^{3}+\frac{18\cdots 50}{18\cdots 07}a^{2}-\frac{60\cdots 86}{18\cdots 07}a+\frac{74\cdots 54}{18\cdots 07}$, $\frac{15\cdots 03}{40\cdots 75}a^{23}+\frac{43\cdots 14}{40\cdots 75}a^{22}+\frac{60\cdots 44}{27\cdots 05}a^{21}-\frac{72\cdots 94}{40\cdots 75}a^{20}-\frac{12\cdots 92}{40\cdots 75}a^{19}+\frac{70\cdots 07}{13\cdots 25}a^{18}+\frac{99\cdots 32}{40\cdots 75}a^{17}+\frac{24\cdots 52}{81\cdots 15}a^{16}-\frac{39\cdots 07}{40\cdots 75}a^{15}-\frac{45\cdots 31}{40\cdots 75}a^{14}-\frac{21\cdots 66}{40\cdots 75}a^{13}-\frac{23\cdots 84}{13\cdots 25}a^{12}+\frac{19\cdots 36}{81\cdots 15}a^{11}+\frac{16\cdots 57}{40\cdots 75}a^{10}+\frac{84\cdots 16}{40\cdots 75}a^{9}+\frac{73\cdots 59}{81\cdots 15}a^{8}+\frac{68\cdots 24}{27\cdots 05}a^{7}+\frac{56\cdots 18}{90\cdots 35}a^{6}+\frac{94\cdots 31}{81\cdots 15}a^{5}+\frac{96\cdots 61}{81\cdots 15}a^{4}+\frac{29\cdots 81}{16\cdots 63}a^{3}+\frac{12\cdots 43}{16\cdots 63}a^{2}-\frac{55\cdots 83}{54\cdots 21}a+\frac{34\cdots 23}{18\cdots 07}$, $\frac{38\cdots 98}{13\cdots 25}a^{23}+\frac{18\cdots 67}{13\cdots 25}a^{22}+\frac{48\cdots 53}{45\cdots 75}a^{21}-\frac{30\cdots 26}{27\cdots 05}a^{20}-\frac{10\cdots 24}{13\cdots 25}a^{19}+\frac{28\cdots 26}{13\cdots 25}a^{18}+\frac{33\cdots 71}{13\cdots 25}a^{17}+\frac{82\cdots 79}{45\cdots 75}a^{16}-\frac{76\cdots 47}{27\cdots 05}a^{15}-\frac{98\cdots 42}{13\cdots 25}a^{14}-\frac{54\cdots 71}{13\cdots 25}a^{13}-\frac{38\cdots 72}{45\cdots 75}a^{12}+\frac{17\cdots 61}{45\cdots 75}a^{11}+\frac{31\cdots 88}{90\cdots 35}a^{10}+\frac{20\cdots 82}{13\cdots 25}a^{9}+\frac{10\cdots 36}{27\cdots 05}a^{8}+\frac{65\cdots 76}{90\cdots 35}a^{7}+\frac{20\cdots 27}{27\cdots 05}a^{6}-\frac{13\cdots 43}{90\cdots 35}a^{5}-\frac{21\cdots 51}{90\cdots 35}a^{4}-\frac{15\cdots 08}{54\cdots 21}a^{3}-\frac{15\cdots 69}{54\cdots 21}a^{2}+\frac{19\cdots 77}{18\cdots 07}a-\frac{46\cdots 09}{18\cdots 07}$, $\frac{16\cdots 82}{40\cdots 75}a^{23}-\frac{13\cdots 38}{81\cdots 15}a^{22}+\frac{61\cdots 94}{13\cdots 25}a^{21}-\frac{64\cdots 34}{40\cdots 75}a^{20}-\frac{62\cdots 44}{40\cdots 75}a^{19}+\frac{41\cdots 46}{45\cdots 75}a^{18}+\frac{26\cdots 17}{81\cdots 15}a^{17}+\frac{20\cdots 11}{40\cdots 75}a^{16}+\frac{45\cdots 33}{40\cdots 75}a^{15}-\frac{72\cdots 97}{40\cdots 75}a^{14}-\frac{40\cdots 89}{40\cdots 75}a^{13}-\frac{10\cdots 98}{27\cdots 05}a^{12}-\frac{45\cdots 66}{40\cdots 75}a^{11}+\frac{30\cdots 77}{40\cdots 75}a^{10}+\frac{16\cdots 02}{40\cdots 75}a^{9}+\frac{28\cdots 05}{16\cdots 63}a^{8}+\frac{13\cdots 21}{27\cdots 05}a^{7}+\frac{61\cdots 69}{54\cdots 21}a^{6}+\frac{16\cdots 97}{81\cdots 15}a^{5}+\frac{15\cdots 92}{81\cdots 15}a^{4}+\frac{45\cdots 28}{16\cdots 63}a^{3}+\frac{39\cdots 62}{16\cdots 63}a^{2}-\frac{12\cdots 46}{54\cdots 21}a+\frac{61\cdots 24}{18\cdots 07}$, $\frac{46\cdots 92}{32\cdots 75}a^{23}-\frac{42\cdots 72}{32\cdots 75}a^{22}+\frac{43\cdots 19}{10\cdots 25}a^{21}+\frac{66\cdots 26}{10\cdots 25}a^{20}-\frac{30\cdots 81}{10\cdots 25}a^{19}-\frac{10\cdots 34}{32\cdots 75}a^{18}-\frac{29\cdots 91}{32\cdots 75}a^{17}-\frac{17\cdots 34}{32\cdots 75}a^{16}+\frac{11\cdots 74}{32\cdots 75}a^{15}+\frac{55\cdots 46}{32\cdots 75}a^{14}+\frac{22\cdots 73}{24\cdots 75}a^{13}+\frac{29\cdots 51}{32\cdots 75}a^{12}-\frac{43\cdots 61}{32\cdots 75}a^{11}-\frac{21\cdots 44}{32\cdots 75}a^{10}-\frac{11\cdots 96}{32\cdots 75}a^{9}-\frac{69\cdots 29}{64\cdots 15}a^{8}-\frac{62\cdots 44}{21\cdots 05}a^{7}-\frac{40\cdots 38}{64\cdots 15}a^{6}-\frac{42\cdots 94}{64\cdots 15}a^{5}-\frac{68\cdots 91}{64\cdots 15}a^{4}-\frac{36\cdots 14}{43\cdots 21}a^{3}+\frac{76\cdots 90}{43\cdots 21}a^{2}-\frac{43\cdots 92}{43\cdots 21}a-\frac{87\cdots 33}{33\cdots 17}$, $\frac{10\cdots 93}{21\cdots 05}a^{23}-\frac{36\cdots 63}{82\cdots 25}a^{22}+\frac{86\cdots 54}{32\cdots 75}a^{21}+\frac{65\cdots 89}{32\cdots 75}a^{20}-\frac{31\cdots 72}{32\cdots 75}a^{19}+\frac{11\cdots 57}{64\cdots 15}a^{18}-\frac{97\cdots 41}{32\cdots 75}a^{17}-\frac{53\cdots 73}{32\cdots 75}a^{16}+\frac{13\cdots 99}{10\cdots 25}a^{15}+\frac{55\cdots 73}{10\cdots 25}a^{14}+\frac{59\cdots 67}{21\cdots 05}a^{13}+\frac{22\cdots 22}{10\cdots 25}a^{12}-\frac{48\cdots 44}{10\cdots 25}a^{11}-\frac{63\cdots 62}{32\cdots 75}a^{10}-\frac{36\cdots 39}{32\cdots 75}a^{9}-\frac{67\cdots 38}{21\cdots 05}a^{8}-\frac{54\cdots 29}{64\cdots 15}a^{7}-\frac{11\cdots 46}{64\cdots 15}a^{6}-\frac{66\cdots 34}{43\cdots 21}a^{5}-\frac{17\cdots 19}{64\cdots 15}a^{4}-\frac{69\cdots 36}{33\cdots 17}a^{3}+\frac{94\cdots 32}{12\cdots 63}a^{2}-\frac{15\cdots 93}{43\cdots 21}a+\frac{16\cdots 45}{43\cdots 21}$, $\frac{36\cdots 24}{32\cdots 75}a^{23}+\frac{12\cdots 44}{10\cdots 25}a^{22}-\frac{86\cdots 93}{64\cdots 15}a^{21}-\frac{15\cdots 62}{32\cdots 75}a^{20}+\frac{29\cdots 03}{10\cdots 25}a^{19}-\frac{60\cdots 12}{32\cdots 75}a^{18}+\frac{22\cdots 86}{32\cdots 75}a^{17}+\frac{22\cdots 33}{64\cdots 15}a^{16}-\frac{10\cdots 96}{32\cdots 75}a^{15}-\frac{33\cdots 88}{32\cdots 75}a^{14}-\frac{13\cdots 61}{24\cdots 75}a^{13}-\frac{19\cdots 16}{32\cdots 75}a^{12}+\frac{74\cdots 57}{64\cdots 15}a^{11}+\frac{12\cdots 81}{32\cdots 75}a^{10}+\frac{25\cdots 31}{10\cdots 25}a^{9}+\frac{71\cdots 98}{12\cdots 63}a^{8}+\frac{29\cdots 23}{21\cdots 05}a^{7}+\frac{17\cdots 74}{64\cdots 15}a^{6}-\frac{34\cdots 41}{64\cdots 15}a^{5}+\frac{65\cdots 51}{21\cdots 05}a^{4}+\frac{13\cdots 50}{12\cdots 63}a^{3}-\frac{33\cdots 94}{12\cdots 63}a^{2}+\frac{46\cdots 82}{43\cdots 21}a-\frac{27\cdots 12}{33\cdots 17}$, $\frac{22\cdots 84}{96\cdots 25}a^{23}+\frac{35\cdots 04}{96\cdots 25}a^{22}-\frac{90\cdots 88}{32\cdots 75}a^{21}-\frac{59\cdots 46}{96\cdots 25}a^{20}+\frac{16\cdots 96}{96\cdots 25}a^{19}-\frac{13\cdots 29}{32\cdots 75}a^{18}-\frac{16\cdots 93}{96\cdots 25}a^{17}-\frac{20\cdots 92}{96\cdots 25}a^{16}-\frac{14\cdots 88}{96\cdots 25}a^{15}+\frac{85\cdots 38}{96\cdots 25}a^{14}+\frac{61\cdots 12}{96\cdots 25}a^{13}+\frac{74\cdots 76}{32\cdots 75}a^{12}-\frac{41\cdots 58}{96\cdots 25}a^{11}-\frac{41\cdots 29}{74\cdots 25}a^{10}-\frac{22\cdots 88}{96\cdots 25}a^{9}-\frac{13\cdots 14}{19\cdots 45}a^{8}-\frac{81\cdots 06}{64\cdots 15}a^{7}-\frac{71\cdots 46}{64\cdots 15}a^{6}-\frac{44\cdots 73}{38\cdots 89}a^{5}+\frac{42\cdots 77}{19\cdots 45}a^{4}-\frac{38\cdots 32}{38\cdots 89}a^{3}-\frac{11\cdots 20}{38\cdots 89}a^{2}+\frac{22\cdots 79}{12\cdots 63}a-\frac{12\cdots 76}{43\cdots 21}$, $\frac{22\cdots 16}{96\cdots 25}a^{23}+\frac{20\cdots 08}{96\cdots 25}a^{22}-\frac{69\cdots 01}{12\cdots 63}a^{21}-\frac{97\cdots 53}{96\cdots 25}a^{20}+\frac{43\cdots 81}{96\cdots 25}a^{19}+\frac{36\cdots 24}{32\cdots 75}a^{18}+\frac{14\cdots 94}{96\cdots 25}a^{17}+\frac{32\cdots 27}{38\cdots 89}a^{16}-\frac{45\cdots 48}{74\cdots 25}a^{15}-\frac{27\cdots 17}{96\cdots 25}a^{14}-\frac{13\cdots 72}{96\cdots 25}a^{13}-\frac{34\cdots 83}{32\cdots 75}a^{12}+\frac{35\cdots 21}{14\cdots 65}a^{11}+\frac{10\cdots 64}{96\cdots 25}a^{10}+\frac{55\cdots 12}{96\cdots 25}a^{9}+\frac{30\cdots 07}{19\cdots 45}a^{8}+\frac{48\cdots 38}{12\cdots 63}a^{7}+\frac{67\cdots 82}{99\cdots 51}a^{6}+\frac{76\cdots 36}{19\cdots 45}a^{5}-\frac{17\cdots 38}{19\cdots 45}a^{4}-\frac{13\cdots 51}{38\cdots 89}a^{3}-\frac{43\cdots 00}{38\cdots 89}a^{2}+\frac{96\cdots 45}{12\cdots 63}a-\frac{58\cdots 18}{43\cdots 21}$, $\frac{48\cdots 33}{96\cdots 25}a^{23}+\frac{14\cdots 39}{96\cdots 25}a^{22}+\frac{27\cdots 66}{12\cdots 63}a^{21}-\frac{11\cdots 38}{74\cdots 25}a^{20}-\frac{67\cdots 42}{96\cdots 25}a^{19}-\frac{89\cdots 48}{32\cdots 75}a^{18}+\frac{25\cdots 97}{96\cdots 25}a^{17}+\frac{11\cdots 19}{38\cdots 89}a^{16}+\frac{78\cdots 23}{96\cdots 25}a^{15}+\frac{18\cdots 94}{96\cdots 25}a^{14}-\frac{23\cdots 26}{96\cdots 25}a^{13}-\frac{17\cdots 48}{10\cdots 25}a^{12}-\frac{11\cdots 81}{19\cdots 45}a^{11}-\frac{23\cdots 03}{96\cdots 25}a^{10}-\frac{46\cdots 59}{96\cdots 25}a^{9}-\frac{21\cdots 11}{19\cdots 45}a^{8}-\frac{22\cdots 43}{21\cdots 05}a^{7}+\frac{11\cdots 97}{43\cdots 21}a^{6}+\frac{15\cdots 98}{19\cdots 45}a^{5}+\frac{16\cdots 51}{19\cdots 45}a^{4}+\frac{57\cdots 82}{38\cdots 89}a^{3}-\frac{41\cdots 84}{38\cdots 89}a^{2}+\frac{75\cdots 66}{12\cdots 63}a-\frac{41\cdots 98}{43\cdots 21}$, $\frac{21\cdots 13}{74\cdots 25}a^{23}+\frac{41\cdots 07}{96\cdots 25}a^{22}-\frac{14\cdots 39}{64\cdots 15}a^{21}-\frac{21\cdots 97}{96\cdots 25}a^{20}+\frac{74\cdots 49}{96\cdots 25}a^{19}-\frac{42\cdots 53}{10\cdots 25}a^{18}+\frac{35\cdots 56}{96\cdots 25}a^{17}-\frac{14\cdots 33}{19\cdots 45}a^{16}-\frac{46\cdots 76}{96\cdots 25}a^{15}-\frac{63\cdots 18}{96\cdots 25}a^{14}-\frac{12\cdots 48}{96\cdots 25}a^{13}+\frac{13\cdots 46}{24\cdots 75}a^{12}+\frac{82\cdots 97}{19\cdots 45}a^{11}+\frac{88\cdots 86}{96\cdots 25}a^{10}+\frac{22\cdots 48}{96\cdots 25}a^{9}+\frac{11\cdots 18}{19\cdots 45}a^{8}+\frac{37\cdots 03}{21\cdots 05}a^{7}+\frac{15\cdots 89}{21\cdots 05}a^{6}+\frac{26\cdots 84}{19\cdots 45}a^{5}+\frac{12\cdots 33}{19\cdots 45}a^{4}-\frac{95\cdots 08}{38\cdots 89}a^{3}+\frac{23\cdots 05}{38\cdots 89}a^{2}-\frac{39\cdots 62}{12\cdots 63}a+\frac{21\cdots 19}{43\cdots 21}$
|
| |
| Regulator: | \( 1278504647863611700000 \) (assuming GRH) |
| |
| Unit signature rank: | \( 3 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 1278504647863611700000 \cdot 4}{2\cdot\sqrt{270761008401829353605241639483649123576469719409942626953125}}\cr\approx \mathstrut & 7.53976676560766 \end{aligned}\] (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 | ${\href{/padicField/2.8.0.1}{8} }^{3}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{5}{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | $24$ | ${\href{/padicField/19.3.0.1}{3} }^{8}$ | $24$ | R | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}{,}\,{\href{/padicField/37.1.0.1}{1} }^{4}$ | $20{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.8.0.1}{8} }^{3}$ | $24$ | $24$ | ${\href{/padicField/59.2.0.1}{2} }^{10}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.1.5.9a1.3 | $x^{5} + 50 x + 5$ | $5$ | $1$ | $9$ | $F_5$ | $$[\frac{9}{4}]_{4}$$ | |
| 5.1.5.9a1.3 | $x^{5} + 50 x + 5$ | $5$ | $1$ | $9$ | $F_5$ | $$[\frac{9}{4}]_{4}$$ | |
| 5.1.5.9a1.3 | $x^{5} + 50 x + 5$ | $5$ | $1$ | $9$ | $F_5$ | $$[\frac{9}{4}]_{4}$$ | |
| 5.1.5.9a1.3 | $x^{5} + 50 x + 5$ | $5$ | $1$ | $9$ | $F_5$ | $$[\frac{9}{4}]_{4}$$ | |
|
\(29\)
| 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 | $$[\ ]_{20}^{2}$$ |