Normalized defining polynomial
\( x^{24} - 4 x^{23} + 6 x^{22} - 4 x^{21} - 1304 x^{20} + 2465 x^{19} + 2465 x^{18} - 1885 x^{17} + \cdots - 6585670043891 \)
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: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $\frac{1}{29}a^{14}-\frac{4}{29}a^{13}+\frac{6}{29}a^{12}-\frac{4}{29}a^{11}+\frac{1}{29}a^{10}$, $\frac{1}{29}a^{15}-\frac{10}{29}a^{13}-\frac{9}{29}a^{12}+\frac{14}{29}a^{11}+\frac{4}{29}a^{10}$, $\frac{1}{29}a^{16}+\frac{9}{29}a^{13}-\frac{13}{29}a^{12}-\frac{7}{29}a^{11}+\frac{10}{29}a^{10}$, $\frac{1}{29}a^{17}-\frac{6}{29}a^{13}-\frac{3}{29}a^{12}-\frac{12}{29}a^{11}-\frac{9}{29}a^{10}$, $\frac{1}{145}a^{18}+\frac{1}{145}a^{17}+\frac{2}{145}a^{16}-\frac{2}{145}a^{15}+\frac{63}{145}a^{13}+\frac{13}{145}a^{12}-\frac{1}{5}a^{11}+\frac{9}{145}a^{10}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{3}-\frac{2}{5}a^{2}+\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{145}a^{19}+\frac{1}{145}a^{17}+\frac{1}{145}a^{16}+\frac{2}{145}a^{15}-\frac{2}{145}a^{14}-\frac{7}{29}a^{13}-\frac{62}{145}a^{12}-\frac{27}{145}a^{11}-\frac{24}{145}a^{10}+\frac{1}{5}a^{9}+\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}-\frac{2}{5}a^{4}-\frac{2}{5}a^{2}-\frac{2}{5}a+\frac{1}{5}$, $\frac{1}{145}a^{20}+\frac{5}{29}a^{13}+\frac{5}{29}a^{12}+\frac{2}{29}a^{11}+\frac{11}{29}a^{10}+\frac{1}{5}$, $\frac{1}{145}a^{21}-\frac{4}{29}a^{13}+\frac{1}{29}a^{12}+\frac{2}{29}a^{11}-\frac{5}{29}a^{10}+\frac{1}{5}a$, $\frac{1}{725}a^{22}-\frac{2}{725}a^{21}+\frac{1}{725}a^{20}+\frac{2}{145}a^{17}-\frac{2}{145}a^{16}+\frac{2}{145}a^{15}+\frac{2}{145}a^{14}-\frac{31}{145}a^{13}-\frac{3}{29}a^{12}+\frac{16}{145}a^{11}-\frac{3}{145}a^{10}+\frac{1}{5}a^{9}-\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{11}{25}a^{2}-\frac{12}{25}a-\frac{9}{25}$, $\frac{1}{55\cdots 75}a^{23}+\frac{23\cdots 52}{55\cdots 75}a^{22}+\frac{46\cdots 43}{55\cdots 75}a^{21}+\frac{60\cdots 04}{55\cdots 75}a^{20}-\frac{24\cdots 61}{11\cdots 75}a^{19}-\frac{20\cdots 39}{65\cdots 75}a^{18}-\frac{16\cdots 36}{13\cdots 55}a^{17}+\frac{45\cdots 94}{16\cdots 25}a^{16}+\frac{15\cdots 08}{11\cdots 75}a^{15}-\frac{15\cdots 41}{11\cdots 75}a^{14}+\frac{32\cdots 76}{11\cdots 75}a^{13}+\frac{17\cdots 98}{11\cdots 75}a^{12}-\frac{28\cdots 62}{11\cdots 75}a^{11}+\frac{85\cdots 01}{11\cdots 75}a^{10}-\frac{71\cdots 69}{38\cdots 75}a^{9}-\frac{93\cdots 22}{38\cdots 75}a^{8}+\frac{18\cdots 09}{38\cdots 75}a^{7}+\frac{80\cdots 08}{38\cdots 75}a^{6}-\frac{10\cdots 98}{38\cdots 75}a^{5}-\frac{39\cdots 69}{38\cdots 75}a^{4}-\frac{39\cdots 54}{19\cdots 75}a^{3}-\frac{31\cdots 83}{19\cdots 75}a^{2}+\frac{87\cdots 28}{19\cdots 75}a+\frac{20\cdots 77}{11\cdots 75}$
| 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: |
$\frac{88\cdots 05}{36\cdots 09}a^{23}-\frac{79\cdots 51}{62\cdots 05}a^{22}+\frac{68\cdots 94}{18\cdots 45}a^{21}-\frac{28\cdots 69}{36\cdots 09}a^{20}-\frac{55\cdots 83}{18\cdots 45}a^{19}+\frac{19\cdots 47}{18\cdots 45}a^{18}-\frac{39\cdots 36}{18\cdots 45}a^{17}+\frac{87\cdots 71}{18\cdots 45}a^{16}-\frac{53\cdots 24}{36\cdots 09}a^{15}+\frac{25\cdots 01}{18\cdots 45}a^{14}+\frac{60\cdots 01}{18\cdots 45}a^{13}-\frac{14\cdots 33}{18\cdots 45}a^{12}-\frac{73\cdots 03}{62\cdots 05}a^{11}-\frac{38\cdots 05}{12\cdots 21}a^{10}+\frac{50\cdots 07}{62\cdots 05}a^{9}+\frac{70\cdots 87}{62\cdots 05}a^{8}-\frac{30\cdots 31}{62\cdots 05}a^{7}+\frac{23\cdots 11}{62\cdots 05}a^{6}+\frac{95\cdots 16}{12\cdots 21}a^{5}-\frac{10\cdots 04}{62\cdots 05}a^{4}-\frac{40\cdots 79}{62\cdots 05}a^{3}-\frac{47\cdots 77}{62\cdots 05}a^{2}+\frac{19\cdots 42}{62\cdots 05}a+\frac{77\cdots 61}{12\cdots 21}$, $\frac{34\cdots 74}{19\cdots 75}a^{23}+\frac{21\cdots 63}{19\cdots 75}a^{22}-\frac{31\cdots 24}{19\cdots 75}a^{21}+\frac{27\cdots 86}{38\cdots 95}a^{20}+\frac{90\cdots 93}{38\cdots 95}a^{19}-\frac{36\cdots 37}{38\cdots 95}a^{18}-\frac{30\cdots 13}{38\cdots 95}a^{17}+\frac{92\cdots 41}{57\cdots 85}a^{16}+\frac{42\cdots 56}{38\cdots 95}a^{15}-\frac{16\cdots 93}{13\cdots 55}a^{14}-\frac{83\cdots 52}{38\cdots 95}a^{13}+\frac{64\cdots 48}{38\cdots 95}a^{12}+\frac{48\cdots 62}{38\cdots 95}a^{11}-\frac{14\cdots 86}{13\cdots 55}a^{10}-\frac{18\cdots 29}{13\cdots 55}a^{9}-\frac{22\cdots 33}{13\cdots 55}a^{8}+\frac{20\cdots 45}{26\cdots 51}a^{7}-\frac{41\cdots 31}{26\cdots 51}a^{6}-\frac{87\cdots 22}{13\cdots 55}a^{5}+\frac{20\cdots 97}{13\cdots 55}a^{4}+\frac{74\cdots 51}{66\cdots 75}a^{3}+\frac{53\cdots 38}{66\cdots 75}a^{2}-\frac{40\cdots 34}{66\cdots 75}a-\frac{13\cdots 32}{13\cdots 55}$, $\frac{16\cdots 49}{19\cdots 75}a^{23}+\frac{63\cdots 78}{19\cdots 75}a^{22}+\frac{50\cdots 11}{19\cdots 75}a^{21}-\frac{22\cdots 53}{38\cdots 95}a^{20}+\frac{81\cdots 95}{77\cdots 79}a^{19}-\frac{79\cdots 01}{38\cdots 95}a^{18}-\frac{90\cdots 89}{77\cdots 79}a^{17}-\frac{36\cdots 07}{57\cdots 85}a^{16}+\frac{22\cdots 73}{38\cdots 95}a^{15}-\frac{16\cdots 01}{38\cdots 95}a^{14}-\frac{85\cdots 64}{38\cdots 95}a^{13}+\frac{17\cdots 87}{38\cdots 95}a^{12}+\frac{29\cdots 19}{38\cdots 95}a^{11}+\frac{11\cdots 78}{13\cdots 55}a^{10}-\frac{11\cdots 62}{13\cdots 55}a^{9}-\frac{42\cdots 82}{13\cdots 55}a^{8}+\frac{37\cdots 03}{13\cdots 55}a^{7}+\frac{41\cdots 84}{13\cdots 55}a^{6}+\frac{78\cdots 33}{26\cdots 51}a^{5}-\frac{16\cdots 87}{13\cdots 55}a^{4}-\frac{17\cdots 59}{66\cdots 75}a^{3}+\frac{12\cdots 48}{66\cdots 75}a^{2}+\frac{37\cdots 36}{66\cdots 75}a-\frac{43\cdots 08}{26\cdots 51}$, $\frac{17\cdots 63}{19\cdots 75}a^{23}-\frac{64\cdots 31}{19\cdots 75}a^{22}-\frac{12\cdots 52}{19\cdots 75}a^{21}+\frac{65\cdots 48}{77\cdots 79}a^{20}-\frac{58\cdots 52}{38\cdots 95}a^{19}+\frac{93\cdots 93}{38\cdots 95}a^{18}+\frac{68\cdots 39}{38\cdots 95}a^{17}-\frac{54\cdots 36}{57\cdots 85}a^{16}-\frac{12\cdots 62}{38\cdots 95}a^{15}+\frac{18\cdots 76}{38\cdots 95}a^{14}+\frac{94\cdots 86}{38\cdots 95}a^{13}-\frac{35\cdots 32}{38\cdots 95}a^{12}-\frac{19\cdots 78}{38\cdots 95}a^{11}+\frac{56\cdots 82}{13\cdots 55}a^{10}+\frac{76\cdots 77}{13\cdots 55}a^{9}-\frac{47\cdots 37}{26\cdots 51}a^{8}-\frac{34\cdots 02}{13\cdots 55}a^{7}+\frac{10\cdots 27}{13\cdots 55}a^{6}+\frac{50\cdots 69}{13\cdots 55}a^{5}-\frac{74\cdots 17}{13\cdots 55}a^{4}-\frac{25\cdots 77}{66\cdots 75}a^{3}+\frac{93\cdots 89}{66\cdots 75}a^{2}+\frac{64\cdots 23}{66\cdots 75}a-\frac{44\cdots 69}{13\cdots 55}$, $\frac{14\cdots 38}{18\cdots 45}a^{23}-\frac{84\cdots 19}{18\cdots 45}a^{22}+\frac{45\cdots 49}{36\cdots 09}a^{21}-\frac{42\cdots 27}{18\cdots 45}a^{20}-\frac{37\cdots 81}{36\cdots 09}a^{19}+\frac{69\cdots 27}{18\cdots 45}a^{18}-\frac{73\cdots 58}{18\cdots 45}a^{17}+\frac{10\cdots 84}{18\cdots 45}a^{16}-\frac{91\cdots 44}{18\cdots 45}a^{15}+\frac{19\cdots 42}{36\cdots 09}a^{14}+\frac{14\cdots 46}{18\cdots 45}a^{13}-\frac{66\cdots 09}{18\cdots 45}a^{12}-\frac{27\cdots 27}{62\cdots 05}a^{11}+\frac{14\cdots 97}{62\cdots 05}a^{10}+\frac{35\cdots 92}{12\cdots 21}a^{9}+\frac{23\cdots 87}{62\cdots 05}a^{8}-\frac{11\cdots 93}{62\cdots 05}a^{7}+\frac{19\cdots 34}{62\cdots 05}a^{6}+\frac{15\cdots 31}{62\cdots 05}a^{5}-\frac{41\cdots 87}{12\cdots 21}a^{4}-\frac{12\cdots 21}{62\cdots 05}a^{3}-\frac{14\cdots 38}{62\cdots 05}a^{2}+\frac{67\cdots 67}{62\cdots 05}a-\frac{17\cdots 39}{62\cdots 05}$, $\frac{15\cdots 21}{19\cdots 75}a^{23}-\frac{86\cdots 77}{19\cdots 75}a^{22}+\frac{74\cdots 86}{19\cdots 75}a^{21}+\frac{50\cdots 04}{77\cdots 79}a^{20}-\frac{53\cdots 96}{38\cdots 95}a^{19}+\frac{16\cdots 27}{38\cdots 95}a^{18}+\frac{38\cdots 07}{38\cdots 95}a^{17}-\frac{63\cdots 82}{57\cdots 85}a^{16}-\frac{10\cdots 44}{77\cdots 79}a^{15}+\frac{71\cdots 74}{13\cdots 55}a^{14}+\frac{43\cdots 94}{77\cdots 79}a^{13}-\frac{34\cdots 52}{38\cdots 95}a^{12}-\frac{82\cdots 36}{38\cdots 95}a^{11}+\frac{87\cdots 22}{13\cdots 55}a^{10}+\frac{27\cdots 07}{13\cdots 55}a^{9}-\frac{62\cdots 09}{13\cdots 55}a^{8}-\frac{10\cdots 70}{26\cdots 51}a^{7}+\frac{95\cdots 59}{13\cdots 55}a^{6}+\frac{11\cdots 67}{13\cdots 55}a^{5}-\frac{15\cdots 67}{26\cdots 51}a^{4}-\frac{77\cdots 69}{66\cdots 75}a^{3}+\frac{87\cdots 73}{66\cdots 75}a^{2}+\frac{20\cdots 11}{66\cdots 75}a-\frac{97\cdots 61}{13\cdots 55}$, $\frac{14\cdots 66}{11\cdots 75}a^{23}+\frac{32\cdots 41}{11\cdots 75}a^{22}-\frac{31\cdots 14}{11\cdots 75}a^{21}-\frac{17\cdots 61}{11\cdots 75}a^{20}+\frac{13\cdots 22}{76\cdots 15}a^{19}-\frac{22\cdots 76}{13\cdots 55}a^{18}-\frac{45\cdots 41}{13\cdots 55}a^{17}-\frac{11\cdots 11}{33\cdots 05}a^{16}+\frac{17\cdots 37}{22\cdots 35}a^{15}-\frac{12\cdots 82}{22\cdots 35}a^{14}-\frac{83\cdots 54}{22\cdots 35}a^{13}-\frac{75\cdots 97}{22\cdots 35}a^{12}+\frac{16\cdots 76}{22\cdots 35}a^{11}+\frac{45\cdots 72}{22\cdots 35}a^{10}-\frac{84\cdots 36}{76\cdots 15}a^{9}-\frac{11\cdots 69}{76\cdots 15}a^{8}-\frac{22\cdots 09}{76\cdots 15}a^{7}-\frac{20\cdots 73}{76\cdots 15}a^{6}-\frac{44\cdots 63}{76\cdots 15}a^{5}-\frac{94\cdots 98}{76\cdots 15}a^{4}+\frac{90\cdots 59}{38\cdots 75}a^{3}+\frac{52\cdots 36}{38\cdots 75}a^{2}+\frac{52\cdots 01}{38\cdots 75}a-\frac{11\cdots 58}{22\cdots 75}$, $\frac{12\cdots 29}{11\cdots 75}a^{23}-\frac{94\cdots 44}{38\cdots 75}a^{22}+\frac{51\cdots 16}{22\cdots 35}a^{21}-\frac{25\cdots 88}{11\cdots 75}a^{20}-\frac{64\cdots 63}{44\cdots 07}a^{19}+\frac{18\cdots 93}{13\cdots 55}a^{18}+\frac{38\cdots 47}{13\cdots 55}a^{17}+\frac{19\cdots 40}{66\cdots 21}a^{16}-\frac{14\cdots 94}{22\cdots 35}a^{15}+\frac{21\cdots 18}{44\cdots 07}a^{14}+\frac{70\cdots 57}{22\cdots 35}a^{13}+\frac{64\cdots 18}{22\cdots 35}a^{12}-\frac{13\cdots 67}{22\cdots 35}a^{11}-\frac{38\cdots 43}{22\cdots 35}a^{10}+\frac{70\cdots 39}{76\cdots 15}a^{9}+\frac{92\cdots 78}{76\cdots 15}a^{8}+\frac{17\cdots 29}{76\cdots 15}a^{7}+\frac{17\cdots 78}{76\cdots 15}a^{6}+\frac{74\cdots 39}{15\cdots 83}a^{5}+\frac{15\cdots 08}{15\cdots 83}a^{4}-\frac{76\cdots 31}{38\cdots 75}a^{3}-\frac{44\cdots 51}{38\cdots 75}a^{2}-\frac{88\cdots 73}{76\cdots 15}a+\frac{93\cdots 31}{22\cdots 75}$, $\frac{12\cdots 73}{55\cdots 75}a^{23}-\frac{44\cdots 59}{55\cdots 75}a^{22}-\frac{87\cdots 51}{55\cdots 75}a^{21}-\frac{59\cdots 13}{55\cdots 75}a^{20}-\frac{37\cdots 78}{11\cdots 75}a^{19}+\frac{13\cdots 43}{65\cdots 75}a^{18}+\frac{40\cdots 29}{13\cdots 55}a^{17}+\frac{26\cdots 72}{16\cdots 25}a^{16}-\frac{57\cdots 86}{11\cdots 75}a^{15}+\frac{17\cdots 72}{11\cdots 75}a^{14}+\frac{90\cdots 68}{11\cdots 75}a^{13}+\frac{13\cdots 19}{11\cdots 75}a^{12}-\frac{22\cdots 26}{11\cdots 75}a^{11}-\frac{79\cdots 42}{11\cdots 75}a^{10}-\frac{30\cdots 42}{38\cdots 75}a^{9}+\frac{29\cdots 59}{38\cdots 75}a^{8}-\frac{46\cdots 43}{38\cdots 75}a^{7}+\frac{41\cdots 89}{38\cdots 75}a^{6}+\frac{62\cdots 11}{38\cdots 75}a^{5}+\frac{17\cdots 93}{38\cdots 75}a^{4}+\frac{30\cdots 33}{19\cdots 75}a^{3}-\frac{32\cdots 14}{19\cdots 75}a^{2}-\frac{40\cdots 96}{19\cdots 75}a+\frac{83\cdots 56}{11\cdots 75}$, $\frac{88\cdots 29}{22\cdots 35}a^{23}+\frac{10\cdots 58}{11\cdots 75}a^{22}-\frac{95\cdots 46}{11\cdots 75}a^{21}-\frac{58\cdots 02}{11\cdots 75}a^{20}+\frac{11\cdots 53}{22\cdots 35}a^{19}-\frac{15\cdots 98}{13\cdots 55}a^{18}-\frac{14\cdots 06}{13\cdots 55}a^{17}-\frac{15\cdots 21}{66\cdots 21}a^{16}+\frac{51\cdots 64}{22\cdots 35}a^{15}-\frac{79\cdots 07}{44\cdots 07}a^{14}-\frac{24\cdots 21}{22\cdots 35}a^{13}-\frac{19\cdots 49}{22\cdots 35}a^{12}+\frac{48\cdots 91}{22\cdots 35}a^{11}+\frac{26\cdots 60}{44\cdots 07}a^{10}-\frac{29\cdots 54}{76\cdots 15}a^{9}-\frac{32\cdots 72}{76\cdots 15}a^{8}-\frac{11\cdots 49}{15\cdots 83}a^{7}-\frac{68\cdots 96}{76\cdots 15}a^{6}-\frac{13\cdots 12}{76\cdots 15}a^{5}-\frac{27\cdots 89}{76\cdots 15}a^{4}+\frac{56\cdots 02}{76\cdots 15}a^{3}+\frac{15\cdots 43}{38\cdots 75}a^{2}+\frac{15\cdots 64}{38\cdots 75}a-\frac{32\cdots 11}{22\cdots 75}$, $\frac{15\cdots 04}{38\cdots 75}a^{23}-\frac{43\cdots 77}{11\cdots 75}a^{22}+\frac{19\cdots 61}{11\cdots 75}a^{21}-\frac{97\cdots 42}{22\cdots 35}a^{20}-\frac{11\cdots 53}{22\cdots 35}a^{19}+\frac{57\cdots 01}{13\cdots 55}a^{18}-\frac{38\cdots 67}{26\cdots 71}a^{17}+\frac{59\cdots 77}{33\cdots 05}a^{16}-\frac{38\cdots 04}{22\cdots 35}a^{15}+\frac{13\cdots 86}{44\cdots 07}a^{14}-\frac{29\cdots 16}{44\cdots 07}a^{13}-\frac{51\cdots 83}{22\cdots 35}a^{12}-\frac{13\cdots 57}{22\cdots 35}a^{11}+\frac{12\cdots 46}{22\cdots 35}a^{10}-\frac{30\cdots 66}{15\cdots 83}a^{9}-\frac{23\cdots 34}{76\cdots 15}a^{8}+\frac{28\cdots 03}{76\cdots 15}a^{7}+\frac{23\cdots 82}{76\cdots 15}a^{6}-\frac{13\cdots 52}{76\cdots 15}a^{5}-\frac{28\cdots 91}{76\cdots 15}a^{4}+\frac{13\cdots 76}{38\cdots 75}a^{3}+\frac{42\cdots 68}{38\cdots 75}a^{2}-\frac{59\cdots 94}{38\cdots 75}a+\frac{30\cdots 91}{89\cdots 99}$, $\frac{15\cdots 57}{55\cdots 75}a^{23}-\frac{15\cdots 81}{55\cdots 75}a^{22}+\frac{44\cdots 41}{55\cdots 75}a^{21}+\frac{80\cdots 33}{55\cdots 75}a^{20}-\frac{39\cdots 22}{11\cdots 75}a^{19}-\frac{24\cdots 33}{65\cdots 75}a^{18}-\frac{11\cdots 04}{26\cdots 71}a^{17}-\frac{33\cdots 77}{16\cdots 25}a^{16}-\frac{18\cdots 59}{11\cdots 75}a^{15}+\frac{11\cdots 88}{11\cdots 75}a^{14}+\frac{95\cdots 42}{11\cdots 75}a^{13}+\frac{20\cdots 16}{11\cdots 75}a^{12}-\frac{11\cdots 59}{11\cdots 75}a^{11}-\frac{52\cdots 33}{11\cdots 75}a^{10}-\frac{16\cdots 73}{38\cdots 75}a^{9}+\frac{48\cdots 41}{38\cdots 75}a^{8}-\frac{33\cdots 97}{38\cdots 75}a^{7}+\frac{83\cdots 26}{38\cdots 75}a^{6}+\frac{44\cdots 14}{38\cdots 75}a^{5}+\frac{14\cdots 77}{38\cdots 75}a^{4}+\frac{39\cdots 97}{19\cdots 75}a^{3}-\frac{25\cdots 76}{19\cdots 75}a^{2}-\frac{34\cdots 14}{19\cdots 75}a+\frac{67\cdots 04}{11\cdots 75}$, $\frac{25\cdots 24}{19\cdots 75}a^{23}-\frac{25\cdots 78}{19\cdots 75}a^{22}+\frac{11\cdots 64}{19\cdots 75}a^{21}-\frac{56\cdots 42}{38\cdots 95}a^{20}-\frac{65\cdots 92}{38\cdots 95}a^{19}+\frac{56\cdots 74}{38\cdots 95}a^{18}-\frac{18\cdots 39}{38\cdots 95}a^{17}+\frac{33\cdots 89}{57\cdots 85}a^{16}-\frac{21\cdots 83}{38\cdots 95}a^{15}+\frac{79\cdots 91}{77\cdots 79}a^{14}-\frac{87\cdots 72}{38\cdots 95}a^{13}-\frac{29\cdots 49}{38\cdots 95}a^{12}-\frac{68\cdots 72}{38\cdots 95}a^{11}+\frac{25\cdots 22}{13\cdots 55}a^{10}-\frac{22\cdots 64}{26\cdots 51}a^{9}-\frac{27\cdots 85}{26\cdots 51}a^{8}+\frac{17\cdots 13}{13\cdots 55}a^{7}+\frac{26\cdots 86}{26\cdots 51}a^{6}-\frac{19\cdots 51}{26\cdots 51}a^{5}-\frac{16\cdots 79}{13\cdots 55}a^{4}+\frac{88\cdots 79}{66\cdots 75}a^{3}+\frac{24\cdots 67}{66\cdots 75}a^{2}-\frac{35\cdots 76}{66\cdots 75}a+\frac{31\cdots 99}{26\cdots 51}$
|
| |
| Regulator: | \( 81380503790989600000 \) (assuming GRH) |
| |
| Unit signature rank: | \( 3 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 81380503790989600000 \cdot 8}{2\cdot\sqrt{270761008401829353605241639483649123576469719409942626953125}}\cr\approx \mathstrut & 0.959855748474632 \end{aligned}\] (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 | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{4}$ | ${\href{/padicField/11.12.0.1}{12} }^{2}$ | $24$ | ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }^{4}$ | $24$ | R | $20{,}\,{\href{/padicField/31.4.0.1}{4} }$ | ${\href{/padicField/37.8.0.1}{8} }^{3}$ | $20{,}\,{\href{/padicField/41.4.0.1}{4} }$ | $24$ | ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.1.0.1}{1} }^{4}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.12.0.1}{12} }^{2}$ |
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 | $x^{20} + 116$ | $20$ | $1$ | $19$ | 20T6 | $$[\ ]_{20}^{2}$$ |