Normalized defining polynomial
\( x^{24} - 4 x^{23} + 6 x^{22} - 4 x^{21} + x^{20} + 2320 x^{19} - 17980 x^{18} + 129920 x^{17} + \cdots + 212661967555 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $(4, 10)$ |
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| Discriminant: |
\(270761008401829353605241639483649123576469719409942626953125\)
\(\medspace = 5^{39}\cdot 29^{22}\)
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| |
| Root discriminant: | \(299.47\) |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $\frac{1}{5}a^{21}-\frac{1}{5}a^{20}$, $\frac{1}{875}a^{22}+\frac{8}{875}a^{21}+\frac{66}{875}a^{20}-\frac{3}{7}a^{19}+\frac{1}{7}a^{18}-\frac{36}{175}a^{17}-\frac{4}{25}a^{16}-\frac{6}{175}a^{15}+\frac{12}{35}a^{14}+\frac{16}{35}a^{13}-\frac{37}{175}a^{12}-\frac{11}{175}a^{11}+\frac{53}{175}a^{10}-\frac{9}{35}a^{9}+\frac{2}{35}a^{8}+\frac{13}{175}a^{7}+\frac{74}{175}a^{6}+\frac{23}{175}a^{5}-\frac{4}{35}a^{4}+\frac{1}{7}a^{3}+\frac{36}{175}a^{2}+\frac{3}{175}a+\frac{76}{175}$, $\frac{1}{65\cdots 25}a^{23}-\frac{24\cdots 98}{65\cdots 25}a^{22}-\frac{45\cdots 82}{65\cdots 25}a^{21}-\frac{22\cdots 46}{65\cdots 25}a^{20}-\frac{35\cdots 89}{10\cdots 25}a^{19}-\frac{17\cdots 86}{13\cdots 25}a^{18}+\frac{14\cdots 13}{13\cdots 25}a^{17}-\frac{24\cdots 38}{13\cdots 25}a^{16}-\frac{70\cdots 54}{13\cdots 25}a^{15}+\frac{13\cdots 37}{37\cdots 75}a^{14}+\frac{20\cdots 83}{13\cdots 25}a^{13}-\frac{51\cdots 14}{13\cdots 25}a^{12}+\frac{39\cdots 42}{18\cdots 75}a^{11}+\frac{59\cdots 37}{13\cdots 25}a^{10}-\frac{11\cdots 67}{37\cdots 75}a^{9}-\frac{52\cdots 47}{13\cdots 25}a^{8}-\frac{21\cdots 29}{13\cdots 25}a^{7}+\frac{50\cdots 54}{13\cdots 25}a^{6}+\frac{33\cdots 92}{13\cdots 25}a^{5}-\frac{75\cdots 21}{26\cdots 25}a^{4}-\frac{55\cdots 14}{13\cdots 25}a^{3}-\frac{56\cdots 88}{13\cdots 25}a^{2}+\frac{62\cdots 83}{13\cdots 25}a+\frac{25\cdots 19}{13\cdots 25}$
| 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: |
$\frac{76\cdots 66}{40\cdots 75}a^{23}-\frac{71\cdots 32}{40\cdots 75}a^{22}+\frac{48\cdots 87}{40\cdots 75}a^{21}-\frac{12\cdots 39}{40\cdots 75}a^{20}+\frac{27\cdots 72}{64\cdots 79}a^{19}-\frac{37\cdots 74}{80\cdots 75}a^{18}-\frac{20\cdots 08}{80\cdots 75}a^{17}+\frac{22\cdots 33}{80\cdots 75}a^{16}-\frac{21\cdots 11}{80\cdots 75}a^{15}+\frac{54\cdots 94}{32\cdots 75}a^{14}-\frac{62\cdots 28}{80\cdots 75}a^{13}+\frac{24\cdots 74}{80\cdots 75}a^{12}-\frac{13\cdots 97}{11\cdots 25}a^{11}+\frac{34\cdots 08}{80\cdots 75}a^{10}-\frac{33\cdots 78}{23\cdots 25}a^{9}+\frac{36\cdots 27}{80\cdots 75}a^{8}-\frac{84\cdots 61}{80\cdots 75}a^{7}+\frac{16\cdots 61}{80\cdots 75}a^{6}-\frac{13\cdots 97}{80\cdots 75}a^{5}-\frac{33\cdots 39}{16\cdots 75}a^{4}+\frac{31\cdots 99}{80\cdots 75}a^{3}+\frac{63\cdots 58}{80\cdots 75}a^{2}-\frac{34\cdots 28}{80\cdots 75}a-\frac{30\cdots 79}{80\cdots 75}$, $\frac{20\cdots 39}{23\cdots 31}a^{23}-\frac{87\cdots 80}{23\cdots 31}a^{22}+\frac{11\cdots 50}{23\cdots 31}a^{21}+\frac{88\cdots 81}{23\cdots 31}a^{20}-\frac{52\cdots 64}{23\cdots 31}a^{19}+\frac{48\cdots 96}{23\cdots 31}a^{18}-\frac{38\cdots 57}{23\cdots 31}a^{17}+\frac{26\cdots 77}{23\cdots 31}a^{16}-\frac{13\cdots 99}{23\cdots 31}a^{15}+\frac{57\cdots 21}{23\cdots 31}a^{14}-\frac{20\cdots 06}{23\cdots 31}a^{13}+\frac{73\cdots 63}{23\cdots 31}a^{12}-\frac{25\cdots 18}{23\cdots 31}a^{11}+\frac{77\cdots 71}{23\cdots 31}a^{10}-\frac{20\cdots 33}{23\cdots 31}a^{9}+\frac{30\cdots 28}{23\cdots 31}a^{8}-\frac{12\cdots 64}{23\cdots 31}a^{7}-\frac{16\cdots 77}{23\cdots 31}a^{6}+\frac{51\cdots 41}{23\cdots 31}a^{5}-\frac{23\cdots 52}{23\cdots 31}a^{4}-\frac{12\cdots 00}{23\cdots 31}a^{3}-\frac{20\cdots 65}{23\cdots 31}a^{2}+\frac{27\cdots 60}{23\cdots 31}a+\frac{37\cdots 79}{23\cdots 31}$, $\frac{59\cdots 95}{23\cdots 31}a^{23}+\frac{48\cdots 22}{23\cdots 31}a^{22}-\frac{14\cdots 28}{23\cdots 31}a^{21}+\frac{26\cdots 02}{23\cdots 31}a^{20}-\frac{35\cdots 52}{23\cdots 31}a^{19}-\frac{13\cdots 24}{23\cdots 31}a^{18}+\frac{16\cdots 31}{23\cdots 31}a^{17}-\frac{12\cdots 20}{23\cdots 31}a^{16}+\frac{75\cdots 53}{23\cdots 31}a^{15}-\frac{36\cdots 67}{23\cdots 31}a^{14}+\frac{14\cdots 17}{23\cdots 31}a^{13}-\frac{56\cdots 60}{23\cdots 31}a^{12}+\frac{20\cdots 92}{23\cdots 31}a^{11}-\frac{71\cdots 02}{23\cdots 31}a^{10}+\frac{22\cdots 93}{23\cdots 31}a^{9}-\frac{58\cdots 88}{23\cdots 31}a^{8}+\frac{12\cdots 72}{23\cdots 31}a^{7}-\frac{19\cdots 99}{23\cdots 31}a^{6}+\frac{15\cdots 67}{23\cdots 31}a^{5}-\frac{18\cdots 39}{23\cdots 31}a^{4}+\frac{56\cdots 75}{23\cdots 31}a^{3}-\frac{88\cdots 25}{23\cdots 31}a^{2}-\frac{12\cdots 60}{23\cdots 31}a-\frac{17\cdots 66}{23\cdots 31}$, $\frac{15\cdots 47}{20\cdots 75}a^{23}+\frac{54\cdots 56}{20\cdots 75}a^{22}+\frac{20\cdots 79}{20\cdots 75}a^{21}-\frac{95\cdots 88}{20\cdots 75}a^{20}+\frac{33\cdots 93}{32\cdots 95}a^{19}-\frac{78\cdots 33}{40\cdots 75}a^{18}+\frac{30\cdots 39}{40\cdots 75}a^{17}-\frac{16\cdots 64}{40\cdots 75}a^{16}+\frac{22\cdots 13}{40\cdots 75}a^{15}+\frac{80\cdots 73}{16\cdots 75}a^{14}-\frac{19\cdots 01}{40\cdots 75}a^{13}+\frac{88\cdots 08}{40\cdots 75}a^{12}-\frac{55\cdots 99}{57\cdots 25}a^{11}+\frac{16\cdots 61}{40\cdots 75}a^{10}-\frac{17\cdots 26}{11\cdots 25}a^{9}+\frac{24\cdots 09}{40\cdots 75}a^{8}-\frac{69\cdots 37}{40\cdots 75}a^{7}+\frac{18\cdots 87}{40\cdots 75}a^{6}-\frac{28\cdots 49}{40\cdots 75}a^{5}+\frac{51\cdots 37}{80\cdots 75}a^{4}+\frac{11\cdots 33}{40\cdots 75}a^{3}+\frac{53\cdots 86}{40\cdots 75}a^{2}-\frac{30\cdots 76}{40\cdots 75}a+\frac{37\cdots 32}{40\cdots 75}$, $\frac{30\cdots 13}{11\cdots 25}a^{23}+\frac{59\cdots 74}{11\cdots 25}a^{22}-\frac{18\cdots 84}{11\cdots 25}a^{21}+\frac{35\cdots 73}{11\cdots 25}a^{20}+\frac{24\cdots 56}{92\cdots 97}a^{19}-\frac{14\cdots 07}{23\cdots 25}a^{18}+\frac{72\cdots 06}{23\cdots 25}a^{17}-\frac{68\cdots 06}{23\cdots 25}a^{16}+\frac{33\cdots 77}{23\cdots 25}a^{15}-\frac{36\cdots 91}{65\cdots 55}a^{14}+\frac{46\cdots 21}{23\cdots 25}a^{13}-\frac{18\cdots 18}{23\cdots 25}a^{12}+\frac{88\cdots 29}{32\cdots 75}a^{11}-\frac{18\cdots 06}{23\cdots 25}a^{10}+\frac{15\cdots 81}{65\cdots 55}a^{9}-\frac{70\cdots 89}{23\cdots 25}a^{8}+\frac{14\cdots 02}{23\cdots 25}a^{7}+\frac{28\cdots 98}{23\cdots 25}a^{6}-\frac{39\cdots 21}{23\cdots 25}a^{5}-\frac{52\cdots 97}{46\cdots 85}a^{4}-\frac{25\cdots 18}{23\cdots 25}a^{3}+\frac{39\cdots 44}{23\cdots 25}a^{2}+\frac{85\cdots 46}{23\cdots 25}a+\frac{60\cdots 78}{23\cdots 25}$, $\frac{10\cdots 33}{40\cdots 75}a^{23}-\frac{18\cdots 59}{40\cdots 75}a^{22}-\frac{20\cdots 06}{40\cdots 75}a^{21}+\frac{22\cdots 32}{40\cdots 75}a^{20}+\frac{48\cdots 65}{64\cdots 79}a^{19}+\frac{47\cdots 12}{80\cdots 75}a^{18}-\frac{37\cdots 21}{80\cdots 75}a^{17}+\frac{11\cdots 96}{80\cdots 75}a^{16}-\frac{35\cdots 07}{80\cdots 75}a^{15}+\frac{60\cdots 78}{32\cdots 75}a^{14}-\frac{52\cdots 86}{80\cdots 75}a^{13}+\frac{17\cdots 88}{80\cdots 75}a^{12}-\frac{82\cdots 64}{11\cdots 25}a^{11}+\frac{71\cdots 96}{80\cdots 75}a^{10}-\frac{15\cdots 36}{23\cdots 25}a^{9}-\frac{14\cdots 26}{80\cdots 75}a^{8}+\frac{19\cdots 18}{80\cdots 75}a^{7}+\frac{52\cdots 57}{80\cdots 75}a^{6}-\frac{56\cdots 14}{80\cdots 75}a^{5}-\frac{64\cdots 43}{16\cdots 75}a^{4}-\frac{11\cdots 62}{80\cdots 75}a^{3}+\frac{77\cdots 96}{80\cdots 75}a^{2}+\frac{12\cdots 39}{80\cdots 75}a+\frac{68\cdots 27}{80\cdots 75}$, $\frac{14\cdots 54}{13\cdots 25}a^{23}+\frac{56\cdots 67}{13\cdots 25}a^{22}-\frac{29\cdots 72}{13\cdots 25}a^{21}+\frac{13\cdots 09}{13\cdots 25}a^{20}-\frac{47\cdots 39}{20\cdots 85}a^{19}-\frac{72\cdots 31}{26\cdots 25}a^{18}+\frac{62\cdots 73}{26\cdots 25}a^{17}-\frac{52\cdots 48}{26\cdots 25}a^{16}+\frac{30\cdots 16}{26\cdots 25}a^{15}-\frac{43\cdots 23}{74\cdots 75}a^{14}+\frac{65\cdots 93}{26\cdots 25}a^{13}-\frac{24\cdots 94}{26\cdots 25}a^{12}+\frac{11\cdots 32}{37\cdots 75}a^{11}-\frac{20\cdots 98}{26\cdots 25}a^{10}+\frac{17\cdots 49}{10\cdots 25}a^{9}-\frac{49\cdots 37}{26\cdots 25}a^{8}+\frac{66\cdots 91}{26\cdots 25}a^{7}+\frac{95\cdots 84}{26\cdots 25}a^{6}+\frac{67\cdots 07}{26\cdots 25}a^{5}-\frac{10\cdots 91}{52\cdots 25}a^{4}-\frac{13\cdots 94}{26\cdots 25}a^{3}+\frac{31\cdots 52}{26\cdots 25}a^{2}+\frac{50\cdots 43}{26\cdots 25}a+\frac{34\cdots 74}{26\cdots 25}$, $\frac{48\cdots 67}{65\cdots 25}a^{23}-\frac{57\cdots 91}{65\cdots 25}a^{22}-\frac{16\cdots 69}{65\cdots 25}a^{21}-\frac{43\cdots 07}{65\cdots 25}a^{20}+\frac{23\cdots 62}{10\cdots 25}a^{19}+\frac{37\cdots 13}{13\cdots 25}a^{18}-\frac{26\cdots 54}{13\cdots 25}a^{17}+\frac{12\cdots 29}{13\cdots 25}a^{16}-\frac{10\cdots 43}{13\cdots 25}a^{15}+\frac{70\cdots 04}{37\cdots 75}a^{14}-\frac{12\cdots 14}{13\cdots 25}a^{13}+\frac{48\cdots 62}{13\cdots 25}a^{12}-\frac{17\cdots 11}{18\cdots 75}a^{11}+\frac{61\cdots 29}{13\cdots 25}a^{10}-\frac{34\cdots 27}{53\cdots 25}a^{9}+\frac{26\cdots 26}{13\cdots 25}a^{8}-\frac{84\cdots 18}{13\cdots 25}a^{7}-\frac{37\cdots 32}{13\cdots 25}a^{6}+\frac{91\cdots 89}{13\cdots 25}a^{5}+\frac{12\cdots 43}{26\cdots 25}a^{4}+\frac{77\cdots 87}{13\cdots 25}a^{3}-\frac{47\cdots 96}{13\cdots 25}a^{2}-\frac{15\cdots 14}{13\cdots 25}a-\frac{11\cdots 52}{13\cdots 25}$, $\frac{69\cdots 88}{65\cdots 25}a^{23}-\frac{36\cdots 49}{65\cdots 25}a^{22}+\frac{90\cdots 34}{65\cdots 25}a^{21}-\frac{14\cdots 73}{65\cdots 25}a^{20}+\frac{31\cdots 43}{10\cdots 25}a^{19}+\frac{31\cdots 57}{13\cdots 25}a^{18}-\frac{29\cdots 06}{13\cdots 25}a^{17}+\frac{21\cdots 56}{13\cdots 25}a^{16}-\frac{12\cdots 52}{13\cdots 25}a^{15}+\frac{16\cdots 81}{37\cdots 75}a^{14}-\frac{22\cdots 71}{13\cdots 25}a^{13}+\frac{85\cdots 93}{13\cdots 25}a^{12}-\frac{44\cdots 79}{18\cdots 75}a^{11}+\frac{10\cdots 06}{13\cdots 25}a^{10}-\frac{90\cdots 46}{37\cdots 75}a^{9}+\frac{75\cdots 39}{13\cdots 25}a^{8}-\frac{15\cdots 77}{13\cdots 25}a^{7}+\frac{20\cdots 77}{13\cdots 25}a^{6}-\frac{17\cdots 29}{13\cdots 25}a^{5}+\frac{67\cdots 52}{26\cdots 25}a^{4}-\frac{10\cdots 57}{13\cdots 25}a^{3}+\frac{31\cdots 06}{13\cdots 25}a^{2}+\frac{10\cdots 79}{13\cdots 25}a+\frac{22\cdots 72}{13\cdots 25}$, $\frac{12\cdots 36}{76\cdots 25}a^{23}-\frac{10\cdots 78}{76\cdots 25}a^{22}+\frac{43\cdots 98}{76\cdots 25}a^{21}-\frac{68\cdots 06}{76\cdots 25}a^{20}-\frac{18\cdots 04}{12\cdots 05}a^{19}+\frac{65\cdots 04}{15\cdots 25}a^{18}-\frac{67\cdots 57}{15\cdots 25}a^{17}+\frac{57\cdots 32}{15\cdots 25}a^{16}-\frac{35\cdots 44}{15\cdots 25}a^{15}+\frac{48\cdots 57}{43\cdots 75}a^{14}-\frac{67\cdots 37}{15\cdots 25}a^{13}+\frac{24\cdots 71}{15\cdots 25}a^{12}-\frac{12\cdots 38}{21\cdots 75}a^{11}+\frac{31\cdots 07}{15\cdots 25}a^{10}-\frac{28\cdots 87}{43\cdots 75}a^{9}+\frac{23\cdots 58}{15\cdots 25}a^{8}-\frac{42\cdots 44}{15\cdots 25}a^{7}+\frac{48\cdots 44}{15\cdots 25}a^{6}-\frac{54\cdots 63}{15\cdots 25}a^{5}+\frac{26\cdots 69}{30\cdots 25}a^{4}-\frac{23\cdots 54}{15\cdots 25}a^{3}+\frac{15\cdots 32}{15\cdots 25}a^{2}+\frac{96\cdots 88}{15\cdots 25}a+\frac{48\cdots 84}{15\cdots 25}$, $\frac{30\cdots 92}{65\cdots 25}a^{23}-\frac{60\cdots 91}{65\cdots 25}a^{22}+\frac{64\cdots 31}{65\cdots 25}a^{21}-\frac{46\cdots 57}{65\cdots 25}a^{20}+\frac{62\cdots 62}{10\cdots 25}a^{19}+\frac{14\cdots 38}{13\cdots 25}a^{18}-\frac{80\cdots 29}{13\cdots 25}a^{17}+\frac{62\cdots 04}{13\cdots 25}a^{16}-\frac{29\cdots 93}{13\cdots 25}a^{15}+\frac{33\cdots 04}{37\cdots 75}a^{14}-\frac{41\cdots 14}{13\cdots 25}a^{13}+\frac{16\cdots 62}{13\cdots 25}a^{12}-\frac{78\cdots 11}{18\cdots 75}a^{11}+\frac{16\cdots 29}{13\cdots 25}a^{10}-\frac{13\cdots 39}{37\cdots 75}a^{9}+\frac{59\cdots 76}{13\cdots 25}a^{8}-\frac{12\cdots 18}{13\cdots 25}a^{7}-\frac{21\cdots 57}{13\cdots 25}a^{6}-\frac{24\cdots 36}{13\cdots 25}a^{5}+\frac{92\cdots 43}{26\cdots 25}a^{4}-\frac{18\cdots 63}{13\cdots 25}a^{3}-\frac{83\cdots 21}{13\cdots 25}a^{2}-\frac{10\cdots 39}{13\cdots 25}a-\frac{62\cdots 52}{13\cdots 25}$, $\frac{28\cdots 17}{13\cdots 25}a^{23}-\frac{80\cdots 16}{13\cdots 25}a^{22}-\frac{21\cdots 19}{13\cdots 25}a^{21}-\frac{11\cdots 57}{13\cdots 25}a^{20}-\frac{53\cdots 28}{20\cdots 85}a^{19}+\frac{12\cdots 38}{26\cdots 25}a^{18}-\frac{85\cdots 04}{26\cdots 25}a^{17}+\frac{58\cdots 54}{26\cdots 25}a^{16}-\frac{29\cdots 18}{26\cdots 25}a^{15}+\frac{44\cdots 22}{10\cdots 25}a^{14}-\frac{40\cdots 89}{26\cdots 25}a^{13}+\frac{15\cdots 12}{26\cdots 25}a^{12}-\frac{76\cdots 11}{37\cdots 75}a^{11}+\frac{15\cdots 54}{26\cdots 25}a^{10}-\frac{12\cdots 64}{74\cdots 75}a^{9}+\frac{54\cdots 76}{26\cdots 25}a^{8}-\frac{69\cdots 43}{26\cdots 25}a^{7}-\frac{43\cdots 82}{26\cdots 25}a^{6}+\frac{68\cdots 89}{26\cdots 25}a^{5}+\frac{51\cdots 18}{52\cdots 25}a^{4}+\frac{27\cdots 62}{26\cdots 25}a^{3}-\frac{21\cdots 46}{26\cdots 25}a^{2}-\frac{58\cdots 39}{26\cdots 25}a-\frac{44\cdots 52}{26\cdots 25}$, $\frac{19\cdots 97}{40\cdots 75}a^{23}-\frac{30\cdots 19}{40\cdots 75}a^{22}-\frac{60\cdots 21}{40\cdots 75}a^{21}+\frac{54\cdots 87}{40\cdots 75}a^{20}+\frac{57\cdots 07}{64\cdots 79}a^{19}-\frac{86\cdots 83}{80\cdots 75}a^{18}+\frac{19\cdots 39}{80\cdots 75}a^{17}-\frac{25\cdots 64}{80\cdots 75}a^{16}+\frac{67\cdots 13}{80\cdots 75}a^{15}-\frac{10\cdots 02}{32\cdots 75}a^{14}+\frac{89\cdots 24}{80\cdots 75}a^{13}-\frac{38\cdots 67}{80\cdots 75}a^{12}+\frac{13\cdots 26}{11\cdots 25}a^{11}-\frac{24\cdots 39}{80\cdots 75}a^{10}+\frac{20\cdots 99}{23\cdots 25}a^{9}+\frac{26\cdots 59}{80\cdots 75}a^{8}+\frac{68\cdots 13}{80\cdots 75}a^{7}+\frac{33\cdots 87}{80\cdots 75}a^{6}+\frac{64\cdots 01}{80\cdots 75}a^{5}-\frac{23\cdots 38}{16\cdots 75}a^{4}-\frac{30\cdots 42}{80\cdots 75}a^{3}-\frac{60\cdots 64}{80\cdots 75}a^{2}-\frac{53\cdots 01}{80\cdots 75}a-\frac{21\cdots 43}{80\cdots 75}$
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| Regulator: | \( 81380503790989600000 \) (assuming GRH) |
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| 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, 12.4.8024353662494678497314453125.2 |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 5.1.20.39a1.3 | $x^{20} + 25 x^{4} + 5$ | $20$ | $1$ | $39$ | 20T5 | $$[\frac{9}{4}]_{4}$$ | |
|
\(29\)
| 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 | $$[\ ]_{20}^{2}$$ |