Normalized defining polynomial
\( x^{24} - 4 x^{23} + 6 x^{22} - 4 x^{21} + 3116 x^{20} + 6764 x^{19} - 132076 x^{18} + \cdots + 34\!\cdots\!81 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $(4, 10)$ |
| |
| Discriminant: |
\(2295251366305746200663037028252313182580209076404571533203125\)
\(\medspace = 5^{25}\cdot 89^{22}\)
|
| |
| Root discriminant: | \(327.37\) |
| |
| Galois root discriminant: | $5^{23/20}89^{19/20}\approx 452.62236684313837$ | ||
| Ramified primes: |
\(5\), \(89\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{5}a^{9}-\frac{2}{5}a^{7}+\frac{1}{5}a^{5}+\frac{1}{5}a^{4}-\frac{2}{5}a^{2}+\frac{1}{5}$, $\frac{1}{5}a^{10}-\frac{2}{5}a^{8}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{2}{5}a^{3}+\frac{1}{5}a$, $\frac{1}{5}a^{11}+\frac{2}{5}a^{7}+\frac{1}{5}a^{6}+\frac{2}{5}a^{5}+\frac{2}{5}a^{2}+\frac{2}{5}$, $\frac{1}{5}a^{12}+\frac{2}{5}a^{8}+\frac{1}{5}a^{7}+\frac{2}{5}a^{6}+\frac{2}{5}a^{3}+\frac{2}{5}a$, $\frac{1}{5}a^{13}+\frac{1}{5}a^{8}+\frac{1}{5}a^{7}-\frac{2}{5}a^{5}+\frac{1}{5}a^{2}-\frac{2}{5}$, $\frac{1}{5}a^{14}+\frac{1}{5}a^{8}+\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{15}+\frac{2}{5}a^{8}-\frac{1}{5}a^{6}-\frac{2}{5}a^{5}+\frac{2}{5}a^{3}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{5}a^{16}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{2}-\frac{1}{5}a-\frac{2}{5}$, $\frac{1}{5}a^{17}-\frac{2}{5}a^{8}-\frac{2}{5}a^{7}-\frac{2}{5}a^{6}-\frac{2}{5}a^{3}-\frac{1}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{25}a^{18}+\frac{1}{25}a^{16}+\frac{1}{25}a^{14}+\frac{2}{25}a^{13}+\frac{1}{25}a^{12}+\frac{2}{25}a^{11}+\frac{1}{25}a^{10}+\frac{2}{25}a^{9}+\frac{6}{25}a^{8}-\frac{8}{25}a^{7}-\frac{9}{25}a^{6}+\frac{7}{25}a^{5}+\frac{1}{25}a^{4}+\frac{1}{5}a^{3}-\frac{9}{25}a^{2}-\frac{2}{5}a+\frac{6}{25}$, $\frac{1}{25}a^{19}+\frac{1}{25}a^{17}+\frac{1}{25}a^{15}+\frac{2}{25}a^{14}+\frac{1}{25}a^{13}+\frac{2}{25}a^{12}+\frac{1}{25}a^{11}+\frac{2}{25}a^{10}+\frac{1}{25}a^{9}-\frac{8}{25}a^{8}+\frac{1}{25}a^{7}+\frac{7}{25}a^{6}-\frac{4}{25}a^{5}-\frac{9}{25}a^{3}+\frac{6}{25}a-\frac{1}{5}$, $\frac{1}{25}a^{20}+\frac{2}{25}a^{15}-\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}+\frac{3}{25}a^{5}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{1}{5}a+\frac{4}{25}$, $\frac{1}{25}a^{21}+\frac{2}{25}a^{16}-\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{3}{25}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{3}-\frac{1}{5}a^{2}+\frac{4}{25}a+\frac{1}{5}$, $\frac{1}{25}a^{22}+\frac{2}{25}a^{17}-\frac{1}{5}a^{8}-\frac{7}{25}a^{7}+\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{3}-\frac{6}{25}a^{2}+\frac{1}{5}a+\frac{1}{5}$, $\frac{1}{48\cdots 25}a^{23}-\frac{65\cdots 52}{48\cdots 25}a^{22}-\frac{66\cdots 26}{48\cdots 25}a^{21}+\frac{13\cdots 87}{69\cdots 75}a^{20}-\frac{69\cdots 54}{48\cdots 25}a^{19}-\frac{17\cdots 96}{96\cdots 45}a^{18}-\frac{67\cdots 38}{48\cdots 25}a^{17}+\frac{21\cdots 76}{48\cdots 25}a^{16}+\frac{73\cdots 29}{48\cdots 25}a^{15}+\frac{58\cdots 76}{96\cdots 45}a^{14}-\frac{76\cdots 36}{10\cdots 75}a^{13}-\frac{55\cdots 13}{96\cdots 45}a^{12}-\frac{32\cdots 93}{48\cdots 25}a^{11}-\frac{11\cdots 46}{96\cdots 45}a^{10}-\frac{20\cdots 13}{48\cdots 25}a^{9}+\frac{86\cdots 43}{48\cdots 25}a^{8}-\frac{22\cdots 34}{48\cdots 25}a^{7}-\frac{14\cdots 83}{48\cdots 25}a^{6}+\frac{64\cdots 29}{48\cdots 25}a^{5}-\frac{20\cdots 72}{48\cdots 25}a^{4}+\frac{45\cdots 96}{96\cdots 45}a^{3}-\frac{60\cdots 13}{13\cdots 35}a^{2}+\frac{18\cdots 62}{48\cdots 25}a+\frac{22\cdots 79}{48\cdots 25}$
| 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: | $C_{2}\times C_{4}$, which has order $8$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{52\cdots 93}{37\cdots 75}a^{23}-\frac{10\cdots 41}{37\cdots 75}a^{22}+\frac{42\cdots 69}{37\cdots 75}a^{21}-\frac{27\cdots 87}{37\cdots 75}a^{20}+\frac{33\cdots 69}{74\cdots 55}a^{19}+\frac{68\cdots 41}{37\cdots 75}a^{18}-\frac{46\cdots 57}{37\cdots 75}a^{17}+\frac{39\cdots 08}{37\cdots 75}a^{16}+\frac{11\cdots 56}{37\cdots 75}a^{15}+\frac{38\cdots 46}{14\cdots 71}a^{14}-\frac{11\cdots 78}{16\cdots 85}a^{13}+\frac{15\cdots 68}{74\cdots 55}a^{12}+\frac{58\cdots 86}{74\cdots 55}a^{11}-\frac{40\cdots 17}{14\cdots 71}a^{10}+\frac{26\cdots 88}{74\cdots 55}a^{9}+\frac{11\cdots 09}{37\cdots 75}a^{8}-\frac{10\cdots 13}{37\cdots 75}a^{7}+\frac{13\cdots 37}{37\cdots 75}a^{6}-\frac{36\cdots 91}{37\cdots 75}a^{5}-\frac{20\cdots 98}{74\cdots 55}a^{4}+\frac{35\cdots 02}{37\cdots 75}a^{3}+\frac{22\cdots 81}{37\cdots 75}a^{2}-\frac{20\cdots 29}{37\cdots 75}a-\frac{31\cdots 48}{37\cdots 75}$, $\frac{10\cdots 86}{91\cdots 55}a^{23}+\frac{40\cdots 58}{45\cdots 75}a^{22}-\frac{11\cdots 31}{45\cdots 75}a^{21}+\frac{26\cdots 31}{45\cdots 75}a^{20}-\frac{32\cdots 37}{91\cdots 55}a^{19}+\frac{57\cdots 72}{91\cdots 55}a^{18}+\frac{80\cdots 36}{45\cdots 75}a^{17}-\frac{67\cdots 37}{45\cdots 75}a^{16}+\frac{12\cdots 07}{45\cdots 75}a^{15}-\frac{14\cdots 33}{18\cdots 91}a^{14}+\frac{15\cdots 94}{91\cdots 55}a^{13}-\frac{51\cdots 78}{91\cdots 55}a^{12}-\frac{48\cdots 31}{91\cdots 55}a^{11}+\frac{52\cdots 71}{91\cdots 55}a^{10}-\frac{15\cdots 19}{91\cdots 55}a^{9}-\frac{75\cdots 61}{91\cdots 55}a^{8}+\frac{17\cdots 09}{45\cdots 75}a^{7}-\frac{76\cdots 58}{45\cdots 75}a^{6}+\frac{12\cdots 68}{45\cdots 75}a^{5}-\frac{14\cdots 61}{91\cdots 55}a^{4}-\frac{20\cdots 82}{91\cdots 55}a^{3}+\frac{20\cdots 52}{45\cdots 75}a^{2}+\frac{39\cdots 16}{45\cdots 75}a-\frac{16\cdots 36}{45\cdots 75}$, $\frac{11\cdots 24}{37\cdots 75}a^{23}+\frac{94\cdots 94}{37\cdots 75}a^{22}+\frac{68\cdots 23}{37\cdots 75}a^{21}+\frac{77\cdots 58}{37\cdots 75}a^{20}-\frac{36\cdots 36}{37\cdots 75}a^{19}-\frac{79\cdots 33}{14\cdots 71}a^{18}+\frac{10\cdots 47}{37\cdots 75}a^{17}-\frac{57\cdots 01}{37\cdots 75}a^{16}-\frac{75\cdots 57}{74\cdots 55}a^{15}-\frac{21\cdots 59}{37\cdots 75}a^{14}+\frac{28\cdots 52}{16\cdots 85}a^{13}+\frac{13\cdots 31}{37\cdots 75}a^{12}-\frac{13\cdots 71}{74\cdots 55}a^{11}+\frac{14\cdots 26}{37\cdots 75}a^{10}+\frac{81\cdots 71}{74\cdots 55}a^{9}-\frac{27\cdots 56}{37\cdots 75}a^{8}+\frac{18\cdots 47}{37\cdots 75}a^{7}+\frac{21\cdots 01}{74\cdots 55}a^{6}-\frac{20\cdots 01}{37\cdots 75}a^{5}+\frac{12\cdots 53}{37\cdots 75}a^{4}-\frac{14\cdots 37}{37\cdots 75}a^{3}-\frac{40\cdots 56}{37\cdots 75}a^{2}+\frac{39\cdots 36}{37\cdots 75}a+\frac{20\cdots 71}{74\cdots 55}$, $\frac{40\cdots 76}{45\cdots 75}a^{23}-\frac{88\cdots 81}{45\cdots 75}a^{22}+\frac{20\cdots 74}{45\cdots 75}a^{21}-\frac{69\cdots 26}{45\cdots 75}a^{20}+\frac{12\cdots 46}{45\cdots 75}a^{19}+\frac{49\cdots 68}{45\cdots 75}a^{18}-\frac{40\cdots 06}{45\cdots 75}a^{17}+\frac{26\cdots 09}{45\cdots 75}a^{16}+\frac{81\cdots 09}{45\cdots 75}a^{15}+\frac{67\cdots 93}{45\cdots 75}a^{14}-\frac{28\cdots 47}{45\cdots 75}a^{13}+\frac{12\cdots 73}{45\cdots 75}a^{12}+\frac{21\cdots 33}{45\cdots 75}a^{11}-\frac{80\cdots 22}{45\cdots 75}a^{10}+\frac{31\cdots 23}{45\cdots 75}a^{9}+\frac{71\cdots 46}{45\cdots 75}a^{8}-\frac{15\cdots 93}{91\cdots 55}a^{7}+\frac{19\cdots 41}{91\cdots 55}a^{6}-\frac{50\cdots 52}{91\cdots 55}a^{5}-\frac{59\cdots 24}{45\cdots 75}a^{4}+\frac{10\cdots 77}{91\cdots 55}a^{3}+\frac{62\cdots 87}{45\cdots 75}a^{2}-\frac{73\cdots 93}{45\cdots 75}a+\frac{19\cdots 47}{45\cdots 75}$, $\frac{15\cdots 43}{74\cdots 55}a^{23}-\frac{41\cdots 69}{14\cdots 71}a^{22}+\frac{29\cdots 86}{37\cdots 75}a^{21}+\frac{20\cdots 96}{37\cdots 75}a^{20}-\frac{23\cdots 59}{37\cdots 75}a^{19}-\frac{48\cdots 16}{37\cdots 75}a^{18}-\frac{58\cdots 94}{37\cdots 75}a^{17}+\frac{15\cdots 01}{37\cdots 75}a^{16}-\frac{57\cdots 07}{37\cdots 75}a^{15}-\frac{61\cdots 19}{37\cdots 75}a^{14}-\frac{16\cdots 42}{80\cdots 25}a^{13}+\frac{17\cdots 26}{37\cdots 75}a^{12}+\frac{20\cdots 44}{37\cdots 75}a^{11}-\frac{70\cdots 94}{37\cdots 75}a^{10}+\frac{15\cdots 44}{37\cdots 75}a^{9}+\frac{97\cdots 51}{37\cdots 75}a^{8}-\frac{17\cdots 06}{37\cdots 75}a^{7}+\frac{56\cdots 54}{37\cdots 75}a^{6}-\frac{14\cdots 43}{37\cdots 75}a^{5}-\frac{12\cdots 96}{37\cdots 75}a^{4}+\frac{18\cdots 71}{37\cdots 75}a^{3}+\frac{74\cdots 74}{37\cdots 75}a^{2}-\frac{38\cdots 70}{14\cdots 71}a-\frac{23\cdots 27}{37\cdots 75}$, $\frac{29\cdots 72}{37\cdots 75}a^{23}+\frac{24\cdots 59}{74\cdots 55}a^{22}-\frac{80\cdots 88}{37\cdots 75}a^{21}-\frac{80\cdots 41}{37\cdots 75}a^{20}-\frac{76\cdots 01}{37\cdots 75}a^{19}-\frac{25\cdots 72}{37\cdots 75}a^{18}+\frac{44\cdots 69}{37\cdots 75}a^{17}-\frac{28\cdots 14}{37\cdots 75}a^{16}-\frac{77\cdots 68}{37\cdots 75}a^{15}-\frac{51\cdots 63}{74\cdots 55}a^{14}+\frac{16\cdots 11}{80\cdots 25}a^{13}+\frac{89\cdots 32}{74\cdots 55}a^{12}-\frac{20\cdots 17}{37\cdots 75}a^{11}+\frac{53\cdots 68}{14\cdots 71}a^{10}-\frac{21\cdots 22}{37\cdots 75}a^{9}-\frac{13\cdots 06}{37\cdots 75}a^{8}+\frac{14\cdots 53}{37\cdots 75}a^{7}-\frac{55\cdots 44}{37\cdots 75}a^{6}+\frac{22\cdots 46}{74\cdots 55}a^{5}-\frac{79\cdots 33}{37\cdots 75}a^{4}-\frac{54\cdots 74}{37\cdots 75}a^{3}+\frac{16\cdots 92}{37\cdots 75}a^{2}-\frac{45\cdots 78}{37\cdots 75}a-\frac{28\cdots 47}{37\cdots 75}$, $\frac{10\cdots 33}{48\cdots 25}a^{23}+\frac{76\cdots 38}{48\cdots 25}a^{22}-\frac{36\cdots 67}{48\cdots 25}a^{21}-\frac{10\cdots 74}{69\cdots 75}a^{20}-\frac{64\cdots 11}{96\cdots 45}a^{19}-\frac{17\cdots 28}{48\cdots 25}a^{18}+\frac{79\cdots 31}{48\cdots 25}a^{17}-\frac{62\cdots 61}{48\cdots 25}a^{16}-\frac{33\cdots 06}{48\cdots 25}a^{15}-\frac{20\cdots 37}{48\cdots 25}a^{14}+\frac{72\cdots 52}{10\cdots 75}a^{13}-\frac{39\cdots 07}{48\cdots 25}a^{12}-\frac{59\cdots 84}{48\cdots 25}a^{11}+\frac{12\cdots 43}{48\cdots 25}a^{10}+\frac{11\cdots 16}{48\cdots 25}a^{9}-\frac{28\cdots 51}{48\cdots 25}a^{8}+\frac{36\cdots 53}{96\cdots 45}a^{7}+\frac{64\cdots 52}{48\cdots 25}a^{6}+\frac{84\cdots 62}{48\cdots 25}a^{5}+\frac{35\cdots 23}{48\cdots 25}a^{4}-\frac{39\cdots 47}{48\cdots 25}a^{3}-\frac{51\cdots 46}{13\cdots 35}a^{2}+\frac{55\cdots 42}{48\cdots 25}a+\frac{11\cdots 56}{48\cdots 25}$, $\frac{45\cdots 76}{96\cdots 45}a^{23}+\frac{18\cdots 26}{48\cdots 25}a^{22}-\frac{50\cdots 46}{48\cdots 25}a^{21}+\frac{17\cdots 61}{69\cdots 75}a^{20}-\frac{14\cdots 04}{96\cdots 45}a^{19}+\frac{13\cdots 98}{48\cdots 25}a^{18}+\frac{36\cdots 12}{48\cdots 25}a^{17}-\frac{29\cdots 94}{48\cdots 25}a^{16}+\frac{56\cdots 59}{48\cdots 25}a^{15}-\frac{13\cdots 87}{48\cdots 25}a^{14}+\frac{75\cdots 97}{10\cdots 75}a^{13}-\frac{11\cdots 57}{48\cdots 25}a^{12}-\frac{10\cdots 54}{48\cdots 25}a^{11}+\frac{11\cdots 73}{48\cdots 25}a^{10}-\frac{34\cdots 54}{48\cdots 25}a^{9}-\frac{15\cdots 77}{48\cdots 25}a^{8}+\frac{77\cdots 69}{48\cdots 25}a^{7}-\frac{68\cdots 42}{96\cdots 45}a^{6}+\frac{55\cdots 92}{48\cdots 25}a^{5}-\frac{30\cdots 87}{48\cdots 25}a^{4}-\frac{91\cdots 02}{96\cdots 45}a^{3}+\frac{13\cdots 16}{69\cdots 75}a^{2}+\frac{18\cdots 46}{48\cdots 25}a-\frac{73\cdots 49}{48\cdots 25}$, $\frac{29\cdots 23}{48\cdots 25}a^{23}-\frac{87\cdots 66}{48\cdots 25}a^{22}-\frac{62\cdots 33}{96\cdots 45}a^{21}+\frac{15\cdots 89}{69\cdots 75}a^{20}+\frac{94\cdots 03}{48\cdots 25}a^{19}+\frac{55\cdots 92}{96\cdots 45}a^{18}-\frac{48\cdots 24}{48\cdots 25}a^{17}+\frac{17\cdots 64}{48\cdots 25}a^{16}+\frac{12\cdots 39}{48\cdots 25}a^{15}+\frac{33\cdots 40}{19\cdots 29}a^{14}-\frac{87\cdots 63}{10\cdots 75}a^{13}-\frac{10\cdots 39}{19\cdots 29}a^{12}+\frac{21\cdots 06}{48\cdots 25}a^{11}-\frac{14\cdots 72}{96\cdots 45}a^{10}-\frac{27\cdots 84}{48\cdots 25}a^{9}+\frac{18\cdots 84}{48\cdots 25}a^{8}-\frac{36\cdots 52}{48\cdots 25}a^{7}-\frac{31\cdots 33}{96\cdots 45}a^{6}+\frac{26\cdots 06}{19\cdots 29}a^{5}+\frac{88\cdots 69}{48\cdots 25}a^{4}-\frac{82\cdots 92}{96\cdots 45}a^{3}-\frac{77\cdots 56}{13\cdots 35}a^{2}+\frac{12\cdots 33}{48\cdots 25}a+\frac{11\cdots 01}{48\cdots 25}$, $\frac{33\cdots 76}{96\cdots 45}a^{23}+\frac{65\cdots 21}{48\cdots 25}a^{22}-\frac{13\cdots 86}{48\cdots 25}a^{21}+\frac{18\cdots 43}{69\cdots 75}a^{20}-\frac{10\cdots 82}{96\cdots 45}a^{19}-\frac{11\cdots 99}{48\cdots 25}a^{18}+\frac{24\cdots 27}{48\cdots 25}a^{17}-\frac{12\cdots 36}{48\cdots 25}a^{16}-\frac{26\cdots 13}{48\cdots 25}a^{15}-\frac{11\cdots 24}{48\cdots 25}a^{14}+\frac{41\cdots 39}{10\cdots 75}a^{13}-\frac{44\cdots 54}{48\cdots 25}a^{12}-\frac{95\cdots 23}{48\cdots 25}a^{11}+\frac{50\cdots 96}{48\cdots 25}a^{10}-\frac{10\cdots 23}{48\cdots 25}a^{9}-\frac{56\cdots 19}{48\cdots 25}a^{8}+\frac{79\cdots 67}{96\cdots 45}a^{7}-\frac{64\cdots 92}{48\cdots 25}a^{6}-\frac{28\cdots 14}{96\cdots 45}a^{5}+\frac{22\cdots 66}{48\cdots 25}a^{4}-\frac{51\cdots 64}{96\cdots 45}a^{3}-\frac{75\cdots 68}{13\cdots 35}a^{2}+\frac{23\cdots 31}{48\cdots 25}a+\frac{68\cdots 04}{96\cdots 45}$, $\frac{23\cdots 31}{96\cdots 45}a^{23}-\frac{42\cdots 94}{48\cdots 25}a^{22}+\frac{31\cdots 31}{96\cdots 45}a^{21}+\frac{21\cdots 17}{69\cdots 75}a^{20}-\frac{27\cdots 68}{48\cdots 25}a^{19}-\frac{35\cdots 92}{48\cdots 25}a^{18}-\frac{17\cdots 31}{48\cdots 25}a^{17}+\frac{10\cdots 23}{48\cdots 25}a^{16}-\frac{41\cdots 59}{19\cdots 29}a^{15}-\frac{29\cdots 28}{48\cdots 25}a^{14}-\frac{18\cdots 39}{10\cdots 75}a^{13}+\frac{23\cdots 72}{48\cdots 25}a^{12}-\frac{23\cdots 57}{48\cdots 25}a^{11}+\frac{26\cdots 32}{48\cdots 25}a^{10}+\frac{53\cdots 03}{48\cdots 25}a^{9}-\frac{67\cdots 33}{48\cdots 25}a^{8}+\frac{40\cdots 36}{48\cdots 25}a^{7}+\frac{46\cdots 07}{48\cdots 25}a^{6}-\frac{64\cdots 59}{96\cdots 45}a^{5}+\frac{47\cdots 13}{48\cdots 25}a^{4}+\frac{18\cdots 37}{48\cdots 25}a^{3}-\frac{27\cdots 39}{69\cdots 75}a^{2}+\frac{32\cdots 57}{48\cdots 25}a+\frac{14\cdots 54}{48\cdots 25}$, $\frac{16\cdots 41}{48\cdots 25}a^{23}-\frac{18\cdots 21}{48\cdots 25}a^{22}+\frac{91\cdots 13}{48\cdots 25}a^{21}+\frac{41\cdots 13}{69\cdots 75}a^{20}+\frac{10\cdots 34}{96\cdots 45}a^{19}+\frac{31\cdots 07}{48\cdots 25}a^{18}-\frac{95\cdots 47}{48\cdots 25}a^{17}+\frac{10\cdots 71}{48\cdots 25}a^{16}+\frac{61\cdots 02}{48\cdots 25}a^{15}+\frac{15\cdots 62}{19\cdots 29}a^{14}-\frac{28\cdots 23}{20\cdots 15}a^{13}+\frac{43\cdots 54}{96\cdots 45}a^{12}+\frac{20\cdots 07}{96\cdots 45}a^{11}-\frac{46\cdots 69}{19\cdots 29}a^{10}+\frac{65\cdots 18}{96\cdots 45}a^{9}+\frac{53\cdots 73}{48\cdots 25}a^{8}-\frac{22\cdots 23}{48\cdots 25}a^{7}-\frac{36\cdots 41}{48\cdots 25}a^{6}+\frac{87\cdots 98}{48\cdots 25}a^{5}-\frac{97\cdots 18}{96\cdots 45}a^{4}+\frac{59\cdots 09}{48\cdots 25}a^{3}+\frac{43\cdots 58}{69\cdots 75}a^{2}-\frac{65\cdots 93}{48\cdots 25}a-\frac{14\cdots 91}{48\cdots 25}$, $\frac{31\cdots 78}{48\cdots 25}a^{23}-\frac{19\cdots 69}{48\cdots 25}a^{22}+\frac{17\cdots 48}{48\cdots 25}a^{21}+\frac{81\cdots 33}{69\cdots 75}a^{20}+\frac{10\cdots 46}{48\cdots 25}a^{19}+\frac{61\cdots 66}{48\cdots 25}a^{18}-\frac{17\cdots 52}{48\cdots 25}a^{17}+\frac{20\cdots 01}{48\cdots 25}a^{16}+\frac{12\cdots 23}{48\cdots 25}a^{15}+\frac{76\cdots 42}{48\cdots 25}a^{14}-\frac{15\cdots 98}{10\cdots 75}a^{13}+\frac{43\cdots 37}{48\cdots 25}a^{12}+\frac{19\cdots 86}{48\cdots 25}a^{11}-\frac{21\cdots 63}{48\cdots 25}a^{10}+\frac{75\cdots 76}{48\cdots 25}a^{9}+\frac{10\cdots 96}{48\cdots 25}a^{8}-\frac{42\cdots 11}{48\cdots 25}a^{7}-\frac{66\cdots 39}{48\cdots 25}a^{6}+\frac{15\cdots 44}{48\cdots 25}a^{5}-\frac{18\cdots 99}{96\cdots 45}a^{4}+\frac{11\cdots 83}{48\cdots 25}a^{3}+\frac{83\cdots 92}{69\cdots 75}a^{2}-\frac{12\cdots 72}{48\cdots 25}a-\frac{27\cdots 26}{48\cdots 25}$
|
| |
| Regulator: | \( 53912689245734216000 \) (assuming GRH) |
| |
| Unit signature rank: | \( 4 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 53912689245734216000 \cdot 8}{2\cdot\sqrt{2295251366305746200663037028252313182580209076404571533203125}}\cr\approx \mathstrut & 0.218400951392628 \end{aligned}\] (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
| 6.2.196069503125.2, 12.4.1522544918455380058642578125.2 |
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 | $24$ | $24$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }^{4}$ | $24$ | $24$ | ${\href{/padicField/19.3.0.1}{3} }^{8}$ | $24$ | ${\href{/padicField/29.5.0.1}{5} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.12.0.1}{12} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }^{3}$ | ${\href{/padicField/41.12.0.1}{12} }^{2}$ | $24$ | ${\href{/padicField/47.4.0.1}{4} }^{5}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | ${\href{/padicField/59.4.0.1}{4} }^{6}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.2.2.2a1.1 | $x^{4} + 8 x^{3} + 20 x^{2} + 21 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ |
| 5.1.20.23a2.2 | $x^{20} + 20 x^{4} + 10$ | $20$ | $1$ | $23$ | 20T20 | $not computed$ | |
|
\(89\)
| 89.1.4.3a1.1 | $x^{4} + 89$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 89.1.20.19a1.3 | $x^{20} + 801$ | $20$ | $1$ | $19$ | 20T6 | $$[\ ]_{20}^{2}$$ |