Normalized defining polynomial
\( x^{25} - 25 x^{23} - 10 x^{22} + 250 x^{21} + 5 x^{20} - 1250 x^{19} - 50 x^{18} + 5085 x^{17} + \cdots + 1 \)
Invariants
| Degree: | $25$ |
| |
| Signature: | $(5, 10)$ |
| |
| Discriminant: |
\(29802322387695312500000000000000000000\)
\(\medspace = 2^{20}\cdot 5^{45}\)
|
| |
| Root discriminant: | \(31.55\) |
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_5$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $\frac{1}{7}a^{20}-\frac{2}{7}a^{19}+\frac{3}{7}a^{17}+\frac{1}{7}a^{15}+\frac{3}{7}a^{14}-\frac{3}{7}a^{13}-\frac{3}{7}a^{12}+\frac{2}{7}a^{11}+\frac{3}{7}a^{10}+\frac{2}{7}a^{8}+\frac{1}{7}a^{7}+\frac{2}{7}a^{6}+\frac{2}{7}a^{5}-\frac{1}{7}a^{4}-\frac{3}{7}a^{3}+\frac{1}{7}a^{2}-\frac{1}{7}a-\frac{1}{7}$, $\frac{1}{7}a^{21}+\frac{3}{7}a^{19}+\frac{3}{7}a^{18}-\frac{1}{7}a^{17}+\frac{1}{7}a^{16}-\frac{2}{7}a^{15}+\frac{3}{7}a^{14}-\frac{2}{7}a^{13}+\frac{3}{7}a^{12}-\frac{1}{7}a^{10}+\frac{2}{7}a^{9}-\frac{2}{7}a^{8}-\frac{3}{7}a^{7}-\frac{1}{7}a^{6}+\frac{3}{7}a^{5}+\frac{2}{7}a^{4}+\frac{2}{7}a^{3}+\frac{1}{7}a^{2}-\frac{3}{7}a-\frac{2}{7}$, $\frac{1}{7}a^{22}+\frac{2}{7}a^{19}-\frac{1}{7}a^{18}-\frac{1}{7}a^{17}-\frac{2}{7}a^{16}+\frac{3}{7}a^{14}-\frac{2}{7}a^{13}+\frac{2}{7}a^{12}-\frac{2}{7}a^{9}-\frac{2}{7}a^{8}+\frac{3}{7}a^{7}-\frac{3}{7}a^{6}+\frac{3}{7}a^{5}-\frac{2}{7}a^{4}+\frac{3}{7}a^{3}+\frac{1}{7}a^{2}+\frac{1}{7}a+\frac{3}{7}$, $\frac{1}{1057}a^{23}-\frac{33}{1057}a^{22}+\frac{9}{151}a^{21}-\frac{11}{1057}a^{20}+\frac{64}{1057}a^{19}-\frac{458}{1057}a^{18}+\frac{83}{1057}a^{17}-\frac{200}{1057}a^{16}+\frac{25}{1057}a^{15}-\frac{48}{151}a^{14}+\frac{450}{1057}a^{13}+\frac{218}{1057}a^{12}+\frac{373}{1057}a^{11}-\frac{503}{1057}a^{10}-\frac{223}{1057}a^{9}-\frac{132}{1057}a^{8}-\frac{458}{1057}a^{7}-\frac{29}{1057}a^{6}+\frac{272}{1057}a^{5}+\frac{334}{1057}a^{4}-\frac{164}{1057}a^{3}-\frac{297}{1057}a^{2}-\frac{465}{1057}a-\frac{240}{1057}$, $\frac{1}{13\cdots 57}a^{24}-\frac{59\cdots 10}{13\cdots 57}a^{23}-\frac{83\cdots 98}{13\cdots 57}a^{22}-\frac{60\cdots 18}{13\cdots 57}a^{21}+\frac{73\cdots 80}{13\cdots 57}a^{20}+\frac{50\cdots 02}{13\cdots 57}a^{19}+\frac{13\cdots 75}{13\cdots 57}a^{18}+\frac{59\cdots 02}{13\cdots 57}a^{17}-\frac{45\cdots 65}{13\cdots 57}a^{16}-\frac{98\cdots 72}{13\cdots 57}a^{15}-\frac{17\cdots 26}{18\cdots 51}a^{14}-\frac{24\cdots 97}{13\cdots 57}a^{13}-\frac{10\cdots 79}{18\cdots 51}a^{12}+\frac{36\cdots 43}{13\cdots 57}a^{11}-\frac{36\cdots 58}{13\cdots 57}a^{10}+\frac{25\cdots 08}{13\cdots 57}a^{9}+\frac{14\cdots 12}{13\cdots 57}a^{8}-\frac{19\cdots 30}{13\cdots 57}a^{7}-\frac{64\cdots 05}{13\cdots 57}a^{6}+\frac{39\cdots 36}{13\cdots 57}a^{5}-\frac{39\cdots 34}{13\cdots 57}a^{4}-\frac{50\cdots 12}{13\cdots 57}a^{3}+\frac{21\cdots 35}{13\cdots 57}a^{2}+\frac{68\cdots 61}{18\cdots 51}a+\frac{32\cdots 51}{13\cdots 57}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | Trivial group, which has order $1$ (assuming GRH) |
|
Unit group
| Rank: | $14$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a$, $\frac{32\cdots 45}{13\cdots 57}a^{24}+\frac{14\cdots 72}{13\cdots 57}a^{23}-\frac{79\cdots 13}{13\cdots 57}a^{22}-\frac{68\cdots 86}{13\cdots 57}a^{21}+\frac{77\cdots 23}{13\cdots 57}a^{20}+\frac{36\cdots 84}{13\cdots 57}a^{19}-\frac{38\cdots 92}{13\cdots 57}a^{18}-\frac{19\cdots 33}{13\cdots 57}a^{17}+\frac{22\cdots 23}{18\cdots 51}a^{16}+\frac{14\cdots 24}{13\cdots 57}a^{15}-\frac{73\cdots 27}{18\cdots 51}a^{14}+\frac{45\cdots 17}{13\cdots 57}a^{13}+\frac{53\cdots 27}{13\cdots 57}a^{12}-\frac{11\cdots 16}{13\cdots 57}a^{11}+\frac{49\cdots 54}{13\cdots 57}a^{10}+\frac{38\cdots 37}{13\cdots 57}a^{9}-\frac{62\cdots 86}{18\cdots 51}a^{8}+\frac{43\cdots 66}{13\cdots 57}a^{7}+\frac{87\cdots 86}{13\cdots 57}a^{6}-\frac{19\cdots 32}{13\cdots 57}a^{5}-\frac{10\cdots 05}{13\cdots 57}a^{4}+\frac{25\cdots 72}{13\cdots 57}a^{3}+\frac{73\cdots 69}{13\cdots 57}a^{2}+\frac{44\cdots 02}{13\cdots 57}a-\frac{56\cdots 75}{13\cdots 57}$, $\frac{68\cdots 10}{13\cdots 57}a^{24}-\frac{30\cdots 45}{13\cdots 57}a^{23}+\frac{17\cdots 06}{13\cdots 57}a^{22}+\frac{14\cdots 68}{13\cdots 57}a^{21}-\frac{16\cdots 89}{13\cdots 57}a^{20}-\frac{77\cdots 02}{13\cdots 57}a^{19}+\frac{82\cdots 60}{13\cdots 57}a^{18}+\frac{40\cdots 71}{13\cdots 57}a^{17}-\frac{47\cdots 34}{18\cdots 51}a^{16}-\frac{28\cdots 97}{13\cdots 57}a^{15}+\frac{15\cdots 86}{18\cdots 51}a^{14}-\frac{96\cdots 10}{13\cdots 57}a^{13}-\frac{11\cdots 09}{13\cdots 57}a^{12}+\frac{24\cdots 48}{13\cdots 57}a^{11}-\frac{10\cdots 12}{13\cdots 57}a^{10}-\frac{82\cdots 71}{13\cdots 57}a^{9}+\frac{13\cdots 60}{18\cdots 51}a^{8}-\frac{91\cdots 02}{13\cdots 57}a^{7}-\frac{18\cdots 28}{13\cdots 57}a^{6}+\frac{42\cdots 91}{13\cdots 57}a^{5}+\frac{22\cdots 90}{13\cdots 57}a^{4}-\frac{55\cdots 15}{13\cdots 57}a^{3}-\frac{15\cdots 28}{13\cdots 57}a^{2}+\frac{28\cdots 06}{13\cdots 57}a+\frac{12\cdots 15}{13\cdots 57}$, $\frac{14\cdots 35}{13\cdots 57}a^{24}+\frac{63\cdots 67}{13\cdots 57}a^{23}-\frac{35\cdots 63}{13\cdots 57}a^{22}-\frac{30\cdots 78}{13\cdots 57}a^{21}+\frac{34\cdots 98}{13\cdots 57}a^{20}+\frac{16\cdots 82}{13\cdots 57}a^{19}-\frac{17\cdots 52}{13\cdots 57}a^{18}-\frac{83\cdots 53}{13\cdots 57}a^{17}+\frac{99\cdots 49}{18\cdots 51}a^{16}+\frac{57\cdots 99}{13\cdots 57}a^{15}-\frac{32\cdots 81}{18\cdots 51}a^{14}+\frac{20\cdots 07}{13\cdots 57}a^{13}+\frac{23\cdots 62}{13\cdots 57}a^{12}-\frac{51\cdots 48}{13\cdots 57}a^{11}+\frac{22\cdots 54}{13\cdots 57}a^{10}+\frac{17\cdots 46}{13\cdots 57}a^{9}-\frac{28\cdots 66}{18\cdots 51}a^{8}+\frac{18\cdots 36}{13\cdots 57}a^{7}+\frac{39\cdots 98}{13\cdots 57}a^{6}-\frac{87\cdots 57}{13\cdots 57}a^{5}-\frac{46\cdots 15}{13\cdots 57}a^{4}+\frac{11\cdots 17}{13\cdots 57}a^{3}+\frac{32\cdots 09}{13\cdots 57}a^{2}-\frac{13\cdots 90}{13\cdots 57}a-\frac{25\cdots 00}{13\cdots 57}$, $\frac{10\cdots 70}{13\cdots 57}a^{24}-\frac{47\cdots 94}{13\cdots 57}a^{23}+\frac{26\cdots 70}{13\cdots 57}a^{22}+\frac{22\cdots 96}{13\cdots 57}a^{21}-\frac{25\cdots 32}{13\cdots 57}a^{20}-\frac{11\cdots 64}{13\cdots 57}a^{19}+\frac{12\cdots 84}{13\cdots 57}a^{18}+\frac{62\cdots 15}{13\cdots 57}a^{17}-\frac{73\cdots 38}{18\cdots 51}a^{16}-\frac{43\cdots 26}{13\cdots 57}a^{15}+\frac{24\cdots 22}{18\cdots 51}a^{14}-\frac{15\cdots 14}{13\cdots 57}a^{13}-\frac{17\cdots 80}{13\cdots 57}a^{12}+\frac{38\cdots 16}{13\cdots 57}a^{11}-\frac{16\cdots 96}{13\cdots 57}a^{10}-\frac{12\cdots 12}{13\cdots 57}a^{9}+\frac{20\cdots 92}{18\cdots 51}a^{8}-\frac{14\cdots 00}{13\cdots 57}a^{7}-\frac{29\cdots 56}{13\cdots 57}a^{6}+\frac{65\cdots 98}{13\cdots 57}a^{5}+\frac{34\cdots 30}{13\cdots 57}a^{4}-\frac{85\cdots 74}{13\cdots 57}a^{3}-\frac{24\cdots 50}{13\cdots 57}a^{2}+\frac{79\cdots 39}{13\cdots 57}a+\frac{18\cdots 60}{13\cdots 57}$, $\frac{30\cdots 16}{13\cdots 57}a^{24}+\frac{83\cdots 01}{13\cdots 57}a^{23}-\frac{11\cdots 33}{18\cdots 51}a^{22}-\frac{34\cdots 42}{18\cdots 51}a^{21}+\frac{77\cdots 38}{13\cdots 57}a^{20}+\frac{22\cdots 82}{13\cdots 57}a^{19}-\frac{47\cdots 65}{13\cdots 57}a^{18}-\frac{11\cdots 05}{13\cdots 57}a^{17}+\frac{19\cdots 75}{13\cdots 57}a^{16}+\frac{40\cdots 45}{13\cdots 57}a^{15}-\frac{11\cdots 27}{18\cdots 51}a^{14}-\frac{10\cdots 98}{18\cdots 51}a^{13}+\frac{27\cdots 95}{13\cdots 57}a^{12}-\frac{10\cdots 23}{13\cdots 57}a^{11}-\frac{34\cdots 17}{13\cdots 57}a^{10}+\frac{43\cdots 15}{13\cdots 57}a^{9}-\frac{10\cdots 03}{18\cdots 51}a^{8}-\frac{19\cdots 29}{13\cdots 57}a^{7}+\frac{13\cdots 02}{13\cdots 57}a^{6}+\frac{34\cdots 21}{13\cdots 57}a^{5}-\frac{27\cdots 89}{13\cdots 57}a^{4}+\frac{63\cdots 74}{13\cdots 57}a^{3}+\frac{11\cdots 78}{13\cdots 57}a^{2}-\frac{74\cdots 14}{18\cdots 51}a+\frac{13\cdots 72}{18\cdots 51}$, $\frac{73\cdots 44}{13\cdots 57}a^{24}+\frac{43\cdots 65}{13\cdots 57}a^{23}-\frac{18\cdots 33}{13\cdots 57}a^{22}-\frac{18\cdots 42}{13\cdots 57}a^{21}+\frac{17\cdots 74}{13\cdots 57}a^{20}+\frac{10\cdots 39}{13\cdots 57}a^{19}-\frac{87\cdots 56}{13\cdots 57}a^{18}-\frac{56\cdots 31}{13\cdots 57}a^{17}+\frac{50\cdots 59}{18\cdots 51}a^{16}+\frac{84\cdots 43}{13\cdots 57}a^{15}-\frac{11\cdots 91}{13\cdots 57}a^{14}+\frac{85\cdots 74}{13\cdots 57}a^{13}+\frac{20\cdots 58}{18\cdots 51}a^{12}-\frac{16\cdots 59}{86\cdots 07}a^{11}+\frac{70\cdots 12}{13\cdots 57}a^{10}+\frac{11\cdots 16}{13\cdots 57}a^{9}-\frac{93\cdots 82}{13\cdots 57}a^{8}-\frac{56\cdots 94}{13\cdots 57}a^{7}+\frac{24\cdots 89}{13\cdots 57}a^{6}-\frac{33\cdots 32}{13\cdots 57}a^{5}-\frac{28\cdots 27}{13\cdots 57}a^{4}+\frac{55\cdots 73}{13\cdots 57}a^{3}+\frac{20\cdots 84}{18\cdots 51}a^{2}-\frac{44\cdots 39}{13\cdots 57}a-\frac{86\cdots 06}{13\cdots 57}$, $\frac{20\cdots 80}{18\cdots 51}a^{24}+\frac{16\cdots 22}{13\cdots 57}a^{23}-\frac{49\cdots 75}{18\cdots 51}a^{22}-\frac{55\cdots 98}{13\cdots 57}a^{21}+\frac{30\cdots 18}{13\cdots 57}a^{20}+\frac{40\cdots 23}{13\cdots 57}a^{19}-\frac{14\cdots 78}{13\cdots 57}a^{18}-\frac{20\cdots 88}{13\cdots 57}a^{17}+\frac{57\cdots 53}{13\cdots 57}a^{16}+\frac{52\cdots 97}{13\cdots 57}a^{15}-\frac{20\cdots 83}{13\cdots 57}a^{14}+\frac{41\cdots 33}{13\cdots 57}a^{13}+\frac{31\cdots 92}{13\cdots 57}a^{12}-\frac{30\cdots 87}{13\cdots 57}a^{11}-\frac{70\cdots 76}{13\cdots 57}a^{10}+\frac{20\cdots 91}{13\cdots 57}a^{9}-\frac{45\cdots 48}{13\cdots 57}a^{8}-\frac{96\cdots 19}{18\cdots 51}a^{7}+\frac{16\cdots 74}{13\cdots 57}a^{6}+\frac{14\cdots 20}{13\cdots 57}a^{5}-\frac{46\cdots 51}{13\cdots 57}a^{4}-\frac{22\cdots 23}{13\cdots 57}a^{3}+\frac{66\cdots 56}{13\cdots 57}a^{2}+\frac{98\cdots 50}{13\cdots 57}a-\frac{83\cdots 92}{13\cdots 57}$, $\frac{44\cdots 54}{13\cdots 57}a^{24}-\frac{54\cdots 73}{13\cdots 57}a^{23}-\frac{11\cdots 61}{13\cdots 57}a^{22}-\frac{31\cdots 60}{13\cdots 57}a^{21}+\frac{11\cdots 23}{13\cdots 57}a^{20}-\frac{11\cdots 08}{13\cdots 57}a^{19}-\frac{54\cdots 94}{13\cdots 57}a^{18}+\frac{46\cdots 07}{13\cdots 57}a^{17}+\frac{22\cdots 67}{13\cdots 57}a^{16}-\frac{10\cdots 76}{13\cdots 57}a^{15}-\frac{68\cdots 72}{13\cdots 57}a^{14}+\frac{10\cdots 57}{13\cdots 57}a^{13}+\frac{23\cdots 68}{13\cdots 57}a^{12}-\frac{18\cdots 90}{13\cdots 57}a^{11}+\frac{17\cdots 99}{13\cdots 57}a^{10}-\frac{22\cdots 82}{13\cdots 57}a^{9}-\frac{66\cdots 24}{13\cdots 57}a^{8}+\frac{45\cdots 20}{13\cdots 57}a^{7}-\frac{67\cdots 62}{13\cdots 57}a^{6}-\frac{39\cdots 77}{13\cdots 57}a^{5}+\frac{18\cdots 74}{13\cdots 57}a^{4}-\frac{28\cdots 72}{13\cdots 57}a^{3}-\frac{10\cdots 54}{13\cdots 57}a^{2}+\frac{96\cdots 28}{13\cdots 57}a-\frac{29\cdots 11}{13\cdots 57}$, $\frac{64\cdots 17}{13\cdots 57}a^{24}-\frac{24\cdots 92}{13\cdots 57}a^{23}+\frac{15\cdots 39}{13\cdots 57}a^{22}+\frac{12\cdots 37}{13\cdots 57}a^{21}-\frac{15\cdots 62}{13\cdots 57}a^{20}-\frac{59\cdots 39}{13\cdots 57}a^{19}+\frac{75\cdots 56}{13\cdots 57}a^{18}+\frac{31\cdots 30}{13\cdots 57}a^{17}-\frac{30\cdots 14}{13\cdots 57}a^{16}+\frac{12\cdots 96}{13\cdots 57}a^{15}+\frac{14\cdots 38}{18\cdots 51}a^{14}-\frac{98\cdots 17}{13\cdots 57}a^{13}-\frac{12\cdots 75}{18\cdots 51}a^{12}+\frac{23\cdots 72}{13\cdots 57}a^{11}-\frac{12\cdots 13}{13\cdots 57}a^{10}-\frac{46\cdots 78}{13\cdots 57}a^{9}+\frac{87\cdots 57}{13\cdots 57}a^{8}-\frac{25\cdots 07}{13\cdots 57}a^{7}-\frac{91\cdots 07}{13\cdots 57}a^{6}+\frac{65\cdots 12}{13\cdots 57}a^{5}-\frac{32\cdots 75}{13\cdots 57}a^{4}-\frac{79\cdots 88}{13\cdots 57}a^{3}+\frac{15\cdots 70}{13\cdots 57}a^{2}+\frac{53\cdots 28}{13\cdots 57}a+\frac{12\cdots 32}{13\cdots 57}$, $\frac{54\cdots 35}{13\cdots 57}a^{24}+\frac{16\cdots 06}{13\cdots 57}a^{23}-\frac{13\cdots 17}{13\cdots 57}a^{22}-\frac{94\cdots 79}{13\cdots 57}a^{21}+\frac{13\cdots 56}{13\cdots 57}a^{20}+\frac{40\cdots 18}{13\cdots 57}a^{19}-\frac{92\cdots 42}{18\cdots 51}a^{18}-\frac{21\cdots 29}{13\cdots 57}a^{17}+\frac{26\cdots 32}{13\cdots 57}a^{16}-\frac{18\cdots 05}{13\cdots 57}a^{15}-\frac{84\cdots 17}{13\cdots 57}a^{14}+\frac{89\cdots 58}{13\cdots 57}a^{13}+\frac{70\cdots 15}{13\cdots 57}a^{12}-\frac{28\cdots 89}{18\cdots 51}a^{11}+\frac{12\cdots 08}{13\cdots 57}a^{10}+\frac{49\cdots 29}{18\cdots 51}a^{9}-\frac{75\cdots 75}{13\cdots 57}a^{8}+\frac{35\cdots 19}{18\cdots 51}a^{7}+\frac{56\cdots 58}{13\cdots 57}a^{6}-\frac{45\cdots 92}{13\cdots 57}a^{5}+\frac{84\cdots 08}{18\cdots 51}a^{4}+\frac{33\cdots 12}{13\cdots 57}a^{3}+\frac{11\cdots 94}{13\cdots 57}a^{2}-\frac{20\cdots 47}{13\cdots 57}a+\frac{29\cdots 67}{13\cdots 57}$, $\frac{39\cdots 00}{13\cdots 57}a^{24}+\frac{14\cdots 01}{13\cdots 57}a^{23}-\frac{10\cdots 53}{13\cdots 57}a^{22}-\frac{76\cdots 82}{13\cdots 57}a^{21}+\frac{14\cdots 59}{18\cdots 51}a^{20}+\frac{39\cdots 97}{13\cdots 57}a^{19}-\frac{51\cdots 70}{13\cdots 57}a^{18}-\frac{20\cdots 49}{13\cdots 57}a^{17}+\frac{21\cdots 64}{13\cdots 57}a^{16}+\frac{59\cdots 33}{13\cdots 57}a^{15}-\frac{70\cdots 05}{13\cdots 57}a^{14}+\frac{88\cdots 83}{18\cdots 51}a^{13}+\frac{83\cdots 21}{13\cdots 57}a^{12}-\frac{17\cdots 60}{13\cdots 57}a^{11}+\frac{53\cdots 03}{13\cdots 57}a^{10}+\frac{13\cdots 31}{18\cdots 51}a^{9}-\frac{88\cdots 17}{13\cdots 57}a^{8}-\frac{42\cdots 75}{18\cdots 51}a^{7}+\frac{45\cdots 46}{18\cdots 51}a^{6}-\frac{87\cdots 09}{13\cdots 57}a^{5}-\frac{40\cdots 12}{13\cdots 57}a^{4}+\frac{19\cdots 33}{13\cdots 57}a^{3}+\frac{18\cdots 78}{18\cdots 51}a^{2}-\frac{16\cdots 27}{13\cdots 57}a+\frac{25\cdots 06}{13\cdots 57}$, $\frac{20\cdots 26}{18\cdots 51}a^{24}+\frac{23\cdots 43}{13\cdots 57}a^{23}-\frac{37\cdots 08}{13\cdots 57}a^{22}-\frac{21\cdots 45}{13\cdots 57}a^{21}+\frac{37\cdots 91}{13\cdots 57}a^{20}+\frac{88\cdots 82}{13\cdots 57}a^{19}-\frac{19\cdots 69}{13\cdots 57}a^{18}-\frac{53\cdots 08}{13\cdots 57}a^{17}+\frac{78\cdots 28}{13\cdots 57}a^{16}-\frac{56\cdots 02}{13\cdots 57}a^{15}-\frac{25\cdots 31}{13\cdots 57}a^{14}+\frac{24\cdots 85}{13\cdots 57}a^{13}+\frac{25\cdots 63}{13\cdots 57}a^{12}-\frac{59\cdots 46}{13\cdots 57}a^{11}+\frac{24\cdots 23}{13\cdots 57}a^{10}+\frac{21\cdots 55}{13\cdots 57}a^{9}-\frac{19\cdots 18}{13\cdots 57}a^{8}-\frac{28\cdots 03}{13\cdots 57}a^{7}+\frac{65\cdots 13}{13\cdots 57}a^{6}+\frac{46\cdots 48}{13\cdots 57}a^{5}-\frac{14\cdots 87}{13\cdots 57}a^{4}+\frac{40\cdots 64}{18\cdots 51}a^{3}+\frac{82\cdots 08}{18\cdots 51}a^{2}-\frac{97\cdots 12}{13\cdots 57}a-\frac{10\cdots 66}{13\cdots 57}$, $\frac{11\cdots 22}{13\cdots 57}a^{24}+\frac{23\cdots 71}{13\cdots 57}a^{23}-\frac{38\cdots 64}{18\cdots 51}a^{22}-\frac{11\cdots 70}{13\cdots 57}a^{21}+\frac{37\cdots 57}{18\cdots 51}a^{20}+\frac{59\cdots 86}{13\cdots 57}a^{19}-\frac{12\cdots 10}{13\cdots 57}a^{18}-\frac{67\cdots 72}{13\cdots 57}a^{17}+\frac{50\cdots 03}{13\cdots 57}a^{16}-\frac{27\cdots 36}{18\cdots 51}a^{15}-\frac{15\cdots 94}{13\cdots 57}a^{14}+\frac{22\cdots 62}{13\cdots 57}a^{13}+\frac{45\cdots 77}{13\cdots 57}a^{12}-\frac{39\cdots 27}{13\cdots 57}a^{11}+\frac{39\cdots 47}{13\cdots 57}a^{10}-\frac{11\cdots 82}{13\cdots 57}a^{9}-\frac{10\cdots 69}{13\cdots 57}a^{8}+\frac{11\cdots 64}{13\cdots 57}a^{7}-\frac{37\cdots 53}{13\cdots 57}a^{6}-\frac{54\cdots 68}{13\cdots 57}a^{5}+\frac{11\cdots 43}{18\cdots 51}a^{4}-\frac{12\cdots 08}{13\cdots 57}a^{3}-\frac{60\cdots 09}{13\cdots 57}a^{2}+\frac{30\cdots 97}{13\cdots 57}a-\frac{80\cdots 62}{13\cdots 57}$
|
| |
| Regulator: | \( 1601939125.473172 \) (assuming GRH) |
| |
| Unit signature rank: | \( 5 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{10}\cdot 1601939125.473172 \cdot 1}{2\cdot\sqrt{29802322387695312500000000000000000000}}\cr\approx \mathstrut & 0.450235381377491 \end{aligned}\] (assuming GRH)
Galois group
$C_5\times F_5$ (as 25T7):
| A solvable group of order 100 |
| The 25 conjugacy class representatives for $C_5\times F_5$ |
| Character table for $C_5\times F_5$ |
Intermediate fields
| 5.5.390625.1, 5.1.50000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 sibling: | data not computed |
| Minimal sibling: | $ x^{20} - 2 x^{15} + 4 x^{10} - 8 x^{5} + 16 $ |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | $20{,}\,{\href{/padicField/3.5.0.1}{5} }$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.1.0.1}{1} }^{5}$ | ${\href{/padicField/11.5.0.1}{5} }^{5}$ | $20{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/23.5.0.1}{5} }$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.5.0.1}{5} }^{5}$ | $20{,}\,{\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.5.0.1}{5} }^{5}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}{,}\,{\href{/padicField/43.1.0.1}{1} }^{5}$ | $20{,}\,{\href{/padicField/47.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/53.5.0.1}{5} }$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| Deg $25$ | $5$ | $5$ | $20$ | |||
|
\(5\)
| Deg $25$ | $25$ | $1$ | $45$ |