Normalized defining polynomial
\( x^{25} - 25 x^{22} + 25 x^{21} + 110 x^{20} - 625 x^{19} + 1250 x^{18} - 3625 x^{17} + 21750 x^{16} + \cdots - 36535 \)
Invariants
| Degree: | $25$ |
| |
| Signature: | $(5, 10)$ |
| |
| Discriminant: |
\(1694065894508600678136645001359283924102783203125\)
\(\medspace = 5^{69}\)
|
| |
| Root discriminant: | \(84.95\) |
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $\frac{1}{7}a^{17}+\frac{3}{7}a^{15}-\frac{1}{7}a^{14}+\frac{1}{7}a^{13}-\frac{3}{7}a^{12}+\frac{1}{7}a^{11}-\frac{3}{7}a^{10}-\frac{3}{7}a^{9}+\frac{2}{7}a^{8}-\frac{3}{7}a^{7}+\frac{2}{7}a^{6}+\frac{1}{7}a^{5}-\frac{3}{7}a^{4}-\frac{2}{7}a^{3}+\frac{2}{7}a^{2}-\frac{3}{7}a+\frac{1}{7}$, $\frac{1}{7}a^{18}+\frac{3}{7}a^{16}-\frac{1}{7}a^{15}+\frac{1}{7}a^{14}-\frac{3}{7}a^{13}+\frac{1}{7}a^{12}-\frac{3}{7}a^{11}-\frac{3}{7}a^{10}+\frac{2}{7}a^{9}-\frac{3}{7}a^{8}+\frac{2}{7}a^{7}+\frac{1}{7}a^{6}-\frac{3}{7}a^{5}-\frac{2}{7}a^{4}+\frac{2}{7}a^{3}-\frac{3}{7}a^{2}+\frac{1}{7}a$, $\frac{1}{49}a^{19}+\frac{1}{49}a^{18}+\frac{1}{49}a^{17}-\frac{5}{49}a^{16}+\frac{1}{49}a^{15}+\frac{2}{7}a^{14}+\frac{10}{49}a^{13}+\frac{18}{49}a^{12}-\frac{1}{49}a^{11}+\frac{19}{49}a^{10}-\frac{2}{49}a^{9}+\frac{16}{49}a^{8}-\frac{5}{49}a^{7}+\frac{1}{49}a^{6}-\frac{1}{7}a^{5}-\frac{1}{49}a^{4}+\frac{17}{49}a^{3}-\frac{13}{49}a^{2}+\frac{2}{7}a+\frac{19}{49}$, $\frac{1}{49}a^{20}+\frac{1}{49}a^{17}+\frac{6}{49}a^{16}-\frac{15}{49}a^{15}-\frac{11}{49}a^{14}+\frac{15}{49}a^{13}+\frac{9}{49}a^{12}-\frac{22}{49}a^{11}+\frac{1}{7}a^{10}-\frac{3}{49}a^{9}-\frac{1}{7}a^{8}-\frac{15}{49}a^{7}+\frac{6}{49}a^{6}+\frac{13}{49}a^{5}-\frac{3}{49}a^{4}+\frac{5}{49}a^{3}-\frac{8}{49}a^{2}-\frac{16}{49}a-\frac{12}{49}$, $\frac{1}{343}a^{21}+\frac{2}{343}a^{20}-\frac{6}{343}a^{18}-\frac{6}{343}a^{17}-\frac{171}{343}a^{16}-\frac{76}{343}a^{15}-\frac{101}{343}a^{13}-\frac{116}{343}a^{12}+\frac{117}{343}a^{11}-\frac{73}{343}a^{10}-\frac{132}{343}a^{9}-\frac{134}{343}a^{8}+\frac{4}{343}a^{7}-\frac{108}{343}a^{6}-\frac{117}{343}a^{5}+\frac{6}{343}a^{4}-\frac{33}{343}a^{3}+\frac{59}{343}a^{2}+\frac{40}{343}a+\frac{158}{343}$, $\frac{1}{343}a^{22}+\frac{3}{343}a^{20}+\frac{1}{343}a^{19}+\frac{13}{343}a^{18}+\frac{2}{343}a^{17}-\frac{10}{49}a^{16}+\frac{152}{343}a^{15}+\frac{116}{343}a^{14}+\frac{65}{343}a^{13}+\frac{97}{343}a^{12}+\frac{22}{343}a^{11}+\frac{2}{7}a^{10}-\frac{3}{343}a^{9}-\frac{57}{343}a^{8}-\frac{11}{343}a^{7}+\frac{99}{343}a^{6}+\frac{86}{343}a^{5}-\frac{171}{343}a^{4}-\frac{15}{343}a^{3}+\frac{69}{343}a^{2}-\frac{34}{343}a-\frac{120}{343}$, $\frac{1}{343}a^{23}+\frac{2}{343}a^{20}-\frac{1}{343}a^{19}+\frac{6}{343}a^{18}-\frac{10}{343}a^{17}+\frac{13}{49}a^{16}+\frac{29}{343}a^{15}+\frac{86}{343}a^{14}+\frac{71}{343}a^{13}+\frac{34}{343}a^{12}-\frac{1}{343}a^{11}-\frac{148}{343}a^{10}-\frac{144}{343}a^{9}-\frac{127}{343}a^{8}-\frac{95}{343}a^{7}-\frac{150}{343}a^{6}+\frac{75}{343}a^{5}+\frac{156}{343}a^{4}-\frac{19}{49}a^{3}+\frac{13}{343}a^{2}-\frac{9}{343}a-\frac{89}{343}$, $\frac{1}{53\cdots 01}a^{24}+\frac{42\cdots 90}{53\cdots 01}a^{23}+\frac{24\cdots 24}{53\cdots 01}a^{22}-\frac{14\cdots 21}{53\cdots 01}a^{21}+\frac{26\cdots 47}{53\cdots 01}a^{20}+\frac{40\cdots 46}{53\cdots 01}a^{19}-\frac{69\cdots 90}{53\cdots 01}a^{18}-\frac{14\cdots 63}{53\cdots 01}a^{17}-\frac{16\cdots 97}{53\cdots 01}a^{16}+\frac{10\cdots 76}{53\cdots 01}a^{15}-\frac{22\cdots 44}{76\cdots 43}a^{14}-\frac{25\cdots 87}{53\cdots 01}a^{13}-\frac{11\cdots 69}{53\cdots 01}a^{12}-\frac{24\cdots 12}{53\cdots 01}a^{11}-\frac{19\cdots 63}{53\cdots 01}a^{10}+\frac{19\cdots 90}{53\cdots 01}a^{9}-\frac{20\cdots 82}{53\cdots 01}a^{8}-\frac{33\cdots 35}{53\cdots 01}a^{7}+\frac{53\cdots 67}{53\cdots 01}a^{6}+\frac{23\cdots 85}{53\cdots 01}a^{5}-\frac{21\cdots 73}{53\cdots 01}a^{4}-\frac{23\cdots 06}{53\cdots 01}a^{3}+\frac{74\cdots 84}{53\cdots 01}a^{2}+\frac{46\cdots 93}{53\cdots 01}a+\frac{31\cdots 21}{53\cdots 01}$
| 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: |
$\frac{90\cdots 75}{75\cdots 51}a^{24}+\frac{41\cdots 00}{75\cdots 51}a^{23}+\frac{16\cdots 35}{75\cdots 51}a^{22}-\frac{22\cdots 35}{75\cdots 51}a^{21}+\frac{12\cdots 39}{75\cdots 51}a^{20}+\frac{10\cdots 00}{75\cdots 51}a^{19}-\frac{51\cdots 00}{75\cdots 51}a^{18}+\frac{88\cdots 05}{75\cdots 51}a^{17}-\frac{28\cdots 30}{75\cdots 51}a^{16}+\frac{18\cdots 70}{75\cdots 51}a^{15}-\frac{61\cdots 00}{10\cdots 93}a^{14}+\frac{81\cdots 25}{75\cdots 51}a^{13}-\frac{17\cdots 55}{75\cdots 51}a^{12}+\frac{32\cdots 05}{75\cdots 51}a^{11}-\frac{86\cdots 38}{75\cdots 51}a^{10}+\frac{11\cdots 75}{75\cdots 51}a^{9}-\frac{35\cdots 75}{75\cdots 51}a^{8}-\frac{81\cdots 95}{75\cdots 51}a^{7}+\frac{34\cdots 20}{75\cdots 51}a^{6}-\frac{15\cdots 83}{75\cdots 51}a^{5}+\frac{62\cdots 25}{75\cdots 51}a^{4}-\frac{68\cdots 25}{75\cdots 51}a^{3}+\frac{85\cdots 00}{75\cdots 51}a^{2}-\frac{40\cdots 25}{75\cdots 51}a+\frac{78\cdots 24}{75\cdots 51}$, $\frac{12\cdots 50}{75\cdots 51}a^{24}-\frac{56\cdots 00}{75\cdots 51}a^{23}-\frac{23\cdots 90}{75\cdots 51}a^{22}+\frac{30\cdots 65}{75\cdots 51}a^{21}-\frac{16\cdots 27}{75\cdots 51}a^{20}-\frac{14\cdots 00}{75\cdots 51}a^{19}+\frac{70\cdots 50}{75\cdots 51}a^{18}-\frac{12\cdots 90}{75\cdots 51}a^{17}+\frac{39\cdots 40}{75\cdots 51}a^{16}-\frac{25\cdots 22}{75\cdots 51}a^{15}+\frac{84\cdots 25}{10\cdots 93}a^{14}-\frac{11\cdots 25}{75\cdots 51}a^{13}+\frac{24\cdots 70}{75\cdots 51}a^{12}-\frac{44\cdots 20}{75\cdots 51}a^{11}+\frac{11\cdots 21}{75\cdots 51}a^{10}-\frac{16\cdots 25}{75\cdots 51}a^{9}+\frac{49\cdots 25}{75\cdots 51}a^{8}+\frac{11\cdots 20}{75\cdots 51}a^{7}-\frac{47\cdots 20}{75\cdots 51}a^{6}+\frac{21\cdots 63}{75\cdots 51}a^{5}-\frac{86\cdots 00}{75\cdots 51}a^{4}+\frac{94\cdots 75}{75\cdots 51}a^{3}-\frac{11\cdots 75}{75\cdots 51}a^{2}+\frac{55\cdots 50}{75\cdots 51}a-\frac{90\cdots 79}{75\cdots 51}$, $\frac{10\cdots 25}{75\cdots 51}a^{24}-\frac{47\cdots 50}{75\cdots 51}a^{23}-\frac{19\cdots 50}{75\cdots 51}a^{22}+\frac{25\cdots 50}{75\cdots 51}a^{21}-\frac{14\cdots 70}{75\cdots 51}a^{20}-\frac{12\cdots 50}{75\cdots 51}a^{19}+\frac{59\cdots 75}{75\cdots 51}a^{18}-\frac{10\cdots 85}{75\cdots 51}a^{17}+\frac{32\cdots 35}{75\cdots 51}a^{16}-\frac{21\cdots 21}{75\cdots 51}a^{15}+\frac{70\cdots 00}{10\cdots 93}a^{14}-\frac{93\cdots 25}{75\cdots 51}a^{13}+\frac{20\cdots 15}{75\cdots 51}a^{12}-\frac{36\cdots 65}{75\cdots 51}a^{11}+\frac{99\cdots 94}{75\cdots 51}a^{10}-\frac{13\cdots 50}{75\cdots 51}a^{9}+\frac{41\cdots 50}{75\cdots 51}a^{8}+\frac{93\cdots 85}{75\cdots 51}a^{7}-\frac{39\cdots 85}{75\cdots 51}a^{6}+\frac{18\cdots 19}{75\cdots 51}a^{5}-\frac{71\cdots 00}{75\cdots 51}a^{4}+\frac{79\cdots 00}{75\cdots 51}a^{3}-\frac{98\cdots 00}{75\cdots 51}a^{2}+\frac{46\cdots 00}{75\cdots 51}a-\frac{94\cdots 26}{75\cdots 51}$, $\frac{22\cdots 75}{75\cdots 51}a^{24}+\frac{10\cdots 50}{75\cdots 51}a^{23}+\frac{42\cdots 40}{75\cdots 51}a^{22}-\frac{56\cdots 15}{75\cdots 51}a^{21}+\frac{30\cdots 97}{75\cdots 51}a^{20}+\frac{26\cdots 50}{75\cdots 51}a^{19}-\frac{13\cdots 25}{75\cdots 51}a^{18}+\frac{22\cdots 75}{75\cdots 51}a^{17}-\frac{72\cdots 75}{75\cdots 51}a^{16}+\frac{46\cdots 43}{75\cdots 51}a^{15}-\frac{15\cdots 25}{10\cdots 93}a^{14}+\frac{20\cdots 50}{75\cdots 51}a^{13}-\frac{45\cdots 85}{75\cdots 51}a^{12}+\frac{81\cdots 85}{75\cdots 51}a^{11}-\frac{21\cdots 15}{75\cdots 51}a^{10}+\frac{29\cdots 75}{75\cdots 51}a^{9}-\frac{90\cdots 75}{75\cdots 51}a^{8}-\frac{20\cdots 05}{75\cdots 51}a^{7}+\frac{87\cdots 05}{75\cdots 51}a^{6}-\frac{40\cdots 82}{75\cdots 51}a^{5}+\frac{15\cdots 00}{75\cdots 51}a^{4}-\frac{17\cdots 75}{75\cdots 51}a^{3}+\frac{21\cdots 75}{75\cdots 51}a^{2}-\frac{10\cdots 50}{75\cdots 51}a+\frac{19\cdots 56}{75\cdots 51}$, $\frac{29\cdots 59}{53\cdots 01}a^{24}-\frac{12\cdots 52}{53\cdots 01}a^{23}-\frac{22\cdots 97}{53\cdots 01}a^{22}+\frac{73\cdots 13}{53\cdots 01}a^{21}-\frac{41\cdots 67}{53\cdots 01}a^{20}-\frac{34\cdots 06}{53\cdots 01}a^{19}+\frac{16\cdots 56}{53\cdots 01}a^{18}-\frac{29\cdots 69}{53\cdots 01}a^{17}+\frac{92\cdots 05}{53\cdots 01}a^{16}-\frac{59\cdots 58}{53\cdots 01}a^{15}+\frac{20\cdots 81}{76\cdots 43}a^{14}-\frac{26\cdots 46}{53\cdots 01}a^{13}+\frac{58\cdots 44}{53\cdots 01}a^{12}-\frac{10\cdots 53}{53\cdots 01}a^{11}+\frac{28\cdots 95}{53\cdots 01}a^{10}-\frac{38\cdots 66}{53\cdots 01}a^{9}+\frac{10\cdots 36}{53\cdots 01}a^{8}+\frac{28\cdots 77}{53\cdots 01}a^{7}-\frac{11\cdots 01}{53\cdots 01}a^{6}+\frac{52\cdots 26}{53\cdots 01}a^{5}+\frac{57\cdots 01}{53\cdots 01}a^{4}+\frac{22\cdots 81}{53\cdots 01}a^{3}-\frac{30\cdots 16}{53\cdots 01}a^{2}+\frac{10\cdots 98}{53\cdots 01}a-\frac{14\cdots 74}{53\cdots 01}$, $\frac{30\cdots 06}{53\cdots 01}a^{24}-\frac{10\cdots 83}{53\cdots 01}a^{23}+\frac{89\cdots 24}{53\cdots 01}a^{22}+\frac{77\cdots 00}{53\cdots 01}a^{21}-\frac{51\cdots 56}{53\cdots 01}a^{20}-\frac{36\cdots 76}{53\cdots 01}a^{19}+\frac{18\cdots 52}{53\cdots 01}a^{18}-\frac{32\cdots 76}{53\cdots 01}a^{17}+\frac{99\cdots 36}{53\cdots 01}a^{16}-\frac{63\cdots 93}{53\cdots 01}a^{15}+\frac{22\cdots 48}{76\cdots 43}a^{14}-\frac{28\cdots 83}{53\cdots 01}a^{13}+\frac{63\cdots 94}{53\cdots 01}a^{12}-\frac{11\cdots 28}{53\cdots 01}a^{11}+\frac{30\cdots 53}{53\cdots 01}a^{10}-\frac{42\cdots 05}{53\cdots 01}a^{9}+\frac{13\cdots 13}{53\cdots 01}a^{8}+\frac{30\cdots 24}{53\cdots 01}a^{7}-\frac{12\cdots 54}{53\cdots 01}a^{6}+\frac{65\cdots 14}{53\cdots 01}a^{5}+\frac{82\cdots 24}{53\cdots 01}a^{4}+\frac{22\cdots 28}{53\cdots 01}a^{3}-\frac{31\cdots 95}{53\cdots 01}a^{2}+\frac{13\cdots 05}{53\cdots 01}a-\frac{21\cdots 79}{53\cdots 01}$, $\frac{43\cdots 86}{53\cdots 01}a^{24}-\frac{19\cdots 71}{53\cdots 01}a^{23}-\frac{98\cdots 77}{53\cdots 01}a^{22}+\frac{10\cdots 60}{53\cdots 01}a^{21}-\frac{59\cdots 04}{53\cdots 01}a^{20}-\frac{50\cdots 89}{53\cdots 01}a^{19}+\frac{24\cdots 07}{53\cdots 01}a^{18}-\frac{43\cdots 16}{53\cdots 01}a^{17}+\frac{13\cdots 30}{53\cdots 01}a^{16}-\frac{88\cdots 87}{53\cdots 01}a^{15}+\frac{29\cdots 35}{76\cdots 43}a^{14}-\frac{39\cdots 51}{53\cdots 01}a^{13}+\frac{87\cdots 95}{53\cdots 01}a^{12}-\frac{15\cdots 78}{53\cdots 01}a^{11}+\frac{42\cdots 43}{53\cdots 01}a^{10}-\frac{57\cdots 39}{53\cdots 01}a^{9}+\frac{18\cdots 09}{53\cdots 01}a^{8}+\frac{38\cdots 89}{53\cdots 01}a^{7}-\frac{16\cdots 60}{53\cdots 01}a^{6}+\frac{79\cdots 01}{53\cdots 01}a^{5}-\frac{87\cdots 60}{53\cdots 01}a^{4}+\frac{31\cdots 78}{53\cdots 01}a^{3}-\frac{42\cdots 13}{53\cdots 01}a^{2}+\frac{20\cdots 80}{53\cdots 01}a-\frac{40\cdots 71}{53\cdots 01}$, $\frac{44\cdots 19}{53\cdots 01}a^{24}+\frac{88\cdots 91}{53\cdots 01}a^{23}-\frac{41\cdots 21}{53\cdots 01}a^{22}-\frac{11\cdots 79}{53\cdots 01}a^{21}+\frac{90\cdots 67}{53\cdots 01}a^{20}+\frac{52\cdots 56}{53\cdots 01}a^{19}-\frac{27\cdots 94}{53\cdots 01}a^{18}+\frac{50\cdots 79}{53\cdots 01}a^{17}-\frac{14\cdots 64}{53\cdots 01}a^{16}+\frac{94\cdots 04}{53\cdots 01}a^{15}-\frac{33\cdots 74}{76\cdots 43}a^{14}+\frac{44\cdots 43}{53\cdots 01}a^{13}-\frac{96\cdots 38}{53\cdots 01}a^{12}+\frac{17\cdots 89}{53\cdots 01}a^{11}-\frac{46\cdots 10}{53\cdots 01}a^{10}+\frac{67\cdots 70}{53\cdots 01}a^{9}-\frac{26\cdots 16}{53\cdots 01}a^{8}-\frac{43\cdots 89}{53\cdots 01}a^{7}+\frac{18\cdots 22}{53\cdots 01}a^{6}-\frac{11\cdots 69}{53\cdots 01}a^{5}-\frac{19\cdots 18}{53\cdots 01}a^{4}-\frac{30\cdots 39}{53\cdots 01}a^{3}+\frac{55\cdots 30}{53\cdots 01}a^{2}-\frac{24\cdots 60}{53\cdots 01}a+\frac{33\cdots 51}{53\cdots 01}$, $\frac{62\cdots 19}{53\cdots 01}a^{24}+\frac{19\cdots 03}{53\cdots 01}a^{23}+\frac{81\cdots 46}{53\cdots 01}a^{22}+\frac{15\cdots 28}{53\cdots 01}a^{21}-\frac{63\cdots 47}{53\cdots 01}a^{20}-\frac{41\cdots 26}{53\cdots 01}a^{19}+\frac{63\cdots 51}{53\cdots 01}a^{18}-\frac{19\cdots 58}{53\cdots 01}a^{17}+\frac{41\cdots 19}{53\cdots 01}a^{16}-\frac{19\cdots 40}{53\cdots 01}a^{15}+\frac{10\cdots 78}{76\cdots 43}a^{14}-\frac{16\cdots 80}{53\cdots 01}a^{13}+\frac{31\cdots 20}{53\cdots 01}a^{12}-\frac{61\cdots 99}{53\cdots 01}a^{11}+\frac{12\cdots 87}{53\cdots 01}a^{10}-\frac{27\cdots 82}{53\cdots 01}a^{9}+\frac{27\cdots 70}{53\cdots 01}a^{8}+\frac{67\cdots 06}{53\cdots 01}a^{7}-\frac{63\cdots 38}{53\cdots 01}a^{6}+\frac{11\cdots 09}{53\cdots 01}a^{5}-\frac{43\cdots 87}{53\cdots 01}a^{4}-\frac{29\cdots 22}{53\cdots 01}a^{3}+\frac{34\cdots 59}{53\cdots 01}a^{2}-\frac{13\cdots 38}{53\cdots 01}a+\frac{18\cdots 46}{53\cdots 01}$, $\frac{43\cdots 09}{53\cdots 01}a^{24}-\frac{99\cdots 12}{53\cdots 01}a^{23}-\frac{10\cdots 85}{53\cdots 01}a^{22}-\frac{11\cdots 41}{53\cdots 01}a^{21}+\frac{35\cdots 24}{53\cdots 01}a^{20}+\frac{49\cdots 35}{53\cdots 01}a^{19}-\frac{38\cdots 80}{53\cdots 01}a^{18}+\frac{10\cdots 99}{53\cdots 01}a^{17}-\frac{22\cdots 85}{53\cdots 01}a^{16}+\frac{12\cdots 58}{53\cdots 01}a^{15}-\frac{62\cdots 62}{76\cdots 43}a^{14}+\frac{85\cdots 25}{53\cdots 01}a^{13}-\frac{17\cdots 49}{53\cdots 01}a^{12}+\frac{34\cdots 54}{53\cdots 01}a^{11}-\frac{74\cdots 29}{53\cdots 01}a^{10}+\frac{15\cdots 98}{53\cdots 01}a^{9}-\frac{12\cdots 86}{53\cdots 01}a^{8}-\frac{45\cdots 72}{53\cdots 01}a^{7}+\frac{27\cdots 43}{53\cdots 01}a^{6}-\frac{48\cdots 21}{53\cdots 01}a^{5}+\frac{94\cdots 75}{53\cdots 01}a^{4}-\frac{37\cdots 99}{53\cdots 01}a^{3}+\frac{12\cdots 65}{53\cdots 01}a^{2}-\frac{91\cdots 65}{53\cdots 01}a+\frac{19\cdots 74}{53\cdots 01}$, $\frac{10\cdots 80}{53\cdots 01}a^{24}-\frac{47\cdots 57}{53\cdots 01}a^{23}+\frac{42\cdots 39}{53\cdots 01}a^{22}-\frac{25\cdots 84}{53\cdots 01}a^{21}+\frac{13\cdots 08}{53\cdots 01}a^{20}-\frac{12\cdots 82}{53\cdots 01}a^{19}-\frac{10\cdots 79}{53\cdots 01}a^{18}+\frac{48\cdots 73}{53\cdots 01}a^{17}-\frac{12\cdots 71}{53\cdots 01}a^{16}+\frac{43\cdots 49}{53\cdots 01}a^{15}-\frac{24\cdots 37}{76\cdots 43}a^{14}+\frac{47\cdots 47}{53\cdots 01}a^{13}-\frac{10\cdots 52}{53\cdots 01}a^{12}+\frac{19\cdots 32}{53\cdots 01}a^{11}-\frac{40\cdots 66}{53\cdots 01}a^{10}+\frac{86\cdots 50}{53\cdots 01}a^{9}-\frac{12\cdots 82}{53\cdots 01}a^{8}+\frac{93\cdots 90}{53\cdots 01}a^{7}+\frac{83\cdots 99}{53\cdots 01}a^{6}-\frac{31\cdots 74}{53\cdots 01}a^{5}+\frac{35\cdots 59}{53\cdots 01}a^{4}-\frac{16\cdots 15}{53\cdots 01}a^{3}+\frac{24\cdots 70}{53\cdots 01}a^{2}+\frac{66\cdots 76}{53\cdots 01}a-\frac{20\cdots 01}{53\cdots 01}$, $\frac{15\cdots 33}{53\cdots 01}a^{24}-\frac{76\cdots 99}{53\cdots 01}a^{23}-\frac{82\cdots 05}{53\cdots 01}a^{22}-\frac{38\cdots 15}{53\cdots 01}a^{21}+\frac{57\cdots 08}{53\cdots 01}a^{20}+\frac{16\cdots 21}{53\cdots 01}a^{19}-\frac{10\cdots 31}{53\cdots 01}a^{18}+\frac{22\cdots 26}{53\cdots 01}a^{17}-\frac{59\cdots 34}{53\cdots 01}a^{16}+\frac{34\cdots 20}{53\cdots 01}a^{15}-\frac{14\cdots 77}{76\cdots 43}a^{14}+\frac{19\cdots 82}{53\cdots 01}a^{13}-\frac{41\cdots 80}{53\cdots 01}a^{12}+\frac{79\cdots 86}{53\cdots 01}a^{11}-\frac{18\cdots 01}{53\cdots 01}a^{10}+\frac{32\cdots 21}{53\cdots 01}a^{9}-\frac{20\cdots 32}{53\cdots 01}a^{8}-\frac{13\cdots 91}{53\cdots 01}a^{7}+\frac{72\cdots 46}{53\cdots 01}a^{6}-\frac{78\cdots 24}{53\cdots 01}a^{5}+\frac{99\cdots 46}{53\cdots 01}a^{4}-\frac{49\cdots 62}{53\cdots 01}a^{3}+\frac{28\cdots 13}{53\cdots 01}a^{2}-\frac{17\cdots 16}{53\cdots 01}a+\frac{28\cdots 24}{53\cdots 01}$, $\frac{67\cdots 83}{76\cdots 43}a^{24}+\frac{40\cdots 20}{76\cdots 43}a^{23}+\frac{21\cdots 25}{76\cdots 43}a^{22}-\frac{16\cdots 09}{76\cdots 43}a^{21}+\frac{66\cdots 68}{76\cdots 43}a^{20}+\frac{78\cdots 96}{76\cdots 43}a^{19}-\frac{37\cdots 48}{76\cdots 43}a^{18}+\frac{60\cdots 59}{76\cdots 43}a^{17}-\frac{20\cdots 55}{76\cdots 43}a^{16}+\frac{13\cdots 58}{76\cdots 43}a^{15}-\frac{43\cdots 67}{10\cdots 49}a^{14}+\frac{56\cdots 97}{76\cdots 43}a^{13}-\frac{12\cdots 33}{76\cdots 43}a^{12}+\frac{22\cdots 51}{76\cdots 43}a^{11}-\frac{61\cdots 00}{76\cdots 43}a^{10}+\frac{78\cdots 70}{76\cdots 43}a^{9}-\frac{16\cdots 28}{76\cdots 43}a^{8}-\frac{60\cdots 41}{76\cdots 43}a^{7}+\frac{25\cdots 58}{76\cdots 43}a^{6}-\frac{84\cdots 46}{76\cdots 43}a^{5}+\frac{14\cdots 77}{76\cdots 43}a^{4}-\frac{50\cdots 30}{76\cdots 43}a^{3}+\frac{53\cdots 60}{76\cdots 43}a^{2}-\frac{23\cdots 55}{76\cdots 43}a+\frac{36\cdots 03}{76\cdots 43}$, $\frac{23\cdots 82}{76\cdots 43}a^{24}-\frac{78\cdots 92}{76\cdots 43}a^{23}-\frac{25\cdots 75}{76\cdots 43}a^{22}+\frac{58\cdots 73}{76\cdots 43}a^{21}-\frac{39\cdots 92}{76\cdots 43}a^{20}-\frac{27\cdots 91}{76\cdots 43}a^{19}+\frac{13\cdots 81}{76\cdots 43}a^{18}-\frac{24\cdots 45}{76\cdots 43}a^{17}+\frac{77\cdots 91}{76\cdots 43}a^{16}-\frac{48\cdots 69}{76\cdots 43}a^{15}+\frac{16\cdots 96}{10\cdots 49}a^{14}-\frac{22\cdots 35}{76\cdots 43}a^{13}+\frac{49\cdots 35}{76\cdots 43}a^{12}-\frac{89\cdots 61}{76\cdots 43}a^{11}+\frac{23\cdots 39}{76\cdots 43}a^{10}-\frac{33\cdots 13}{76\cdots 43}a^{9}+\frac{12\cdots 63}{76\cdots 43}a^{8}+\frac{20\cdots 44}{76\cdots 43}a^{7}-\frac{93\cdots 39}{76\cdots 43}a^{6}+\frac{52\cdots 57}{76\cdots 43}a^{5}-\frac{47\cdots 63}{76\cdots 43}a^{4}+\frac{16\cdots 59}{76\cdots 43}a^{3}-\frac{24\cdots 92}{76\cdots 43}a^{2}+\frac{12\cdots 35}{76\cdots 43}a-\frac{25\cdots 07}{76\cdots 43}$
|
| |
| Regulator: | \( 475377324722890.7 \) (assuming GRH) |
| |
| Unit signature rank: | \( 5 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{10}\cdot 475377324722890.7 \cdot 1}{2\cdot\sqrt{1694065894508600678136645001359283924102783203125}}\cr\approx \mathstrut & 0.560392067220791 \end{aligned}\] (assuming GRH)
Galois group
$A_5^5.C_{10}$ (as 25T187):
| A non-solvable group of order 7776000000 |
| The 419 conjugacy class representatives for $A_5^5.C_{10}$ |
| Character table for $A_5^5.C_{10}$ |
Intermediate fields
| 5.5.390625.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 30 sibling: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20{,}\,{\href{/padicField/2.5.0.1}{5} }$ | ${\href{/padicField/3.10.0.1}{10} }{,}\,{\href{/padicField/3.5.0.1}{5} }^{3}$ | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }^{4}{,}\,{\href{/padicField/7.1.0.1}{1} }^{7}$ | ${\href{/padicField/11.10.0.1}{10} }^{2}{,}\,{\href{/padicField/11.5.0.1}{5} }$ | $15{,}\,{\href{/padicField/13.10.0.1}{10} }$ | ${\href{/padicField/17.10.0.1}{10} }{,}\,{\href{/padicField/17.5.0.1}{5} }^{3}$ | $25$ | $15{,}\,{\href{/padicField/23.10.0.1}{10} }$ | $25$ | $25$ | $20{,}\,{\href{/padicField/37.5.0.1}{5} }$ | $15{,}\,{\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{3}{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | $15{,}\,{\href{/padicField/47.10.0.1}{10} }$ | $20{,}\,{\href{/padicField/53.5.0.1}{5} }$ | $15{,}\,{\href{/padicField/59.5.0.1}{5} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| Deg $25$ | $25$ | $1$ | $69$ |
Spectrum of ring of integers
Additional information
This was the first (historically) explicit field which was ramified at only one prime less than $11$ whose defining polynomial has non-solvable Galois group [10.1142/S1793042111004113, MR:2782660].