Normalized defining polynomial
\( x^{25} - 10 x^{24} + 35 x^{23} - 30 x^{22} - 110 x^{21} + 255 x^{20} + 120 x^{19} - 1090 x^{18} + \cdots + 1 \)
Invariants
| Degree: | $25$ |
| |
| Signature: | $(5, 10)$ |
| |
| Discriminant: |
\(47683715820312500000000000000000000\)
\(\medspace = 2^{20}\cdot 5^{41}\)
|
| |
| Root discriminant: | \(24.39\) |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $\frac{1}{27\cdots 03}a^{24}+\frac{28\cdots 07}{27\cdots 03}a^{23}+\frac{13\cdots 81}{27\cdots 03}a^{22}-\frac{17\cdots 21}{27\cdots 03}a^{21}+\frac{86\cdots 45}{27\cdots 03}a^{20}+\frac{10\cdots 41}{27\cdots 03}a^{19}+\frac{31\cdots 48}{27\cdots 03}a^{18}-\frac{10\cdots 80}{27\cdots 03}a^{17}-\frac{45\cdots 74}{27\cdots 03}a^{16}+\frac{21\cdots 24}{27\cdots 03}a^{15}-\frac{64\cdots 10}{27\cdots 03}a^{14}-\frac{30\cdots 67}{27\cdots 03}a^{13}-\frac{34\cdots 11}{27\cdots 03}a^{12}-\frac{37\cdots 83}{27\cdots 03}a^{11}+\frac{69\cdots 13}{27\cdots 03}a^{10}+\frac{17\cdots 66}{27\cdots 03}a^{9}-\frac{75\cdots 64}{27\cdots 03}a^{8}+\frac{76\cdots 41}{27\cdots 03}a^{7}-\frac{87\cdots 94}{27\cdots 03}a^{6}+\frac{11\cdots 57}{27\cdots 03}a^{5}-\frac{74\cdots 75}{27\cdots 03}a^{4}+\frac{15\cdots 59}{27\cdots 03}a^{3}-\frac{13\cdots 08}{27\cdots 03}a^{2}+\frac{13\cdots 04}{27\cdots 03}a-\frac{22\cdots 70}{27\cdots 03}$
| 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: |
$\frac{13\cdots 83}{27\cdots 03}a^{24}+\frac{13\cdots 05}{27\cdots 03}a^{23}-\frac{45\cdots 37}{27\cdots 03}a^{22}+\frac{34\cdots 96}{27\cdots 03}a^{21}+\frac{15\cdots 26}{27\cdots 03}a^{20}-\frac{32\cdots 45}{27\cdots 03}a^{19}-\frac{20\cdots 34}{27\cdots 03}a^{18}+\frac{14\cdots 86}{27\cdots 03}a^{17}-\frac{21\cdots 85}{27\cdots 03}a^{16}+\frac{19\cdots 57}{27\cdots 03}a^{15}-\frac{28\cdots 40}{27\cdots 03}a^{14}+\frac{68\cdots 40}{27\cdots 03}a^{13}-\frac{13\cdots 20}{27\cdots 03}a^{12}+\frac{19\cdots 92}{27\cdots 03}a^{11}-\frac{17\cdots 56}{27\cdots 03}a^{10}+\frac{87\cdots 02}{27\cdots 03}a^{9}+\frac{19\cdots 67}{27\cdots 03}a^{8}-\frac{41\cdots 02}{27\cdots 03}a^{7}+\frac{34\cdots 31}{27\cdots 03}a^{6}-\frac{22\cdots 40}{27\cdots 03}a^{5}+\frac{10\cdots 73}{27\cdots 03}a^{4}-\frac{35\cdots 76}{27\cdots 03}a^{3}+\frac{14\cdots 40}{27\cdots 03}a^{2}-\frac{21\cdots 01}{27\cdots 03}a+\frac{88\cdots 44}{27\cdots 03}$, $\frac{50\cdots 57}{27\cdots 03}a^{24}+\frac{51\cdots 22}{27\cdots 03}a^{23}-\frac{18\cdots 91}{27\cdots 03}a^{22}+\frac{17\cdots 56}{27\cdots 03}a^{21}+\frac{56\cdots 97}{27\cdots 03}a^{20}-\frac{14\cdots 28}{27\cdots 03}a^{19}-\frac{47\cdots 86}{27\cdots 03}a^{18}+\frac{58\cdots 98}{27\cdots 03}a^{17}-\frac{95\cdots 56}{27\cdots 03}a^{16}+\frac{88\cdots 91}{27\cdots 03}a^{15}-\frac{11\cdots 70}{27\cdots 03}a^{14}+\frac{27\cdots 50}{27\cdots 03}a^{13}-\frac{58\cdots 53}{27\cdots 03}a^{12}+\frac{86\cdots 63}{27\cdots 03}a^{11}-\frac{81\cdots 96}{27\cdots 03}a^{10}+\frac{41\cdots 72}{27\cdots 03}a^{9}+\frac{24\cdots 87}{27\cdots 03}a^{8}-\frac{21\cdots 39}{27\cdots 03}a^{7}+\frac{17\cdots 99}{27\cdots 03}a^{6}-\frac{96\cdots 65}{27\cdots 03}a^{5}+\frac{40\cdots 62}{27\cdots 03}a^{4}-\frac{11\cdots 76}{27\cdots 03}a^{3}+\frac{28\cdots 33}{27\cdots 03}a^{2}-\frac{11\cdots 61}{27\cdots 03}a-\frac{21\cdots 43}{27\cdots 03}$, $\frac{57\cdots 67}{27\cdots 03}a^{24}-\frac{55\cdots 17}{27\cdots 03}a^{23}+\frac{18\cdots 49}{27\cdots 03}a^{22}-\frac{10\cdots 01}{27\cdots 03}a^{21}-\frac{68\cdots 17}{27\cdots 03}a^{20}+\frac{12\cdots 73}{27\cdots 03}a^{19}+\frac{11\cdots 04}{27\cdots 03}a^{18}-\frac{60\cdots 26}{27\cdots 03}a^{17}+\frac{77\cdots 46}{27\cdots 03}a^{16}-\frac{62\cdots 48}{27\cdots 03}a^{15}+\frac{10\cdots 98}{27\cdots 03}a^{14}-\frac{27\cdots 32}{27\cdots 03}a^{13}+\frac{54\cdots 63}{27\cdots 03}a^{12}-\frac{72\cdots 32}{27\cdots 03}a^{11}+\frac{58\cdots 03}{27\cdots 03}a^{10}-\frac{21\cdots 81}{27\cdots 03}a^{9}-\frac{80\cdots 95}{27\cdots 03}a^{8}+\frac{16\cdots 53}{27\cdots 03}a^{7}-\frac{99\cdots 44}{27\cdots 03}a^{6}+\frac{66\cdots 42}{27\cdots 03}a^{5}-\frac{28\cdots 72}{27\cdots 03}a^{4}+\frac{10\cdots 82}{27\cdots 03}a^{3}-\frac{47\cdots 09}{27\cdots 03}a^{2}+\frac{40\cdots 70}{27\cdots 03}a-\frac{22\cdots 73}{27\cdots 03}$, $\frac{99\cdots 39}{27\cdots 03}a^{24}+\frac{65\cdots 72}{27\cdots 03}a^{23}-\frac{13\cdots 36}{27\cdots 03}a^{22}-\frac{91\cdots 12}{27\cdots 03}a^{21}+\frac{20\cdots 27}{27\cdots 03}a^{20}+\frac{16\cdots 27}{27\cdots 03}a^{19}-\frac{10\cdots 79}{27\cdots 03}a^{18}+\frac{56\cdots 06}{27\cdots 03}a^{17}+\frac{23\cdots 05}{27\cdots 03}a^{16}-\frac{42\cdots 12}{27\cdots 03}a^{15}+\frac{23\cdots 43}{27\cdots 03}a^{14}-\frac{97\cdots 68}{27\cdots 03}a^{13}+\frac{63\cdots 77}{27\cdots 03}a^{12}-\frac{20\cdots 76}{27\cdots 03}a^{11}+\frac{36\cdots 85}{27\cdots 03}a^{10}-\frac{34\cdots 05}{27\cdots 03}a^{9}+\frac{13\cdots 10}{27\cdots 03}a^{8}+\frac{75\cdots 33}{27\cdots 03}a^{7}-\frac{13\cdots 10}{27\cdots 03}a^{6}+\frac{63\cdots 68}{27\cdots 03}a^{5}-\frac{19\cdots 06}{27\cdots 03}a^{4}+\frac{56\cdots 60}{27\cdots 03}a^{3}+\frac{22\cdots 23}{27\cdots 03}a^{2}+\frac{36\cdots 24}{27\cdots 03}a+\frac{22\cdots 25}{27\cdots 03}$, $\frac{53\cdots 15}{27\cdots 03}a^{24}-\frac{49\cdots 33}{27\cdots 03}a^{23}+\frac{14\cdots 78}{27\cdots 03}a^{22}-\frac{32\cdots 22}{27\cdots 03}a^{21}-\frac{66\cdots 82}{27\cdots 03}a^{20}+\frac{87\cdots 06}{27\cdots 03}a^{19}+\frac{15\cdots 97}{27\cdots 03}a^{18}-\frac{49\cdots 84}{27\cdots 03}a^{17}+\frac{48\cdots 73}{27\cdots 03}a^{16}-\frac{33\cdots 81}{27\cdots 03}a^{15}+\frac{85\cdots 78}{27\cdots 03}a^{14}-\frac{22\cdots 15}{27\cdots 03}a^{13}+\frac{40\cdots 71}{27\cdots 03}a^{12}-\frac{49\cdots 13}{27\cdots 03}a^{11}+\frac{32\cdots 03}{27\cdots 03}a^{10}-\frac{72\cdots 31}{27\cdots 03}a^{9}-\frac{55\cdots 61}{27\cdots 03}a^{8}+\frac{68\cdots 38}{27\cdots 03}a^{7}-\frac{41\cdots 86}{27\cdots 03}a^{6}+\frac{59\cdots 84}{27\cdots 03}a^{5}-\frac{24\cdots 18}{27\cdots 03}a^{4}+\frac{43\cdots 19}{27\cdots 03}a^{3}-\frac{35\cdots 00}{27\cdots 03}a^{2}-\frac{64\cdots 79}{27\cdots 03}a-\frac{68\cdots 11}{27\cdots 03}$, $\frac{54\cdots 44}{27\cdots 03}a^{24}-\frac{53\cdots 70}{27\cdots 03}a^{23}+\frac{18\cdots 56}{27\cdots 03}a^{22}-\frac{15\cdots 94}{27\cdots 03}a^{21}-\frac{60\cdots 99}{27\cdots 03}a^{20}+\frac{13\cdots 89}{27\cdots 03}a^{19}+\frac{72\cdots 83}{27\cdots 03}a^{18}-\frac{59\cdots 77}{27\cdots 03}a^{17}+\frac{90\cdots 13}{27\cdots 03}a^{16}-\frac{83\cdots 19}{27\cdots 03}a^{15}+\frac{12\cdots 76}{27\cdots 03}a^{14}-\frac{28\cdots 14}{27\cdots 03}a^{13}+\frac{59\cdots 62}{27\cdots 03}a^{12}-\frac{84\cdots 00}{27\cdots 03}a^{11}+\frac{78\cdots 26}{27\cdots 03}a^{10}-\frac{40\cdots 86}{27\cdots 03}a^{9}+\frac{28\cdots 07}{27\cdots 03}a^{8}+\frac{14\cdots 66}{27\cdots 03}a^{7}-\frac{13\cdots 90}{27\cdots 03}a^{6}+\frac{10\cdots 25}{27\cdots 03}a^{5}-\frac{56\cdots 67}{27\cdots 03}a^{4}+\frac{22\cdots 13}{27\cdots 03}a^{3}-\frac{86\cdots 80}{27\cdots 03}a^{2}+\frac{19\cdots 79}{27\cdots 03}a-\frac{31\cdots 46}{27\cdots 03}$, $\frac{96\cdots 58}{27\cdots 03}a^{24}-\frac{95\cdots 89}{27\cdots 03}a^{23}+\frac{32\cdots 59}{27\cdots 03}a^{22}-\frac{23\cdots 73}{27\cdots 03}a^{21}-\frac{11\cdots 89}{27\cdots 03}a^{20}+\frac{23\cdots 59}{27\cdots 03}a^{19}+\frac{15\cdots 76}{27\cdots 03}a^{18}-\frac{10\cdots 67}{27\cdots 03}a^{17}+\frac{15\cdots 75}{27\cdots 03}a^{16}-\frac{12\cdots 29}{27\cdots 03}a^{15}+\frac{18\cdots 06}{27\cdots 03}a^{14}-\frac{47\cdots 45}{27\cdots 03}a^{13}+\frac{98\cdots 61}{27\cdots 03}a^{12}-\frac{13\cdots 15}{27\cdots 03}a^{11}+\frac{11\cdots 55}{27\cdots 03}a^{10}-\frac{49\cdots 71}{27\cdots 03}a^{9}-\frac{13\cdots 15}{27\cdots 03}a^{8}+\frac{35\cdots 75}{27\cdots 03}a^{7}-\frac{23\cdots 32}{27\cdots 03}a^{6}+\frac{12\cdots 75}{27\cdots 03}a^{5}-\frac{46\cdots 28}{27\cdots 03}a^{4}+\frac{12\cdots 33}{27\cdots 03}a^{3}-\frac{61\cdots 18}{27\cdots 03}a^{2}+\frac{47\cdots 36}{27\cdots 03}a-\frac{11\cdots 62}{27\cdots 03}$, $\frac{87\cdots 01}{27\cdots 03}a^{24}+\frac{85\cdots 96}{27\cdots 03}a^{23}-\frac{28\cdots 45}{27\cdots 03}a^{22}+\frac{17\cdots 87}{27\cdots 03}a^{21}+\frac{10\cdots 02}{27\cdots 03}a^{20}-\frac{20\cdots 90}{27\cdots 03}a^{19}-\frac{17\cdots 90}{27\cdots 03}a^{18}+\frac{95\cdots 33}{27\cdots 03}a^{17}-\frac{12\cdots 59}{27\cdots 03}a^{16}+\frac{94\cdots 95}{27\cdots 03}a^{15}-\frac{15\cdots 69}{27\cdots 03}a^{14}+\frac{41\cdots 34}{27\cdots 03}a^{13}-\frac{84\cdots 75}{27\cdots 03}a^{12}+\frac{11\cdots 04}{27\cdots 03}a^{11}-\frac{88\cdots 72}{27\cdots 03}a^{10}+\frac{26\cdots 93}{27\cdots 03}a^{9}+\frac{20\cdots 74}{27\cdots 03}a^{8}-\frac{29\cdots 78}{27\cdots 03}a^{7}+\frac{14\cdots 71}{27\cdots 03}a^{6}-\frac{74\cdots 51}{27\cdots 03}a^{5}+\frac{28\cdots 26}{27\cdots 03}a^{4}-\frac{75\cdots 73}{27\cdots 03}a^{3}+\frac{57\cdots 27}{27\cdots 03}a^{2}-\frac{19\cdots 76}{27\cdots 03}a+\frac{15\cdots 53}{27\cdots 03}$, $\frac{94\cdots 20}{51\cdots 51}a^{24}+\frac{91\cdots 50}{51\cdots 51}a^{23}-\frac{29\cdots 12}{51\cdots 51}a^{22}+\frac{15\cdots 36}{51\cdots 51}a^{21}+\frac{11\cdots 33}{51\cdots 51}a^{20}-\frac{20\cdots 14}{51\cdots 51}a^{19}-\frac{21\cdots 91}{51\cdots 51}a^{18}+\frac{10\cdots 43}{51\cdots 51}a^{17}-\frac{12\cdots 42}{51\cdots 51}a^{16}+\frac{85\cdots 84}{51\cdots 51}a^{15}-\frac{15\cdots 56}{51\cdots 51}a^{14}+\frac{43\cdots 59}{51\cdots 51}a^{13}-\frac{85\cdots 54}{51\cdots 51}a^{12}+\frac{11\cdots 63}{51\cdots 51}a^{11}-\frac{81\cdots 85}{51\cdots 51}a^{10}+\frac{17\cdots 21}{51\cdots 51}a^{9}+\frac{25\cdots 65}{51\cdots 51}a^{8}-\frac{28\cdots 75}{51\cdots 51}a^{7}+\frac{11\cdots 24}{51\cdots 51}a^{6}-\frac{61\cdots 07}{51\cdots 51}a^{5}+\frac{26\cdots 96}{51\cdots 51}a^{4}-\frac{60\cdots 41}{51\cdots 51}a^{3}+\frac{47\cdots 00}{51\cdots 51}a^{2}+\frac{66\cdots 22}{51\cdots 51}a+\frac{45\cdots 74}{51\cdots 51}$, $\frac{26\cdots 54}{27\cdots 03}a^{24}-\frac{28\cdots 70}{27\cdots 03}a^{23}+\frac{10\cdots 60}{27\cdots 03}a^{22}-\frac{12\cdots 85}{27\cdots 03}a^{21}-\frac{28\cdots 45}{27\cdots 03}a^{20}+\frac{90\cdots 73}{27\cdots 03}a^{19}+\frac{42\cdots 32}{27\cdots 03}a^{18}-\frac{34\cdots 76}{27\cdots 03}a^{17}+\frac{62\cdots 13}{27\cdots 03}a^{16}-\frac{57\cdots 80}{27\cdots 03}a^{15}+\frac{65\cdots 48}{27\cdots 03}a^{14}-\frac{16\cdots 52}{27\cdots 03}a^{13}+\frac{35\cdots 51}{27\cdots 03}a^{12}-\frac{54\cdots 29}{27\cdots 03}a^{11}+\frac{52\cdots 64}{27\cdots 03}a^{10}-\frac{25\cdots 58}{27\cdots 03}a^{9}-\frac{40\cdots 70}{27\cdots 03}a^{8}+\frac{16\cdots 58}{27\cdots 03}a^{7}-\frac{12\cdots 94}{27\cdots 03}a^{6}+\frac{50\cdots 14}{27\cdots 03}a^{5}-\frac{19\cdots 45}{27\cdots 03}a^{4}+\frac{29\cdots 41}{27\cdots 03}a^{3}-\frac{10\cdots 09}{27\cdots 03}a^{2}+\frac{38\cdots 99}{27\cdots 03}a+\frac{15\cdots 49}{27\cdots 03}$, $\frac{20\cdots 32}{27\cdots 03}a^{24}+\frac{20\cdots 03}{27\cdots 03}a^{23}-\frac{67\cdots 49}{27\cdots 03}a^{22}+\frac{48\cdots 41}{27\cdots 03}a^{21}+\frac{23\cdots 60}{27\cdots 03}a^{20}-\frac{49\cdots 80}{27\cdots 03}a^{19}-\frac{34\cdots 16}{27\cdots 03}a^{18}+\frac{22\cdots 52}{27\cdots 03}a^{17}-\frac{31\cdots 11}{27\cdots 03}a^{16}+\frac{26\cdots 11}{27\cdots 03}a^{15}-\frac{38\cdots 95}{27\cdots 03}a^{14}+\frac{10\cdots 76}{27\cdots 03}a^{13}-\frac{20\cdots 40}{27\cdots 03}a^{12}+\frac{28\cdots 84}{27\cdots 03}a^{11}-\frac{24\cdots 32}{27\cdots 03}a^{10}+\frac{95\cdots 72}{27\cdots 03}a^{9}+\frac{33\cdots 10}{27\cdots 03}a^{8}-\frac{73\cdots 77}{27\cdots 03}a^{7}+\frac{46\cdots 45}{27\cdots 03}a^{6}-\frac{24\cdots 38}{27\cdots 03}a^{5}+\frac{94\cdots 61}{27\cdots 03}a^{4}-\frac{25\cdots 97}{27\cdots 03}a^{3}+\frac{12\cdots 06}{27\cdots 03}a^{2}-\frac{54\cdots 50}{27\cdots 03}a+\frac{58\cdots 92}{27\cdots 03}$, $\frac{12\cdots 04}{27\cdots 03}a^{24}+\frac{12\cdots 66}{27\cdots 03}a^{23}-\frac{41\cdots 72}{27\cdots 03}a^{22}+\frac{33\cdots 56}{27\cdots 03}a^{21}+\frac{13\cdots 16}{27\cdots 03}a^{20}-\frac{30\cdots 03}{27\cdots 03}a^{19}-\frac{16\cdots 33}{27\cdots 03}a^{18}+\frac{13\cdots 92}{27\cdots 03}a^{17}-\frac{19\cdots 48}{27\cdots 03}a^{16}+\frac{18\cdots 38}{27\cdots 03}a^{15}-\frac{25\cdots 67}{27\cdots 03}a^{14}+\frac{62\cdots 30}{27\cdots 03}a^{13}-\frac{12\cdots 04}{27\cdots 03}a^{12}+\frac{18\cdots 16}{27\cdots 03}a^{11}-\frac{16\cdots 46}{27\cdots 03}a^{10}+\frac{80\cdots 50}{27\cdots 03}a^{9}+\frac{60\cdots 65}{27\cdots 03}a^{8}-\frac{43\cdots 93}{27\cdots 03}a^{7}+\frac{34\cdots 80}{27\cdots 03}a^{6}-\frac{20\cdots 07}{27\cdots 03}a^{5}+\frac{86\cdots 07}{27\cdots 03}a^{4}-\frac{25\cdots 00}{27\cdots 03}a^{3}+\frac{95\cdots 81}{27\cdots 03}a^{2}-\frac{78\cdots 46}{27\cdots 03}a+\frac{37\cdots 27}{27\cdots 03}$, $a$, $\frac{44\cdots 78}{27\cdots 03}a^{24}-\frac{41\cdots 86}{27\cdots 03}a^{23}+\frac{12\cdots 28}{27\cdots 03}a^{22}-\frac{32\cdots 27}{27\cdots 03}a^{21}-\frac{57\cdots 52}{27\cdots 03}a^{20}+\frac{79\cdots 28}{27\cdots 03}a^{19}+\frac{12\cdots 72}{27\cdots 03}a^{18}-\frac{44\cdots 24}{27\cdots 03}a^{17}+\frac{43\cdots 16}{27\cdots 03}a^{16}-\frac{25\cdots 02}{27\cdots 03}a^{15}+\frac{63\cdots 69}{27\cdots 03}a^{14}-\frac{18\cdots 44}{27\cdots 03}a^{13}+\frac{34\cdots 56}{27\cdots 03}a^{12}-\frac{41\cdots 28}{27\cdots 03}a^{11}+\frac{25\cdots 98}{27\cdots 03}a^{10}-\frac{19\cdots 79}{27\cdots 03}a^{9}-\frac{11\cdots 86}{27\cdots 03}a^{8}+\frac{87\cdots 01}{27\cdots 03}a^{7}-\frac{21\cdots 04}{27\cdots 03}a^{6}+\frac{24\cdots 30}{27\cdots 03}a^{5}-\frac{10\cdots 01}{27\cdots 03}a^{4}+\frac{23\cdots 39}{27\cdots 03}a^{3}-\frac{18\cdots 35}{27\cdots 03}a^{2}-\frac{89\cdots 83}{27\cdots 03}a-\frac{37\cdots 16}{27\cdots 03}$
|
| |
| Regulator: | \( 34092554.06508241 \) (assuming GRH) |
| |
| Unit signature rank: | \( 5 \) (assuming GRH) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{5}\cdot(2\pi)^{10}\cdot 34092554.06508241 \cdot 1}{2\cdot\sqrt{47683715820312500000000000000000000}}\cr\approx \mathstrut & 0.239548336100025 \end{aligned}\] (assuming GRH)
Galois group
$C_5^2:F_5$ (as 25T35):
| A solvable group of order 500 |
| The 26 conjugacy class representatives for $C_5^2:F_5$ |
| Character table for $C_5^2:F_5$ |
Intermediate fields
| 5.1.50000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 25 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 | R | $20{,}\,{\href{/padicField/3.5.0.1}{5} }$ | R | $20{,}\,{\href{/padicField/7.5.0.1}{5} }$ | ${\href{/padicField/11.5.0.1}{5} }^{5}$ | $20{,}\,{\href{/padicField/13.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/17.5.0.1}{5} }$ | ${\href{/padicField/19.10.0.1}{10} }^{2}{,}\,{\href{/padicField/19.5.0.1}{5} }$ | ${\href{/padicField/23.4.0.1}{4} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }^{5}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}{,}\,{\href{/padicField/29.5.0.1}{5} }$ | ${\href{/padicField/31.5.0.1}{5} }^{5}$ | $20{,}\,{\href{/padicField/37.5.0.1}{5} }$ | ${\href{/padicField/41.5.0.1}{5} }^{5}$ | $20{,}\,{\href{/padicField/43.5.0.1}{5} }$ | $20{,}\,{\href{/padicField/47.5.0.1}{5} }$ | ${\href{/padicField/53.4.0.1}{4} }^{5}{,}\,{\href{/padicField/53.1.0.1}{1} }^{5}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}{,}\,{\href{/padicField/59.5.0.1}{5} }$ |
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 $25$ | $25$ | $1$ | $41$ |