Normalized defining polynomial
\( x^{26} - 11 x^{25} + 45 x^{24} - 240 x^{23} + 1425 x^{22} - 4005 x^{21} + 12885 x^{20} + \cdots - 14641021707 \)
Invariants
| Degree: | $26$ |
| |
| Signature: | $[2, 12]$ |
| |
| Discriminant: |
\(167794323485736669509699424789101816713809967041015625\)
\(\medspace = 5^{30}\cdot 41^{20}\)
|
| |
| Root discriminant: | \(111.46\) |
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(5\), \(41\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $a^{18}$, $a^{19}$, $a^{20}$, $a^{21}$, $a^{22}$, $a^{23}$, $a^{24}$, $\frac{1}{44\cdots 73}a^{25}-\frac{28\cdots 66}{44\cdots 73}a^{24}-\frac{16\cdots 25}{44\cdots 73}a^{23}-\frac{20\cdots 44}{44\cdots 73}a^{22}+\frac{94\cdots 22}{44\cdots 73}a^{21}-\frac{22\cdots 72}{44\cdots 73}a^{20}+\frac{89\cdots 61}{44\cdots 73}a^{19}+\frac{11\cdots 69}{44\cdots 73}a^{18}-\frac{11\cdots 93}{44\cdots 73}a^{17}+\frac{71\cdots 50}{44\cdots 73}a^{16}-\frac{17\cdots 84}{44\cdots 73}a^{15}+\frac{78\cdots 27}{44\cdots 73}a^{14}+\frac{21\cdots 35}{44\cdots 73}a^{13}-\frac{13\cdots 32}{44\cdots 73}a^{12}-\frac{13\cdots 80}{44\cdots 73}a^{11}+\frac{58\cdots 63}{44\cdots 73}a^{10}+\frac{85\cdots 75}{44\cdots 73}a^{9}-\frac{91\cdots 67}{44\cdots 73}a^{8}+\frac{17\cdots 50}{44\cdots 73}a^{7}-\frac{14\cdots 63}{44\cdots 73}a^{6}-\frac{63\cdots 33}{44\cdots 73}a^{5}+\frac{49\cdots 13}{44\cdots 73}a^{4}+\frac{14\cdots 39}{44\cdots 73}a^{3}-\frac{43\cdots 20}{44\cdots 73}a^{2}+\frac{19\cdots 25}{44\cdots 73}a+\frac{47\cdots 09}{44\cdots 73}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{6}$, which has order $6$ (assuming GRH) |
| |
| Narrow class group: | $C_{6}$, which has order $6$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{87\cdots 97}{44\cdots 73}a^{25}+\frac{84\cdots 82}{44\cdots 73}a^{24}-\frac{25\cdots 40}{44\cdots 73}a^{23}+\frac{14\cdots 98}{44\cdots 73}a^{22}-\frac{91\cdots 75}{44\cdots 73}a^{21}+\frac{16\cdots 36}{44\cdots 73}a^{20}-\frac{55\cdots 51}{44\cdots 73}a^{19}+\frac{24\cdots 82}{44\cdots 73}a^{18}+\frac{23\cdots 84}{44\cdots 73}a^{17}+\frac{31\cdots 54}{44\cdots 73}a^{16}-\frac{21\cdots 80}{44\cdots 73}a^{15}-\frac{16\cdots 44}{44\cdots 73}a^{14}+\frac{24\cdots 97}{44\cdots 73}a^{13}+\frac{51\cdots 90}{44\cdots 73}a^{12}+\frac{26\cdots 86}{44\cdots 73}a^{11}-\frac{29\cdots 84}{44\cdots 73}a^{10}-\frac{18\cdots 72}{44\cdots 73}a^{9}-\frac{49\cdots 58}{44\cdots 73}a^{8}+\frac{36\cdots 91}{44\cdots 73}a^{7}+\frac{52\cdots 50}{44\cdots 73}a^{6}+\frac{11\cdots 56}{44\cdots 73}a^{5}+\frac{23\cdots 68}{44\cdots 73}a^{4}+\frac{19\cdots 73}{44\cdots 73}a^{3}+\frac{83\cdots 13}{44\cdots 73}a^{2}+\frac{41\cdots 12}{44\cdots 73}a-\frac{48\cdots 33}{44\cdots 73}$, $\frac{30\cdots 81}{44\cdots 73}a^{25}-\frac{37\cdots 91}{44\cdots 73}a^{24}+\frac{18\cdots 04}{44\cdots 73}a^{23}-\frac{94\cdots 31}{44\cdots 73}a^{22}+\frac{55\cdots 16}{44\cdots 73}a^{21}-\frac{18\cdots 13}{44\cdots 73}a^{20}+\frac{60\cdots 46}{44\cdots 73}a^{19}-\frac{22\cdots 39}{44\cdots 73}a^{18}+\frac{42\cdots 45}{44\cdots 73}a^{17}-\frac{90\cdots 06}{44\cdots 73}a^{16}+\frac{26\cdots 74}{44\cdots 73}a^{15}+\frac{16\cdots 68}{44\cdots 73}a^{14}-\frac{57\cdots 21}{44\cdots 73}a^{13}-\frac{91\cdots 19}{44\cdots 73}a^{12}-\frac{62\cdots 99}{44\cdots 73}a^{11}+\frac{29\cdots 83}{44\cdots 73}a^{10}+\frac{57\cdots 29}{44\cdots 73}a^{9}+\frac{12\cdots 88}{44\cdots 73}a^{8}-\frac{52\cdots 91}{44\cdots 73}a^{7}-\frac{82\cdots 36}{44\cdots 73}a^{6}-\frac{10\cdots 32}{44\cdots 73}a^{5}-\frac{30\cdots 21}{44\cdots 73}a^{4}+\frac{62\cdots 44}{44\cdots 73}a^{3}-\frac{75\cdots 31}{44\cdots 73}a^{2}+\frac{15\cdots 24}{44\cdots 73}a-\frac{80\cdots 29}{44\cdots 73}$, $\frac{57\cdots 26}{44\cdots 73}a^{25}+\frac{52\cdots 69}{44\cdots 73}a^{24}-\frac{13\cdots 69}{44\cdots 73}a^{23}+\frac{90\cdots 53}{44\cdots 73}a^{22}-\frac{56\cdots 54}{44\cdots 73}a^{21}+\frac{77\cdots 56}{44\cdots 73}a^{20}-\frac{32\cdots 04}{44\cdots 73}a^{19}+\frac{16\cdots 09}{44\cdots 73}a^{18}+\frac{22\cdots 18}{44\cdots 73}a^{17}+\frac{31\cdots 45}{44\cdots 73}a^{16}-\frac{17\cdots 66}{44\cdots 73}a^{15}-\frac{12\cdots 95}{44\cdots 73}a^{14}-\frac{40\cdots 70}{44\cdots 73}a^{13}+\frac{49\cdots 55}{44\cdots 73}a^{12}+\frac{19\cdots 79}{44\cdots 73}a^{11}-\frac{96\cdots 06}{44\cdots 73}a^{10}-\frac{16\cdots 53}{44\cdots 73}a^{9}-\frac{43\cdots 49}{44\cdots 73}a^{8}+\frac{26\cdots 94}{44\cdots 73}a^{7}+\frac{39\cdots 96}{44\cdots 73}a^{6}+\frac{10\cdots 95}{44\cdots 73}a^{5}+\frac{15\cdots 97}{44\cdots 73}a^{4}+\frac{94\cdots 19}{44\cdots 73}a^{3}-\frac{14\cdots 13}{44\cdots 73}a^{2}-\frac{63\cdots 27}{44\cdots 73}a-\frac{29\cdots 36}{44\cdots 73}$, $\frac{16\cdots 02}{44\cdots 73}a^{25}+\frac{15\cdots 56}{44\cdots 73}a^{24}-\frac{50\cdots 32}{44\cdots 73}a^{23}+\frac{30\cdots 75}{44\cdots 73}a^{22}-\frac{18\cdots 52}{44\cdots 73}a^{21}+\frac{37\cdots 61}{44\cdots 73}a^{20}-\frac{14\cdots 04}{44\cdots 73}a^{19}+\frac{58\cdots 93}{44\cdots 73}a^{18}+\frac{50\cdots 86}{44\cdots 73}a^{17}+\frac{20\cdots 80}{44\cdots 73}a^{16}-\frac{50\cdots 30}{44\cdots 73}a^{15}-\frac{27\cdots 20}{44\cdots 73}a^{14}-\frac{46\cdots 61}{44\cdots 73}a^{13}+\frac{56\cdots 07}{44\cdots 73}a^{12}+\frac{53\cdots 45}{44\cdots 73}a^{11}-\frac{54\cdots 93}{44\cdots 73}a^{10}-\frac{24\cdots 92}{44\cdots 73}a^{9}-\frac{11\cdots 39}{44\cdots 73}a^{8}+\frac{52\cdots 13}{44\cdots 73}a^{7}+\frac{78\cdots 80}{44\cdots 73}a^{6}+\frac{23\cdots 19}{44\cdots 73}a^{5}+\frac{54\cdots 80}{44\cdots 73}a^{4}+\frac{67\cdots 90}{44\cdots 73}a^{3}+\frac{80\cdots 43}{44\cdots 73}a^{2}+\frac{59\cdots 64}{44\cdots 73}a+\frac{35\cdots 95}{44\cdots 73}$, $\frac{36\cdots 50}{44\cdots 73}a^{25}+\frac{36\cdots 29}{44\cdots 73}a^{24}-\frac{12\cdots 43}{44\cdots 73}a^{23}+\frac{74\cdots 35}{44\cdots 73}a^{22}-\frac{44\cdots 55}{44\cdots 73}a^{21}+\frac{97\cdots 33}{44\cdots 73}a^{20}-\frac{37\cdots 25}{44\cdots 73}a^{19}+\frac{15\cdots 23}{44\cdots 73}a^{18}-\frac{34\cdots 70}{44\cdots 73}a^{17}+\frac{50\cdots 00}{44\cdots 73}a^{16}-\frac{14\cdots 52}{44\cdots 73}a^{15}-\frac{62\cdots 89}{44\cdots 73}a^{14}+\frac{51\cdots 35}{44\cdots 73}a^{13}+\frac{14\cdots 30}{44\cdots 73}a^{12}+\frac{13\cdots 03}{44\cdots 73}a^{11}-\frac{14\cdots 75}{44\cdots 73}a^{10}-\frac{63\cdots 40}{44\cdots 73}a^{9}-\frac{26\cdots 16}{44\cdots 73}a^{8}+\frac{19\cdots 20}{44\cdots 73}a^{7}+\frac{20\cdots 75}{44\cdots 73}a^{6}+\frac{51\cdots 72}{44\cdots 73}a^{5}+\frac{11\cdots 13}{44\cdots 73}a^{4}+\frac{75\cdots 98}{44\cdots 73}a^{3}+\frac{12\cdots 77}{44\cdots 73}a^{2}-\frac{10\cdots 12}{44\cdots 73}a-\frac{11\cdots 63}{44\cdots 73}$, $\frac{14\cdots 41}{44\cdots 73}a^{25}+\frac{15\cdots 68}{44\cdots 73}a^{24}-\frac{65\cdots 81}{44\cdots 73}a^{23}+\frac{36\cdots 86}{44\cdots 73}a^{22}-\frac{21\cdots 16}{44\cdots 73}a^{21}+\frac{58\cdots 05}{44\cdots 73}a^{20}-\frac{20\cdots 28}{44\cdots 73}a^{19}+\frac{74\cdots 23}{44\cdots 73}a^{18}-\frac{71\cdots 01}{44\cdots 73}a^{17}+\frac{23\cdots 98}{44\cdots 73}a^{16}-\frac{59\cdots 78}{44\cdots 73}a^{15}-\frac{20\cdots 39}{44\cdots 73}a^{14}+\frac{13\cdots 77}{44\cdots 73}a^{13}+\frac{37\cdots 43}{44\cdots 73}a^{12}+\frac{44\cdots 16}{44\cdots 73}a^{11}-\frac{79\cdots 55}{44\cdots 73}a^{10}-\frac{14\cdots 40}{44\cdots 73}a^{9}-\frac{89\cdots 07}{44\cdots 73}a^{8}+\frac{10\cdots 26}{44\cdots 73}a^{7}+\frac{68\cdots 36}{44\cdots 73}a^{6}+\frac{14\cdots 12}{44\cdots 73}a^{5}+\frac{37\cdots 89}{44\cdots 73}a^{4}+\frac{13\cdots 93}{44\cdots 73}a^{3}+\frac{36\cdots 29}{44\cdots 73}a^{2}-\frac{13\cdots 58}{44\cdots 73}a-\frac{42\cdots 69}{44\cdots 73}$, $\frac{21\cdots 21}{44\cdots 73}a^{25}-\frac{24\cdots 55}{44\cdots 73}a^{24}+\frac{99\cdots 74}{44\cdots 73}a^{23}-\frac{47\cdots 62}{44\cdots 73}a^{22}+\frac{28\cdots 91}{44\cdots 73}a^{21}-\frac{79\cdots 12}{44\cdots 73}a^{20}+\frac{20\cdots 66}{44\cdots 73}a^{19}-\frac{83\cdots 19}{44\cdots 73}a^{18}+\frac{55\cdots 30}{44\cdots 73}a^{17}-\frac{11\cdots 79}{44\cdots 73}a^{16}+\frac{64\cdots 15}{44\cdots 73}a^{15}+\frac{26\cdots 26}{44\cdots 73}a^{14}-\frac{68\cdots 68}{44\cdots 73}a^{13}-\frac{11\cdots 58}{44\cdots 73}a^{12}-\frac{28\cdots 45}{44\cdots 73}a^{11}+\frac{17\cdots 48}{44\cdots 73}a^{10}+\frac{24\cdots 36}{44\cdots 73}a^{9}+\frac{36\cdots 86}{44\cdots 73}a^{8}-\frac{26\cdots 08}{44\cdots 73}a^{7}-\frac{91\cdots 50}{44\cdots 73}a^{6}-\frac{34\cdots 30}{44\cdots 73}a^{5}-\frac{16\cdots 19}{44\cdots 73}a^{4}+\frac{86\cdots 19}{44\cdots 73}a^{3}+\frac{20\cdots 62}{44\cdots 73}a^{2}+\frac{52\cdots 97}{44\cdots 73}a-\frac{45\cdots 04}{44\cdots 73}$, $\frac{63\cdots 45}{44\cdots 73}a^{25}-\frac{75\cdots 86}{44\cdots 73}a^{24}+\frac{33\cdots 37}{44\cdots 73}a^{23}-\frac{15\cdots 57}{44\cdots 73}a^{22}+\frac{92\cdots 11}{44\cdots 73}a^{21}-\frac{27\cdots 20}{44\cdots 73}a^{20}+\frac{73\cdots 86}{44\cdots 73}a^{19}-\frac{28\cdots 69}{44\cdots 73}a^{18}+\frac{30\cdots 36}{44\cdots 73}a^{17}-\frac{13\cdots 88}{44\cdots 73}a^{16}+\frac{20\cdots 29}{44\cdots 73}a^{15}+\frac{63\cdots 66}{44\cdots 73}a^{14}-\frac{24\cdots 82}{44\cdots 73}a^{13}-\frac{22\cdots 41}{44\cdots 73}a^{12}-\frac{63\cdots 55}{44\cdots 73}a^{11}+\frac{57\cdots 33}{44\cdots 73}a^{10}+\frac{40\cdots 00}{44\cdots 73}a^{9}+\frac{62\cdots 84}{44\cdots 73}a^{8}-\frac{87\cdots 19}{44\cdots 73}a^{7}-\frac{22\cdots 69}{44\cdots 73}a^{6}+\frac{43\cdots 25}{44\cdots 73}a^{5}-\frac{46\cdots 18}{44\cdots 73}a^{4}+\frac{22\cdots 87}{44\cdots 73}a^{3}+\frac{43\cdots 81}{44\cdots 73}a^{2}-\frac{17\cdots 75}{44\cdots 73}a-\frac{70\cdots 29}{44\cdots 73}$, $\frac{19\cdots 15}{44\cdots 73}a^{25}-\frac{22\cdots 20}{44\cdots 73}a^{24}+\frac{10\cdots 42}{44\cdots 73}a^{23}-\frac{54\cdots 58}{44\cdots 73}a^{22}+\frac{31\cdots 43}{44\cdots 73}a^{21}-\frac{10\cdots 98}{44\cdots 73}a^{20}+\frac{32\cdots 25}{44\cdots 73}a^{19}-\frac{12\cdots 27}{44\cdots 73}a^{18}+\frac{19\cdots 18}{44\cdots 73}a^{17}-\frac{43\cdots 78}{44\cdots 73}a^{16}+\frac{13\cdots 49}{44\cdots 73}a^{15}+\frac{12\cdots 13}{44\cdots 73}a^{14}-\frac{44\cdots 67}{44\cdots 73}a^{13}-\frac{33\cdots 47}{44\cdots 73}a^{12}-\frac{49\cdots 58}{44\cdots 73}a^{11}+\frac{18\cdots 85}{44\cdots 73}a^{10}+\frac{69\cdots 13}{44\cdots 73}a^{9}+\frac{86\cdots 00}{44\cdots 73}a^{8}-\frac{29\cdots 58}{44\cdots 73}a^{7}-\frac{62\cdots 94}{44\cdots 73}a^{6}-\frac{91\cdots 77}{44\cdots 73}a^{5}-\frac{26\cdots 66}{44\cdots 73}a^{4}+\frac{26\cdots 21}{44\cdots 73}a^{3}-\frac{51\cdots 07}{44\cdots 73}a^{2}+\frac{82\cdots 66}{44\cdots 73}a-\frac{36\cdots 92}{44\cdots 73}$, $\frac{79\cdots 20}{44\cdots 73}a^{25}+\frac{84\cdots 43}{44\cdots 73}a^{24}-\frac{32\cdots 59}{44\cdots 73}a^{23}+\frac{17\cdots 97}{44\cdots 73}a^{22}-\frac{10\cdots 85}{44\cdots 73}a^{21}+\frac{27\cdots 57}{44\cdots 73}a^{20}-\frac{89\cdots 42}{44\cdots 73}a^{19}+\frac{36\cdots 43}{44\cdots 73}a^{18}-\frac{25\cdots 40}{44\cdots 73}a^{17}+\frac{98\cdots 92}{44\cdots 73}a^{16}-\frac{39\cdots 32}{44\cdots 73}a^{15}-\frac{11\cdots 72}{44\cdots 73}a^{14}+\frac{85\cdots 16}{44\cdots 73}a^{13}+\frac{33\cdots 18}{44\cdots 73}a^{12}+\frac{24\cdots 98}{44\cdots 73}a^{11}-\frac{48\cdots 19}{44\cdots 73}a^{10}-\frac{11\cdots 94}{44\cdots 73}a^{9}-\frac{40\cdots 93}{44\cdots 73}a^{8}+\frac{86\cdots 16}{44\cdots 73}a^{7}+\frac{47\cdots 78}{44\cdots 73}a^{6}+\frac{96\cdots 03}{44\cdots 73}a^{5}+\frac{21\cdots 07}{44\cdots 73}a^{4}+\frac{12\cdots 23}{44\cdots 73}a^{3}+\frac{21\cdots 10}{44\cdots 73}a^{2}-\frac{34\cdots 13}{44\cdots 73}a-\frac{16\cdots 94}{44\cdots 73}$, $\frac{39\cdots 44}{44\cdots 73}a^{25}-\frac{58\cdots 35}{44\cdots 73}a^{24}+\frac{31\cdots 72}{44\cdots 73}a^{23}-\frac{13\cdots 62}{44\cdots 73}a^{22}+\frac{80\cdots 61}{44\cdots 73}a^{21}-\frac{31\cdots 69}{44\cdots 73}a^{20}+\frac{79\cdots 03}{44\cdots 73}a^{19}-\frac{29\cdots 46}{44\cdots 73}a^{18}+\frac{68\cdots 64}{44\cdots 73}a^{17}-\frac{33\cdots 55}{44\cdots 73}a^{16}+\frac{24\cdots 79}{44\cdots 73}a^{15}-\frac{15\cdots 01}{44\cdots 73}a^{14}-\frac{32\cdots 88}{44\cdots 73}a^{13}-\frac{11\cdots 62}{44\cdots 73}a^{12}+\frac{35\cdots 49}{44\cdots 73}a^{11}+\frac{74\cdots 23}{44\cdots 73}a^{10}-\frac{41\cdots 14}{44\cdots 73}a^{9}-\frac{20\cdots 55}{44\cdots 73}a^{8}-\frac{12\cdots 99}{44\cdots 73}a^{7}-\frac{92\cdots 40}{44\cdots 73}a^{6}+\frac{74\cdots 33}{44\cdots 73}a^{5}+\frac{98\cdots 18}{44\cdots 73}a^{4}+\frac{18\cdots 25}{44\cdots 73}a^{3}-\frac{18\cdots 06}{44\cdots 73}a^{2}-\frac{17\cdots 34}{44\cdots 73}a+\frac{48\cdots 32}{44\cdots 73}$, $\frac{30\cdots 78}{44\cdots 73}a^{25}-\frac{34\cdots 45}{44\cdots 73}a^{24}+\frac{15\cdots 34}{44\cdots 73}a^{23}-\frac{80\cdots 87}{44\cdots 73}a^{22}+\frac{47\cdots 39}{44\cdots 73}a^{21}-\frac{14\cdots 05}{44\cdots 73}a^{20}+\frac{47\cdots 83}{44\cdots 73}a^{19}-\frac{17\cdots 15}{44\cdots 73}a^{18}+\frac{24\cdots 02}{44\cdots 73}a^{17}-\frac{62\cdots 15}{44\cdots 73}a^{16}+\frac{19\cdots 72}{44\cdots 73}a^{15}+\frac{28\cdots 33}{44\cdots 73}a^{14}-\frac{49\cdots 12}{44\cdots 73}a^{13}-\frac{84\cdots 94}{44\cdots 73}a^{12}-\frac{90\cdots 56}{44\cdots 73}a^{11}+\frac{23\cdots 93}{44\cdots 73}a^{10}+\frac{25\cdots 26}{44\cdots 73}a^{9}+\frac{16\cdots 87}{44\cdots 73}a^{8}-\frac{38\cdots 65}{44\cdots 73}a^{7}-\frac{12\cdots 69}{44\cdots 73}a^{6}-\frac{23\cdots 55}{44\cdots 73}a^{5}-\frac{54\cdots 64}{44\cdots 73}a^{4}+\frac{35\cdots 16}{44\cdots 73}a^{3}-\frac{45\cdots 15}{44\cdots 73}a^{2}+\frac{11\cdots 10}{44\cdots 73}a-\frac{58\cdots 29}{44\cdots 73}$, $\frac{43\cdots 39}{44\cdots 73}a^{25}+\frac{50\cdots 99}{44\cdots 73}a^{24}-\frac{23\cdots 77}{44\cdots 73}a^{23}+\frac{12\cdots 03}{44\cdots 73}a^{22}-\frac{73\cdots 02}{44\cdots 73}a^{21}+\frac{23\cdots 02}{44\cdots 73}a^{20}-\frac{83\cdots 88}{44\cdots 73}a^{19}+\frac{31\cdots 13}{44\cdots 73}a^{18}-\frac{55\cdots 08}{44\cdots 73}a^{17}+\frac{15\cdots 01}{44\cdots 73}a^{16}-\frac{40\cdots 88}{44\cdots 73}a^{15}-\frac{19\cdots 14}{44\cdots 73}a^{14}-\frac{59\cdots 30}{44\cdots 73}a^{13}+\frac{14\cdots 51}{44\cdots 73}a^{12}+\frac{12\cdots 33}{44\cdots 73}a^{11}-\frac{37\cdots 12}{44\cdots 73}a^{10}+\frac{43\cdots 01}{44\cdots 73}a^{9}-\frac{32\cdots 19}{44\cdots 73}a^{8}+\frac{72\cdots 20}{44\cdots 73}a^{7}+\frac{82\cdots 93}{44\cdots 73}a^{6}+\frac{48\cdots 05}{44\cdots 73}a^{5}+\frac{55\cdots 57}{44\cdots 73}a^{4}+\frac{57\cdots 14}{44\cdots 73}a^{3}+\frac{58\cdots 50}{44\cdots 73}a^{2}-\frac{43\cdots 83}{44\cdots 73}a-\frac{39\cdots 65}{44\cdots 73}$
|
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| Regulator: | \( 9851869726433920.0 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{12}\cdot 9851869726433920.0 \cdot 6}{2\cdot\sqrt{167794323485736669509699424789101816713809967041015625}}\cr\approx \mathstrut & 1.09262138041333 \end{aligned}\] (assuming GRH)
Galois group
$\PSL(2,25)$ (as 26T42):
| A non-solvable group of order 7800 |
| The 15 conjugacy class representatives for $\PSL(2,25)$ |
| Character table for $\PSL(2,25)$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.13.0.1}{13} }^{2}$ | ${\href{/padicField/3.12.0.1}{12} }^{2}{,}\,{\href{/padicField/3.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/7.13.0.1}{13} }^{2}$ | ${\href{/padicField/11.2.0.1}{2} }^{12}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.12.0.1}{12} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{6}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.5.0.1}{5} }^{5}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.13.0.1}{13} }^{2}$ | ${\href{/padicField/31.12.0.1}{12} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.13.0.1}{13} }^{2}$ | R | ${\href{/padicField/43.13.0.1}{13} }^{2}$ | ${\href{/padicField/47.13.0.1}{13} }^{2}$ | ${\href{/padicField/53.13.0.1}{13} }^{2}$ | ${\href{/padicField/59.12.0.1}{12} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 | $$[\ ]$$ |
| Deg $25$ | $25$ | $1$ | $30$ | ||||
|
\(41\)
| $\Q_{41}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| Deg $25$ | $5$ | $5$ | $20$ |