Normalized defining polynomial
\( x^{24} - 35 x^{22} - 55 x^{21} + 955 x^{20} + 505 x^{19} - 13360 x^{18} - 20130 x^{17} + \cdots - 4828553680 \)
Invariants
| Degree: | $24$ |
| |
| Signature: | $[4, 10]$ |
| |
| Discriminant: |
\(61343664831565854732948944775316051495075225830078125\)
\(\medspace = 5^{23}\cdot 197^{16}\)
|
| |
| Root discriminant: | \(158.30\) |
| |
| Galois root discriminant: | $5^{23/24}197^{4/5}\approx 320.19845723009803$ | ||
| Ramified primes: |
\(5\), \(197\)
|
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{6}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{7}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{8}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a$, $\frac{1}{4}a^{17}-\frac{1}{4}a^{15}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{2}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{16}-\frac{1}{4}a^{12}-\frac{1}{4}a^{10}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}$, $\frac{1}{4}a^{19}-\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{788}a^{20}+\frac{15}{788}a^{19}-\frac{31}{788}a^{18}-\frac{63}{788}a^{17}-\frac{4}{197}a^{16}-\frac{29}{394}a^{15}-\frac{185}{788}a^{14}+\frac{157}{788}a^{13}+\frac{89}{788}a^{12}+\frac{103}{788}a^{11}-\frac{25}{788}a^{10}+\frac{173}{788}a^{9}-\frac{169}{394}a^{8}+\frac{155}{394}a^{7}-\frac{115}{788}a^{6}-\frac{261}{788}a^{5}+\frac{191}{788}a^{4}+\frac{323}{788}a^{3}+\frac{31}{394}a^{2}-\frac{69}{394}a+\frac{84}{197}$, $\frac{1}{788}a^{21}-\frac{59}{788}a^{19}+\frac{2}{197}a^{18}-\frac{14}{197}a^{17}+\frac{91}{394}a^{16}-\frac{103}{788}a^{15}+\frac{87}{394}a^{14}-\frac{99}{788}a^{13}-\frac{25}{394}a^{12}-\frac{191}{788}a^{11}+\frac{77}{394}a^{10}+\frac{11}{394}a^{9}-\frac{34}{197}a^{8}-\frac{37}{788}a^{7}-\frac{28}{197}a^{6}-\frac{31}{788}a^{5}-\frac{89}{394}a^{4}-\frac{63}{197}a^{3}+\frac{57}{394}a^{2}-\frac{88}{197}a-\frac{78}{197}$, $\frac{1}{371936}a^{22}-\frac{3}{11623}a^{21}+\frac{3}{6304}a^{20}-\frac{7927}{371936}a^{19}+\frac{26335}{371936}a^{18}-\frac{44179}{371936}a^{17}-\frac{7357}{92984}a^{16}-\frac{5359}{185968}a^{15}-\frac{68737}{371936}a^{14}+\frac{78617}{371936}a^{13}-\frac{20585}{185968}a^{12}+\frac{76485}{371936}a^{11}-\frac{43317}{371936}a^{10}+\frac{45081}{371936}a^{9}-\frac{19289}{185968}a^{8}-\frac{39909}{185968}a^{7}+\frac{25739}{371936}a^{6}+\frac{147067}{371936}a^{5}+\frac{178733}{371936}a^{4}-\frac{39139}{371936}a^{3}+\frac{98487}{371936}a^{2}-\frac{25185}{92984}a+\frac{29085}{92984}$, $\frac{1}{89\cdots 92}a^{23}+\frac{23\cdots 79}{44\cdots 96}a^{22}-\frac{45\cdots 43}{89\cdots 92}a^{21}+\frac{28\cdots 07}{89\cdots 92}a^{20}-\frac{60\cdots 27}{89\cdots 92}a^{19}-\frac{40\cdots 29}{89\cdots 92}a^{18}+\frac{46\cdots 97}{44\cdots 96}a^{17}-\frac{93\cdots 99}{44\cdots 96}a^{16}+\frac{17\cdots 95}{89\cdots 92}a^{15}+\frac{57\cdots 71}{89\cdots 92}a^{14}+\frac{12\cdots 03}{52\cdots 36}a^{13}+\frac{26\cdots 99}{15\cdots 88}a^{12}+\frac{68\cdots 01}{89\cdots 92}a^{11}+\frac{24\cdots 23}{89\cdots 92}a^{10}+\frac{38\cdots 71}{52\cdots 36}a^{9}+\frac{14\cdots 41}{44\cdots 96}a^{8}-\frac{78\cdots 13}{89\cdots 92}a^{7}+\frac{25\cdots 57}{89\cdots 92}a^{6}+\frac{18\cdots 23}{89\cdots 92}a^{5}-\frac{16\cdots 85}{89\cdots 92}a^{4}-\frac{39\cdots 11}{89\cdots 92}a^{3}+\frac{79\cdots 11}{44\cdots 96}a^{2}+\frac{14\cdots 11}{22\cdots 48}a+\frac{40\cdots 27}{11\cdots 24}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ (assuming GRH) |
| |
| Narrow class group: | $C_{4}$, which has order $4$ (assuming GRH) |
|
Unit group
| Rank: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{67\cdots 99}{23\cdots 14}a^{23}-\frac{10\cdots 97}{47\cdots 28}a^{22}+\frac{48\cdots 31}{47\cdots 28}a^{21}+\frac{11\cdots 45}{47\cdots 28}a^{20}-\frac{31\cdots 93}{11\cdots 07}a^{19}-\frac{95\cdots 39}{23\cdots 14}a^{18}+\frac{18\cdots 71}{47\cdots 28}a^{17}+\frac{45\cdots 03}{47\cdots 28}a^{16}-\frac{26\cdots 19}{47\cdots 28}a^{15}-\frac{77\cdots 89}{47\cdots 28}a^{14}+\frac{21\cdots 25}{23\cdots 14}a^{13}-\frac{63\cdots 65}{47\cdots 28}a^{12}-\frac{25\cdots 85}{11\cdots 07}a^{11}-\frac{75\cdots 37}{11\cdots 07}a^{10}+\frac{82\cdots 75}{47\cdots 28}a^{9}+\frac{16\cdots 65}{47\cdots 28}a^{8}-\frac{68\cdots 25}{47\cdots 28}a^{7}+\frac{53\cdots 85}{47\cdots 28}a^{6}-\frac{13\cdots 55}{11\cdots 07}a^{5}-\frac{74\cdots 55}{23\cdots 14}a^{4}+\frac{14\cdots 25}{47\cdots 28}a^{3}-\frac{79\cdots 80}{11\cdots 07}a^{2}+\frac{53\cdots 15}{11\cdots 07}a-\frac{68\cdots 57}{11\cdots 07}$, $\frac{10\cdots 23}{11\cdots 07}a^{23}-\frac{46\cdots 57}{47\cdots 28}a^{22}+\frac{14\cdots 77}{47\cdots 28}a^{21}+\frac{20\cdots 29}{23\cdots 14}a^{20}-\frac{94\cdots 50}{11\cdots 07}a^{19}-\frac{69\cdots 89}{47\cdots 28}a^{18}+\frac{53\cdots 39}{47\cdots 28}a^{17}+\frac{15\cdots 45}{47\cdots 28}a^{16}-\frac{77\cdots 87}{47\cdots 28}a^{15}-\frac{13\cdots 39}{23\cdots 14}a^{14}+\frac{63\cdots 91}{23\cdots 14}a^{13}+\frac{58\cdots 35}{23\cdots 14}a^{12}-\frac{14\cdots 45}{23\cdots 14}a^{11}-\frac{10\cdots 99}{47\cdots 28}a^{10}+\frac{21\cdots 65}{47\cdots 28}a^{9}+\frac{56\cdots 47}{47\cdots 28}a^{8}-\frac{18\cdots 49}{47\cdots 28}a^{7}+\frac{31\cdots 94}{11\cdots 07}a^{6}-\frac{43\cdots 08}{11\cdots 07}a^{5}-\frac{13\cdots 87}{47\cdots 28}a^{4}+\frac{44\cdots 51}{47\cdots 28}a^{3}-\frac{21\cdots 94}{11\cdots 07}a^{2}+\frac{13\cdots 05}{11\cdots 07}a-\frac{11\cdots 77}{11\cdots 07}$, $\frac{23\cdots 89}{14\cdots 78}a^{23}-\frac{13\cdots 15}{14\cdots 78}a^{22}+\frac{42\cdots 94}{70\cdots 89}a^{21}+\frac{92\cdots 98}{70\cdots 89}a^{20}-\frac{44\cdots 67}{28\cdots 56}a^{19}-\frac{58\cdots 41}{28\cdots 56}a^{18}+\frac{64\cdots 17}{28\cdots 56}a^{17}+\frac{14\cdots 27}{28\cdots 56}a^{16}-\frac{48\cdots 25}{14\cdots 78}a^{15}-\frac{12\cdots 97}{14\cdots 78}a^{14}+\frac{36\cdots 99}{65\cdots 92}a^{13}-\frac{34\cdots 75}{28\cdots 56}a^{12}-\frac{37\cdots 73}{28\cdots 56}a^{11}-\frac{10\cdots 89}{28\cdots 56}a^{10}+\frac{71\cdots 43}{65\cdots 92}a^{9}+\frac{57\cdots 11}{28\cdots 56}a^{8}-\frac{59\cdots 79}{70\cdots 89}a^{7}+\frac{94\cdots 75}{14\cdots 78}a^{6}-\frac{20\cdots 49}{28\cdots 56}a^{5}+\frac{11\cdots 85}{28\cdots 56}a^{4}+\frac{49\cdots 65}{28\cdots 56}a^{3}-\frac{11\cdots 35}{28\cdots 56}a^{2}+\frac{18\cdots 25}{70\cdots 89}a-\frac{33\cdots 46}{70\cdots 89}$, $\frac{28\cdots 83}{56\cdots 12}a^{23}-\frac{10\cdots 97}{56\cdots 12}a^{22}-\frac{13\cdots 73}{56\cdots 12}a^{21}+\frac{43\cdots 55}{14\cdots 78}a^{20}+\frac{54\cdots 29}{70\cdots 89}a^{19}-\frac{28\cdots 21}{28\cdots 56}a^{18}-\frac{69\cdots 05}{56\cdots 12}a^{17}+\frac{15\cdots 03}{28\cdots 56}a^{16}+\frac{11\cdots 23}{56\cdots 12}a^{15}-\frac{17\cdots 87}{28\cdots 56}a^{14}-\frac{44\cdots 89}{13\cdots 84}a^{13}+\frac{29\cdots 55}{56\cdots 12}a^{12}+\frac{68\cdots 25}{70\cdots 89}a^{11}+\frac{10\cdots 07}{28\cdots 56}a^{10}-\frac{12\cdots 57}{13\cdots 84}a^{9}-\frac{22\cdots 37}{14\cdots 78}a^{8}+\frac{33\cdots 71}{56\cdots 12}a^{7}-\frac{56\cdots 17}{70\cdots 89}a^{6}+\frac{16\cdots 64}{70\cdots 89}a^{5}-\frac{13\cdots 71}{14\cdots 78}a^{4}-\frac{48\cdots 61}{70\cdots 89}a^{3}+\frac{18\cdots 29}{56\cdots 12}a^{2}-\frac{20\cdots 07}{70\cdots 89}a+\frac{86\cdots 37}{14\cdots 78}$, $\frac{59\cdots 75}{22\cdots 48}a^{23}-\frac{16\cdots 87}{11\cdots 24}a^{22}+\frac{20\cdots 13}{22\cdots 48}a^{21}+\frac{43\cdots 35}{22\cdots 48}a^{20}-\frac{54\cdots 79}{22\cdots 48}a^{19}-\frac{58\cdots 29}{22\cdots 48}a^{18}+\frac{38\cdots 95}{11\cdots 24}a^{17}+\frac{80\cdots 21}{11\cdots 24}a^{16}-\frac{11\cdots 77}{22\cdots 48}a^{15}-\frac{27\cdots 37}{22\cdots 48}a^{14}+\frac{56\cdots 75}{65\cdots 92}a^{13}-\frac{11\cdots 03}{22\cdots 48}a^{12}-\frac{35\cdots 47}{22\cdots 48}a^{11}-\frac{10\cdots 73}{22\cdots 48}a^{10}+\frac{11\cdots 11}{65\cdots 92}a^{9}+\frac{26\cdots 65}{11\cdots 24}a^{8}-\frac{30\cdots 13}{22\cdots 48}a^{7}+\frac{33\cdots 13}{22\cdots 48}a^{6}-\frac{44\cdots 73}{22\cdots 48}a^{5}+\frac{43\cdots 07}{22\cdots 48}a^{4}+\frac{60\cdots 17}{22\cdots 48}a^{3}-\frac{77\cdots 99}{11\cdots 24}a^{2}+\frac{42\cdots 83}{56\cdots 12}a-\frac{51\cdots 51}{28\cdots 56}$, $\frac{42\cdots 45}{22\cdots 48}a^{23}+\frac{63\cdots 11}{11\cdots 24}a^{22}-\frac{14\cdots 71}{22\cdots 48}a^{21}-\frac{27\cdots 53}{22\cdots 48}a^{20}+\frac{40\cdots 53}{22\cdots 48}a^{19}+\frac{34\cdots 87}{22\cdots 48}a^{18}-\frac{28\cdots 03}{11\cdots 24}a^{17}-\frac{51\cdots 27}{11\cdots 24}a^{16}+\frac{86\cdots 55}{22\cdots 48}a^{15}+\frac{30\cdots 17}{38\cdots 72}a^{14}-\frac{84\cdots 91}{13\cdots 84}a^{13}+\frac{11\cdots 61}{22\cdots 48}a^{12}+\frac{23\cdots 09}{22\cdots 48}a^{11}+\frac{66\cdots 39}{22\cdots 48}a^{10}-\frac{17\cdots 17}{13\cdots 84}a^{9}-\frac{12\cdots 83}{11\cdots 24}a^{8}+\frac{22\cdots 39}{22\cdots 48}a^{7}-\frac{33\cdots 23}{22\cdots 48}a^{6}+\frac{33\cdots 63}{22\cdots 48}a^{5}+\frac{83\cdots 11}{22\cdots 48}a^{4}-\frac{42\cdots 23}{22\cdots 48}a^{3}+\frac{58\cdots 87}{11\cdots 24}a^{2}-\frac{36\cdots 53}{56\cdots 12}a+\frac{77\cdots 79}{28\cdots 56}$, $\frac{16\cdots 09}{22\cdots 48}a^{23}-\frac{62\cdots 43}{11\cdots 24}a^{22}-\frac{68\cdots 91}{22\cdots 48}a^{21}-\frac{74\cdots 61}{22\cdots 48}a^{20}+\frac{20\cdots 85}{22\cdots 48}a^{19}+\frac{14\cdots 83}{22\cdots 48}a^{18}-\frac{15\cdots 45}{11\cdots 24}a^{17}-\frac{25\cdots 35}{11\cdots 24}a^{16}+\frac{47\cdots 07}{22\cdots 48}a^{15}+\frac{10\cdots 63}{22\cdots 48}a^{14}-\frac{21\cdots 07}{65\cdots 92}a^{13}-\frac{55\cdots 91}{22\cdots 48}a^{12}+\frac{24\cdots 09}{22\cdots 48}a^{11}+\frac{69\cdots 91}{22\cdots 48}a^{10}-\frac{40\cdots 17}{65\cdots 92}a^{9}-\frac{20\cdots 27}{11\cdots 24}a^{8}+\frac{72\cdots 07}{22\cdots 48}a^{7}-\frac{48\cdots 67}{22\cdots 48}a^{6}+\frac{19\cdots 15}{22\cdots 48}a^{5}+\frac{15\cdots 83}{22\cdots 48}a^{4}-\frac{20\cdots 51}{22\cdots 48}a^{3}+\frac{20\cdots 45}{11\cdots 24}a^{2}-\frac{65\cdots 21}{56\cdots 12}a+\frac{57\cdots 01}{28\cdots 56}$, $\frac{99\cdots 19}{22\cdots 48}a^{23}-\frac{96\cdots 91}{11\cdots 24}a^{22}-\frac{12\cdots 09}{22\cdots 48}a^{21}+\frac{95\cdots 57}{22\cdots 48}a^{20}+\frac{37\cdots 63}{22\cdots 48}a^{19}-\frac{52\cdots 63}{22\cdots 48}a^{18}+\frac{16\cdots 59}{11\cdots 24}a^{17}+\frac{44\cdots 87}{11\cdots 24}a^{16}+\frac{45\cdots 77}{22\cdots 48}a^{15}-\frac{15\cdots 35}{22\cdots 48}a^{14}-\frac{10\cdots 01}{13\cdots 84}a^{13}+\frac{34\cdots 59}{22\cdots 48}a^{12}-\frac{41\cdots 97}{22\cdots 48}a^{11}-\frac{17\cdots 75}{22\cdots 48}a^{10}+\frac{70\cdots 37}{13\cdots 84}a^{9}+\frac{58\cdots 83}{11\cdots 24}a^{8}-\frac{20\cdots 95}{22\cdots 48}a^{7}-\frac{81\cdots 45}{22\cdots 48}a^{6}+\frac{10\cdots 77}{22\cdots 48}a^{5}+\frac{17\cdots 49}{22\cdots 48}a^{4}-\frac{57\cdots 45}{22\cdots 48}a^{3}+\frac{29\cdots 17}{11\cdots 24}a^{2}-\frac{57\cdots 31}{56\cdots 12}a+\frac{25\cdots 89}{28\cdots 56}$, $\frac{11\cdots 03}{11\cdots 24}a^{23}+\frac{28\cdots 67}{28\cdots 56}a^{22}-\frac{44\cdots 69}{11\cdots 24}a^{21}-\frac{12\cdots 37}{11\cdots 24}a^{20}+\frac{10\cdots 61}{11\cdots 24}a^{19}+\frac{22\cdots 27}{11\cdots 24}a^{18}-\frac{17\cdots 09}{14\cdots 78}a^{17}-\frac{23\cdots 93}{56\cdots 12}a^{16}+\frac{20\cdots 17}{11\cdots 24}a^{15}+\frac{79\cdots 59}{11\cdots 24}a^{14}-\frac{38\cdots 37}{13\cdots 84}a^{13}-\frac{94\cdots 77}{11\cdots 24}a^{12}+\frac{45\cdots 01}{11\cdots 24}a^{11}+\frac{27\cdots 79}{11\cdots 24}a^{10}-\frac{56\cdots 99}{13\cdots 84}a^{9}-\frac{68\cdots 31}{56\cdots 12}a^{8}+\frac{49\cdots 77}{11\cdots 24}a^{7}-\frac{29\cdots 99}{11\cdots 24}a^{6}+\frac{96\cdots 23}{11\cdots 24}a^{5}+\frac{13\cdots 51}{11\cdots 24}a^{4}-\frac{17\cdots 21}{19\cdots 36}a^{3}+\frac{14\cdots 39}{70\cdots 89}a^{2}-\frac{21\cdots 33}{14\cdots 48}a+\frac{22\cdots 18}{70\cdots 89}$, $\frac{43\cdots 47}{14\cdots 78}a^{23}-\frac{35\cdots 37}{14\cdots 78}a^{22}+\frac{74\cdots 54}{70\cdots 89}a^{21}+\frac{72\cdots 73}{28\cdots 56}a^{20}-\frac{76\cdots 59}{28\cdots 56}a^{19}-\frac{10\cdots 55}{28\cdots 56}a^{18}+\frac{10\cdots 43}{28\cdots 56}a^{17}+\frac{65\cdots 00}{70\cdots 89}a^{16}-\frac{39\cdots 20}{70\cdots 89}a^{15}-\frac{44\cdots 99}{28\cdots 56}a^{14}+\frac{61\cdots 11}{65\cdots 92}a^{13}-\frac{11\cdots 29}{28\cdots 56}a^{12}-\frac{45\cdots 57}{28\cdots 56}a^{11}-\frac{16\cdots 19}{28\cdots 56}a^{10}+\frac{11\cdots 31}{65\cdots 92}a^{9}+\frac{38\cdots 25}{14\cdots 78}a^{8}-\frac{20\cdots 67}{14\cdots 78}a^{7}+\frac{47\cdots 67}{28\cdots 56}a^{6}-\frac{54\cdots 49}{28\cdots 56}a^{5}-\frac{21\cdots 75}{28\cdots 56}a^{4}+\frac{83\cdots 47}{28\cdots 56}a^{3}-\frac{50\cdots 80}{70\cdots 89}a^{2}+\frac{54\cdots 97}{70\cdots 89}a-\frac{12\cdots 49}{70\cdots 89}$, $\frac{85\cdots 87}{28\cdots 56}a^{23}+\frac{19\cdots 27}{14\cdots 78}a^{22}-\frac{14\cdots 49}{23\cdots 42}a^{21}-\frac{34\cdots 14}{70\cdots 89}a^{20}+\frac{14\cdots 57}{14\cdots 78}a^{19}+\frac{59\cdots 20}{70\cdots 89}a^{18}-\frac{14\cdots 49}{14\cdots 78}a^{17}-\frac{20\cdots 97}{14\cdots 78}a^{16}+\frac{57\cdots 72}{70\cdots 89}a^{15}+\frac{29\cdots 13}{14\cdots 78}a^{14}-\frac{14\cdots 55}{65\cdots 92}a^{13}-\frac{11\cdots 53}{14\cdots 78}a^{12}-\frac{19\cdots 69}{70\cdots 89}a^{11}+\frac{38\cdots 17}{70\cdots 89}a^{10}+\frac{18\cdots 22}{16\cdots 23}a^{9}-\frac{49\cdots 91}{14\cdots 78}a^{8}+\frac{21\cdots 37}{70\cdots 89}a^{7}+\frac{10\cdots 76}{70\cdots 89}a^{6}+\frac{11\cdots 03}{14\cdots 78}a^{5}+\frac{69\cdots 85}{14\cdots 78}a^{4}-\frac{39\cdots 61}{28\cdots 56}a^{3}+\frac{17\cdots 97}{14\cdots 78}a^{2}+\frac{92\cdots 21}{70\cdots 89}a-\frac{16\cdots 21}{11\cdots 71}$, $\frac{17\cdots 73}{22\cdots 48}a^{23}+\frac{57\cdots 71}{28\cdots 56}a^{22}+\frac{22\cdots 07}{22\cdots 48}a^{21}-\frac{11\cdots 21}{22\cdots 48}a^{20}-\frac{93\cdots 31}{22\cdots 48}a^{19}+\frac{21\cdots 91}{22\cdots 48}a^{18}+\frac{40\cdots 61}{56\cdots 12}a^{17}-\frac{10\cdots 69}{11\cdots 24}a^{16}-\frac{14\cdots 63}{11\cdots 84}a^{15}+\frac{18\cdots 71}{22\cdots 48}a^{14}+\frac{84\cdots 13}{44\cdots 52}a^{13}-\frac{53\cdots 33}{22\cdots 48}a^{12}-\frac{17\cdots 39}{22\cdots 48}a^{11}-\frac{11\cdots 77}{22\cdots 48}a^{10}+\frac{13\cdots 95}{26\cdots 68}a^{9}+\frac{23\cdots 01}{11\cdots 24}a^{8}-\frac{67\cdots 75}{22\cdots 48}a^{7}+\frac{65\cdots 97}{22\cdots 48}a^{6}+\frac{25\cdots 63}{22\cdots 48}a^{5}+\frac{31\cdots 39}{22\cdots 48}a^{4}+\frac{10\cdots 65}{22\cdots 48}a^{3}-\frac{80\cdots 25}{56\cdots 12}a^{2}+\frac{55\cdots 11}{56\cdots 12}a-\frac{12\cdots 01}{70\cdots 89}$, $\frac{13\cdots 95}{23\cdots 42}a^{23}+\frac{73\cdots 57}{28\cdots 56}a^{22}-\frac{60\cdots 51}{28\cdots 56}a^{21}-\frac{22\cdots 25}{47\cdots 84}a^{20}+\frac{81\cdots 07}{14\cdots 78}a^{19}+\frac{66\cdots 84}{70\cdots 89}a^{18}-\frac{23\cdots 71}{28\cdots 56}a^{17}-\frac{64\cdots 57}{28\cdots 56}a^{16}+\frac{34\cdots 73}{28\cdots 56}a^{15}+\frac{11\cdots 51}{28\cdots 56}a^{14}-\frac{62\cdots 15}{32\cdots 46}a^{13}-\frac{26\cdots 71}{28\cdots 56}a^{12}+\frac{34\cdots 97}{70\cdots 89}a^{11}+\frac{30\cdots 77}{14\cdots 78}a^{10}-\frac{15\cdots 17}{65\cdots 92}a^{9}-\frac{31\cdots 03}{28\cdots 56}a^{8}+\frac{44\cdots 61}{28\cdots 56}a^{7}-\frac{44\cdots 91}{28\cdots 56}a^{6}+\frac{69\cdots 91}{14\cdots 78}a^{5}+\frac{51\cdots 80}{70\cdots 89}a^{4}-\frac{15\cdots 15}{28\cdots 56}a^{3}+\frac{13\cdots 83}{14\cdots 78}a^{2}-\frac{44\cdots 67}{70\cdots 89}a+\frac{81\cdots 73}{70\cdots 89}$
|
| |
| Regulator: | \( 591498666797754600 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{10}\cdot 591498666797754600 \cdot 2}{2\cdot\sqrt{61343664831565854732948944775316051495075225830078125}}\cr\approx \mathstrut & 3.66427092990403 \end{aligned}\] (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
| 6.2.4706682753125.1, 12.4.110764312692821648486328125.1 |
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 | ${\href{/padicField/2.4.0.1}{4} }^{5}{,}\,{\href{/padicField/2.1.0.1}{1} }^{4}$ | ${\href{/padicField/3.8.0.1}{8} }^{3}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{8}$ | $24$ | ${\href{/padicField/17.4.0.1}{4} }^{5}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.12.0.1}{12} }^{2}$ | $24$ | ${\href{/padicField/29.12.0.1}{12} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{6}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{8}$ | ${\href{/padicField/43.4.0.1}{4} }^{5}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | $24$ | ${\href{/padicField/53.8.0.1}{8} }^{3}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}{,}\,{\href{/padicField/59.1.0.1}{1} }^{4}$ |
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 $24$ | $24$ | $1$ | $23$ | |||
|
\(197\)
| 197.4.1.0a1.1 | $x^{4} + 16 x^{2} + 124 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 197.2.5.8a1.1 | $x^{10} + 960 x^{9} + 368650 x^{8} + 70786560 x^{7} + 6796984360 x^{6} + 261202401792 x^{5} + 13593968720 x^{4} + 283146240 x^{3} + 2949200 x^{2} + 15360 x + 229$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ | |
| 197.2.5.8a1.1 | $x^{10} + 960 x^{9} + 368650 x^{8} + 70786560 x^{7} + 6796984360 x^{6} + 261202401792 x^{5} + 13593968720 x^{4} + 283146240 x^{3} + 2949200 x^{2} + 15360 x + 229$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ |