Normalized defining polynomial
\( x^{20} - x^{19} - x^{18} + 16 x^{17} - 324 x^{16} + 428 x^{15} + 1518 x^{14} - 3988 x^{13} + \cdots + 302044 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $[12, 4]$ |
| |
| Discriminant: |
\(477051862117015968785053941760000000000\)
\(\medspace = 2^{24}\cdot 5^{10}\cdot 13^{6}\cdot 29^{4}\cdot 31^{8}\)
|
| |
| Root discriminant: | \(85.89\) |
| |
| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(5\), \(13\), \(29\), \(31\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{4}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{5}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{6}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{15}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a$, $\frac{1}{2}a^{16}-\frac{1}{2}a^{10}-\frac{1}{2}a^{8}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{10}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{17}-\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{2}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}$, $\frac{1}{27\cdots 44}a^{19}-\frac{14\cdots 71}{13\cdots 22}a^{18}-\frac{37\cdots 15}{27\cdots 44}a^{17}-\frac{55\cdots 49}{27\cdots 44}a^{16}+\frac{65\cdots 07}{27\cdots 44}a^{15}-\frac{85\cdots 81}{27\cdots 44}a^{14}-\frac{56\cdots 79}{27\cdots 44}a^{13}+\frac{47\cdots 03}{27\cdots 44}a^{12}-\frac{81\cdots 89}{27\cdots 44}a^{11}-\frac{13\cdots 59}{27\cdots 44}a^{10}-\frac{42\cdots 57}{27\cdots 44}a^{9}-\frac{59\cdots 57}{27\cdots 44}a^{8}+\frac{87\cdots 91}{27\cdots 44}a^{7}+\frac{99\cdots 11}{27\cdots 44}a^{6}+\frac{39\cdots 53}{27\cdots 44}a^{5}-\frac{93\cdots 81}{27\cdots 44}a^{4}+\frac{49\cdots 23}{13\cdots 22}a^{3}+\frac{16\cdots 07}{27\cdots 44}a^{2}+\frac{14\cdots 59}{13\cdots 22}a+\frac{19\cdots 53}{69\cdots 11}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}\times C_{2}\times C_{2}$, which has order $8$ (assuming GRH) |
|
Unit group
| Rank: | $15$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{11\cdots 59}{82\cdots 56}a^{19}-\frac{27\cdots 09}{82\cdots 56}a^{18}+\frac{12\cdots 97}{41\cdots 78}a^{17}+\frac{15\cdots 95}{82\cdots 56}a^{16}-\frac{20\cdots 17}{41\cdots 78}a^{15}+\frac{10\cdots 01}{82\cdots 56}a^{14}+\frac{97\cdots 51}{20\cdots 89}a^{13}-\frac{51\cdots 65}{82\cdots 56}a^{12}+\frac{91\cdots 72}{20\cdots 89}a^{11}-\frac{79\cdots 99}{82\cdots 56}a^{10}-\frac{52\cdots 75}{20\cdots 89}a^{9}+\frac{50\cdots 43}{82\cdots 56}a^{8}+\frac{32\cdots 81}{41\cdots 78}a^{7}-\frac{19\cdots 87}{82\cdots 56}a^{6}-\frac{54\cdots 16}{20\cdots 89}a^{5}-\frac{27\cdots 93}{82\cdots 56}a^{4}+\frac{19\cdots 09}{82\cdots 56}a^{3}+\frac{24\cdots 47}{41\cdots 78}a^{2}+\frac{11\cdots 41}{41\cdots 78}a-\frac{27\cdots 90}{20\cdots 89}$, $\frac{51\cdots 49}{13\cdots 22}a^{19}-\frac{97\cdots 81}{13\cdots 22}a^{18}+\frac{17\cdots 40}{69\cdots 11}a^{17}+\frac{39\cdots 38}{69\cdots 11}a^{16}-\frac{86\cdots 75}{69\cdots 11}a^{15}+\frac{18\cdots 31}{69\cdots 11}a^{14}+\frac{22\cdots 32}{69\cdots 11}a^{13}-\frac{12\cdots 82}{69\cdots 11}a^{12}+\frac{15\cdots 21}{13\cdots 22}a^{11}-\frac{13\cdots 81}{69\cdots 11}a^{10}-\frac{11\cdots 89}{13\cdots 22}a^{9}+\frac{10\cdots 36}{69\cdots 11}a^{8}+\frac{19\cdots 34}{69\cdots 11}a^{7}-\frac{38\cdots 09}{69\cdots 11}a^{6}-\frac{47\cdots 02}{69\cdots 11}a^{5}-\frac{79\cdots 17}{69\cdots 11}a^{4}+\frac{27\cdots 09}{69\cdots 11}a^{3}+\frac{27\cdots 45}{13\cdots 22}a^{2}+\frac{15\cdots 19}{13\cdots 22}a+\frac{88\cdots 27}{69\cdots 11}$, $\frac{22\cdots 71}{27\cdots 44}a^{19}-\frac{42\cdots 99}{27\cdots 44}a^{18}+\frac{24\cdots 99}{69\cdots 11}a^{17}+\frac{35\cdots 41}{27\cdots 44}a^{16}-\frac{38\cdots 55}{13\cdots 22}a^{15}+\frac{16\cdots 93}{27\cdots 44}a^{14}+\frac{54\cdots 86}{69\cdots 11}a^{13}-\frac{11\cdots 83}{27\cdots 44}a^{12}+\frac{33\cdots 21}{13\cdots 22}a^{11}-\frac{11\cdots 75}{27\cdots 44}a^{10}-\frac{26\cdots 77}{13\cdots 22}a^{9}+\frac{88\cdots 53}{27\cdots 44}a^{8}+\frac{92\cdots 41}{13\cdots 22}a^{7}-\frac{34\cdots 79}{27\cdots 44}a^{6}-\frac{10\cdots 71}{69\cdots 11}a^{5}-\frac{73\cdots 47}{27\cdots 44}a^{4}+\frac{22\cdots 03}{27\cdots 44}a^{3}+\frac{32\cdots 63}{69\cdots 11}a^{2}+\frac{37\cdots 33}{13\cdots 22}a+\frac{19\cdots 34}{69\cdots 11}$, $\frac{70\cdots 37}{13\cdots 22}a^{19}-\frac{65\cdots 32}{69\cdots 11}a^{18}+\frac{15\cdots 76}{69\cdots 11}a^{17}+\frac{55\cdots 33}{69\cdots 11}a^{16}-\frac{11\cdots 52}{69\cdots 11}a^{15}+\frac{50\cdots 63}{13\cdots 22}a^{14}+\frac{33\cdots 85}{69\cdots 11}a^{13}-\frac{17\cdots 04}{69\cdots 11}a^{12}+\frac{20\cdots 53}{13\cdots 22}a^{11}-\frac{17\cdots 45}{69\cdots 11}a^{10}-\frac{16\cdots 35}{13\cdots 22}a^{9}+\frac{13\cdots 48}{69\cdots 11}a^{8}+\frac{28\cdots 68}{69\cdots 11}a^{7}-\frac{10\cdots 93}{13\cdots 22}a^{6}-\frac{67\cdots 32}{69\cdots 11}a^{5}-\frac{11\cdots 75}{69\cdots 11}a^{4}+\frac{35\cdots 68}{69\cdots 11}a^{3}+\frac{19\cdots 66}{69\cdots 11}a^{2}+\frac{23\cdots 47}{13\cdots 22}a+\frac{11\cdots 72}{69\cdots 11}$, $\frac{48\cdots 87}{27\cdots 44}a^{19}-\frac{92\cdots 51}{27\cdots 44}a^{18}+\frac{19\cdots 63}{13\cdots 22}a^{17}+\frac{73\cdots 49}{27\cdots 44}a^{16}-\frac{81\cdots 81}{13\cdots 22}a^{15}+\frac{35\cdots 99}{27\cdots 44}a^{14}+\frac{99\cdots 43}{69\cdots 11}a^{13}-\frac{22\cdots 65}{27\cdots 44}a^{12}+\frac{35\cdots 14}{69\cdots 11}a^{11}-\frac{25\cdots 33}{27\cdots 44}a^{10}-\frac{26\cdots 29}{69\cdots 11}a^{9}+\frac{18\cdots 49}{27\cdots 44}a^{8}+\frac{17\cdots 21}{13\cdots 22}a^{7}-\frac{73\cdots 13}{27\cdots 44}a^{6}-\frac{21\cdots 28}{69\cdots 11}a^{5}-\frac{14\cdots 25}{27\cdots 44}a^{4}+\frac{53\cdots 41}{27\cdots 44}a^{3}+\frac{12\cdots 15}{13\cdots 22}a^{2}+\frac{69\cdots 19}{13\cdots 22}a+\frac{38\cdots 45}{69\cdots 11}$, $\frac{81\cdots 75}{27\cdots 44}a^{19}-\frac{15\cdots 61}{27\cdots 44}a^{18}+\frac{23\cdots 41}{13\cdots 22}a^{17}+\frac{12\cdots 39}{27\cdots 44}a^{16}-\frac{68\cdots 56}{69\cdots 11}a^{15}+\frac{58\cdots 59}{27\cdots 44}a^{14}+\frac{36\cdots 47}{13\cdots 22}a^{13}-\frac{38\cdots 69}{27\cdots 44}a^{12}+\frac{59\cdots 62}{69\cdots 11}a^{11}-\frac{41\cdots 59}{27\cdots 44}a^{10}-\frac{45\cdots 26}{69\cdots 11}a^{9}+\frac{31\cdots 47}{27\cdots 44}a^{8}+\frac{15\cdots 73}{69\cdots 11}a^{7}-\frac{12\cdots 89}{27\cdots 44}a^{6}-\frac{76\cdots 69}{13\cdots 22}a^{5}-\frac{25\cdots 01}{27\cdots 44}a^{4}+\frac{84\cdots 45}{27\cdots 44}a^{3}+\frac{22\cdots 65}{13\cdots 22}a^{2}+\frac{12\cdots 89}{13\cdots 22}a+\frac{73\cdots 94}{69\cdots 11}$, $\frac{55\cdots 50}{69\cdots 11}a^{19}-\frac{39\cdots 18}{69\cdots 11}a^{18}-\frac{13\cdots 60}{69\cdots 11}a^{17}+\frac{10\cdots 81}{69\cdots 11}a^{16}-\frac{17\cdots 25}{69\cdots 11}a^{15}+\frac{17\cdots 50}{69\cdots 11}a^{14}+\frac{11\cdots 62}{69\cdots 11}a^{13}-\frac{26\cdots 69}{69\cdots 11}a^{12}+\frac{26\cdots 29}{13\cdots 22}a^{11}-\frac{14\cdots 13}{13\cdots 22}a^{10}-\frac{34\cdots 57}{13\cdots 22}a^{9}+\frac{11\cdots 57}{69\cdots 11}a^{8}+\frac{75\cdots 33}{69\cdots 11}a^{7}+\frac{12\cdots 30}{69\cdots 11}a^{6}-\frac{13\cdots 13}{69\cdots 11}a^{5}-\frac{25\cdots 60}{69\cdots 11}a^{4}-\frac{78\cdots 63}{13\cdots 22}a^{3}+\frac{88\cdots 77}{13\cdots 22}a^{2}+\frac{69\cdots 45}{13\cdots 22}a+\frac{49\cdots 23}{69\cdots 11}$, $\frac{15\cdots 55}{69\cdots 11}a^{19}-\frac{29\cdots 47}{69\cdots 11}a^{18}+\frac{11\cdots 06}{69\cdots 11}a^{17}+\frac{23\cdots 69}{69\cdots 11}a^{16}-\frac{52\cdots 51}{69\cdots 11}a^{15}+\frac{22\cdots 13}{13\cdots 22}a^{14}+\frac{13\cdots 79}{69\cdots 11}a^{13}-\frac{14\cdots 73}{13\cdots 22}a^{12}+\frac{91\cdots 25}{13\cdots 22}a^{11}-\frac{81\cdots 53}{69\cdots 11}a^{10}-\frac{68\cdots 43}{13\cdots 22}a^{9}+\frac{60\cdots 71}{69\cdots 11}a^{8}+\frac{11\cdots 19}{69\cdots 11}a^{7}-\frac{46\cdots 43}{13\cdots 22}a^{6}-\frac{28\cdots 76}{69\cdots 11}a^{5}-\frac{94\cdots 09}{13\cdots 22}a^{4}+\frac{33\cdots 11}{13\cdots 22}a^{3}+\frac{81\cdots 97}{69\cdots 11}a^{2}+\frac{92\cdots 07}{13\cdots 22}a+\frac{50\cdots 16}{69\cdots 11}$, $\frac{10\cdots 39}{27\cdots 44}a^{19}-\frac{19\cdots 53}{27\cdots 44}a^{18}+\frac{34\cdots 83}{13\cdots 22}a^{17}+\frac{15\cdots 57}{27\cdots 44}a^{16}-\frac{17\cdots 33}{13\cdots 22}a^{15}+\frac{75\cdots 23}{27\cdots 44}a^{14}+\frac{22\cdots 81}{69\cdots 11}a^{13}-\frac{49\cdots 63}{27\cdots 44}a^{12}+\frac{15\cdots 77}{13\cdots 22}a^{11}-\frac{53\cdots 25}{27\cdots 44}a^{10}-\frac{11\cdots 39}{13\cdots 22}a^{9}+\frac{40\cdots 01}{27\cdots 44}a^{8}+\frac{39\cdots 87}{13\cdots 22}a^{7}-\frac{15\cdots 49}{27\cdots 44}a^{6}-\frac{47\cdots 86}{69\cdots 11}a^{5}-\frac{31\cdots 75}{27\cdots 44}a^{4}+\frac{11\cdots 27}{27\cdots 44}a^{3}+\frac{13\cdots 49}{69\cdots 11}a^{2}+\frac{78\cdots 89}{69\cdots 11}a+\frac{88\cdots 17}{69\cdots 11}$, $\frac{11\cdots 40}{69\cdots 11}a^{19}-\frac{37\cdots 68}{69\cdots 11}a^{18}-\frac{83\cdots 69}{13\cdots 22}a^{17}+\frac{15\cdots 71}{69\cdots 11}a^{16}-\frac{82\cdots 41}{13\cdots 22}a^{15}+\frac{12\cdots 52}{69\cdots 11}a^{14}+\frac{10\cdots 65}{69\cdots 11}a^{13}-\frac{71\cdots 71}{69\cdots 11}a^{12}+\frac{73\cdots 01}{13\cdots 22}a^{11}-\frac{95\cdots 80}{69\cdots 11}a^{10}-\frac{26\cdots 41}{69\cdots 11}a^{9}+\frac{80\cdots 84}{69\cdots 11}a^{8}+\frac{22\cdots 71}{13\cdots 22}a^{7}-\frac{10\cdots 44}{69\cdots 11}a^{6}-\frac{40\cdots 41}{69\cdots 11}a^{5}-\frac{47\cdots 09}{69\cdots 11}a^{4}+\frac{80\cdots 09}{13\cdots 22}a^{3}+\frac{11\cdots 17}{69\cdots 11}a^{2}+\frac{13\cdots 21}{13\cdots 22}a+\frac{73\cdots 30}{69\cdots 11}$, $\frac{10\cdots 52}{69\cdots 11}a^{19}-\frac{38\cdots 27}{27\cdots 44}a^{18}+\frac{14\cdots 09}{27\cdots 44}a^{17}-\frac{62\cdots 43}{13\cdots 22}a^{16}-\frac{15\cdots 55}{27\cdots 44}a^{15}+\frac{71\cdots 33}{13\cdots 22}a^{14}-\frac{47\cdots 65}{27\cdots 44}a^{13}+\frac{11\cdots 69}{13\cdots 22}a^{12}+\frac{25\cdots 73}{27\cdots 44}a^{11}-\frac{69\cdots 73}{13\cdots 22}a^{10}+\frac{33\cdots 87}{27\cdots 44}a^{9}+\frac{14\cdots 75}{13\cdots 22}a^{8}-\frac{20\cdots 35}{27\cdots 44}a^{7}+\frac{10\cdots 11}{13\cdots 22}a^{6}+\frac{18\cdots 67}{27\cdots 44}a^{5}+\frac{30\cdots 75}{13\cdots 22}a^{4}+\frac{28\cdots 45}{27\cdots 44}a^{3}-\frac{12\cdots 35}{27\cdots 44}a^{2}-\frac{52\cdots 51}{13\cdots 22}a-\frac{31\cdots 00}{69\cdots 11}$, $\frac{22\cdots 35}{27\cdots 44}a^{19}-\frac{42\cdots 01}{27\cdots 44}a^{18}+\frac{34\cdots 93}{69\cdots 11}a^{17}+\frac{35\cdots 05}{27\cdots 44}a^{16}-\frac{38\cdots 19}{13\cdots 22}a^{15}+\frac{16\cdots 23}{27\cdots 44}a^{14}+\frac{10\cdots 13}{13\cdots 22}a^{13}-\frac{10\cdots 93}{27\cdots 44}a^{12}+\frac{33\cdots 95}{13\cdots 22}a^{11}-\frac{11\cdots 65}{27\cdots 44}a^{10}-\frac{12\cdots 27}{69\cdots 11}a^{9}+\frac{89\cdots 25}{27\cdots 44}a^{8}+\frac{88\cdots 27}{13\cdots 22}a^{7}-\frac{32\cdots 17}{27\cdots 44}a^{6}-\frac{21\cdots 87}{13\cdots 22}a^{5}-\frac{71\cdots 69}{27\cdots 44}a^{4}+\frac{23\cdots 03}{27\cdots 44}a^{3}+\frac{31\cdots 73}{69\cdots 11}a^{2}+\frac{17\cdots 05}{69\cdots 11}a+\frac{20\cdots 81}{69\cdots 11}$, $\frac{33\cdots 83}{69\cdots 11}a^{19}-\frac{58\cdots 08}{69\cdots 11}a^{18}+\frac{10\cdots 78}{69\cdots 11}a^{17}+\frac{52\cdots 32}{69\cdots 11}a^{16}-\frac{11\cdots 50}{69\cdots 11}a^{15}+\frac{45\cdots 63}{13\cdots 22}a^{14}+\frac{66\cdots 91}{13\cdots 22}a^{13}-\frac{31\cdots 85}{13\cdots 22}a^{12}+\frac{95\cdots 42}{69\cdots 11}a^{11}-\frac{15\cdots 39}{69\cdots 11}a^{10}-\frac{77\cdots 95}{69\cdots 11}a^{9}+\frac{12\cdots 52}{69\cdots 11}a^{8}+\frac{27\cdots 26}{69\cdots 11}a^{7}-\frac{63\cdots 09}{13\cdots 22}a^{6}-\frac{12\cdots 81}{13\cdots 22}a^{5}-\frac{21\cdots 05}{13\cdots 22}a^{4}+\frac{26\cdots 40}{69\cdots 11}a^{3}+\frac{18\cdots 65}{69\cdots 11}a^{2}+\frac{11\cdots 88}{69\cdots 11}a+\frac{14\cdots 73}{69\cdots 11}$, $\frac{45\cdots 12}{69\cdots 11}a^{19}-\frac{34\cdots 11}{27\cdots 44}a^{18}+\frac{13\cdots 69}{27\cdots 44}a^{17}+\frac{13\cdots 49}{13\cdots 22}a^{16}-\frac{61\cdots 19}{27\cdots 44}a^{15}+\frac{66\cdots 63}{13\cdots 22}a^{14}+\frac{15\cdots 75}{27\cdots 44}a^{13}-\frac{21\cdots 90}{69\cdots 11}a^{12}+\frac{53\cdots 73}{27\cdots 44}a^{11}-\frac{47\cdots 75}{13\cdots 22}a^{10}-\frac{39\cdots 07}{27\cdots 44}a^{9}+\frac{35\cdots 79}{13\cdots 22}a^{8}+\frac{13\cdots 17}{27\cdots 44}a^{7}-\frac{14\cdots 69}{13\cdots 22}a^{6}-\frac{33\cdots 17}{27\cdots 44}a^{5}-\frac{13\cdots 09}{69\cdots 11}a^{4}+\frac{20\cdots 05}{27\cdots 44}a^{3}+\frac{95\cdots 25}{27\cdots 44}a^{2}+\frac{13\cdots 73}{69\cdots 11}a+\frac{15\cdots 87}{69\cdots 11}$, $\frac{41\cdots 05}{13\cdots 22}a^{19}-\frac{14\cdots 43}{27\cdots 44}a^{18}+\frac{22\cdots 21}{27\cdots 44}a^{17}+\frac{33\cdots 12}{69\cdots 11}a^{16}-\frac{27\cdots 55}{27\cdots 44}a^{15}+\frac{13\cdots 75}{69\cdots 11}a^{14}+\frac{85\cdots 65}{27\cdots 44}a^{13}-\frac{19\cdots 91}{13\cdots 22}a^{12}+\frac{23\cdots 95}{27\cdots 44}a^{11}-\frac{19\cdots 23}{13\cdots 22}a^{10}-\frac{19\cdots 59}{27\cdots 44}a^{9}+\frac{76\cdots 99}{69\cdots 11}a^{8}+\frac{69\cdots 97}{27\cdots 44}a^{7}-\frac{18\cdots 80}{69\cdots 11}a^{6}-\frac{15\cdots 51}{27\cdots 44}a^{5}-\frac{13\cdots 01}{13\cdots 22}a^{4}+\frac{62\cdots 01}{27\cdots 44}a^{3}+\frac{47\cdots 25}{27\cdots 44}a^{2}+\frac{74\cdots 61}{69\cdots 11}a+\frac{85\cdots 94}{69\cdots 11}$
|
| |
| Regulator: | \( 2524911578570 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{12}\cdot(2\pi)^{4}\cdot 2524911578570 \cdot 1}{2\cdot\sqrt{477051862117015968785053941760000000000}}\cr\approx \mathstrut & 0.368988475278619 \end{aligned}\] (assuming GRH)
Galois group
$C_2^8.\POPlus(4,5)$ (as 20T1009):
| A non-solvable group of order 3686400 |
| The 114 conjugacy class representatives for $C_2^8.\POPlus(4,5)$ |
| Character table for $C_2^8.\POPlus(4,5)$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 10.10.109268775200000.1 |
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 |
| Degree 32 sibling: | data not computed |
| Degree 40 siblings: | 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 | ${\href{/padicField/3.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/7.10.0.1}{10} }^{2}$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}$ | R | ${\href{/padicField/17.12.0.1}{12} }{,}\,{\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.6.0.1}{6} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | R | R | ${\href{/padicField/37.6.0.1}{6} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.12.0.1}{12} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.12.0.1}{12} }{,}\,{\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 2.2.8.24b3.5 | $x^{16} + 8 x^{15} + 38 x^{14} + 126 x^{13} + 322 x^{12} + 658 x^{11} + 1108 x^{10} + 1558 x^{9} + 1851 x^{8} + 1862 x^{7} + 1590 x^{6} + 1146 x^{5} + 694 x^{4} + 346 x^{3} + 138 x^{2} + 40 x + 9$ | $8$ | $2$ | $24$ | 16T1286 | $$[\frac{4}{3}, \frac{4}{3}, \frac{4}{3}, \frac{4}{3}, 2, 2]_{3}^{6}$$ | |
|
\(5\)
| 5.2.2.2a1.2 | $x^{4} + 8 x^{3} + 20 x^{2} + 16 x + 9$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 5.4.2.4a1.2 | $x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ | |
| 5.4.2.4a1.2 | $x^{8} + 8 x^{6} + 8 x^{5} + 20 x^{4} + 32 x^{3} + 32 x^{2} + 16 x + 9$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ | |
|
\(13\)
| 13.6.1.0a1.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |
| 13.6.1.0a1.1 | $x^{6} + 10 x^{3} + 11 x^{2} + 11 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | |
| 13.2.4.6a1.3 | $x^{8} + 48 x^{7} + 872 x^{6} + 7200 x^{5} + 24216 x^{4} + 14400 x^{3} + 3488 x^{2} + 397 x + 159$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ | |
|
\(29\)
| 29.4.1.0a1.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 29.5.1.0a1.1 | $x^{5} + 3 x + 27$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ | |
| 29.5.1.0a1.1 | $x^{5} + 3 x + 27$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ | |
| 29.2.3.4a1.2 | $x^{6} + 72 x^{5} + 1734 x^{4} + 14112 x^{3} + 3468 x^{2} + 288 x + 37$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 31.2.1.0a1.1 | $x^{2} + 29 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 31.2.3.4a1.3 | $x^{6} + 87 x^{5} + 2532 x^{4} + 24911 x^{3} + 7596 x^{2} + 845 x + 895$ | $3$ | $2$ | $4$ | $C_6$ | $$[\ ]_{3}^{2}$$ | |
| 31.4.2.4a1.1 | $x^{8} + 6 x^{6} + 32 x^{5} + 15 x^{4} + 96 x^{3} + 274 x^{2} + 127 x + 9$ | $2$ | $4$ | $4$ | $C_8$ | $$[\ ]_{2}^{4}$$ |