Normalized defining polynomial
\( x^{21} - 42 x^{19} - 252 x^{18} - 126 x^{17} + 1078 x^{16} + 8876 x^{15} + 112790 x^{14} + \cdots - 82335377944 \)
Invariants
| Degree: | $21$ |
| |
| Signature: | $[7, 7]$ |
| |
| Discriminant: |
\(-19124990112791859831305063356808458501281808384\)
\(\medspace = -\,2^{18}\cdot 7^{36}\cdot 31^{7}\)
|
| |
| Root discriminant: | \(159.91\) |
| |
| Galois root discriminant: | $2^{6/7}7^{96/49}31^{1/2}\approx 456.46642613478804$ | ||
| Ramified primes: |
\(2\), \(7\), \(31\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-31}) \) | ||
| $\Aut(K/\Q)$: | $C_7$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{2}a^{12}$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{14}$, $\frac{1}{4}a^{15}$, $\frac{1}{20}a^{16}-\frac{1}{10}a^{15}-\frac{1}{20}a^{14}-\frac{1}{10}a^{12}+\frac{1}{5}a^{11}+\frac{1}{10}a^{10}+\frac{1}{10}a^{9}-\frac{1}{10}a^{8}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}+\frac{1}{5}a^{3}+\frac{1}{5}a^{2}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{20}a^{17}-\frac{1}{10}a^{14}-\frac{1}{10}a^{13}-\frac{1}{5}a^{10}+\frac{1}{10}a^{9}-\frac{1}{5}a^{8}-\frac{1}{5}a^{7}-\frac{1}{5}a^{6}+\frac{1}{5}a^{5}-\frac{1}{5}a^{4}-\frac{2}{5}a^{3}+\frac{1}{5}a^{2}+\frac{2}{5}a-\frac{2}{5}$, $\frac{1}{20}a^{18}-\frac{1}{10}a^{15}-\frac{1}{10}a^{14}-\frac{1}{5}a^{11}+\frac{1}{10}a^{10}-\frac{1}{5}a^{9}-\frac{1}{5}a^{8}-\frac{1}{5}a^{7}+\frac{1}{5}a^{6}-\frac{1}{5}a^{5}-\frac{2}{5}a^{4}+\frac{1}{5}a^{3}+\frac{2}{5}a^{2}-\frac{2}{5}a$, $\frac{1}{40}a^{19}+\frac{1}{10}a^{15}-\frac{1}{20}a^{14}-\frac{1}{5}a^{12}-\frac{1}{5}a^{8}+\frac{1}{10}a^{7}+\frac{1}{5}a^{6}-\frac{1}{2}a^{5}+\frac{2}{5}a^{4}+\frac{2}{5}a^{3}-\frac{1}{5}a-\frac{1}{5}$, $\frac{1}{11\cdots 00}a^{20}+\frac{16\cdots 04}{14\cdots 25}a^{19}+\frac{19\cdots 11}{58\cdots 00}a^{18}-\frac{55\cdots 79}{58\cdots 00}a^{17}-\frac{12\cdots 71}{58\cdots 00}a^{16}-\frac{66\cdots 27}{14\cdots 25}a^{15}+\frac{25\cdots 18}{14\cdots 25}a^{14}+\frac{32\cdots 29}{58\cdots 00}a^{13}+\frac{36\cdots 82}{14\cdots 25}a^{12}-\frac{14\cdots 59}{14\cdots 25}a^{11}-\frac{56\cdots 39}{29\cdots 50}a^{10}+\frac{14\cdots 82}{14\cdots 25}a^{9}-\frac{66\cdots 79}{29\cdots 50}a^{8}+\frac{45\cdots 26}{29\cdots 45}a^{7}-\frac{15\cdots 23}{29\cdots 45}a^{6}+\frac{46\cdots 34}{14\cdots 25}a^{5}+\frac{15\cdots 91}{14\cdots 25}a^{4}+\frac{13\cdots 42}{29\cdots 45}a^{3}-\frac{63\cdots 26}{14\cdots 25}a^{2}+\frac{56\cdots 49}{29\cdots 45}a+\frac{91\cdots 64}{14\cdots 25}$
| 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: | $13$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{11\cdots 13}{34\cdots 00}a^{20}-\frac{22\cdots 59}{85\cdots 75}a^{19}+\frac{77\cdots 47}{17\cdots 50}a^{18}+\frac{28\cdots 77}{17\cdots 50}a^{17}+\frac{35\cdots 21}{34\cdots 00}a^{16}+\frac{53\cdots 09}{17\cdots 50}a^{15}+\frac{42\cdots 79}{17\cdots 50}a^{14}-\frac{18\cdots 29}{34\cdots 00}a^{13}-\frac{18\cdots 17}{85\cdots 75}a^{12}+\frac{71\cdots 83}{17\cdots 50}a^{11}+\frac{40\cdots 69}{17\cdots 50}a^{10}-\frac{14\cdots 62}{85\cdots 75}a^{9}-\frac{14\cdots 01}{17\cdots 50}a^{8}-\frac{23\cdots 01}{17\cdots 95}a^{7}-\frac{41\cdots 83}{34\cdots 90}a^{6}+\frac{26\cdots 36}{85\cdots 75}a^{5}+\frac{18\cdots 69}{85\cdots 75}a^{4}-\frac{34\cdots 47}{17\cdots 95}a^{3}-\frac{89\cdots 14}{85\cdots 75}a^{2}+\frac{78\cdots 72}{17\cdots 95}a+\frac{66\cdots 26}{85\cdots 75}$, $\frac{97\cdots 19}{46\cdots 52}a^{20}-\frac{40\cdots 95}{58\cdots 69}a^{19}-\frac{46\cdots 09}{11\cdots 80}a^{18}+\frac{87\cdots 69}{23\cdots 76}a^{17}+\frac{18\cdots 51}{11\cdots 80}a^{16}+\frac{66\cdots 81}{11\cdots 80}a^{15}-\frac{20\cdots 73}{11\cdots 80}a^{14}-\frac{12\cdots 21}{23\cdots 76}a^{13}-\frac{82\cdots 93}{29\cdots 45}a^{12}-\frac{26\cdots 37}{11\cdots 38}a^{11}+\frac{67\cdots 26}{29\cdots 45}a^{10}+\frac{32\cdots 77}{29\cdots 45}a^{9}-\frac{11\cdots 74}{29\cdots 45}a^{8}+\frac{93\cdots 43}{58\cdots 90}a^{7}+\frac{25\cdots 64}{58\cdots 69}a^{6}-\frac{47\cdots 45}{58\cdots 69}a^{5}-\frac{57\cdots 78}{29\cdots 45}a^{4}+\frac{25\cdots 52}{29\cdots 45}a^{3}-\frac{25\cdots 27}{29\cdots 45}a^{2}-\frac{13\cdots 88}{29\cdots 45}a+\frac{23\cdots 94}{29\cdots 45}$, $\frac{71\cdots 53}{23\cdots 60}a^{20}+\frac{31\cdots 41}{23\cdots 60}a^{19}-\frac{19\cdots 21}{29\cdots 45}a^{18}-\frac{12\cdots 11}{11\cdots 80}a^{17}-\frac{64\cdots 17}{11\cdots 80}a^{16}-\frac{12\cdots 31}{58\cdots 90}a^{15}-\frac{59\cdots 33}{11\cdots 80}a^{14}+\frac{46\cdots 61}{23\cdots 76}a^{13}-\frac{71\cdots 52}{29\cdots 45}a^{12}-\frac{11\cdots 09}{29\cdots 45}a^{11}+\frac{11\cdots 82}{58\cdots 69}a^{10}+\frac{53\cdots 63}{58\cdots 90}a^{9}-\frac{58\cdots 51}{58\cdots 90}a^{8}+\frac{73\cdots 91}{29\cdots 45}a^{7}+\frac{24\cdots 71}{29\cdots 45}a^{6}-\frac{26\cdots 53}{58\cdots 90}a^{5}+\frac{38\cdots 55}{58\cdots 69}a^{4}+\frac{52\cdots 52}{29\cdots 45}a^{3}-\frac{29\cdots 32}{58\cdots 69}a^{2}+\frac{25\cdots 54}{58\cdots 69}a-\frac{54\cdots 66}{29\cdots 45}$, $\frac{23\cdots 61}{29\cdots 50}a^{20}+\frac{26\cdots 27}{11\cdots 00}a^{19}+\frac{13\cdots 61}{58\cdots 00}a^{18}-\frac{35\cdots 89}{58\cdots 00}a^{17}-\frac{30\cdots 31}{58\cdots 00}a^{16}-\frac{94\cdots 02}{14\cdots 25}a^{15}-\frac{65\cdots 79}{29\cdots 50}a^{14}+\frac{16\cdots 29}{58\cdots 00}a^{13}+\frac{80\cdots 69}{29\cdots 50}a^{12}-\frac{27\cdots 83}{29\cdots 50}a^{11}-\frac{31\cdots 89}{29\cdots 50}a^{10}+\frac{33\cdots 99}{29\cdots 50}a^{9}-\frac{20\cdots 99}{29\cdots 50}a^{8}-\frac{34\cdots 54}{29\cdots 45}a^{7}+\frac{31\cdots 89}{58\cdots 90}a^{6}-\frac{57\cdots 47}{29\cdots 50}a^{5}-\frac{71\cdots 54}{14\cdots 25}a^{4}+\frac{94\cdots 36}{58\cdots 69}a^{3}-\frac{26\cdots 36}{14\cdots 25}a^{2}-\frac{35\cdots 11}{29\cdots 45}a+\frac{33\cdots 34}{14\cdots 25}$, $\frac{39\cdots 11}{11\cdots 80}a^{20}-\frac{21\cdots 43}{46\cdots 52}a^{19}-\frac{13\cdots 27}{11\cdots 80}a^{18}-\frac{86\cdots 69}{11\cdots 80}a^{17}+\frac{36\cdots 26}{29\cdots 45}a^{16}+\frac{18\cdots 47}{11\cdots 80}a^{15}+\frac{23\cdots 61}{58\cdots 90}a^{14}+\frac{82\cdots 25}{23\cdots 76}a^{13}-\frac{21\cdots 65}{11\cdots 38}a^{12}-\frac{13\cdots 73}{29\cdots 45}a^{11}+\frac{51\cdots 61}{58\cdots 90}a^{10}-\frac{28\cdots 89}{58\cdots 90}a^{9}-\frac{21\cdots 53}{29\cdots 45}a^{8}+\frac{23\cdots 48}{29\cdots 45}a^{7}-\frac{13\cdots 21}{58\cdots 90}a^{6}-\frac{17\cdots 61}{58\cdots 90}a^{5}+\frac{61\cdots 47}{29\cdots 45}a^{4}-\frac{94\cdots 59}{29\cdots 45}a^{3}+\frac{12\cdots 88}{58\cdots 69}a^{2}+\frac{53\cdots 24}{29\cdots 45}a-\frac{87\cdots 47}{29\cdots 45}$, $\frac{11\cdots 93}{14\cdots 25}a^{20}+\frac{32\cdots 49}{58\cdots 00}a^{19}-\frac{67\cdots 51}{58\cdots 00}a^{18}-\frac{20\cdots 11}{58\cdots 00}a^{17}-\frac{12\cdots 09}{58\cdots 00}a^{16}-\frac{21\cdots 41}{29\cdots 50}a^{15}-\frac{10\cdots 47}{58\cdots 00}a^{14}+\frac{28\cdots 21}{58\cdots 00}a^{13}+\frac{25\cdots 73}{14\cdots 25}a^{12}-\frac{41\cdots 17}{29\cdots 50}a^{11}-\frac{72\cdots 91}{29\cdots 50}a^{10}+\frac{13\cdots 58}{14\cdots 25}a^{9}-\frac{27\cdots 53}{14\cdots 25}a^{8}-\frac{19\cdots 91}{58\cdots 90}a^{7}+\frac{29\cdots 61}{58\cdots 90}a^{6}-\frac{11\cdots 39}{14\cdots 25}a^{5}-\frac{48\cdots 51}{14\cdots 25}a^{4}+\frac{40\cdots 89}{29\cdots 45}a^{3}-\frac{67\cdots 34}{14\cdots 25}a^{2}-\frac{37\cdots 27}{58\cdots 69}a+\frac{10\cdots 56}{14\cdots 25}$, $\frac{50\cdots 37}{58\cdots 00}a^{20}-\frac{19\cdots 29}{58\cdots 00}a^{19}+\frac{42\cdots 79}{14\cdots 25}a^{18}+\frac{11\cdots 51}{58\cdots 00}a^{17}+\frac{47\cdots 46}{14\cdots 25}a^{16}+\frac{13\cdots 17}{58\cdots 00}a^{15}+\frac{29\cdots 81}{29\cdots 50}a^{14}-\frac{23\cdots 81}{58\cdots 00}a^{13}+\frac{62\cdots 07}{14\cdots 25}a^{12}+\frac{35\cdots 57}{29\cdots 50}a^{11}-\frac{13\cdots 32}{14\cdots 25}a^{10}-\frac{41\cdots 11}{29\cdots 50}a^{9}+\frac{17\cdots 01}{29\cdots 50}a^{8}-\frac{93\cdots 65}{58\cdots 69}a^{7}+\frac{16\cdots 11}{58\cdots 90}a^{6}+\frac{42\cdots 64}{14\cdots 25}a^{5}-\frac{12\cdots 84}{14\cdots 25}a^{4}+\frac{10\cdots 96}{29\cdots 45}a^{3}+\frac{81\cdots 69}{14\cdots 25}a^{2}-\frac{30\cdots 14}{29\cdots 45}a-\frac{96\cdots 36}{14\cdots 25}$, $\frac{14\cdots 87}{11\cdots 80}a^{20}+\frac{30\cdots 09}{23\cdots 60}a^{19}+\frac{14\cdots 26}{29\cdots 45}a^{18}+\frac{29\cdots 69}{11\cdots 80}a^{17}-\frac{13\cdots 01}{11\cdots 80}a^{16}-\frac{13\cdots 75}{11\cdots 38}a^{15}-\frac{28\cdots 48}{29\cdots 45}a^{14}-\frac{72\cdots 01}{58\cdots 90}a^{13}+\frac{80\cdots 35}{11\cdots 38}a^{12}+\frac{40\cdots 21}{58\cdots 90}a^{11}-\frac{50\cdots 67}{58\cdots 90}a^{10}+\frac{45\cdots 51}{58\cdots 90}a^{9}+\frac{76\cdots 91}{11\cdots 38}a^{8}-\frac{37\cdots 57}{11\cdots 38}a^{7}+\frac{11\cdots 42}{29\cdots 45}a^{6}+\frac{17\cdots 63}{58\cdots 90}a^{5}-\frac{38\cdots 39}{29\cdots 45}a^{4}+\frac{37\cdots 96}{29\cdots 45}a^{3}+\frac{13\cdots 53}{29\cdots 45}a^{2}-\frac{42\cdots 32}{29\cdots 45}a+\frac{30\cdots 86}{29\cdots 45}$, $\frac{15\cdots 65}{23\cdots 76}a^{20}+\frac{73\cdots 79}{23\cdots 60}a^{19}+\frac{21\cdots 19}{11\cdots 80}a^{18}+\frac{27\cdots 19}{23\cdots 76}a^{17}-\frac{71\cdots 93}{11\cdots 80}a^{16}-\frac{18\cdots 19}{11\cdots 80}a^{15}-\frac{11\cdots 89}{11\cdots 80}a^{14}-\frac{13\cdots 18}{58\cdots 69}a^{13}+\frac{37\cdots 55}{58\cdots 69}a^{12}-\frac{58\cdots 39}{58\cdots 90}a^{11}+\frac{22\cdots 61}{58\cdots 90}a^{10}-\frac{22\cdots 31}{58\cdots 90}a^{9}+\frac{23\cdots 61}{29\cdots 45}a^{8}+\frac{86\cdots 41}{58\cdots 90}a^{7}-\frac{52\cdots 29}{29\cdots 45}a^{6}+\frac{45\cdots 21}{58\cdots 90}a^{5}-\frac{42\cdots 02}{29\cdots 45}a^{4}-\frac{37\cdots 01}{29\cdots 45}a^{3}+\frac{77\cdots 62}{58\cdots 69}a^{2}-\frac{77\cdots 39}{29\cdots 45}a+\frac{48\cdots 49}{29\cdots 45}$, $\frac{16\cdots 93}{46\cdots 52}a^{20}+\frac{61\cdots 21}{23\cdots 60}a^{19}-\frac{18\cdots 59}{11\cdots 80}a^{18}-\frac{64\cdots 27}{58\cdots 90}a^{17}-\frac{14\cdots 31}{11\cdots 80}a^{16}+\frac{45\cdots 51}{58\cdots 90}a^{15}+\frac{41\cdots 13}{58\cdots 90}a^{14}+\frac{16\cdots 42}{29\cdots 45}a^{13}-\frac{34\cdots 08}{29\cdots 45}a^{12}-\frac{20\cdots 47}{29\cdots 45}a^{11}+\frac{48\cdots 33}{58\cdots 90}a^{10}+\frac{96\cdots 83}{58\cdots 90}a^{9}-\frac{17\cdots 27}{11\cdots 38}a^{8}+\frac{18\cdots 76}{29\cdots 45}a^{7}+\frac{34\cdots 69}{58\cdots 90}a^{6}-\frac{48\cdots 47}{58\cdots 90}a^{5}+\frac{10\cdots 61}{58\cdots 69}a^{4}+\frac{13\cdots 87}{58\cdots 69}a^{3}-\frac{29\cdots 08}{29\cdots 45}a^{2}+\frac{63\cdots 51}{58\cdots 69}a+\frac{24\cdots 23}{29\cdots 45}$, $\frac{13\cdots 71}{58\cdots 00}a^{20}+\frac{50\cdots 03}{14\cdots 25}a^{19}-\frac{54\cdots 43}{58\cdots 00}a^{18}-\frac{20\cdots 79}{29\cdots 50}a^{17}-\frac{18\cdots 28}{14\cdots 25}a^{16}+\frac{57\cdots 69}{58\cdots 00}a^{15}+\frac{12\cdots 89}{58\cdots 00}a^{14}+\frac{17\cdots 73}{58\cdots 00}a^{13}-\frac{95\cdots 96}{14\cdots 25}a^{12}-\frac{59\cdots 03}{14\cdots 25}a^{11}+\frac{16\cdots 41}{14\cdots 25}a^{10}+\frac{23\cdots 14}{14\cdots 25}a^{9}-\frac{36\cdots 33}{29\cdots 50}a^{8}+\frac{96\cdots 24}{29\cdots 45}a^{7}+\frac{26\cdots 31}{58\cdots 90}a^{6}-\frac{10\cdots 37}{14\cdots 25}a^{5}+\frac{18\cdots 37}{14\cdots 25}a^{4}+\frac{12\cdots 02}{58\cdots 69}a^{3}-\frac{16\cdots 97}{14\cdots 25}a^{2}+\frac{24\cdots 74}{29\cdots 45}a+\frac{23\cdots 43}{14\cdots 25}$, $\frac{50\cdots 59}{58\cdots 00}a^{20}+\frac{13\cdots 31}{11\cdots 00}a^{19}-\frac{23\cdots 37}{58\cdots 00}a^{18}-\frac{33\cdots 13}{14\cdots 25}a^{17}+\frac{20\cdots 57}{58\cdots 00}a^{16}+\frac{12\cdots 91}{58\cdots 00}a^{15}+\frac{58\cdots 61}{58\cdots 00}a^{14}+\frac{13\cdots 33}{14\cdots 25}a^{13}-\frac{13\cdots 93}{29\cdots 50}a^{12}-\frac{23\cdots 67}{14\cdots 25}a^{11}+\frac{99\cdots 79}{14\cdots 25}a^{10}+\frac{27\cdots 77}{29\cdots 50}a^{9}-\frac{85\cdots 56}{14\cdots 25}a^{8}+\frac{36\cdots 19}{29\cdots 45}a^{7}+\frac{91\cdots 49}{29\cdots 45}a^{6}-\frac{84\cdots 71}{29\cdots 50}a^{5}+\frac{10\cdots 88}{14\cdots 25}a^{4}+\frac{64\cdots 80}{58\cdots 69}a^{3}-\frac{74\cdots 58}{14\cdots 25}a^{2}+\frac{52\cdots 16}{29\cdots 45}a+\frac{79\cdots 72}{14\cdots 25}$, $\frac{15\cdots 69}{11\cdots 00}a^{20}-\frac{84\cdots 97}{11\cdots 00}a^{19}-\frac{32\cdots 31}{58\cdots 00}a^{18}-\frac{18\cdots 91}{58\cdots 00}a^{17}-\frac{10\cdots 09}{58\cdots 00}a^{16}+\frac{64\cdots 19}{29\cdots 50}a^{15}+\frac{48\cdots 59}{29\cdots 50}a^{14}+\frac{21\cdots 89}{14\cdots 25}a^{13}-\frac{20\cdots 39}{29\cdots 50}a^{12}-\frac{44\cdots 07}{29\cdots 50}a^{11}+\frac{21\cdots 89}{29\cdots 50}a^{10}+\frac{49\cdots 71}{29\cdots 50}a^{9}-\frac{61\cdots 48}{14\cdots 25}a^{8}+\frac{69\cdots 27}{29\cdots 45}a^{7}-\frac{25\cdots 37}{58\cdots 90}a^{6}-\frac{77\cdots 03}{29\cdots 50}a^{5}+\frac{13\cdots 99}{14\cdots 25}a^{4}-\frac{58\cdots 48}{29\cdots 45}a^{3}-\frac{21\cdots 99}{14\cdots 25}a^{2}+\frac{20\cdots 74}{29\cdots 45}a+\frac{81\cdots 86}{14\cdots 25}$
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| Regulator: | \( 814410889718241.8 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{7}\cdot 814410889718241.8 \cdot 1}{2\cdot\sqrt{19124990112791859831305063356808458501281808384}}\cr\approx \mathstrut & 0.145707536289881 \end{aligned}\] (assuming GRH)
Galois group
$C_7^2:S_3$ (as 21T18):
| A solvable group of order 294 |
| The 20 conjugacy class representatives for $C_7^2:S_3$ |
| Character table for $C_7^2:S_3$ |
Intermediate fields
| 3.1.31.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 14 sibling: | data not computed |
| Degree 21 sibling: | data not computed |
| Degree 42 siblings: | data not computed |
| Minimal sibling: | 14.0.337337631352558562817384077600704.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.7.0.1}{7} }$ | ${\href{/padicField/5.3.0.1}{3} }^{7}$ | R | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.7.0.1}{7} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.7.0.1}{7} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.7.0.1}{7} }$ | ${\href{/padicField/19.3.0.1}{3} }^{7}$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.7.0.1}{7} }$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.7.0.1}{7} }$ | R | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.7.0.1}{7} }$ | ${\href{/padicField/41.3.0.1}{3} }^{7}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.7.0.1}{7} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{7}$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.7.0.1}{7} }$ | ${\href{/padicField/59.3.0.1}{3} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.21.18.1 | $x^{21} + 7 x^{19} + 7 x^{18} + 21 x^{17} + 42 x^{16} + 56 x^{15} + 105 x^{14} + 140 x^{13} + 175 x^{12} + 231 x^{11} + 245 x^{10} + 252 x^{9} + 252 x^{8} + 211 x^{7} + 168 x^{6} + 126 x^{5} + 77 x^{4} + 42 x^{3} + 21 x^{2} + 7 x + 3$ | $7$ | $3$ | $18$ | 21T2 | $$[\ ]_{7}^{3}$$ |
|
\(7\)
| 7.21.36.382 | $x^{21} + 56 x^{20} + 1288 x^{19} + 16198 x^{18} + 123816 x^{17} + 604968 x^{16} + 1968960 x^{15} + 4611648 x^{14} + 8772288 x^{13} + 14035616 x^{12} + 18602752 x^{11} + 23631104 x^{10} + 21996800 x^{9} + 24890880 x^{8} + 15267840 x^{7} + 16955904 x^{6} + 6150144 x^{5} + 7268352 x^{4} + 1318912 x^{3} + 1777664 x^{2} + 114786 x + 188423$ | $7$ | $3$ | $36$ | not computed | not computed |
|
\(31\)
| 31.7.0.1 | $x^{7} + x + 28$ | $1$ | $7$ | $0$ | $C_7$ | $$[\ ]^{7}$$ |
| 31.14.7.2 | $x^{14} + 2 x^{8} + 56 x^{7} + x^{2} + 56 x + 815$ | $2$ | $7$ | $7$ | $C_{14}$ | $$[\ ]_{2}^{7}$$ |