Normalized defining polynomial
\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 29 x^{17} + 52 x^{16} - 373 x^{15} + 248 x^{14} + \cdots + 569387 \)
Invariants
| Degree: | $21$ |
| |
| Signature: | $[3, 9]$ |
| |
| Discriminant: |
\(-48958786234221774012025092444897160075327\)
\(\medspace = -\,7^{17}\cdot 29^{18}\)
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| |
| Root discriminant: | \(86.62\) |
| |
| Galois root discriminant: | $7^{5/6}29^{6/7}\approx 90.72612557219027$ | ||
| Ramified primes: |
\(7\), \(29\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\Aut(K/\Q)$: | $C_3$ |
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| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{4}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}+\frac{1}{4}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}a-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a$, $\frac{1}{4}a^{14}-\frac{1}{4}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{13}-\frac{1}{8}a^{11}-\frac{1}{8}a^{8}+\frac{1}{8}a^{7}-\frac{1}{8}a^{6}-\frac{1}{2}a^{5}+\frac{1}{8}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}-\frac{1}{4}a-\frac{1}{8}$, $\frac{1}{16}a^{16}-\frac{1}{16}a^{15}+\frac{1}{16}a^{14}-\frac{1}{16}a^{13}-\frac{1}{16}a^{12}-\frac{1}{16}a^{11}-\frac{1}{16}a^{9}-\frac{5}{16}a^{6}+\frac{5}{16}a^{5}-\frac{1}{16}a^{4}-\frac{1}{2}a^{3}+\frac{7}{16}a+\frac{5}{16}$, $\frac{1}{16}a^{17}-\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{16}a^{11}-\frac{1}{16}a^{10}-\frac{1}{16}a^{9}+\frac{3}{16}a^{7}+\frac{1}{4}a^{5}+\frac{7}{16}a^{4}-\frac{1}{2}a^{3}+\frac{7}{16}a^{2}-\frac{1}{4}a-\frac{3}{16}$, $\frac{1}{1392}a^{18}+\frac{7}{348}a^{17}+\frac{2}{87}a^{16}+\frac{1}{348}a^{15}+\frac{19}{696}a^{14}+\frac{41}{696}a^{13}-\frac{73}{1392}a^{12}-\frac{5}{48}a^{11}+\frac{43}{1392}a^{10}-\frac{53}{348}a^{9}+\frac{239}{1392}a^{8}-\frac{1}{6}a^{7}+\frac{1}{87}a^{6}+\frac{583}{1392}a^{5}+\frac{9}{29}a^{4}-\frac{665}{1392}a^{3}+\frac{10}{87}a^{2}-\frac{379}{1392}a-\frac{17}{87}$, $\frac{1}{5568}a^{19}-\frac{1}{5568}a^{18}+\frac{15}{928}a^{17}-\frac{47}{1856}a^{16}+\frac{3}{1856}a^{15}+\frac{37}{1856}a^{14}-\frac{23}{464}a^{13}+\frac{5}{192}a^{12}-\frac{121}{1856}a^{11}+\frac{1151}{5568}a^{10}-\frac{23}{928}a^{9}+\frac{35}{192}a^{8}+\frac{5}{58}a^{7}-\frac{941}{2784}a^{6}+\frac{257}{696}a^{5}-\frac{67}{174}a^{4}+\frac{1001}{5568}a^{3}+\frac{241}{1856}a^{2}+\frac{161}{464}a-\frac{61}{192}$, $\frac{1}{60\cdots 36}a^{20}-\frac{24\cdots 05}{50\cdots 28}a^{19}+\frac{12\cdots 71}{20\cdots 12}a^{18}-\frac{16\cdots 91}{60\cdots 36}a^{17}+\frac{84\cdots 29}{37\cdots 96}a^{16}+\frac{67\cdots 37}{15\cdots 84}a^{15}+\frac{10\cdots 11}{11\cdots 92}a^{14}-\frac{72\cdots 35}{60\cdots 36}a^{13}+\frac{12\cdots 33}{10\cdots 56}a^{12}-\frac{12\cdots 51}{18\cdots 48}a^{11}-\frac{33\cdots 77}{20\cdots 12}a^{10}+\frac{10\cdots 45}{46\cdots 72}a^{9}+\frac{53\cdots 49}{20\cdots 12}a^{8}+\frac{73\cdots 25}{30\cdots 68}a^{7}+\frac{33\cdots 67}{30\cdots 68}a^{6}+\frac{19\cdots 03}{50\cdots 28}a^{5}-\frac{98\cdots 21}{20\cdots 12}a^{4}-\frac{12\cdots 85}{50\cdots 28}a^{3}+\frac{11\cdots 87}{60\cdots 36}a^{2}+\frac{64\cdots 65}{20\cdots 12}a+\frac{25\cdots 47}{46\cdots 72}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{7}\times C_{7}$, which has order $49$ (assuming GRH) |
| |
| Narrow class group: | $C_{7}\times C_{7}$, which has order $49$ (assuming GRH) |
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Unit group
| Rank: | $11$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{16\cdots 21}{39\cdots 64}a^{20}-\frac{26\cdots 63}{98\cdots 16}a^{19}+\frac{28\cdots 05}{39\cdots 64}a^{18}-\frac{40\cdots 83}{39\cdots 64}a^{17}+\frac{16\cdots 11}{24\cdots 04}a^{16}+\frac{24\cdots 53}{98\cdots 16}a^{15}-\frac{56\cdots 73}{39\cdots 64}a^{14}+\frac{14\cdots 33}{39\cdots 64}a^{13}-\frac{29\cdots 17}{19\cdots 32}a^{12}+\frac{21\cdots 47}{12\cdots 52}a^{11}-\frac{24\cdots 51}{39\cdots 64}a^{10}+\frac{24\cdots 77}{39\cdots 64}a^{9}+\frac{19\cdots 31}{39\cdots 64}a^{8}+\frac{13\cdots 81}{19\cdots 32}a^{7}-\frac{25\cdots 93}{19\cdots 32}a^{6}+\frac{13\cdots 29}{98\cdots 16}a^{5}+\frac{54\cdots 29}{39\cdots 64}a^{4}+\frac{16\cdots 81}{98\cdots 16}a^{3}-\frac{20\cdots 97}{39\cdots 64}a^{2}-\frac{64\cdots 85}{39\cdots 64}a+\frac{34\cdots 75}{39\cdots 64}$, $\frac{16\cdots 21}{39\cdots 64}a^{20}-\frac{26\cdots 63}{98\cdots 16}a^{19}+\frac{28\cdots 05}{39\cdots 64}a^{18}-\frac{40\cdots 83}{39\cdots 64}a^{17}+\frac{16\cdots 11}{24\cdots 04}a^{16}+\frac{24\cdots 53}{98\cdots 16}a^{15}-\frac{56\cdots 73}{39\cdots 64}a^{14}+\frac{14\cdots 33}{39\cdots 64}a^{13}-\frac{29\cdots 17}{19\cdots 32}a^{12}+\frac{21\cdots 47}{12\cdots 52}a^{11}-\frac{24\cdots 51}{39\cdots 64}a^{10}+\frac{24\cdots 77}{39\cdots 64}a^{9}+\frac{19\cdots 31}{39\cdots 64}a^{8}+\frac{13\cdots 81}{19\cdots 32}a^{7}-\frac{25\cdots 93}{19\cdots 32}a^{6}+\frac{13\cdots 29}{98\cdots 16}a^{5}+\frac{54\cdots 29}{39\cdots 64}a^{4}+\frac{16\cdots 81}{98\cdots 16}a^{3}-\frac{20\cdots 97}{39\cdots 64}a^{2}-\frac{64\cdots 85}{39\cdots 64}a+\frac{73\cdots 39}{39\cdots 64}$, $\frac{39\cdots 03}{14\cdots 96}a^{20}-\frac{16\cdots 05}{12\cdots 08}a^{19}+\frac{40\cdots 75}{14\cdots 96}a^{18}-\frac{11\cdots 23}{49\cdots 32}a^{17}-\frac{17\cdots 91}{61\cdots 04}a^{16}+\frac{21\cdots 55}{12\cdots 08}a^{15}-\frac{25\cdots 81}{37\cdots 64}a^{14}-\frac{12\cdots 33}{14\cdots 96}a^{13}-\frac{77\cdots 87}{73\cdots 48}a^{12}+\frac{17\cdots 91}{18\cdots 12}a^{11}-\frac{38\cdots 89}{14\cdots 96}a^{10}-\frac{65\cdots 49}{11\cdots 92}a^{9}+\frac{61\cdots 25}{14\cdots 96}a^{8}+\frac{69\cdots 15}{73\cdots 48}a^{7}-\frac{17\cdots 81}{24\cdots 16}a^{6}-\frac{27\cdots 97}{36\cdots 24}a^{5}+\frac{15\cdots 37}{16\cdots 08}a^{4}+\frac{77\cdots 17}{36\cdots 24}a^{3}-\frac{17\cdots 05}{49\cdots 32}a^{2}-\frac{13\cdots 07}{14\cdots 96}a-\frac{16\cdots 15}{11\cdots 92}$, $\frac{18\cdots 57}{62\cdots 16}a^{20}-\frac{48\cdots 19}{25\cdots 64}a^{19}+\frac{13\cdots 27}{25\cdots 64}a^{18}-\frac{35\cdots 97}{43\cdots 08}a^{17}+\frac{17\cdots 23}{25\cdots 64}a^{16}+\frac{34\cdots 09}{25\cdots 64}a^{15}-\frac{48\cdots 57}{47\cdots 08}a^{14}+\frac{23\cdots 71}{62\cdots 16}a^{13}-\frac{25\cdots 99}{25\cdots 64}a^{12}+\frac{30\cdots 65}{25\cdots 64}a^{11}-\frac{11\cdots 97}{25\cdots 64}a^{10}+\frac{48\cdots 39}{96\cdots 64}a^{9}+\frac{53\cdots 91}{25\cdots 64}a^{8}+\frac{19\cdots 19}{31\cdots 08}a^{7}-\frac{11\cdots 75}{12\cdots 32}a^{6}+\frac{69\cdots 15}{62\cdots 16}a^{5}+\frac{12\cdots 99}{31\cdots 08}a^{4}+\frac{20\cdots 41}{25\cdots 64}a^{3}-\frac{10\cdots 33}{25\cdots 64}a^{2}+\frac{39\cdots 01}{62\cdots 16}a-\frac{34\cdots 77}{19\cdots 28}$, $\frac{36\cdots 95}{20\cdots 12}a^{20}+\frac{65\cdots 99}{15\cdots 84}a^{19}-\frac{48\cdots 83}{60\cdots 36}a^{18}+\frac{57\cdots 55}{20\cdots 12}a^{17}-\frac{10\cdots 01}{25\cdots 64}a^{16}+\frac{81\cdots 73}{50\cdots 28}a^{15}+\frac{49\cdots 49}{37\cdots 64}a^{14}-\frac{15\cdots 17}{20\cdots 12}a^{13}-\frac{62\cdots 97}{30\cdots 68}a^{12}+\frac{59\cdots 63}{25\cdots 64}a^{11}+\frac{24\cdots 09}{60\cdots 36}a^{10}-\frac{37\cdots 89}{15\cdots 24}a^{9}+\frac{20\cdots 91}{60\cdots 36}a^{8}+\frac{37\cdots 91}{10\cdots 56}a^{7}-\frac{37\cdots 01}{30\cdots 68}a^{6}-\frac{53\cdots 91}{15\cdots 84}a^{5}+\frac{54\cdots 17}{60\cdots 36}a^{4}+\frac{51\cdots 39}{15\cdots 84}a^{3}-\frac{72\cdots 47}{69\cdots 28}a^{2}-\frac{60\cdots 59}{20\cdots 12}a-\frac{25\cdots 73}{46\cdots 72}$, $\frac{17\cdots 27}{25\cdots 64}a^{20}-\frac{16\cdots 37}{25\cdots 64}a^{19}+\frac{12\cdots 15}{47\cdots 12}a^{18}-\frac{43\cdots 69}{75\cdots 92}a^{17}+\frac{51\cdots 15}{75\cdots 92}a^{16}+\frac{28\cdots 29}{75\cdots 92}a^{15}-\frac{34\cdots 65}{70\cdots 12}a^{14}+\frac{73\cdots 53}{75\cdots 92}a^{13}-\frac{23\cdots 77}{75\cdots 92}a^{12}+\frac{26\cdots 05}{75\cdots 92}a^{11}-\frac{42\cdots 61}{23\cdots 06}a^{10}+\frac{24\cdots 55}{58\cdots 84}a^{9}-\frac{12\cdots 21}{37\cdots 96}a^{8}-\frac{10\cdots 39}{37\cdots 96}a^{7}-\frac{23\cdots 11}{18\cdots 48}a^{6}+\frac{13\cdots 01}{18\cdots 48}a^{5}-\frac{22\cdots 01}{25\cdots 64}a^{4}+\frac{14\cdots 41}{75\cdots 92}a^{3}+\frac{26\cdots 29}{37\cdots 96}a^{2}+\frac{68\cdots 11}{75\cdots 92}a-\frac{69\cdots 89}{29\cdots 92}$, $\frac{84\cdots 73}{20\cdots 12}a^{20}-\frac{14\cdots 41}{50\cdots 28}a^{19}+\frac{16\cdots 41}{20\cdots 12}a^{18}-\frac{24\cdots 71}{20\cdots 12}a^{17}+\frac{22\cdots 03}{25\cdots 64}a^{16}+\frac{12\cdots 71}{50\cdots 28}a^{15}-\frac{58\cdots 25}{37\cdots 64}a^{14}+\frac{13\cdots 89}{20\cdots 12}a^{13}-\frac{15\cdots 57}{10\cdots 56}a^{12}+\frac{44\cdots 07}{25\cdots 64}a^{11}-\frac{13\cdots 95}{20\cdots 12}a^{10}+\frac{12\cdots 37}{15\cdots 24}a^{9}+\frac{73\cdots 07}{20\cdots 12}a^{8}+\frac{47\cdots 97}{10\cdots 56}a^{7}-\frac{13\cdots 13}{10\cdots 56}a^{6}+\frac{89\cdots 65}{50\cdots 28}a^{5}+\frac{18\cdots 69}{20\cdots 12}a^{4}+\frac{64\cdots 27}{50\cdots 28}a^{3}-\frac{10\cdots 37}{20\cdots 12}a^{2}+\frac{53\cdots 43}{20\cdots 12}a-\frac{29\cdots 93}{15\cdots 24}$, $\frac{61\cdots 93}{73\cdots 48}a^{20}-\frac{10\cdots 47}{18\cdots 12}a^{19}+\frac{12\cdots 05}{73\cdots 48}a^{18}-\frac{19\cdots 71}{73\cdots 48}a^{17}+\frac{11\cdots 85}{57\cdots 66}a^{16}+\frac{86\cdots 61}{18\cdots 12}a^{15}-\frac{17\cdots 25}{56\cdots 96}a^{14}+\frac{11\cdots 13}{73\cdots 48}a^{13}-\frac{11\cdots 17}{36\cdots 24}a^{12}+\frac{81\cdots 97}{23\cdots 64}a^{11}-\frac{10\cdots 79}{73\cdots 48}a^{10}+\frac{95\cdots 17}{56\cdots 96}a^{9}+\frac{42\cdots 91}{73\cdots 48}a^{8}+\frac{12\cdots 81}{12\cdots 56}a^{7}-\frac{98\cdots 49}{36\cdots 24}a^{6}+\frac{71\cdots 09}{18\cdots 12}a^{5}+\frac{11\cdots 81}{73\cdots 48}a^{4}+\frac{38\cdots 17}{18\cdots 12}a^{3}-\frac{84\cdots 73}{73\cdots 48}a^{2}+\frac{82\cdots 29}{24\cdots 16}a+\frac{11\cdots 87}{56\cdots 96}$, $\frac{21\cdots 45}{20\cdots 12}a^{20}-\frac{34\cdots 61}{50\cdots 28}a^{19}+\frac{10\cdots 23}{60\cdots 36}a^{18}-\frac{14\cdots 17}{60\cdots 36}a^{17}+\frac{11\cdots 89}{75\cdots 92}a^{16}+\frac{98\cdots 45}{15\cdots 84}a^{15}-\frac{40\cdots 11}{11\cdots 92}a^{14}+\frac{17\cdots 35}{60\cdots 36}a^{13}-\frac{11\cdots 03}{30\cdots 68}a^{12}+\frac{32\cdots 25}{75\cdots 92}a^{11}-\frac{92\cdots 05}{60\cdots 36}a^{10}+\frac{63\cdots 39}{46\cdots 72}a^{9}+\frac{84\cdots 13}{60\cdots 36}a^{8}+\frac{63\cdots 75}{30\cdots 68}a^{7}-\frac{10\cdots 31}{30\cdots 68}a^{6}+\frac{49\cdots 15}{15\cdots 84}a^{5}+\frac{73\cdots 41}{20\cdots 12}a^{4}+\frac{75\cdots 01}{15\cdots 84}a^{3}-\frac{76\cdots 15}{60\cdots 36}a^{2}-\frac{23\cdots 07}{60\cdots 36}a-\frac{42\cdots 27}{46\cdots 72}$, $\frac{18\cdots 07}{10\cdots 56}a^{20}-\frac{70\cdots 17}{62\cdots 16}a^{19}+\frac{30\cdots 95}{10\cdots 56}a^{18}-\frac{46\cdots 69}{10\cdots 56}a^{17}+\frac{85\cdots 79}{25\cdots 64}a^{16}+\frac{39\cdots 59}{31\cdots 08}a^{15}-\frac{12\cdots 99}{18\cdots 32}a^{14}+\frac{13\cdots 83}{10\cdots 56}a^{13}-\frac{36\cdots 77}{50\cdots 28}a^{12}+\frac{18\cdots 09}{25\cdots 64}a^{11}-\frac{25\cdots 17}{10\cdots 56}a^{10}+\frac{20\cdots 43}{77\cdots 12}a^{9}+\frac{82\cdots 29}{10\cdots 56}a^{8}+\frac{10\cdots 63}{50\cdots 28}a^{7}-\frac{23\cdots 15}{50\cdots 28}a^{6}+\frac{11\cdots 07}{25\cdots 64}a^{5}+\frac{27\cdots 15}{10\cdots 56}a^{4}+\frac{11\cdots 23}{15\cdots 04}a^{3}-\frac{29\cdots 03}{10\cdots 56}a^{2}-\frac{13\cdots 87}{10\cdots 56}a-\frac{30\cdots 23}{77\cdots 12}$, $\frac{27\cdots 61}{10\cdots 56}a^{20}-\frac{10\cdots 93}{75\cdots 92}a^{19}+\frac{97\cdots 39}{30\cdots 68}a^{18}-\frac{36\cdots 19}{10\cdots 56}a^{17}+\frac{94\cdots 71}{62\cdots 16}a^{16}+\frac{42\cdots 21}{25\cdots 64}a^{15}-\frac{13\cdots 57}{18\cdots 32}a^{14}-\frac{58\cdots 47}{10\cdots 56}a^{13}-\frac{16\cdots 59}{15\cdots 84}a^{12}+\frac{30\cdots 27}{31\cdots 08}a^{11}-\frac{88\cdots 97}{30\cdots 68}a^{10}+\frac{82\cdots 65}{77\cdots 12}a^{9}+\frac{93\cdots 41}{30\cdots 68}a^{8}+\frac{41\cdots 25}{50\cdots 28}a^{7}-\frac{11\cdots 19}{15\cdots 84}a^{6}+\frac{11\cdots 51}{75\cdots 92}a^{5}+\frac{20\cdots 23}{30\cdots 68}a^{4}+\frac{13\cdots 47}{75\cdots 92}a^{3}-\frac{83\cdots 73}{10\cdots 56}a^{2}-\frac{29\cdots 97}{10\cdots 56}a-\frac{20\cdots 75}{23\cdots 36}$
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| Regulator: | \( 72630093220.36017 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 72630093220.36017 \cdot 49}{2\cdot\sqrt{48958786234221774012025092444897160075327}}\cr\approx \mathstrut & 0.981920137928192 \end{aligned}\] (assuming GRH)
Galois group
| A solvable group of order 42 |
| The 7 conjugacy class representatives for $F_7$ |
| Character table for $F_7$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\), 7.1.9997195556047.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 42 |
| Degree 7 sibling: | 7.1.9997195556047.1 |
| Degree 14 sibling: | deg 14 |
| Minimal sibling: | 7.1.9997195556047.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.3.0.1}{3} }^{7}$ | ${\href{/padicField/3.6.0.1}{6} }^{3}{,}\,{\href{/padicField/3.3.0.1}{3} }$ | ${\href{/padicField/5.6.0.1}{6} }^{3}{,}\,{\href{/padicField/5.3.0.1}{3} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{7}$ | ${\href{/padicField/13.2.0.1}{2} }^{9}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }^{7}$ | R | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{7}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | ${\href{/padicField/43.7.0.1}{7} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }^{7}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.1.3.2a1.1 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 7.1.6.5a1.1 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ | |
| 7.1.6.5a1.1 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ | |
| 7.1.6.5a1.1 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ | |
|
\(29\)
| 29.1.7.6a1.3 | $x^{7} + 87$ | $7$ | $1$ | $6$ | $C_7$ | $$[\ ]_{7}$$ |
| 29.1.7.6a1.3 | $x^{7} + 87$ | $7$ | $1$ | $6$ | $C_7$ | $$[\ ]_{7}$$ | |
| 29.1.7.6a1.3 | $x^{7} + 87$ | $7$ | $1$ | $6$ | $C_7$ | $$[\ ]_{7}$$ |