Normalized defining polynomial
\( x^{21} - 7 x^{20} + 21 x^{19} - 34 x^{18} + 330 x^{17} - 2629 x^{16} + 9945 x^{15} - 21697 x^{14} + \cdots + 35136283 \)
Invariants
| Degree: | $21$ |
| |
| Signature: | $[3, 9]$ |
| |
| Discriminant: |
\(-58762312802028807390251370834685864006539343\)
\(\medspace = -\,7^{17}\cdot 43^{18}\)
|
| |
| Root discriminant: | \(121.41\) |
| |
| Galois root discriminant: | $7^{5/6}43^{6/7}\approx 127.16400157052036$ | ||
| Ramified primes: |
\(7\), \(43\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-7}) \) | ||
| $\Aut(K/\Q)$: | $C_3$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{7}-\frac{1}{2}$, $\frac{1}{2}a^{8}-\frac{1}{2}a$, $\frac{1}{2}a^{9}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{10}-\frac{1}{2}a^{3}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{6}-\frac{1}{2}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{2}-\frac{1}{4}$, $\frac{1}{4}a^{13}-\frac{1}{4}a^{11}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}+\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{2}a^{2}+\frac{1}{4}a-\frac{1}{2}$, $\frac{1}{4}a^{14}-\frac{1}{4}$, $\frac{1}{8}a^{15}-\frac{1}{8}a^{14}-\frac{1}{8}a^{13}+\frac{1}{8}a^{11}+\frac{1}{8}a^{9}-\frac{1}{4}a^{8}+\frac{1}{8}a^{7}-\frac{1}{8}a^{5}+\frac{1}{4}a^{4}+\frac{3}{8}a^{3}-\frac{1}{4}a^{2}-\frac{1}{2}a-\frac{1}{8}$, $\frac{1}{8}a^{16}-\frac{1}{8}a^{13}-\frac{1}{8}a^{12}+\frac{1}{8}a^{11}-\frac{1}{8}a^{10}-\frac{1}{8}a^{9}+\frac{1}{8}a^{8}+\frac{1}{8}a^{7}+\frac{1}{8}a^{6}-\frac{3}{8}a^{5}-\frac{1}{8}a^{4}-\frac{3}{8}a^{3}-\frac{1}{2}a^{2}+\frac{3}{8}a-\frac{1}{8}$, $\frac{1}{104}a^{17}+\frac{5}{104}a^{16}-\frac{5}{104}a^{15}-\frac{3}{26}a^{14}-\frac{9}{104}a^{13}+\frac{3}{26}a^{12}-\frac{1}{8}a^{11}+\frac{1}{52}a^{10}-\frac{1}{104}a^{9}-\frac{3}{26}a^{8}+\frac{25}{104}a^{7}-\frac{11}{52}a^{6}-\frac{27}{104}a^{5}-\frac{23}{52}a^{4}+\frac{19}{52}a^{3}-\frac{31}{104}a^{2}+\frac{15}{52}a$, $\frac{1}{3255616}a^{18}-\frac{1821}{465088}a^{17}+\frac{291}{17888}a^{16}-\frac{36713}{813904}a^{15}-\frac{2127}{3255616}a^{14}+\frac{205845}{3255616}a^{13}+\frac{24121}{1627808}a^{12}-\frac{178371}{3255616}a^{11}+\frac{34425}{232544}a^{10}+\frac{22441}{250432}a^{9}+\frac{701}{17888}a^{8}+\frac{541193}{3255616}a^{7}+\frac{396313}{1627808}a^{6}+\frac{1116653}{3255616}a^{5}-\frac{1626295}{3255616}a^{4}+\frac{2865}{813904}a^{3}-\frac{2143}{7267}a^{2}+\frac{33379}{465088}a-\frac{10481}{35776}$, $\frac{1}{3255616}a^{19}+\frac{487}{465088}a^{17}+\frac{69717}{1627808}a^{16}+\frac{148541}{3255616}a^{15}+\frac{901}{203476}a^{14}-\frac{228559}{3255616}a^{13}+\frac{202275}{3255616}a^{12}+\frac{61091}{465088}a^{11}-\frac{327697}{3255616}a^{10}-\frac{3833}{35776}a^{9}-\frac{102541}{3255616}a^{8}+\frac{259933}{3255616}a^{7}+\frac{27119}{250432}a^{6}-\frac{33}{2366}a^{5}+\frac{953527}{3255616}a^{4}-\frac{51599}{116272}a^{3}-\frac{196501}{465088}a^{2}-\frac{14403}{116272}a+\frac{4165}{35776}$, $\frac{1}{79\cdots 44}a^{20}-\frac{18\cdots 31}{39\cdots 72}a^{19}-\frac{97\cdots 87}{79\cdots 44}a^{18}+\frac{97\cdots 35}{23\cdots 76}a^{17}-\frac{23\cdots 79}{79\cdots 44}a^{16}+\frac{20\cdots 01}{39\cdots 72}a^{15}+\frac{65\cdots 01}{79\cdots 44}a^{14}-\frac{91\cdots 99}{79\cdots 44}a^{13}-\frac{97\cdots 53}{79\cdots 44}a^{12}+\frac{13\cdots 49}{79\cdots 44}a^{11}+\frac{15\cdots 63}{79\cdots 44}a^{10}-\frac{88\cdots 83}{79\cdots 44}a^{9}+\frac{39\cdots 11}{79\cdots 44}a^{8}-\frac{87\cdots 35}{79\cdots 44}a^{7}-\frac{18\cdots 05}{39\cdots 72}a^{6}-\frac{17\cdots 65}{79\cdots 44}a^{5}+\frac{18\cdots 91}{39\cdots 72}a^{4}+\frac{35\cdots 73}{79\cdots 44}a^{3}-\frac{10\cdots 07}{56\cdots 96}a^{2}+\frac{17\cdots 05}{11\cdots 92}a+\frac{13\cdots 43}{43\cdots 92}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{7}$, which has order $7$ (assuming GRH) |
| |
| Narrow class group: | $C_{7}$, which has order $7$ (assuming GRH) |
|
Unit group
| Rank: | $11$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{21\cdots 57}{63\cdots 28}a^{20}-\frac{16\cdots 87}{79\cdots 16}a^{19}+\frac{93\cdots 33}{18\cdots 08}a^{18}-\frac{84\cdots 03}{12\cdots 56}a^{17}+\frac{33\cdots 59}{31\cdots 64}a^{16}-\frac{35\cdots 27}{45\cdots 52}a^{15}+\frac{33\cdots 55}{12\cdots 56}a^{14}-\frac{62\cdots 65}{12\cdots 56}a^{13}+\frac{51\cdots 57}{31\cdots 64}a^{12}-\frac{12\cdots 05}{12\cdots 56}a^{11}+\frac{29\cdots 43}{79\cdots 16}a^{10}-\frac{12\cdots 73}{12\cdots 56}a^{9}+\frac{18\cdots 93}{79\cdots 16}a^{8}-\frac{12\cdots 19}{18\cdots 08}a^{7}+\frac{14\cdots 81}{63\cdots 28}a^{6}-\frac{80\cdots 69}{12\cdots 56}a^{5}+\frac{14\cdots 65}{12\cdots 56}a^{4}-\frac{98\cdots 03}{90\cdots 04}a^{3}+\frac{15\cdots 13}{56\cdots 94}a^{2}+\frac{46\cdots 65}{18\cdots 08}a-\frac{18\cdots 73}{13\cdots 16}$, $\frac{33\cdots 61}{12\cdots 56}a^{20}-\frac{99\cdots 97}{63\cdots 28}a^{19}+\frac{71\cdots 69}{18\cdots 08}a^{18}-\frac{31\cdots 03}{63\cdots 28}a^{17}+\frac{10\cdots 53}{12\cdots 56}a^{16}-\frac{38\cdots 85}{63\cdots 28}a^{15}+\frac{36\cdots 17}{18\cdots 08}a^{14}-\frac{47\cdots 01}{12\cdots 56}a^{13}+\frac{15\cdots 83}{12\cdots 56}a^{12}-\frac{96\cdots 89}{12\cdots 56}a^{11}+\frac{35\cdots 39}{12\cdots 56}a^{10}-\frac{93\cdots 49}{12\cdots 56}a^{9}+\frac{22\cdots 75}{12\cdots 56}a^{8}-\frac{66\cdots 49}{12\cdots 56}a^{7}+\frac{10\cdots 07}{63\cdots 28}a^{6}-\frac{61\cdots 59}{12\cdots 56}a^{5}+\frac{13\cdots 37}{15\cdots 32}a^{4}-\frac{14\cdots 97}{18\cdots 08}a^{3}+\frac{10\cdots 07}{69\cdots 08}a^{2}+\frac{42\cdots 71}{18\cdots 08}a-\frac{36\cdots 57}{34\cdots 04}$, $\frac{37\cdots 63}{11\cdots 92}a^{20}-\frac{39\cdots 81}{19\cdots 36}a^{19}+\frac{47\cdots 97}{99\cdots 68}a^{18}-\frac{66\cdots 81}{11\cdots 92}a^{17}+\frac{81\cdots 47}{79\cdots 44}a^{16}-\frac{10\cdots 63}{14\cdots 24}a^{15}+\frac{61\cdots 55}{24\cdots 42}a^{14}-\frac{17\cdots 51}{39\cdots 72}a^{13}+\frac{94\cdots 71}{61\cdots 88}a^{12}-\frac{37\cdots 97}{39\cdots 72}a^{11}+\frac{27\cdots 27}{79\cdots 44}a^{10}-\frac{35\cdots 59}{39\cdots 72}a^{9}+\frac{17\cdots 51}{79\cdots 44}a^{8}-\frac{37\cdots 15}{56\cdots 96}a^{7}+\frac{85\cdots 73}{39\cdots 72}a^{6}-\frac{23\cdots 27}{39\cdots 72}a^{5}+\frac{11\cdots 65}{11\cdots 92}a^{4}-\frac{73\cdots 55}{79\cdots 44}a^{3}+\frac{19\cdots 15}{14\cdots 24}a^{2}+\frac{14\cdots 63}{56\cdots 96}a-\frac{85\cdots 03}{87\cdots 84}$, $\frac{27\cdots 71}{39\cdots 72}a^{20}-\frac{47\cdots 95}{11\cdots 92}a^{19}+\frac{83\cdots 99}{79\cdots 44}a^{18}-\frac{26\cdots 45}{19\cdots 36}a^{17}+\frac{19\cdots 13}{92\cdots 04}a^{16}-\frac{18\cdots 59}{11\cdots 92}a^{15}+\frac{42\cdots 47}{79\cdots 44}a^{14}-\frac{30\cdots 91}{30\cdots 44}a^{13}+\frac{26\cdots 49}{79\cdots 44}a^{12}-\frac{79\cdots 57}{39\cdots 72}a^{11}+\frac{85\cdots 75}{11\cdots 92}a^{10}-\frac{77\cdots 09}{39\cdots 72}a^{9}+\frac{38\cdots 13}{79\cdots 44}a^{8}-\frac{79\cdots 01}{56\cdots 96}a^{7}+\frac{36\cdots 51}{79\cdots 44}a^{6}-\frac{10\cdots 49}{79\cdots 44}a^{5}+\frac{71\cdots 95}{30\cdots 44}a^{4}-\frac{85\cdots 65}{39\cdots 72}a^{3}+\frac{52\cdots 49}{11\cdots 92}a^{2}+\frac{65\cdots 53}{11\cdots 92}a-\frac{10\cdots 53}{43\cdots 92}$, $\frac{21\cdots 43}{99\cdots 68}a^{20}-\frac{54\cdots 65}{39\cdots 72}a^{19}+\frac{41\cdots 47}{11\cdots 92}a^{18}-\frac{93\cdots 11}{18\cdots 08}a^{17}+\frac{29\cdots 57}{43\cdots 92}a^{16}-\frac{20\cdots 95}{39\cdots 72}a^{15}+\frac{14\cdots 93}{79\cdots 44}a^{14}-\frac{28\cdots 37}{79\cdots 44}a^{13}+\frac{22\cdots 07}{19\cdots 36}a^{12}-\frac{52\cdots 81}{79\cdots 44}a^{11}+\frac{12\cdots 23}{49\cdots 84}a^{10}-\frac{53\cdots 73}{79\cdots 44}a^{9}+\frac{23\cdots 25}{14\cdots 24}a^{8}-\frac{38\cdots 61}{79\cdots 44}a^{7}+\frac{23\cdots 25}{15\cdots 72}a^{6}-\frac{49\cdots 01}{11\cdots 92}a^{5}+\frac{66\cdots 39}{79\cdots 44}a^{4}-\frac{24\cdots 15}{28\cdots 48}a^{3}+\frac{17\cdots 49}{56\cdots 96}a^{2}+\frac{21\cdots 91}{11\cdots 92}a-\frac{12\cdots 91}{87\cdots 84}$, $\frac{11\cdots 11}{72\cdots 52}a^{20}-\frac{54\cdots 81}{72\cdots 52}a^{19}+\frac{28\cdots 87}{16\cdots 64}a^{18}-\frac{12\cdots 73}{72\cdots 52}a^{17}+\frac{36\cdots 17}{72\cdots 52}a^{16}-\frac{22\cdots 05}{72\cdots 52}a^{15}+\frac{92\cdots 03}{10\cdots 36}a^{14}-\frac{53\cdots 63}{36\cdots 76}a^{13}+\frac{17\cdots 89}{26\cdots 84}a^{12}-\frac{14\cdots 23}{36\cdots 76}a^{11}+\frac{22\cdots 89}{17\cdots 22}a^{10}-\frac{11\cdots 09}{36\cdots 76}a^{9}+\frac{14\cdots 75}{18\cdots 88}a^{8}-\frac{90\cdots 97}{36\cdots 76}a^{7}+\frac{59\cdots 05}{72\cdots 52}a^{6}-\frac{22\cdots 59}{10\cdots 36}a^{5}+\frac{24\cdots 59}{72\cdots 52}a^{4}-\frac{14\cdots 57}{56\cdots 04}a^{3}+\frac{23\cdots 73}{10\cdots 36}a^{2}+\frac{58\cdots 41}{80\cdots 72}a-\frac{15\cdots 23}{61\cdots 44}$, $\frac{56\cdots 83}{79\cdots 44}a^{20}-\frac{17\cdots 09}{39\cdots 72}a^{19}+\frac{10\cdots 03}{99\cdots 68}a^{18}-\frac{11\cdots 11}{79\cdots 44}a^{17}+\frac{17\cdots 17}{79\cdots 44}a^{16}-\frac{66\cdots 47}{39\cdots 72}a^{15}+\frac{54\cdots 97}{99\cdots 68}a^{14}-\frac{72\cdots 91}{70\cdots 12}a^{13}+\frac{27\cdots 39}{79\cdots 44}a^{12}-\frac{41\cdots 27}{19\cdots 36}a^{11}+\frac{61\cdots 71}{79\cdots 44}a^{10}-\frac{28\cdots 05}{14\cdots 24}a^{9}+\frac{39\cdots 23}{79\cdots 44}a^{8}-\frac{28\cdots 77}{19\cdots 36}a^{7}+\frac{94\cdots 71}{19\cdots 36}a^{6}-\frac{52\cdots 41}{39\cdots 72}a^{5}+\frac{27\cdots 65}{11\cdots 92}a^{4}-\frac{17\cdots 93}{79\cdots 44}a^{3}+\frac{27\cdots 91}{56\cdots 96}a^{2}+\frac{33\cdots 13}{56\cdots 96}a-\frac{22\cdots 87}{87\cdots 84}$, $\frac{42\cdots 01}{24\cdots 42}a^{20}-\frac{32\cdots 31}{39\cdots 72}a^{19}+\frac{21\cdots 25}{11\cdots 92}a^{18}-\frac{16\cdots 59}{79\cdots 44}a^{17}+\frac{16\cdots 05}{30\cdots 44}a^{16}-\frac{13\cdots 69}{39\cdots 72}a^{15}+\frac{79\cdots 51}{79\cdots 44}a^{14}-\frac{13\cdots 87}{79\cdots 44}a^{13}+\frac{14\cdots 91}{19\cdots 36}a^{12}-\frac{33\cdots 55}{79\cdots 44}a^{11}+\frac{14\cdots 45}{99\cdots 68}a^{10}-\frac{29\cdots 11}{79\cdots 44}a^{9}+\frac{45\cdots 31}{49\cdots 84}a^{8}-\frac{16\cdots 79}{61\cdots 88}a^{7}+\frac{25\cdots 65}{28\cdots 48}a^{6}-\frac{19\cdots 45}{79\cdots 44}a^{5}+\frac{31\cdots 29}{79\cdots 44}a^{4}-\frac{89\cdots 79}{28\cdots 48}a^{3}+\frac{27\cdots 75}{56\cdots 96}a^{2}+\frac{95\cdots 57}{11\cdots 92}a-\frac{32\cdots 05}{87\cdots 84}$, $\frac{74\cdots 21}{39\cdots 72}a^{20}-\frac{10\cdots 37}{11\cdots 92}a^{19}+\frac{14\cdots 95}{79\cdots 44}a^{18}-\frac{14\cdots 31}{92\cdots 04}a^{17}+\frac{22\cdots 75}{39\cdots 72}a^{16}-\frac{29\cdots 11}{79\cdots 44}a^{15}+\frac{11\cdots 57}{11\cdots 92}a^{14}-\frac{14\cdots 59}{99\cdots 68}a^{13}+\frac{58\cdots 31}{79\cdots 44}a^{12}-\frac{13\cdots 51}{28\cdots 48}a^{11}+\frac{16\cdots 41}{11\cdots 92}a^{10}-\frac{70\cdots 11}{19\cdots 36}a^{9}+\frac{71\cdots 03}{79\cdots 44}a^{8}-\frac{27\cdots 03}{99\cdots 68}a^{7}+\frac{73\cdots 45}{79\cdots 44}a^{6}-\frac{19\cdots 85}{79\cdots 44}a^{5}+\frac{70\cdots 71}{19\cdots 36}a^{4}-\frac{58\cdots 75}{30\cdots 44}a^{3}-\frac{46\cdots 65}{11\cdots 92}a^{2}+\frac{73\cdots 81}{11\cdots 92}a-\frac{75\cdots 73}{54\cdots 24}$, $\frac{82\cdots 35}{79\cdots 44}a^{20}-\frac{64\cdots 17}{99\cdots 68}a^{19}+\frac{13\cdots 65}{79\cdots 44}a^{18}-\frac{43\cdots 67}{19\cdots 36}a^{17}+\frac{25\cdots 11}{79\cdots 44}a^{16}-\frac{24\cdots 73}{99\cdots 68}a^{15}+\frac{67\cdots 01}{79\cdots 44}a^{14}-\frac{29\cdots 23}{18\cdots 08}a^{13}+\frac{40\cdots 59}{79\cdots 44}a^{12}-\frac{24\cdots 53}{79\cdots 44}a^{11}+\frac{94\cdots 83}{79\cdots 44}a^{10}-\frac{24\cdots 05}{79\cdots 44}a^{9}+\frac{60\cdots 35}{79\cdots 44}a^{8}-\frac{17\cdots 73}{79\cdots 44}a^{7}+\frac{11\cdots 49}{15\cdots 72}a^{6}-\frac{12\cdots 85}{61\cdots 88}a^{5}+\frac{15\cdots 53}{39\cdots 72}a^{4}-\frac{21\cdots 21}{61\cdots 88}a^{3}+\frac{24\cdots 05}{28\cdots 48}a^{2}+\frac{10\cdots 89}{11\cdots 92}a-\frac{19\cdots 79}{43\cdots 92}$, $\frac{96\cdots 57}{61\cdots 88}a^{20}-\frac{98\cdots 53}{79\cdots 44}a^{19}+\frac{28\cdots 41}{79\cdots 44}a^{18}-\frac{35\cdots 43}{79\cdots 44}a^{17}+\frac{40\cdots 15}{79\cdots 44}a^{16}-\frac{35\cdots 49}{79\cdots 44}a^{15}+\frac{13\cdots 85}{79\cdots 44}a^{14}-\frac{13\cdots 89}{39\cdots 72}a^{13}+\frac{35\cdots 73}{39\cdots 72}a^{12}-\frac{22\cdots 77}{39\cdots 72}a^{11}+\frac{93\cdots 27}{39\cdots 72}a^{10}-\frac{18\cdots 75}{30\cdots 44}a^{9}+\frac{59\cdots 25}{39\cdots 72}a^{8}-\frac{17\cdots 31}{39\cdots 72}a^{7}+\frac{10\cdots 37}{79\cdots 44}a^{6}-\frac{32\cdots 33}{79\cdots 44}a^{5}+\frac{63\cdots 09}{79\cdots 44}a^{4}-\frac{62\cdots 79}{79\cdots 44}a^{3}+\frac{21\cdots 05}{11\cdots 92}a^{2}+\frac{23\cdots 49}{11\cdots 92}a-\frac{81\cdots 65}{87\cdots 84}$
|
| |
| Regulator: | \( 17109387227548.602 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{3}\cdot(2\pi)^{9}\cdot 17109387227548.602 \cdot 7}{2\cdot\sqrt{58762312802028807390251370834685864006539343}}\cr\approx \mathstrut & 0.953809978949905 \end{aligned}\] (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
| \(\Q(\zeta_{7})^+\), 7.1.106243148764543.1 |
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: | 7.1.106243148764543.1 |
| Degree 14 sibling: | deg 14 |
| Minimal sibling: | 7.1.106243148764543.1 |
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 | ${\href{/padicField/11.3.0.1}{3} }^{7}$ | ${\href{/padicField/13.2.0.1}{2} }^{9}{,}\,{\href{/padicField/13.1.0.1}{1} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }^{3}{,}\,{\href{/padicField/17.3.0.1}{3} }$ | ${\href{/padicField/19.6.0.1}{6} }^{3}{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.3.0.1}{3} }^{7}$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{3}{,}\,{\href{/padicField/31.3.0.1}{3} }$ | ${\href{/padicField/37.3.0.1}{3} }^{7}$ | ${\href{/padicField/41.2.0.1}{2} }^{9}{,}\,{\href{/padicField/41.1.0.1}{1} }^{3}$ | R | ${\href{/padicField/47.6.0.1}{6} }^{3}{,}\,{\href{/padicField/47.3.0.1}{3} }$ | ${\href{/padicField/53.3.0.1}{3} }^{7}$ | ${\href{/padicField/59.6.0.1}{6} }^{3}{,}\,{\href{/padicField/59.3.0.1}{3} }$ |
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 | $x^{6} + 7$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ | |
|
\(43\)
| 43.1.7.6a1.2 | $x^{7} + 129$ | $7$ | $1$ | $6$ | $C_7$ | $$[\ ]_{7}$$ |
| 43.1.7.6a1.2 | $x^{7} + 129$ | $7$ | $1$ | $6$ | $C_7$ | $$[\ ]_{7}$$ | |
| 43.1.7.6a1.2 | $x^{7} + 129$ | $7$ | $1$ | $6$ | $C_7$ | $$[\ ]_{7}$$ |