Normalized defining polynomial
\( x^{21} - 6 x^{19} - 24 x^{18} + 262 x^{17} - 1188 x^{16} - 1310 x^{15} + 16498 x^{14} - 8820 x^{13} + \cdots + 45152 \)
Invariants
| Degree: | $21$ |
| |
| Signature: | $[7, 7]$ |
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| Discriminant: |
\(-8064534907239574094801006114055662600192\)
\(\medspace = -\,2^{18}\cdot 7^{24}\cdot 107^{7}\)
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| |
| Root discriminant: | \(79.49\) |
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| Galois root discriminant: | $2^{6/7}7^{242/147}107^{1/2}\approx 461.2829834966581$ | ||
| Ramified primes: |
\(2\), \(7\), \(107\)
|
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| Discriminant root field: | \(\Q(\sqrt{-107}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{4}a^{11}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{6}$, $\frac{1}{8}a^{14}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{8}a^{15}-\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{8}a^{16}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{2}a^{5}$, $\frac{1}{8}a^{17}-\frac{1}{4}a^{10}-\frac{1}{2}a^{6}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}$, $\frac{1}{16}a^{18}-\frac{1}{8}a^{12}-\frac{1}{8}a^{11}-\frac{1}{4}a^{10}-\frac{1}{4}a^{9}-\frac{1}{4}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}$, $\frac{1}{16}a^{19}-\frac{1}{8}a^{13}-\frac{1}{8}a^{12}-\frac{1}{4}a^{10}-\frac{1}{4}a^{8}-\frac{1}{2}a^{4}$, $\frac{1}{15\cdots 12}a^{20}-\frac{97\cdots 45}{77\cdots 56}a^{19}+\frac{39\cdots 35}{38\cdots 28}a^{18}+\frac{19\cdots 43}{38\cdots 28}a^{17}+\frac{26\cdots 13}{96\cdots 57}a^{16}-\frac{40\cdots 17}{38\cdots 28}a^{15}+\frac{35\cdots 33}{38\cdots 28}a^{14}+\frac{61\cdots 25}{77\cdots 56}a^{13}-\frac{13\cdots 99}{38\cdots 28}a^{12}-\frac{10\cdots 48}{96\cdots 57}a^{11}-\frac{54\cdots 11}{38\cdots 28}a^{10}+\frac{87\cdots 93}{38\cdots 28}a^{9}+\frac{48\cdots 87}{38\cdots 28}a^{8}-\frac{79\cdots 71}{38\cdots 28}a^{7}-\frac{25\cdots 88}{96\cdots 57}a^{6}+\frac{10\cdots 89}{19\cdots 14}a^{5}+\frac{46\cdots 97}{96\cdots 57}a^{4}+\frac{48\cdots 10}{96\cdots 57}a^{3}-\frac{21\cdots 07}{96\cdots 57}a^{2}+\frac{28\cdots 18}{96\cdots 57}a-\frac{26\cdots 17}{96\cdots 57}$
| 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) |
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Unit group
| Rank: | $13$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{14\cdots 97}{59\cdots 52}a^{20}+\frac{59\cdots 73}{23\cdots 08}a^{19}-\frac{25\cdots 45}{23\cdots 08}a^{18}-\frac{19\cdots 11}{29\cdots 26}a^{17}+\frac{64\cdots 59}{11\cdots 04}a^{16}-\frac{26\cdots 11}{11\cdots 04}a^{15}-\frac{15\cdots 05}{29\cdots 26}a^{14}+\frac{38\cdots 11}{11\cdots 04}a^{13}+\frac{61\cdots 81}{59\cdots 52}a^{12}-\frac{20\cdots 31}{11\cdots 04}a^{11}+\frac{33\cdots 15}{59\cdots 52}a^{10}+\frac{25\cdots 71}{59\cdots 52}a^{9}-\frac{53\cdots 81}{14\cdots 63}a^{8}-\frac{24\cdots 77}{59\cdots 52}a^{7}+\frac{15\cdots 11}{29\cdots 26}a^{6}+\frac{49\cdots 94}{14\cdots 63}a^{5}+\frac{19\cdots 59}{29\cdots 26}a^{4}+\frac{66\cdots 38}{14\cdots 63}a^{3}-\frac{10\cdots 16}{14\cdots 63}a^{2}-\frac{81\cdots 36}{14\cdots 63}a+\frac{33\cdots 21}{14\cdots 63}$, $\frac{20\cdots 69}{77\cdots 56}a^{20}-\frac{29\cdots 88}{96\cdots 57}a^{19}-\frac{19\cdots 25}{15\cdots 12}a^{18}-\frac{47\cdots 82}{96\cdots 57}a^{17}+\frac{28\cdots 13}{38\cdots 28}a^{16}-\frac{30\cdots 41}{77\cdots 56}a^{15}+\frac{21\cdots 61}{19\cdots 14}a^{14}+\frac{41\cdots 37}{96\cdots 57}a^{13}-\frac{55\cdots 75}{77\cdots 56}a^{12}-\frac{11\cdots 19}{77\cdots 56}a^{11}+\frac{15\cdots 47}{38\cdots 28}a^{10}+\frac{70\cdots 56}{96\cdots 57}a^{9}-\frac{35\cdots 93}{38\cdots 28}a^{8}+\frac{20\cdots 77}{38\cdots 28}a^{7}+\frac{81\cdots 13}{19\cdots 14}a^{6}+\frac{25\cdots 00}{96\cdots 57}a^{5}-\frac{11\cdots 59}{96\cdots 57}a^{4}-\frac{14\cdots 87}{96\cdots 57}a^{3}+\frac{23\cdots 63}{96\cdots 57}a^{2}+\frac{79\cdots 03}{96\cdots 57}a+\frac{11\cdots 13}{96\cdots 57}$, $\frac{27\cdots 19}{15\cdots 12}a^{20}-\frac{14\cdots 65}{15\cdots 12}a^{19}-\frac{37\cdots 47}{38\cdots 28}a^{18}-\frac{28\cdots 91}{77\cdots 56}a^{17}+\frac{36\cdots 73}{77\cdots 56}a^{16}-\frac{18\cdots 99}{77\cdots 56}a^{15}-\frac{38\cdots 71}{38\cdots 28}a^{14}+\frac{11\cdots 79}{38\cdots 28}a^{13}-\frac{24\cdots 01}{77\cdots 56}a^{12}-\frac{48\cdots 81}{38\cdots 28}a^{11}+\frac{43\cdots 41}{19\cdots 14}a^{10}+\frac{77\cdots 25}{38\cdots 28}a^{9}-\frac{24\cdots 61}{38\cdots 28}a^{8}+\frac{36\cdots 87}{38\cdots 28}a^{7}+\frac{10\cdots 09}{19\cdots 14}a^{6}+\frac{13\cdots 85}{19\cdots 14}a^{5}+\frac{18\cdots 15}{96\cdots 57}a^{4}-\frac{94\cdots 09}{96\cdots 57}a^{3}-\frac{33\cdots 65}{96\cdots 57}a^{2}+\frac{68\cdots 72}{96\cdots 57}a+\frac{10\cdots 12}{96\cdots 57}$, $\frac{40\cdots 55}{77\cdots 56}a^{20}-\frac{19\cdots 65}{15\cdots 12}a^{19}-\frac{19\cdots 53}{77\cdots 56}a^{18}-\frac{49\cdots 83}{77\cdots 56}a^{17}+\frac{15\cdots 34}{96\cdots 57}a^{16}-\frac{73\cdots 25}{77\cdots 56}a^{15}+\frac{37\cdots 47}{38\cdots 28}a^{14}+\frac{71\cdots 83}{77\cdots 56}a^{13}-\frac{18\cdots 57}{77\cdots 56}a^{12}-\frac{90\cdots 83}{38\cdots 28}a^{11}+\frac{24\cdots 99}{19\cdots 14}a^{10}-\frac{12\cdots 45}{38\cdots 28}a^{9}-\frac{29\cdots 26}{96\cdots 57}a^{8}+\frac{53\cdots 55}{19\cdots 14}a^{7}+\frac{23\cdots 13}{96\cdots 57}a^{6}-\frac{50\cdots 89}{19\cdots 14}a^{5}-\frac{26\cdots 37}{19\cdots 14}a^{4}-\frac{19\cdots 85}{96\cdots 57}a^{3}+\frac{22\cdots 35}{96\cdots 57}a^{2}+\frac{34\cdots 58}{96\cdots 57}a-\frac{69\cdots 05}{96\cdots 57}$, $\frac{42\cdots 75}{15\cdots 12}a^{20}-\frac{90\cdots 15}{15\cdots 12}a^{19}-\frac{18\cdots 31}{15\cdots 12}a^{18}-\frac{14\cdots 31}{38\cdots 28}a^{17}+\frac{64\cdots 15}{77\cdots 56}a^{16}-\frac{37\cdots 37}{77\cdots 56}a^{15}+\frac{18\cdots 47}{38\cdots 28}a^{14}+\frac{17\cdots 59}{38\cdots 28}a^{13}-\frac{11\cdots 01}{96\cdots 57}a^{12}-\frac{81\cdots 99}{77\cdots 56}a^{11}+\frac{11\cdots 57}{19\cdots 14}a^{10}-\frac{39\cdots 97}{19\cdots 14}a^{9}-\frac{45\cdots 05}{38\cdots 28}a^{8}+\frac{24\cdots 07}{19\cdots 14}a^{7}+\frac{28\cdots 54}{96\cdots 57}a^{6}-\frac{23\cdots 68}{96\cdots 57}a^{5}-\frac{42\cdots 11}{96\cdots 57}a^{4}-\frac{14\cdots 00}{96\cdots 57}a^{3}+\frac{51\cdots 16}{96\cdots 57}a^{2}+\frac{93\cdots 48}{96\cdots 57}a+\frac{15\cdots 67}{96\cdots 57}$, $\frac{71\cdots 41}{77\cdots 56}a^{20}-\frac{16\cdots 35}{19\cdots 14}a^{19}-\frac{74\cdots 05}{15\cdots 12}a^{18}-\frac{34\cdots 03}{19\cdots 14}a^{17}+\frac{20\cdots 27}{77\cdots 56}a^{16}-\frac{51\cdots 69}{38\cdots 28}a^{15}+\frac{26\cdots 99}{77\cdots 56}a^{14}+\frac{59\cdots 41}{38\cdots 28}a^{13}-\frac{17\cdots 61}{77\cdots 56}a^{12}-\frac{44\cdots 19}{77\cdots 56}a^{11}+\frac{53\cdots 15}{38\cdots 28}a^{10}+\frac{53\cdots 87}{96\cdots 57}a^{9}-\frac{13\cdots 09}{38\cdots 28}a^{8}+\frac{15\cdots 02}{96\cdots 57}a^{7}+\frac{21\cdots 80}{96\cdots 57}a^{6}+\frac{30\cdots 57}{19\cdots 14}a^{5}+\frac{42\cdots 52}{96\cdots 57}a^{4}-\frac{51\cdots 87}{96\cdots 57}a^{3}+\frac{63\cdots 33}{96\cdots 57}a^{2}+\frac{28\cdots 30}{96\cdots 57}a+\frac{41\cdots 73}{96\cdots 57}$, $\frac{28\cdots 07}{15\cdots 12}a^{20}-\frac{36\cdots 91}{15\cdots 12}a^{19}+\frac{53\cdots 47}{77\cdots 56}a^{18}+\frac{18\cdots 59}{77\cdots 56}a^{17}+\frac{40\cdots 73}{77\cdots 56}a^{16}-\frac{98\cdots 42}{96\cdots 57}a^{15}+\frac{37\cdots 79}{77\cdots 56}a^{14}-\frac{96\cdots 31}{19\cdots 14}a^{13}-\frac{32\cdots 73}{77\cdots 56}a^{12}+\frac{13\cdots 03}{96\cdots 57}a^{11}+\frac{25\cdots 69}{38\cdots 28}a^{10}-\frac{77\cdots 10}{96\cdots 57}a^{9}+\frac{14\cdots 95}{38\cdots 28}a^{8}+\frac{18\cdots 29}{96\cdots 57}a^{7}-\frac{12\cdots 27}{96\cdots 57}a^{6}-\frac{13\cdots 59}{96\cdots 57}a^{5}-\frac{79\cdots 80}{96\cdots 57}a^{4}+\frac{58\cdots 79}{96\cdots 57}a^{3}+\frac{29\cdots 36}{96\cdots 57}a^{2}-\frac{48\cdots 73}{96\cdots 57}a-\frac{15\cdots 53}{96\cdots 57}$, $\frac{60\cdots 37}{15\cdots 12}a^{20}-\frac{90\cdots 41}{38\cdots 28}a^{19}-\frac{40\cdots 73}{19\cdots 14}a^{18}-\frac{30\cdots 63}{38\cdots 28}a^{17}+\frac{20\cdots 47}{19\cdots 14}a^{16}-\frac{20\cdots 07}{38\cdots 28}a^{15}-\frac{65\cdots 55}{38\cdots 28}a^{14}+\frac{49\cdots 15}{77\cdots 56}a^{13}-\frac{28\cdots 39}{38\cdots 28}a^{12}-\frac{10\cdots 65}{38\cdots 28}a^{11}+\frac{20\cdots 43}{38\cdots 28}a^{10}+\frac{15\cdots 29}{38\cdots 28}a^{9}-\frac{54\cdots 97}{38\cdots 28}a^{8}+\frac{10\cdots 13}{38\cdots 28}a^{7}+\frac{23\cdots 17}{19\cdots 14}a^{6}+\frac{40\cdots 27}{19\cdots 14}a^{5}+\frac{44\cdots 89}{19\cdots 14}a^{4}-\frac{21\cdots 09}{96\cdots 57}a^{3}-\frac{27\cdots 55}{96\cdots 57}a^{2}+\frac{17\cdots 47}{96\cdots 57}a+\frac{24\cdots 63}{96\cdots 57}$, $\frac{34\cdots 35}{15\cdots 12}a^{20}-\frac{48\cdots 41}{19\cdots 14}a^{19}-\frac{42\cdots 39}{38\cdots 28}a^{18}-\frac{32\cdots 85}{77\cdots 56}a^{17}+\frac{24\cdots 39}{38\cdots 28}a^{16}-\frac{26\cdots 07}{77\cdots 56}a^{15}+\frac{28\cdots 99}{38\cdots 28}a^{14}+\frac{28\cdots 49}{77\cdots 56}a^{13}-\frac{58\cdots 59}{96\cdots 57}a^{12}-\frac{12\cdots 14}{96\cdots 57}a^{11}+\frac{34\cdots 14}{96\cdots 57}a^{10}+\frac{12\cdots 39}{19\cdots 14}a^{9}-\frac{15\cdots 79}{19\cdots 14}a^{8}+\frac{19\cdots 91}{38\cdots 28}a^{7}+\frac{35\cdots 11}{96\cdots 57}a^{6}+\frac{11\cdots 87}{96\cdots 57}a^{5}+\frac{46\cdots 17}{96\cdots 57}a^{4}-\frac{13\cdots 35}{96\cdots 57}a^{3}+\frac{38\cdots 84}{96\cdots 57}a^{2}+\frac{58\cdots 73}{96\cdots 57}a+\frac{90\cdots 90}{96\cdots 57}$, $\frac{90\cdots 81}{15\cdots 12}a^{20}-\frac{86\cdots 59}{15\cdots 12}a^{19}-\frac{11\cdots 71}{38\cdots 28}a^{18}-\frac{10\cdots 28}{96\cdots 57}a^{17}+\frac{12\cdots 37}{77\cdots 56}a^{16}-\frac{66\cdots 05}{77\cdots 56}a^{15}+\frac{35\cdots 99}{77\cdots 56}a^{14}+\frac{37\cdots 63}{38\cdots 28}a^{13}-\frac{11\cdots 69}{77\cdots 56}a^{12}-\frac{13\cdots 41}{38\cdots 28}a^{11}+\frac{17\cdots 51}{19\cdots 14}a^{10}+\frac{12\cdots 43}{38\cdots 28}a^{9}-\frac{41\cdots 91}{19\cdots 14}a^{8}+\frac{99\cdots 67}{96\cdots 57}a^{7}+\frac{24\cdots 03}{19\cdots 14}a^{6}+\frac{20\cdots 32}{96\cdots 57}a^{5}+\frac{31\cdots 03}{96\cdots 57}a^{4}-\frac{35\cdots 01}{96\cdots 57}a^{3}+\frac{29\cdots 27}{96\cdots 57}a^{2}+\frac{20\cdots 95}{96\cdots 57}a+\frac{27\cdots 87}{96\cdots 57}$, $\frac{26\cdots 49}{15\cdots 12}a^{20}-\frac{16\cdots 67}{15\cdots 12}a^{19}-\frac{75\cdots 37}{77\cdots 56}a^{18}-\frac{27\cdots 95}{77\cdots 56}a^{17}+\frac{92\cdots 65}{19\cdots 14}a^{16}-\frac{22\cdots 02}{96\cdots 57}a^{15}-\frac{60\cdots 27}{77\cdots 56}a^{14}+\frac{11\cdots 67}{38\cdots 28}a^{13}-\frac{26\cdots 63}{77\cdots 56}a^{12}-\frac{12\cdots 50}{96\cdots 57}a^{11}+\frac{45\cdots 17}{19\cdots 14}a^{10}+\frac{75\cdots 19}{38\cdots 28}a^{9}-\frac{25\cdots 35}{38\cdots 28}a^{8}+\frac{19\cdots 69}{19\cdots 14}a^{7}+\frac{11\cdots 65}{19\cdots 14}a^{6}+\frac{94\cdots 19}{96\cdots 57}a^{5}+\frac{79\cdots 49}{96\cdots 57}a^{4}-\frac{10\cdots 25}{96\cdots 57}a^{3}-\frac{22\cdots 13}{96\cdots 57}a^{2}+\frac{83\cdots 33}{96\cdots 57}a+\frac{12\cdots 03}{96\cdots 57}$, $\frac{42\cdots 47}{19\cdots 14}a^{20}-\frac{40\cdots 53}{38\cdots 28}a^{19}-\frac{18\cdots 03}{15\cdots 12}a^{18}-\frac{35\cdots 27}{77\cdots 56}a^{17}+\frac{11\cdots 35}{19\cdots 14}a^{16}-\frac{11\cdots 23}{38\cdots 28}a^{15}-\frac{10\cdots 63}{77\cdots 56}a^{14}+\frac{70\cdots 47}{19\cdots 14}a^{13}-\frac{29\cdots 41}{77\cdots 56}a^{12}-\frac{12\cdots 81}{77\cdots 56}a^{11}+\frac{10\cdots 71}{38\cdots 28}a^{10}+\frac{97\cdots 77}{38\cdots 28}a^{9}-\frac{30\cdots 25}{38\cdots 28}a^{8}+\frac{18\cdots 23}{19\cdots 14}a^{7}+\frac{70\cdots 62}{96\cdots 57}a^{6}+\frac{12\cdots 71}{96\cdots 57}a^{5}+\frac{26\cdots 43}{19\cdots 14}a^{4}-\frac{11\cdots 60}{96\cdots 57}a^{3}-\frac{18\cdots 63}{96\cdots 57}a^{2}+\frac{10\cdots 89}{96\cdots 57}a+\frac{22\cdots 19}{96\cdots 57}$, $\frac{50\cdots 33}{77\cdots 56}a^{20}-\frac{17\cdots 15}{38\cdots 28}a^{19}-\frac{54\cdots 11}{15\cdots 12}a^{18}-\frac{25\cdots 33}{19\cdots 14}a^{17}+\frac{17\cdots 50}{96\cdots 57}a^{16}-\frac{69\cdots 11}{77\cdots 56}a^{15}-\frac{15\cdots 39}{77\cdots 56}a^{14}+\frac{41\cdots 75}{38\cdots 28}a^{13}-\frac{10\cdots 95}{77\cdots 56}a^{12}-\frac{34\cdots 45}{77\cdots 56}a^{11}+\frac{17\cdots 11}{19\cdots 14}a^{10}+\frac{12\cdots 29}{19\cdots 14}a^{9}-\frac{23\cdots 76}{96\cdots 57}a^{8}+\frac{51\cdots 99}{96\cdots 57}a^{7}+\frac{39\cdots 85}{19\cdots 14}a^{6}+\frac{37\cdots 55}{96\cdots 57}a^{5}+\frac{37\cdots 05}{19\cdots 14}a^{4}-\frac{38\cdots 24}{96\cdots 57}a^{3}-\frac{55\cdots 80}{96\cdots 57}a^{2}+\frac{29\cdots 71}{96\cdots 57}a+\frac{40\cdots 21}{96\cdots 57}$
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| Regulator: | \( 3153077248290 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{7}\cdot(2\pi)^{7}\cdot 3153077248290 \cdot 1}{2\cdot\sqrt{8064534907239574094801006114055662600192}}\cr\approx \mathstrut & 0.868728361560060 \end{aligned}\] (assuming GRH)
Galois group
$C_7^3:(C_3\times S_3)$ (as 21T40):
| A solvable group of order 6174 |
| The 60 conjugacy class representatives for $C_7^3:(C_3\times S_3)$ |
| Character table for $C_7^3:(C_3\times S_3)$ |
Intermediate fields
| 3.1.107.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 21 siblings: | data not computed |
| Degree 42 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 | $21$ | ${\href{/padicField/5.6.0.1}{6} }^{2}{,}\,{\href{/padicField/5.3.0.1}{3} }^{2}{,}\,{\href{/padicField/5.2.0.1}{2} }{,}\,{\href{/padicField/5.1.0.1}{1} }$ | R | $21$ | $21$ | ${\href{/padicField/17.6.0.1}{6} }^{2}{,}\,{\href{/padicField/17.3.0.1}{3} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/padicField/29.7.0.1}{7} }^{3}$ | ${\href{/padicField/31.6.0.1}{6} }^{2}{,}\,{\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | $21$ | $21$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.7.0.1}{7} }$ | ${\href{/padicField/47.3.0.1}{3} }^{6}{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | $21$ | ${\href{/padicField/59.6.0.1}{6} }^{2}{,}\,{\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.7.6a1.1 | $x^{7} + 2$ | $7$ | $1$ | $6$ | $C_7:C_3$ | $$[\ ]_{7}^{3}$$ |
| 2.2.7.12a1.1 | $x^{14} + 7 x^{13} + 28 x^{12} + 77 x^{11} + 161 x^{10} + 266 x^{9} + 357 x^{8} + 393 x^{7} + 357 x^{6} + 266 x^{5} + 161 x^{4} + 77 x^{3} + 28 x^{2} + 7 x + 3$ | $7$ | $2$ | $12$ | $(C_7:C_3) \times C_2$ | $$[\ ]_{7}^{6}$$ | |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 7.1.3.2a1.1 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 7.1.3.2a1.1 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
| 7.2.7.20a14.1 | $x^{14} + 42 x^{13} + 777 x^{12} + 8316 x^{11} + 56889 x^{10} + 259707 x^{9} + 804363 x^{8} + 1696428 x^{7} + 2443707 x^{6} + 2438100 x^{5} + 1715175 x^{4} + 852768 x^{3} + 288603 x^{2} + 59535 x + 5596$ | $7$ | $2$ | $20$ | 14T14 | $$[\frac{5}{3}, \frac{5}{3}]_{3}^{2}$$ | |
|
\(107\)
| $\Q_{107}$ | $x + 105$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 107.1.2.1a1.1 | $x^{2} + 107$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 107.3.1.0a1.1 | $x^{3} + 5 x + 105$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 107.3.1.0a1.1 | $x^{3} + 5 x + 105$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 107.3.2.3a1.2 | $x^{6} + 10 x^{4} + 210 x^{3} + 25 x^{2} + 1050 x + 11132$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ | |
| 107.3.2.3a1.2 | $x^{6} + 10 x^{4} + 210 x^{3} + 25 x^{2} + 1050 x + 11132$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |