Normalized defining polynomial
\( x^{21} - 154 x^{19} - 350 x^{18} + 9422 x^{17} + 38318 x^{16} - 232820 x^{15} - 1495990 x^{14} + \cdots - 50542023253032 \)
Invariants
| Degree: | $21$ |
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| Signature: | $[1, 10]$ |
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| Discriminant: |
\(569752580450182296234409142462680787211686353567744\)
\(\medspace = 2^{18}\cdot 7^{36}\cdot 31^{10}\)
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| Root discriminant: | \(261.18\) |
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| Galois root discriminant: | $2^{6/7}7^{96/49}31^{1/2}\approx 456.46642613478804$ | ||
| Ramified primes: |
\(2\), \(7\), \(31\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{2}a^{9}$, $\frac{1}{2}a^{10}$, $\frac{1}{2}a^{11}$, $\frac{1}{4}a^{12}-\frac{1}{2}a^{5}$, $\frac{1}{4}a^{13}-\frac{1}{2}a^{6}$, $\frac{1}{4}a^{14}$, $\frac{1}{4}a^{15}$, $\frac{1}{4}a^{16}$, $\frac{1}{8}a^{17}-\frac{1}{2}a^{3}$, $\frac{1}{24}a^{18}-\frac{1}{12}a^{16}+\frac{1}{12}a^{15}-\frac{1}{12}a^{14}-\frac{1}{12}a^{13}+\frac{1}{12}a^{12}-\frac{1}{6}a^{11}+\frac{1}{6}a^{9}-\frac{1}{6}a^{8}+\frac{1}{6}a^{7}+\frac{1}{6}a^{6}-\frac{1}{2}a^{5}+\frac{1}{6}a^{4}+\frac{1}{3}a^{2}-\frac{1}{3}a$, $\frac{1}{408}a^{19}+\frac{1}{68}a^{18}+\frac{2}{51}a^{17}-\frac{7}{102}a^{16}-\frac{13}{204}a^{15}-\frac{4}{51}a^{14}+\frac{1}{51}a^{13}+\frac{5}{102}a^{12}+\frac{3}{17}a^{11}-\frac{4}{51}a^{10}+\frac{5}{102}a^{9}-\frac{23}{102}a^{8}+\frac{11}{51}a^{7}+\frac{6}{17}a^{6}+\frac{49}{102}a^{5}-\frac{2}{17}a^{4}+\frac{16}{51}a^{3}+\frac{5}{51}a^{2}+\frac{7}{17}a-\frac{1}{17}$, $\frac{1}{14\cdots 96}a^{20}+\frac{35\cdots 25}{14\cdots 96}a^{19}+\frac{23\cdots 45}{48\cdots 32}a^{18}+\frac{77\cdots 43}{14\cdots 96}a^{17}+\frac{48\cdots 53}{72\cdots 48}a^{16}-\frac{20\cdots 95}{72\cdots 48}a^{15}-\frac{65\cdots 95}{26\cdots 24}a^{14}+\frac{21\cdots 97}{72\cdots 48}a^{13}-\frac{87\cdots 39}{72\cdots 48}a^{12}+\frac{30\cdots 97}{18\cdots 37}a^{11}-\frac{51\cdots 97}{36\cdots 74}a^{10}-\frac{27\cdots 69}{36\cdots 74}a^{9}-\frac{42\cdots 82}{18\cdots 37}a^{8}+\frac{53\cdots 73}{36\cdots 74}a^{7}-\frac{27\cdots 28}{20\cdots 93}a^{6}+\frac{54\cdots 11}{18\cdots 37}a^{5}+\frac{51\cdots 03}{36\cdots 74}a^{4}-\frac{71\cdots 61}{36\cdots 74}a^{3}+\frac{77\cdots 17}{60\cdots 79}a^{2}-\frac{71\cdots 92}{16\cdots 67}a-\frac{23\cdots 93}{11\cdots 29}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{7}$, which has order $7$ (assuming GRH) |
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| Narrow class group: | $C_{7}$, which has order $7$ (assuming GRH) |
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Unit group
| Rank: | $10$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{26\cdots 65}{46\cdots 54}a^{20}-\frac{45\cdots 29}{18\cdots 16}a^{19}-\frac{72\cdots 73}{92\cdots 08}a^{18}+\frac{24\cdots 69}{18\cdots 16}a^{17}+\frac{44\cdots 99}{92\cdots 08}a^{16}+\frac{12\cdots 27}{92\cdots 08}a^{15}-\frac{32\cdots 29}{23\cdots 77}a^{14}-\frac{61\cdots 17}{23\cdots 77}a^{13}+\frac{51\cdots 91}{46\cdots 54}a^{12}+\frac{21\cdots 27}{46\cdots 54}a^{11}+\frac{10\cdots 17}{46\cdots 54}a^{10}+\frac{15\cdots 08}{23\cdots 77}a^{9}-\frac{67\cdots 94}{23\cdots 77}a^{8}-\frac{53\cdots 63}{46\cdots 54}a^{7}+\frac{88\cdots 60}{23\cdots 77}a^{6}-\frac{41\cdots 09}{46\cdots 54}a^{5}-\frac{45\cdots 94}{23\cdots 77}a^{4}-\frac{23\cdots 61}{46\cdots 54}a^{3}-\frac{58\cdots 04}{23\cdots 77}a^{2}-\frac{17\cdots 51}{23\cdots 77}a-\frac{14\cdots 94}{23\cdots 77}$, $\frac{50\cdots 17}{14\cdots 96}a^{20}+\frac{12\cdots 45}{72\cdots 48}a^{19}-\frac{22\cdots 55}{48\cdots 32}a^{18}-\frac{52\cdots 53}{14\cdots 96}a^{17}+\frac{58\cdots 49}{36\cdots 74}a^{16}+\frac{95\cdots 45}{42\cdots 44}a^{15}+\frac{25\cdots 73}{80\cdots 72}a^{14}-\frac{14\cdots 35}{36\cdots 74}a^{13}-\frac{15\cdots 33}{72\cdots 48}a^{12}-\frac{99\cdots 04}{18\cdots 37}a^{11}-\frac{14\cdots 45}{36\cdots 74}a^{10}+\frac{20\cdots 27}{18\cdots 37}a^{9}+\frac{23\cdots 27}{36\cdots 74}a^{8}+\frac{40\cdots 48}{18\cdots 37}a^{7}+\frac{48\cdots 43}{40\cdots 86}a^{6}+\frac{25\cdots 13}{36\cdots 74}a^{5}+\frac{83\cdots 43}{36\cdots 74}a^{4}+\frac{13\cdots 47}{36\cdots 74}a^{3}+\frac{33\cdots 52}{60\cdots 79}a^{2}+\frac{30\cdots 92}{16\cdots 67}a+\frac{69\cdots 12}{20\cdots 93}$, $\frac{13\cdots 85}{14\cdots 96}a^{20}-\frac{72\cdots 33}{21\cdots 22}a^{19}-\frac{68\cdots 53}{48\cdots 32}a^{18}+\frac{31\cdots 77}{14\cdots 96}a^{17}+\frac{67\cdots 79}{72\cdots 48}a^{16}-\frac{18\cdots 41}{36\cdots 74}a^{15}-\frac{11\cdots 01}{40\cdots 86}a^{14}-\frac{18\cdots 03}{72\cdots 48}a^{13}+\frac{22\cdots 11}{72\cdots 48}a^{12}+\frac{49\cdots 40}{18\cdots 37}a^{11}+\frac{67\cdots 52}{18\cdots 37}a^{10}+\frac{56\cdots 97}{36\cdots 74}a^{9}-\frac{31\cdots 55}{36\cdots 74}a^{8}-\frac{51\cdots 23}{36\cdots 74}a^{7}+\frac{24\cdots 18}{20\cdots 93}a^{6}-\frac{72\cdots 43}{21\cdots 22}a^{5}-\frac{13\cdots 61}{36\cdots 74}a^{4}-\frac{16\cdots 27}{36\cdots 74}a^{3}+\frac{44\cdots 52}{60\cdots 79}a^{2}-\frac{54\cdots 57}{16\cdots 67}a-\frac{10\cdots 26}{20\cdots 93}$, $\frac{49\cdots 13}{72\cdots 48}a^{20}-\frac{28\cdots 45}{14\cdots 96}a^{19}-\frac{25\cdots 33}{24\cdots 16}a^{18}+\frac{39\cdots 83}{36\cdots 74}a^{17}+\frac{13\cdots 97}{21\cdots 22}a^{16}+\frac{35\cdots 27}{72\cdots 48}a^{15}-\frac{24\cdots 87}{13\cdots 62}a^{14}-\frac{16\cdots 49}{36\cdots 74}a^{13}+\frac{66\cdots 67}{42\cdots 44}a^{12}+\frac{24\cdots 73}{36\cdots 74}a^{11}+\frac{98\cdots 29}{36\cdots 74}a^{10}+\frac{40\cdots 95}{36\cdots 74}a^{9}-\frac{64\cdots 04}{18\cdots 37}a^{8}-\frac{31\cdots 59}{18\cdots 37}a^{7}+\frac{10\cdots 31}{20\cdots 93}a^{6}-\frac{15\cdots 72}{18\cdots 37}a^{5}-\frac{48\cdots 84}{18\cdots 37}a^{4}-\frac{13\cdots 19}{18\cdots 37}a^{3}-\frac{94\cdots 60}{60\cdots 79}a^{2}-\frac{21\cdots 37}{16\cdots 67}a-\frac{18\cdots 23}{20\cdots 93}$, $\frac{17\cdots 39}{14\cdots 96}a^{20}+\frac{13\cdots 85}{14\cdots 96}a^{19}-\frac{13\cdots 89}{12\cdots 58}a^{18}-\frac{10\cdots 49}{72\cdots 48}a^{17}+\frac{13\cdots 95}{36\cdots 74}a^{16}+\frac{20\cdots 77}{36\cdots 74}a^{15}+\frac{28\cdots 57}{13\cdots 62}a^{14}-\frac{90\cdots 73}{36\cdots 74}a^{13}-\frac{68\cdots 91}{18\cdots 37}a^{12}-\frac{30\cdots 50}{18\cdots 37}a^{11}-\frac{20\cdots 73}{36\cdots 74}a^{10}-\frac{15\cdots 29}{36\cdots 74}a^{9}+\frac{63\cdots 87}{36\cdots 74}a^{8}-\frac{31\cdots 31}{36\cdots 74}a^{7}+\frac{12\cdots 17}{79\cdots 86}a^{6}+\frac{28\cdots 23}{36\cdots 74}a^{5}+\frac{62\cdots 69}{18\cdots 37}a^{4}+\frac{10\cdots 62}{18\cdots 37}a^{3}+\frac{37\cdots 48}{60\cdots 79}a^{2}+\frac{47\cdots 32}{16\cdots 67}a+\frac{13\cdots 68}{20\cdots 93}$, $\frac{32\cdots 79}{22\cdots 56}a^{20}+\frac{70\cdots 55}{44\cdots 12}a^{19}-\frac{33\cdots 03}{14\cdots 04}a^{18}-\frac{33\cdots 81}{44\cdots 12}a^{17}+\frac{29\cdots 97}{22\cdots 56}a^{16}+\frac{16\cdots 77}{22\cdots 56}a^{15}-\frac{36\cdots 17}{12\cdots 42}a^{14}-\frac{61\cdots 89}{22\cdots 56}a^{13}-\frac{56\cdots 53}{22\cdots 56}a^{12}+\frac{33\cdots 03}{11\cdots 78}a^{11}+\frac{17\cdots 05}{11\cdots 78}a^{10}+\frac{58\cdots 91}{11\cdots 78}a^{9}+\frac{24\cdots 93}{11\cdots 78}a^{8}-\frac{89\cdots 49}{11\cdots 78}a^{7}-\frac{10\cdots 39}{72\cdots 26}a^{6}+\frac{12\cdots 25}{55\cdots 89}a^{5}-\frac{56\cdots 69}{11\cdots 78}a^{4}-\frac{43\cdots 83}{11\cdots 78}a^{3}-\frac{16\cdots 73}{18\cdots 63}a^{2}+\frac{65\cdots 33}{55\cdots 89}a+\frac{11\cdots 82}{61\cdots 21}$, $\frac{27\cdots 55}{80\cdots 72}a^{20}+\frac{17\cdots 33}{26\cdots 24}a^{19}-\frac{84\cdots 73}{16\cdots 44}a^{18}-\frac{17\cdots 35}{80\cdots 72}a^{17}+\frac{24\cdots 23}{80\cdots 72}a^{16}+\frac{39\cdots 84}{20\cdots 93}a^{15}-\frac{42\cdots 13}{80\cdots 72}a^{14}-\frac{55\cdots 05}{80\cdots 72}a^{13}-\frac{31\cdots 19}{26\cdots 24}a^{12}+\frac{11\cdots 10}{20\cdots 93}a^{11}+\frac{16\cdots 47}{40\cdots 86}a^{10}+\frac{32\cdots 65}{20\cdots 93}a^{9}+\frac{29\cdots 35}{20\cdots 93}a^{8}-\frac{23\cdots 03}{13\cdots 62}a^{7}-\frac{11\cdots 47}{23\cdots 58}a^{6}+\frac{14\cdots 15}{13\cdots 62}a^{5}-\frac{48\cdots 41}{40\cdots 86}a^{4}-\frac{20\cdots 56}{20\cdots 93}a^{3}-\frac{19\cdots 90}{67\cdots 31}a^{2}-\frac{16\cdots 01}{61\cdots 21}a+\frac{95\cdots 42}{67\cdots 31}$, $\frac{99\cdots 27}{14\cdots 96}a^{20}-\frac{11\cdots 47}{18\cdots 37}a^{19}-\frac{13\cdots 85}{12\cdots 58}a^{18}-\frac{15\cdots 27}{14\cdots 96}a^{17}+\frac{26\cdots 37}{36\cdots 74}a^{16}+\frac{30\cdots 31}{18\cdots 37}a^{15}-\frac{18\cdots 09}{80\cdots 72}a^{14}-\frac{52\cdots 99}{72\cdots 48}a^{13}+\frac{79\cdots 73}{36\cdots 74}a^{12}+\frac{14\cdots 37}{18\cdots 37}a^{11}+\frac{51\cdots 70}{18\cdots 37}a^{10}+\frac{62\cdots 73}{36\cdots 74}a^{9}-\frac{70\cdots 66}{18\cdots 37}a^{8}-\frac{50\cdots 48}{18\cdots 37}a^{7}+\frac{44\cdots 89}{67\cdots 31}a^{6}+\frac{27\cdots 32}{18\cdots 37}a^{5}-\frac{63\cdots 84}{18\cdots 37}a^{4}-\frac{37\cdots 33}{36\cdots 74}a^{3}-\frac{23\cdots 07}{60\cdots 79}a^{2}-\frac{28\cdots 85}{16\cdots 67}a-\frac{24\cdots 24}{20\cdots 93}$, $\frac{24\cdots 19}{48\cdots 32}a^{20}+\frac{84\cdots 33}{12\cdots 58}a^{19}-\frac{22\cdots 27}{16\cdots 44}a^{18}-\frac{37\cdots 79}{48\cdots 32}a^{17}-\frac{49\cdots 09}{12\cdots 58}a^{16}+\frac{72\cdots 50}{60\cdots 79}a^{15}+\frac{52\cdots 57}{26\cdots 24}a^{14}+\frac{15\cdots 77}{24\cdots 16}a^{13}-\frac{70\cdots 19}{24\cdots 16}a^{12}-\frac{95\cdots 05}{12\cdots 58}a^{11}-\frac{23\cdots 97}{71\cdots 74}a^{10}-\frac{99\cdots 88}{60\cdots 79}a^{9}-\frac{12\cdots 77}{12\cdots 58}a^{8}-\frac{58\cdots 33}{12\cdots 58}a^{7}-\frac{33\cdots 26}{20\cdots 93}a^{6}-\frac{48\cdots 87}{12\cdots 58}a^{5}-\frac{84\cdots 05}{12\cdots 58}a^{4}-\frac{15\cdots 11}{12\cdots 58}a^{3}-\frac{59\cdots 35}{20\cdots 93}a^{2}-\frac{22\cdots 99}{55\cdots 89}a+\frac{11\cdots 88}{67\cdots 31}$, $\frac{42\cdots 39}{14\cdots 96}a^{20}-\frac{62\cdots 35}{72\cdots 48}a^{19}-\frac{10\cdots 81}{24\cdots 16}a^{18}+\frac{26\cdots 11}{72\cdots 48}a^{17}+\frac{20\cdots 57}{72\cdots 48}a^{16}+\frac{14\cdots 89}{72\cdots 48}a^{15}-\frac{59\cdots 12}{67\cdots 31}a^{14}-\frac{11\cdots 73}{72\cdots 48}a^{13}+\frac{62\cdots 33}{72\cdots 48}a^{12}+\frac{41\cdots 57}{21\cdots 22}a^{11}+\frac{21\cdots 42}{18\cdots 37}a^{10}+\frac{10\cdots 69}{21\cdots 22}a^{9}-\frac{74\cdots 39}{36\cdots 74}a^{8}-\frac{24\cdots 09}{36\cdots 74}a^{7}+\frac{19\cdots 87}{67\cdots 31}a^{6}-\frac{18\cdots 97}{36\cdots 74}a^{5}-\frac{22\cdots 42}{18\cdots 37}a^{4}-\frac{46\cdots 00}{18\cdots 37}a^{3}+\frac{17\cdots 69}{60\cdots 79}a^{2}-\frac{15\cdots 91}{16\cdots 67}a-\frac{58\cdots 86}{20\cdots 93}$
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| Regulator: | \( 21312988696200000 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{1}\cdot(2\pi)^{10}\cdot 21312988696200000 \cdot 7}{2\cdot\sqrt{569752580450182296234409142462680787211686353567744}}\cr\approx \mathstrut & 0.599374038453928 \end{aligned}\] (assuming GRH)
Galois group
$C_7^2:S_3$ (as 21T17):
| 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 | ${\href{/padicField/3.14.0.1}{14} }{,}\,{\href{/padicField/3.2.0.1}{2} }^{3}{,}\,{\href{/padicField/3.1.0.1}{1} }$ | ${\href{/padicField/5.3.0.1}{3} }^{7}$ | R | ${\href{/padicField/11.14.0.1}{14} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{3}{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.14.0.1}{14} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.14.0.1}{14} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{3}{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.3.0.1}{3} }^{7}$ | ${\href{/padicField/23.14.0.1}{14} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{3}{,}\,{\href{/padicField/23.1.0.1}{1} }$ | ${\href{/padicField/29.14.0.1}{14} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}{,}\,{\href{/padicField/29.1.0.1}{1} }$ | R | ${\href{/padicField/37.14.0.1}{14} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.3.0.1}{3} }^{7}$ | ${\href{/padicField/43.14.0.1}{14} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{3}{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.7.0.1}{7} }^{3}$ | ${\href{/padicField/53.14.0.1}{14} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{3}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.3.0.1}{3} }^{7}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.7.18a1.2 | $x^{21} + 7 x^{19} + 7 x^{18} + 21 x^{17} + 42 x^{16} + 56 x^{15} + 105 x^{14} + 140 x^{13} + 175 x^{12} + 231 x^{11} + 245 x^{10} + 252 x^{9} + 252 x^{8} + 211 x^{7} + 168 x^{6} + 126 x^{5} + 77 x^{4} + 42 x^{3} + 21 x^{2} + 7 x + 3$ | $7$ | $3$ | $18$ | 21T2 | $$[\ ]_{7}^{3}$$ |
|
\(7\)
| 7.3.7.36a89.6 | $x^{21} + 77 x^{20} + 2016 x^{19} + 26502 x^{18} + 198912 x^{17} + 891576 x^{16} + 2444064 x^{15} + 4572336 x^{14} + 7846272 x^{13} + 11762912 x^{12} + 11791360 x^{11} + 18127872 x^{10} + 9703680 x^{9} + 17552640 x^{8} + 4515840 x^{7} + 10719744 x^{6} + 1118208 x^{5} + 3999744 x^{4} + 114688 x^{3} + 831488 x^{2} + 245 x + 73735$ | $7$ | $3$ | $36$ | not computed | not computed |
|
\(31\)
| $\Q_{31}$ | $x + 28$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 31.1.2.1a1.1 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 31.1.2.1a1.1 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 31.1.2.1a1.1 | $x^{2} + 31$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 31.7.2.7a1.2 | $x^{14} + 2 x^{8} + 56 x^{7} + x^{2} + 56 x + 815$ | $2$ | $7$ | $7$ | $C_{14}$ | $$[\ ]_{2}^{7}$$ |