Normalized defining polynomial
\( x^{20} - 5 x^{19} + 5 x^{18} + 30 x^{17} - 460 x^{16} + 1594 x^{15} - 660 x^{14} - 6250 x^{13} + \cdots + 55238851 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(75045768722673667967319488525390625\)
\(\medspace = 3^{10}\cdot 5^{26}\cdot 31^{8}\)
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| Root discriminant: | \(55.43\) |
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| Galois root discriminant: | $3^{1/2}5^{13/10}31^{4/5}\approx 218.93287896315394$ | ||
| Ramified primes: |
\(3\), \(5\), \(31\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $D_5$ |
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| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}, \sqrt{5})\) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{7}a^{12}-\frac{1}{7}a^{11}+\frac{3}{7}a^{10}-\frac{3}{7}a^{9}+\frac{2}{7}a^{8}+\frac{1}{7}a^{6}-\frac{3}{7}a^{5}-\frac{1}{7}a^{4}+\frac{2}{7}a^{3}-\frac{1}{7}a^{2}+\frac{3}{7}a+\frac{2}{7}$, $\frac{1}{7}a^{13}+\frac{2}{7}a^{11}-\frac{1}{7}a^{9}+\frac{2}{7}a^{8}+\frac{1}{7}a^{7}-\frac{2}{7}a^{6}+\frac{3}{7}a^{5}+\frac{1}{7}a^{4}+\frac{1}{7}a^{3}+\frac{2}{7}a^{2}-\frac{2}{7}a+\frac{2}{7}$, $\frac{1}{21}a^{14}+\frac{1}{21}a^{13}-\frac{10}{21}a^{11}-\frac{1}{3}a^{10}-\frac{8}{21}a^{8}-\frac{8}{21}a^{7}-\frac{8}{21}a^{6}+\frac{1}{7}a^{5}-\frac{10}{21}a^{4}-\frac{1}{21}a^{3}+\frac{3}{7}a^{2}+\frac{1}{21}a-\frac{2}{21}$, $\frac{1}{21}a^{15}-\frac{1}{21}a^{13}-\frac{1}{21}a^{12}-\frac{2}{7}a^{11}-\frac{8}{21}a^{10}+\frac{1}{3}a^{9}-\frac{1}{7}a^{8}-\frac{1}{21}a^{6}+\frac{2}{21}a^{5}+\frac{1}{3}a^{3}+\frac{4}{21}a^{2}+\frac{1}{7}a-\frac{1}{21}$, $\frac{1}{147}a^{16}-\frac{2}{147}a^{15}-\frac{2}{147}a^{14}-\frac{1}{49}a^{13}-\frac{1}{21}a^{12}-\frac{31}{147}a^{11}-\frac{2}{7}a^{10}-\frac{5}{147}a^{9}+\frac{23}{147}a^{8}+\frac{67}{147}a^{7}-\frac{2}{49}a^{6}+\frac{5}{21}a^{5}+\frac{38}{147}a^{4}-\frac{13}{49}a^{3}+\frac{25}{147}a^{2}-\frac{53}{147}a+\frac{55}{147}$, $\frac{1}{147}a^{17}+\frac{1}{147}a^{15}+\frac{8}{147}a^{13}-\frac{10}{147}a^{12}-\frac{23}{49}a^{11}-\frac{68}{147}a^{10}+\frac{62}{147}a^{9}+\frac{5}{49}a^{8}-\frac{18}{49}a^{7}-\frac{40}{147}a^{6}-\frac{67}{147}a^{5}-\frac{18}{49}a^{4}-\frac{53}{147}a^{3}-\frac{59}{147}a^{2}+\frac{61}{147}a+\frac{68}{147}$, $\frac{1}{20839804409151}a^{18}+\frac{659383222}{330790546177}a^{17}+\frac{21749369104}{6946601469717}a^{16}+\frac{86574279254}{6946601469717}a^{15}+\frac{443739573470}{20839804409151}a^{14}-\frac{1296244423021}{20839804409151}a^{13}-\frac{86631466504}{6946601469717}a^{12}+\frac{4611258228013}{20839804409151}a^{11}+\frac{9259984482440}{20839804409151}a^{10}-\frac{1298839885738}{6946601469717}a^{9}-\frac{410893484644}{1603061877627}a^{8}+\frac{1518083742374}{20839804409151}a^{7}-\frac{2251411091155}{20839804409151}a^{6}-\frac{638130180918}{2315533823239}a^{5}-\frac{10361577959318}{20839804409151}a^{4}-\frac{70688592965}{20839804409151}a^{3}-\frac{6780412654415}{20839804409151}a^{2}-\frac{6979638660403}{20839804409151}a+\frac{3840975562373}{20839804409151}$, $\frac{1}{17\cdots 41}a^{19}-\frac{55\cdots 29}{56\cdots 47}a^{18}-\frac{50\cdots 74}{18\cdots 49}a^{17}+\frac{13\cdots 10}{43\cdots 19}a^{16}+\frac{17\cdots 20}{13\cdots 57}a^{15}+\frac{30\cdots 47}{18\cdots 51}a^{14}+\frac{29\cdots 66}{80\cdots 21}a^{13}-\frac{56\cdots 22}{24\cdots 63}a^{12}+\frac{44\cdots 65}{17\cdots 41}a^{11}+\frac{18\cdots 61}{56\cdots 47}a^{10}+\frac{11\cdots 19}{17\cdots 41}a^{9}-\frac{59\cdots 40}{17\cdots 41}a^{8}+\frac{30\cdots 00}{17\cdots 41}a^{7}-\frac{13\cdots 67}{56\cdots 47}a^{6}+\frac{82\cdots 74}{17\cdots 41}a^{5}-\frac{29\cdots 84}{17\cdots 41}a^{4}+\frac{70\cdots 14}{17\cdots 41}a^{3}-\frac{11\cdots 28}{34\cdots 53}a^{2}+\frac{11\cdots 19}{24\cdots 63}a+\frac{91\cdots 33}{56\cdots 47}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | $C_{5}$, which has order $5$ (assuming GRH) |
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| Narrow class group: | $C_{5}$, which has order $5$ (assuming GRH) |
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Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -\frac{365258343734433380611056131441581655}{4999774636833680677039311606729464615097537} a^{19} + \frac{712047643339558004792473396119850055}{1666591545611226892346437202243154871699179} a^{18} - \frac{1281413633132608486730366014714905650}{1666591545611226892346437202243154871699179} a^{17} - \frac{747608263025333709156270329790663580}{555530515203742297448812400747718290566393} a^{16} + \frac{172663743815190190959380176923691697339}{4999774636833680677039311606729464615097537} a^{15} - \frac{733354908380152741141780866342400845055}{4999774636833680677039311606729464615097537} a^{14} + \frac{317276678333288966421967994530033606315}{1666591545611226892346437202243154871699179} a^{13} + \frac{1170828767021066171847377146098030854845}{4999774636833680677039311606729464615097537} a^{12} - \frac{3059754271162097181383794185052988749645}{714253519547668668148473086675637802156791} a^{11} + \frac{2724649732245482602759545520741593801617}{238084506515889556049491028891879267385597} a^{10} - \frac{80244458150637625684414718665717375701820}{4999774636833680677039311606729464615097537} a^{9} + \frac{45771859249144718127136517934808034813150}{4999774636833680677039311606729464615097537} a^{8} + \frac{414057368913300936049660361090776971598160}{4999774636833680677039311606729464615097537} a^{7} - \frac{246447008195094229290939606555893587171645}{1666591545611226892346437202243154871699179} a^{6} + \frac{596946143769814560407387216064008044873078}{4999774636833680677039311606729464615097537} a^{5} - \frac{3415373291894769166531207041353498704940165}{4999774636833680677039311606729464615097537} a^{4} - \frac{60272381008873502409405439885707513777665}{102036217078238381164067583810805400308113} a^{3} + \frac{229863512768314187258456296640082846929110}{70419361082164516578018473334217811480247} a^{2} - \frac{4447289813486818492641475773988419773328055}{4999774636833680677039311606729464615097537} a - \frac{1704574943378131688421234162515889451527461}{555530515203742297448812400747718290566393} \)
(order $6$)
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| Fundamental units: |
$\frac{17\cdots 86}{62\cdots 87}a^{19}-\frac{12\cdots 10}{62\cdots 87}a^{18}+\frac{95\cdots 62}{20\cdots 29}a^{17}+\frac{21\cdots 92}{60\cdots 03}a^{16}-\frac{88\cdots 68}{62\cdots 87}a^{15}+\frac{15\cdots 64}{20\cdots 29}a^{14}-\frac{77\cdots 11}{62\cdots 87}a^{13}-\frac{36\cdots 46}{62\cdots 87}a^{12}+\frac{11\cdots 11}{62\cdots 87}a^{11}-\frac{38\cdots 74}{62\cdots 87}a^{10}+\frac{58\cdots 14}{62\cdots 87}a^{9}-\frac{16\cdots 74}{20\cdots 29}a^{8}-\frac{61\cdots 84}{20\cdots 29}a^{7}+\frac{57\cdots 22}{62\cdots 87}a^{6}-\frac{70\cdots 38}{88\cdots 41}a^{5}+\frac{64\cdots 10}{20\cdots 29}a^{4}-\frac{25\cdots 18}{20\cdots 29}a^{3}-\frac{15\cdots 93}{80\cdots 33}a^{2}+\frac{10\cdots 40}{68\cdots 43}a+\frac{15\cdots 88}{62\cdots 87}$, $\frac{22\cdots 34}{33\cdots 01}a^{19}-\frac{15\cdots 91}{33\cdots 01}a^{18}+\frac{21\cdots 61}{11\cdots 67}a^{17}+\frac{46\cdots 58}{86\cdots 59}a^{16}-\frac{11\cdots 28}{25\cdots 77}a^{15}+\frac{52\cdots 76}{28\cdots 53}a^{14}+\frac{13\cdots 40}{33\cdots 01}a^{13}-\frac{55\cdots 53}{33\cdots 01}a^{12}+\frac{26\cdots 62}{33\cdots 01}a^{11}-\frac{73\cdots 93}{33\cdots 01}a^{10}-\frac{74\cdots 39}{98\cdots 07}a^{9}+\frac{10\cdots 05}{16\cdots 81}a^{8}-\frac{10\cdots 67}{22\cdots 83}a^{7}+\frac{29\cdots 06}{33\cdots 01}a^{6}-\frac{39\cdots 90}{33\cdots 01}a^{5}+\frac{29\cdots 77}{11\cdots 67}a^{4}-\frac{66\cdots 42}{11\cdots 67}a^{3}-\frac{34\cdots 98}{47\cdots 31}a^{2}+\frac{35\cdots 81}{11\cdots 67}a+\frac{20\cdots 79}{33\cdots 01}$, $\frac{88\cdots 23}{18\cdots 49}a^{19}-\frac{29\cdots 41}{17\cdots 41}a^{18}-\frac{28\cdots 90}{56\cdots 47}a^{17}+\frac{27\cdots 45}{18\cdots 49}a^{16}-\frac{80\cdots 94}{56\cdots 47}a^{15}+\frac{57\cdots 17}{17\cdots 41}a^{14}+\frac{31\cdots 87}{17\cdots 41}a^{13}-\frac{58\cdots 31}{18\cdots 49}a^{12}+\frac{12\cdots 29}{17\cdots 41}a^{11}-\frac{82\cdots 68}{17\cdots 41}a^{10}-\frac{73\cdots 22}{52\cdots 83}a^{9}-\frac{93\cdots 76}{17\cdots 41}a^{8}-\frac{19\cdots 19}{17\cdots 41}a^{7}-\frac{53\cdots 41}{17\cdots 41}a^{6}-\frac{49\cdots 12}{56\cdots 47}a^{5}+\frac{14\cdots 55}{24\cdots 63}a^{4}+\frac{99\cdots 97}{13\cdots 57}a^{3}+\frac{10\cdots 39}{23\cdots 71}a^{2}-\frac{12\cdots 87}{17\cdots 41}a-\frac{12\cdots 59}{17\cdots 41}$, $\frac{11\cdots 85}{17\cdots 41}a^{19}-\frac{80\cdots 49}{17\cdots 41}a^{18}+\frac{24\cdots 27}{18\cdots 49}a^{17}-\frac{28\cdots 63}{56\cdots 47}a^{16}-\frac{52\cdots 60}{17\cdots 41}a^{15}+\frac{29\cdots 58}{17\cdots 41}a^{14}-\frac{65\cdots 39}{17\cdots 41}a^{13}+\frac{82\cdots 32}{25\cdots 23}a^{12}+\frac{63\cdots 77}{18\cdots 49}a^{11}-\frac{24\cdots 59}{17\cdots 41}a^{10}+\frac{79\cdots 76}{24\cdots 63}a^{9}-\frac{13\cdots 32}{24\cdots 63}a^{8}+\frac{14\cdots 40}{24\cdots 63}a^{7}+\frac{16\cdots 49}{17\cdots 41}a^{6}-\frac{30\cdots 34}{17\cdots 41}a^{5}+\frac{13\cdots 54}{17\cdots 41}a^{4}-\frac{47\cdots 36}{13\cdots 57}a^{3}-\frac{14\cdots 13}{79\cdots 57}a^{2}+\frac{11\cdots 16}{17\cdots 41}a+\frac{43\cdots 50}{17\cdots 41}$, $\frac{19\cdots 12}{26\cdots 07}a^{19}-\frac{26\cdots 28}{56\cdots 47}a^{18}+\frac{30\cdots 51}{26\cdots 07}a^{17}+\frac{86\cdots 12}{18\cdots 49}a^{16}-\frac{19\cdots 54}{56\cdots 47}a^{15}+\frac{97\cdots 25}{56\cdots 47}a^{14}-\frac{18\cdots 90}{56\cdots 47}a^{13}+\frac{11\cdots 73}{56\cdots 47}a^{12}+\frac{23\cdots 68}{56\cdots 47}a^{11}-\frac{84\cdots 16}{56\cdots 47}a^{10}+\frac{16\cdots 53}{56\cdots 47}a^{9}-\frac{18\cdots 96}{56\cdots 47}a^{8}-\frac{32\cdots 98}{56\cdots 47}a^{7}+\frac{12\cdots 57}{56\cdots 47}a^{6}-\frac{17\cdots 09}{43\cdots 19}a^{5}+\frac{70\cdots 17}{56\cdots 47}a^{4}-\frac{53\cdots 89}{56\cdots 47}a^{3}-\frac{54\cdots 66}{26\cdots 19}a^{2}+\frac{11\cdots 38}{56\cdots 47}a+\frac{10\cdots 96}{56\cdots 47}$, $\frac{67\cdots 04}{18\cdots 49}a^{19}-\frac{62\cdots 13}{17\cdots 41}a^{18}+\frac{69\cdots 02}{56\cdots 47}a^{17}-\frac{47\cdots 49}{56\cdots 47}a^{16}-\frac{10\cdots 57}{56\cdots 47}a^{15}+\frac{23\cdots 59}{17\cdots 41}a^{14}-\frac{63\cdots 86}{17\cdots 41}a^{13}+\frac{52\cdots 65}{18\cdots 49}a^{12}+\frac{78\cdots 08}{36\cdots 89}a^{11}-\frac{24\cdots 13}{18\cdots 51}a^{10}+\frac{15\cdots 90}{56\cdots 47}a^{9}-\frac{69\cdots 24}{17\cdots 41}a^{8}+\frac{19\cdots 58}{17\cdots 41}a^{7}+\frac{28\cdots 57}{17\cdots 41}a^{6}-\frac{77\cdots 63}{56\cdots 47}a^{5}+\frac{86\cdots 60}{17\cdots 41}a^{4}-\frac{33\cdots 73}{24\cdots 63}a^{3}-\frac{75\cdots 96}{23\cdots 71}a^{2}+\frac{64\cdots 44}{17\cdots 41}a+\frac{56\cdots 39}{17\cdots 41}$, $\frac{37\cdots 36}{43\cdots 19}a^{19}-\frac{12\cdots 10}{17\cdots 41}a^{18}+\frac{93\cdots 14}{56\cdots 47}a^{17}+\frac{24\cdots 79}{56\cdots 47}a^{16}-\frac{95\cdots 92}{18\cdots 49}a^{15}+\frac{37\cdots 45}{17\cdots 41}a^{14}-\frac{59\cdots 89}{17\cdots 41}a^{13}-\frac{45\cdots 64}{56\cdots 47}a^{12}+\frac{14\cdots 82}{24\cdots 63}a^{11}-\frac{40\cdots 32}{24\cdots 63}a^{10}+\frac{15\cdots 83}{56\cdots 47}a^{9}+\frac{10\cdots 49}{13\cdots 57}a^{8}-\frac{19\cdots 52}{17\cdots 41}a^{7}+\frac{30\cdots 93}{17\cdots 41}a^{6}-\frac{20\cdots 99}{56\cdots 47}a^{5}+\frac{15\cdots 24}{17\cdots 41}a^{4}+\frac{35\cdots 36}{34\cdots 09}a^{3}-\frac{10\cdots 77}{23\cdots 71}a^{2}+\frac{22\cdots 52}{17\cdots 41}a+\frac{60\cdots 16}{17\cdots 41}$, $\frac{56\cdots 53}{13\cdots 57}a^{19}-\frac{24\cdots 79}{17\cdots 41}a^{18}-\frac{11\cdots 57}{18\cdots 49}a^{17}+\frac{76\cdots 63}{56\cdots 47}a^{16}-\frac{30\cdots 08}{17\cdots 41}a^{15}+\frac{66\cdots 66}{17\cdots 41}a^{14}+\frac{86\cdots 46}{17\cdots 41}a^{13}-\frac{40\cdots 55}{17\cdots 41}a^{12}+\frac{40\cdots 41}{18\cdots 49}a^{11}-\frac{19\cdots 20}{17\cdots 41}a^{10}-\frac{11\cdots 38}{17\cdots 41}a^{9}+\frac{79\cdots 52}{13\cdots 57}a^{8}-\frac{98\cdots 77}{17\cdots 41}a^{7}-\frac{43\cdots 63}{17\cdots 41}a^{6}-\frac{65\cdots 88}{17\cdots 41}a^{5}+\frac{90\cdots 18}{24\cdots 63}a^{4}+\frac{22\cdots 18}{17\cdots 41}a^{3}+\frac{12\cdots 39}{79\cdots 57}a^{2}-\frac{29\cdots 08}{17\cdots 41}a-\frac{17\cdots 33}{17\cdots 41}$, $\frac{36\cdots 17}{18\cdots 49}a^{19}+\frac{25\cdots 10}{17\cdots 41}a^{18}-\frac{68\cdots 80}{56\cdots 47}a^{17}+\frac{21\cdots 21}{80\cdots 21}a^{16}-\frac{41\cdots 29}{56\cdots 47}a^{15}-\frac{13\cdots 11}{17\cdots 41}a^{14}+\frac{82\cdots 41}{17\cdots 41}a^{13}-\frac{12\cdots 64}{18\cdots 49}a^{12}+\frac{14\cdots 18}{17\cdots 41}a^{11}+\frac{21\cdots 74}{17\cdots 41}a^{10}-\frac{91\cdots 62}{18\cdots 49}a^{9}+\frac{70\cdots 04}{17\cdots 41}a^{8}+\frac{48\cdots 08}{17\cdots 41}a^{7}-\frac{65\cdots 86}{17\cdots 41}a^{6}+\frac{81\cdots 38}{80\cdots 21}a^{5}+\frac{14\cdots 75}{13\cdots 57}a^{4}-\frac{48\cdots 15}{17\cdots 41}a^{3}-\frac{43\cdots 17}{23\cdots 71}a^{2}+\frac{55\cdots 75}{17\cdots 41}a+\frac{44\cdots 40}{17\cdots 41}$
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| Regulator: | \( 1625055692.34 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 1625055692.34 \cdot 5}{6\cdot\sqrt{75045768722673667967319488525390625}}\cr\approx \mathstrut & 0.474048286949 \end{aligned}\] (assuming GRH)
Galois group
| A solvable group of order 100 |
| The 16 conjugacy class representatives for $D_5^2$ |
| Character table for $D_5^2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \), \(\Q(\sqrt{-15}) \), \(\Q(\sqrt{5}) \), \(\Q(\sqrt{-3}, \sqrt{5})\), 10.0.54788965576171875.1 x5 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 20 sibling: | data not computed |
| Degree 25 sibling: | data not computed |
| Minimal sibling: | 10.0.54788965576171875.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.10.0.1}{10} }^{2}$ | R | R | ${\href{/padicField/7.2.0.1}{2} }^{10}$ | ${\href{/padicField/11.10.0.1}{10} }^{2}$ | ${\href{/padicField/13.2.0.1}{2} }^{10}$ | ${\href{/padicField/17.2.0.1}{2} }^{10}$ | ${\href{/padicField/19.5.0.1}{5} }^{4}$ | ${\href{/padicField/23.10.0.1}{10} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/37.2.0.1}{2} }^{10}$ | ${\href{/padicField/41.2.0.1}{2} }^{10}$ | ${\href{/padicField/43.2.0.1}{2} }^{10}$ | ${\href{/padicField/47.10.0.1}{10} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.2.10.26a3.1 | $x^{20} + 40 x^{19} + 740 x^{18} + 8400 x^{17} + 65460 x^{16} + 371328 x^{15} + 1586880 x^{14} + 5218560 x^{13} + 13381920 x^{12} + 26970880 x^{11} + 42904960 x^{10} + 53941760 x^{9} + 53527695 x^{8} + 41748720 x^{7} + 25391640 x^{6} + 11887776 x^{5} + 4199400 x^{4} + 1085760 x^{3} + 195680 x^{2} + 22400 x + 1269$ | $10$ | $2$ | $26$ | 20T4 | not computed |
|
\(31\)
| 31.1.5.4a1.5 | $x^{5} + 837$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ |
| 31.5.1.0a1.1 | $x^{5} + 7 x + 28$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ | |
| 31.5.1.0a1.1 | $x^{5} + 7 x + 28$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ | |
| 31.1.5.4a1.5 | $x^{5} + 837$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ |