Normalized defining polynomial
\( x^{20} - x^{19} + 17 x^{18} - 38 x^{17} + 294 x^{16} + 679 x^{15} + 3849 x^{14} + 6496 x^{13} + \cdots + 6561 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(1945771207112214793287128936767578125\)
\(\medspace = 5^{15}\cdot 41^{16}\)
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| Root discriminant: | \(65.23\) |
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| Galois root discriminant: | $5^{3/4}41^{4/5}\approx 65.23108294968821$ | ||
| Ramified primes: |
\(5\), \(41\)
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| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$ $=$ $\Gal(K/\Q)$: | $C_{20}$ |
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| This field is Galois and abelian over $\Q$. | |||
| Conductor: | \(205=5\cdot 41\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{205}(1,·)$, $\chi_{205}(133,·)$, $\chi_{205}(201,·)$, $\chi_{205}(139,·)$, $\chi_{205}(141,·)$, $\chi_{205}(78,·)$, $\chi_{205}(16,·)$, $\chi_{205}(18,·)$, $\chi_{205}(83,·)$, $\chi_{205}(92,·)$, $\chi_{205}(98,·)$, $\chi_{205}(37,·)$, $\chi_{205}(42,·)$, $\chi_{205}(174,·)$, $\chi_{205}(51,·)$, $\chi_{205}(182,·)$, $\chi_{205}(119,·)$, $\chi_{205}(57,·)$, $\chi_{205}(59,·)$, $\chi_{205}(124,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{3}a^{9}+\frac{1}{3}a^{7}+\frac{1}{3}a^{5}+\frac{1}{3}a^{3}+\frac{1}{3}a$, $\frac{1}{3}a^{10}+\frac{1}{3}a^{8}+\frac{1}{3}a^{6}+\frac{1}{3}a^{4}+\frac{1}{3}a^{2}$, $\frac{1}{3}a^{11}-\frac{1}{3}a$, $\frac{1}{3}a^{12}-\frac{1}{3}a^{2}$, $\frac{1}{3}a^{13}-\frac{1}{3}a^{3}$, $\frac{1}{9}a^{14}-\frac{1}{9}a^{13}-\frac{1}{9}a^{12}+\frac{1}{9}a^{11}+\frac{1}{9}a^{9}+\frac{4}{9}a^{7}-\frac{2}{9}a^{5}+\frac{2}{9}a^{4}-\frac{1}{9}a^{3}-\frac{2}{9}a^{2}-\frac{1}{3}a$, $\frac{1}{27}a^{15}-\frac{1}{27}a^{14}-\frac{1}{27}a^{13}-\frac{2}{27}a^{12}-\frac{1}{9}a^{11}+\frac{4}{27}a^{10}-\frac{1}{9}a^{9}-\frac{11}{27}a^{8}-\frac{4}{9}a^{7}+\frac{10}{27}a^{6}+\frac{8}{27}a^{5}+\frac{11}{27}a^{4}-\frac{5}{27}a^{3}-\frac{2}{9}a^{2}+\frac{1}{3}a$, $\frac{1}{243}a^{16}+\frac{2}{243}a^{15}-\frac{13}{243}a^{14}+\frac{40}{243}a^{13}-\frac{4}{27}a^{12}-\frac{5}{243}a^{11}-\frac{1}{9}a^{10}+\frac{34}{243}a^{9}-\frac{2}{9}a^{8}+\frac{28}{243}a^{7}+\frac{110}{243}a^{6}-\frac{100}{243}a^{5}-\frac{53}{243}a^{4}-\frac{4}{81}a^{3}+\frac{4}{27}a^{2}-\frac{4}{9}a+\frac{1}{3}$, $\frac{1}{19\cdots 39}a^{17}+\frac{29\cdots 91}{19\cdots 39}a^{16}+\frac{70\cdots 01}{19\cdots 39}a^{15}+\frac{39\cdots 50}{19\cdots 39}a^{14}+\frac{19\cdots 30}{64\cdots 13}a^{13}+\frac{99\cdots 70}{19\cdots 39}a^{12}+\frac{89\cdots 62}{64\cdots 13}a^{11}+\frac{17\cdots 57}{19\cdots 39}a^{10}+\frac{49\cdots 59}{64\cdots 13}a^{9}-\frac{18\cdots 55}{19\cdots 39}a^{8}-\frac{24\cdots 05}{19\cdots 39}a^{7}+\frac{44\cdots 46}{19\cdots 39}a^{6}+\frac{44\cdots 33}{19\cdots 39}a^{5}-\frac{11\cdots 99}{64\cdots 13}a^{4}+\frac{96\cdots 61}{21\cdots 71}a^{3}+\frac{13\cdots 96}{71\cdots 57}a^{2}-\frac{93\cdots 71}{23\cdots 19}a+\frac{950506816119959}{79\cdots 73}$, $\frac{1}{58\cdots 17}a^{18}-\frac{1}{58\cdots 17}a^{17}-\frac{76\cdots 47}{58\cdots 17}a^{16}-\frac{34\cdots 70}{58\cdots 17}a^{15}-\frac{39\cdots 66}{19\cdots 39}a^{14}+\frac{36\cdots 23}{58\cdots 17}a^{13}-\frac{50\cdots 08}{19\cdots 39}a^{12}+\frac{82\cdots 82}{58\cdots 17}a^{11}+\frac{15\cdots 49}{19\cdots 39}a^{10}+\frac{84\cdots 42}{58\cdots 17}a^{9}+\frac{21\cdots 85}{58\cdots 17}a^{8}+\frac{21\cdots 48}{58\cdots 17}a^{7}-\frac{26\cdots 17}{58\cdots 17}a^{6}+\frac{33\cdots 09}{19\cdots 39}a^{5}+\frac{32\cdots 14}{64\cdots 13}a^{4}-\frac{13\cdots 75}{21\cdots 71}a^{3}-\frac{31\cdots 64}{71\cdots 57}a^{2}-\frac{41\cdots 17}{23\cdots 19}a-\frac{203965088156943}{26\cdots 91}$, $\frac{1}{17\cdots 51}a^{19}-\frac{1}{17\cdots 51}a^{18}-\frac{1}{17\cdots 51}a^{17}-\frac{30\cdots 79}{17\cdots 51}a^{16}+\frac{97\cdots 07}{58\cdots 17}a^{15}-\frac{51\cdots 70}{17\cdots 51}a^{14}-\frac{15\cdots 68}{58\cdots 17}a^{13}-\frac{17\cdots 77}{17\cdots 51}a^{12}+\frac{13\cdots 72}{58\cdots 17}a^{11}-\frac{16\cdots 55}{17\cdots 51}a^{10}+\frac{13\cdots 51}{17\cdots 51}a^{9}+\frac{18\cdots 70}{17\cdots 51}a^{8}+\frac{62\cdots 03}{17\cdots 51}a^{7}-\frac{18\cdots 71}{58\cdots 17}a^{6}+\frac{78\cdots 21}{19\cdots 39}a^{5}+\frac{11\cdots 69}{64\cdots 13}a^{4}-\frac{32\cdots 32}{21\cdots 71}a^{3}-\frac{575421866121556}{23\cdots 19}a^{2}+\frac{11\cdots 94}{23\cdots 19}a+\frac{11\cdots 85}{26\cdots 91}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{505}$, which has order $505$ (assuming GRH) |
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| Narrow class group: | $C_{505}$, which has order $505$ (assuming GRH) |
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| Relative class number: | $505$ (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -\frac{37742873463880}{5820102231020314317} a^{19} - \frac{120777195084416}{5820102231020314317} a^{18} - \frac{415335701549948}{5820102231020314317} a^{17} - \frac{1328549145928576}{5820102231020314317} a^{16} - \frac{1305903421850248}{1940034077006771439} a^{15} - \frac{74813923780102936}{5820102231020314317} a^{14} - \frac{77644639289893936}{1940034077006771439} a^{13} - \frac{809453903593472374}{5820102231020314317} a^{12} - \frac{824455327944994720}{1940034077006771439} a^{11} - \frac{7773205178483628208}{5820102231020314317} a^{10} - \frac{4144831780906988288}{5820102231020314317} a^{9} - \frac{11518392769430976728}{5820102231020314317} a^{8} - \frac{6084705844284598201}{5820102231020314317} a^{7} - \frac{67544646350959648}{23951037987737919} a^{6} - \frac{288453684735049288}{215559341889641271} a^{5} - \frac{289767136731592312}{71853113963213757} a^{4} - \frac{42385246899937240}{23951037987737919} a^{3} - \frac{139191462630118925}{23951037987737919} a^{2} - \frac{822794641512584}{2661226443081991} a - \frac{362331585253248}{2661226443081991} \)
(order $10$)
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| Fundamental units: |
$\frac{450553955257480}{58\cdots 17}a^{19}-\frac{471136255221553}{58\cdots 17}a^{18}+\frac{76\cdots 48}{58\cdots 17}a^{17}-\frac{17\cdots 24}{58\cdots 17}a^{16}+\frac{44\cdots 48}{19\cdots 39}a^{15}+\frac{30\cdots 36}{58\cdots 17}a^{14}+\frac{56\cdots 43}{19\cdots 39}a^{13}+\frac{28\cdots 74}{58\cdots 17}a^{12}+\frac{66\cdots 20}{19\cdots 39}a^{11}+\frac{11\cdots 08}{58\cdots 17}a^{10}+\frac{29\cdots 88}{58\cdots 17}a^{9}+\frac{18\cdots 88}{58\cdots 17}a^{8}+\frac{43\cdots 01}{58\cdots 17}a^{7}+\frac{79\cdots 32}{21\cdots 71}a^{6}+\frac{22\cdots 88}{21\cdots 71}a^{5}+\frac{37\cdots 12}{71\cdots 57}a^{4}+\frac{97\cdots 13}{71\cdots 57}a^{3}+\frac{14\cdots 25}{23\cdots 19}a^{2}+\frac{10\cdots 84}{26\cdots 91}a+\frac{67466526829248}{26\cdots 91}$, $\frac{12831380305367}{64\cdots 13}a^{19}-\frac{57644344335847}{58\cdots 17}a^{18}+\frac{17\cdots 00}{58\cdots 17}a^{17}-\frac{32\cdots 11}{58\cdots 17}a^{16}+\frac{28\cdots 28}{58\cdots 17}a^{15}+\frac{34\cdots 80}{19\cdots 39}a^{14}+\frac{43\cdots 65}{58\cdots 17}a^{13}+\frac{29\cdots 64}{19\cdots 39}a^{12}+\frac{49\cdots 54}{58\cdots 17}a^{11}+\frac{16\cdots 24}{19\cdots 39}a^{10}+\frac{29\cdots 50}{58\cdots 17}a^{9}+\frac{66\cdots 85}{58\cdots 17}a^{8}+\frac{23\cdots 12}{58\cdots 17}a^{7}+\frac{12\cdots 75}{58\cdots 17}a^{6}+\frac{11\cdots 00}{23\cdots 19}a^{5}+\frac{16\cdots 68}{71\cdots 57}a^{4}+\frac{23\cdots 02}{23\cdots 19}a^{3}+\frac{77\cdots 96}{23\cdots 19}a^{2}-\frac{11\cdots 63}{23\cdots 19}a+\frac{16\cdots 92}{26\cdots 91}$, $\frac{106408797536995}{17\cdots 51}a^{19}-\frac{248883467214508}{17\cdots 51}a^{18}+\frac{17\cdots 40}{17\cdots 51}a^{17}-\frac{65\cdots 74}{17\cdots 51}a^{16}+\frac{11\cdots 72}{58\cdots 17}a^{15}+\frac{33\cdots 74}{17\cdots 51}a^{14}+\frac{87\cdots 15}{58\cdots 17}a^{13}-\frac{89\cdots 09}{17\cdots 51}a^{12}+\frac{94\cdots 82}{58\cdots 17}a^{11}-\frac{60\cdots 18}{17\cdots 51}a^{10}-\frac{77\cdots 54}{17\cdots 51}a^{9}-\frac{25\cdots 62}{17\cdots 51}a^{8}-\frac{19\cdots 80}{17\cdots 51}a^{7}-\frac{13\cdots 68}{58\cdots 17}a^{6}-\frac{31\cdots 53}{19\cdots 39}a^{5}-\frac{65\cdots 25}{21\cdots 71}a^{4}-\frac{95\cdots 32}{71\cdots 57}a^{3}-\frac{12\cdots 17}{71\cdots 57}a^{2}+\frac{49\cdots 27}{23\cdots 19}a-\frac{24\cdots 35}{79\cdots 73}$, $\frac{692144770469396}{58\cdots 17}a^{19}-\frac{13\cdots 19}{58\cdots 17}a^{18}+\frac{42\cdots 77}{19\cdots 39}a^{17}-\frac{38\cdots 30}{58\cdots 17}a^{16}+\frac{23\cdots 58}{58\cdots 17}a^{15}+\frac{25\cdots 29}{58\cdots 17}a^{14}+\frac{22\cdots 14}{58\cdots 17}a^{13}+\frac{19\cdots 81}{58\cdots 17}a^{12}+\frac{27\cdots 01}{58\cdots 17}a^{11}-\frac{10\cdots 80}{58\cdots 17}a^{10}+\frac{41\cdots 56}{58\cdots 17}a^{9}-\frac{15\cdots 69}{64\cdots 13}a^{8}+\frac{67\cdots 56}{58\cdots 17}a^{7}-\frac{23\cdots 81}{58\cdots 17}a^{6}+\frac{29\cdots 08}{19\cdots 39}a^{5}-\frac{43\cdots 75}{64\cdots 13}a^{4}+\frac{47\cdots 22}{21\cdots 71}a^{3}-\frac{59\cdots 41}{71\cdots 57}a^{2}+\frac{35\cdots 70}{79\cdots 73}a-\frac{42\cdots 89}{79\cdots 73}$, $\frac{559574343415210}{17\cdots 51}a^{19}-\frac{528348719793586}{17\cdots 51}a^{18}+\frac{87\cdots 21}{17\cdots 51}a^{17}-\frac{19\cdots 57}{17\cdots 51}a^{16}+\frac{50\cdots 70}{58\cdots 17}a^{15}+\frac{42\cdots 16}{17\cdots 51}a^{14}+\frac{65\cdots 62}{58\cdots 17}a^{13}+\frac{34\cdots 87}{17\cdots 51}a^{12}+\frac{77\cdots 23}{58\cdots 17}a^{11}+\frac{14\cdots 83}{17\cdots 51}a^{10}+\frac{15\cdots 52}{17\cdots 51}a^{9}+\frac{36\cdots 34}{17\cdots 51}a^{8}+\frac{38\cdots 58}{17\cdots 51}a^{7}+\frac{52\cdots 83}{58\cdots 17}a^{6}+\frac{59\cdots 34}{19\cdots 39}a^{5}+\frac{90\cdots 04}{21\cdots 71}a^{4}+\frac{13\cdots 08}{71\cdots 57}a^{3}+\frac{20\cdots 12}{71\cdots 57}a^{2}+\frac{15\cdots 87}{23\cdots 19}a+\frac{41\cdots 64}{79\cdots 73}$, $\frac{144412477597537}{58\cdots 17}a^{19}-\frac{283467712368068}{58\cdots 17}a^{18}+\frac{804892791458780}{19\cdots 39}a^{17}-\frac{75\cdots 15}{58\cdots 17}a^{16}+\frac{44\cdots 92}{58\cdots 17}a^{15}+\frac{65\cdots 35}{58\cdots 17}a^{14}+\frac{40\cdots 30}{58\cdots 17}a^{13}+\frac{30\cdots 78}{58\cdots 17}a^{12}+\frac{50\cdots 58}{58\cdots 17}a^{11}-\frac{29\cdots 32}{58\cdots 17}a^{10}-\frac{46\cdots 40}{58\cdots 17}a^{9}-\frac{43\cdots 05}{64\cdots 13}a^{8}+\frac{11\cdots 43}{58\cdots 17}a^{7}-\frac{49\cdots 21}{58\cdots 17}a^{6}+\frac{71\cdots 40}{21\cdots 71}a^{5}-\frac{11\cdots 97}{79\cdots 73}a^{4}-\frac{50\cdots 99}{79\cdots 73}a^{3}-\frac{73\cdots 21}{23\cdots 19}a^{2}-\frac{15\cdots 02}{23\cdots 19}a-\frac{16\cdots 47}{26\cdots 91}$, $\frac{791063729432692}{17\cdots 51}a^{19}-\frac{732857504871319}{17\cdots 51}a^{18}+\frac{13\cdots 47}{17\cdots 51}a^{17}-\frac{29\cdots 97}{17\cdots 51}a^{16}+\frac{78\cdots 90}{58\cdots 17}a^{15}+\frac{53\cdots 42}{17\cdots 51}a^{14}+\frac{10\cdots 45}{58\cdots 17}a^{13}+\frac{56\cdots 85}{17\cdots 51}a^{12}+\frac{12\cdots 72}{58\cdots 17}a^{11}+\frac{26\cdots 59}{17\cdots 51}a^{10}+\frac{75\cdots 27}{17\cdots 51}a^{9}+\frac{42\cdots 68}{17\cdots 51}a^{8}+\frac{85\cdots 88}{17\cdots 51}a^{7}+\frac{12\cdots 44}{58\cdots 17}a^{6}+\frac{53\cdots 55}{64\cdots 13}a^{5}+\frac{34\cdots 21}{64\cdots 13}a^{4}+\frac{23\cdots 10}{21\cdots 71}a^{3}+\frac{19\cdots 52}{71\cdots 57}a^{2}+\frac{95\cdots 61}{79\cdots 73}a+\frac{66\cdots 69}{79\cdots 73}$, $\frac{285890809901222}{17\cdots 51}a^{19}-\frac{446071768784213}{17\cdots 51}a^{18}+\frac{48\cdots 09}{17\cdots 51}a^{17}-\frac{12\cdots 63}{17\cdots 51}a^{16}+\frac{94\cdots 22}{19\cdots 39}a^{15}+\frac{16\cdots 80}{17\cdots 51}a^{14}+\frac{98\cdots 60}{19\cdots 39}a^{13}+\frac{13\cdots 56}{17\cdots 51}a^{12}+\frac{12\cdots 01}{19\cdots 39}a^{11}+\frac{17\cdots 73}{17\cdots 51}a^{10}+\frac{91\cdots 20}{17\cdots 51}a^{9}+\frac{23\cdots 15}{17\cdots 51}a^{8}-\frac{15\cdots 48}{17\cdots 51}a^{7}+\frac{18\cdots 58}{58\cdots 17}a^{6}-\frac{25\cdots 93}{19\cdots 39}a^{5}+\frac{57\cdots 77}{21\cdots 71}a^{4}+\frac{84\cdots 68}{71\cdots 57}a^{3}-\frac{11\cdots 93}{71\cdots 57}a^{2}+\frac{43\cdots 73}{23\cdots 19}a+\frac{137049766155941}{79\cdots 73}$, $\frac{690983413058053}{17\cdots 51}a^{19}-\frac{13\cdots 96}{17\cdots 51}a^{18}+\frac{12\cdots 89}{17\cdots 51}a^{17}-\frac{37\cdots 07}{17\cdots 51}a^{16}+\frac{77\cdots 94}{58\cdots 17}a^{15}+\frac{27\cdots 38}{17\cdots 51}a^{14}+\frac{76\cdots 40}{58\cdots 17}a^{13}+\frac{22\cdots 83}{17\cdots 51}a^{12}+\frac{93\cdots 00}{58\cdots 17}a^{11}-\frac{79\cdots 51}{17\cdots 51}a^{10}+\frac{44\cdots 74}{17\cdots 51}a^{9}-\frac{10\cdots 56}{17\cdots 51}a^{8}+\frac{56\cdots 72}{17\cdots 51}a^{7}-\frac{14\cdots 28}{58\cdots 17}a^{6}+\frac{22\cdots 89}{64\cdots 13}a^{5}+\frac{60\cdots 85}{64\cdots 13}a^{4}+\frac{90\cdots 53}{21\cdots 71}a^{3}-\frac{77\cdots 32}{71\cdots 57}a^{2}-\frac{35\cdots 17}{79\cdots 73}a+\frac{24\cdots 60}{79\cdots 73}$
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| Regulator: | \( 41023218.2567 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 41023218.2567 \cdot 505}{10\cdot\sqrt{1945771207112214793287128936767578125}}\cr\approx \mathstrut & 0.142420958097 \end{aligned}\] (assuming GRH)
Galois group
| A cyclic group of order 20 |
| The 20 conjugacy class representatives for $C_{20}$ |
| Character table for $C_{20}$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.2825761.1, 10.10.24952891341003125.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | $20$ | ${\href{/padicField/3.4.0.1}{4} }^{5}$ | R | $20$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | $20$ | $20$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | $20$ | R | $20$ | $20$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
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\(5\)
| 5.5.4.15a1.4 | $x^{20} + 16 x^{16} + 12 x^{15} + 96 x^{12} + 144 x^{11} + 54 x^{10} + 256 x^{8} + 576 x^{7} + 432 x^{6} + 108 x^{5} + 256 x^{4} + 768 x^{3} + 864 x^{2} + 432 x + 86$ | $4$ | $5$ | $15$ | 20T1 | not computed |
|
\(41\)
| 41.1.5.4a1.1 | $x^{5} + 41$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ |
| 41.1.5.4a1.1 | $x^{5} + 41$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ | |
| 41.1.5.4a1.1 | $x^{5} + 41$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ | |
| 41.1.5.4a1.1 | $x^{5} + 41$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ |