Normalized defining polynomial
\( x^{20} - x^{19} + 13 x^{18} - 4 x^{17} + 138 x^{16} - 481 x^{15} + 2045 x^{14} - 4980 x^{13} + \cdots + 625 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(22199191947851112787734405517578125\)
\(\medspace = 5^{15}\cdot 31^{16}\)
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| Root discriminant: | \(52.16\) |
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| Galois root discriminant: | $5^{3/4}31^{4/5}\approx 52.15751099959023$ | ||
| Ramified primes: |
\(5\), \(31\)
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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: | \(155=5\cdot 31\) | ||
| Dirichlet character group: | $\lbrace$$\chi_{155}(64,·)$, $\chi_{155}(1,·)$, $\chi_{155}(2,·)$, $\chi_{155}(4,·)$, $\chi_{155}(97,·)$, $\chi_{155}(8,·)$, $\chi_{155}(128,·)$, $\chi_{155}(66,·)$, $\chi_{155}(78,·)$, $\chi_{155}(16,·)$, $\chi_{155}(132,·)$, $\chi_{155}(94,·)$, $\chi_{155}(32,·)$, $\chi_{155}(33,·)$, $\chi_{155}(101,·)$, $\chi_{155}(39,·)$, $\chi_{155}(109,·)$, $\chi_{155}(47,·)$, $\chi_{155}(126,·)$, $\chi_{155}(63,·)$$\rbrace$ | ||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $\frac{1}{5}a^{13}+\frac{2}{5}a^{12}+\frac{1}{5}a^{9}+\frac{1}{5}a^{7}+\frac{1}{5}a^{5}-\frac{2}{5}a^{2}+\frac{1}{5}a$, $\frac{1}{5}a^{14}+\frac{1}{5}a^{12}+\frac{1}{5}a^{10}-\frac{2}{5}a^{9}+\frac{1}{5}a^{8}-\frac{2}{5}a^{7}+\frac{1}{5}a^{6}-\frac{2}{5}a^{5}-\frac{2}{5}a^{3}-\frac{2}{5}a$, $\frac{1}{25}a^{15}+\frac{1}{25}a^{13}+\frac{1}{5}a^{12}+\frac{6}{25}a^{11}-\frac{12}{25}a^{10}+\frac{6}{25}a^{9}-\frac{7}{25}a^{8}+\frac{1}{25}a^{7}-\frac{2}{25}a^{6}-\frac{1}{5}a^{5}+\frac{3}{25}a^{4}-\frac{2}{5}a^{3}+\frac{3}{25}a^{2}+\frac{1}{5}a$, $\frac{1}{25}a^{16}+\frac{1}{25}a^{14}-\frac{4}{25}a^{12}-\frac{12}{25}a^{11}+\frac{6}{25}a^{10}-\frac{12}{25}a^{9}+\frac{1}{25}a^{8}-\frac{7}{25}a^{7}-\frac{1}{5}a^{6}-\frac{2}{25}a^{5}-\frac{2}{5}a^{4}+\frac{3}{25}a^{3}-\frac{2}{5}a^{2}-\frac{1}{5}a$, $\frac{1}{22\cdots 25}a^{17}-\frac{294864270357121}{22\cdots 25}a^{16}+\frac{10760498473873}{45\cdots 05}a^{15}+\frac{909205899230709}{22\cdots 25}a^{14}-\frac{210814021047053}{45\cdots 05}a^{13}+\frac{10\cdots 22}{22\cdots 25}a^{12}+\frac{10\cdots 67}{22\cdots 25}a^{11}+\frac{45\cdots 49}{22\cdots 25}a^{10}-\frac{32\cdots 48}{22\cdots 25}a^{9}+\frac{504977962972054}{22\cdots 25}a^{8}-\frac{15\cdots 29}{22\cdots 25}a^{7}+\frac{984027356852601}{45\cdots 05}a^{6}-\frac{55\cdots 73}{22\cdots 25}a^{5}-\frac{563624833687709}{45\cdots 05}a^{4}+\frac{11\cdots 02}{22\cdots 25}a^{3}+\frac{29\cdots 97}{22\cdots 25}a^{2}+\frac{22\cdots 13}{45\cdots 05}a+\frac{190931591142933}{900275927177101}$, $\frac{1}{22\cdots 25}a^{18}+\frac{108192405962256}{22\cdots 25}a^{16}+\frac{440880878229536}{22\cdots 25}a^{15}+\frac{17\cdots 36}{22\cdots 25}a^{14}-\frac{19\cdots 71}{22\cdots 25}a^{13}-\frac{16\cdots 64}{22\cdots 25}a^{12}-\frac{93\cdots 06}{22\cdots 25}a^{11}+\frac{76\cdots 09}{22\cdots 25}a^{10}-\frac{10\cdots 66}{22\cdots 25}a^{9}+\frac{74\cdots 08}{22\cdots 25}a^{8}+\frac{43\cdots 09}{22\cdots 25}a^{7}+\frac{92\cdots 03}{22\cdots 25}a^{6}-\frac{54\cdots 77}{22\cdots 25}a^{5}+\frac{36\cdots 48}{22\cdots 25}a^{4}+\frac{429428454357341}{45\cdots 05}a^{3}-\frac{17\cdots 27}{22\cdots 25}a^{2}-\frac{14\cdots 82}{45\cdots 05}a-\frac{68619080804835}{900275927177101}$, $\frac{1}{11\cdots 25}a^{19}-\frac{1}{11\cdots 25}a^{18}-\frac{2}{11\cdots 25}a^{17}-\frac{533609215018744}{11\cdots 25}a^{16}-\frac{129527128999212}{11\cdots 25}a^{15}-\frac{10\cdots 96}{11\cdots 25}a^{14}+\frac{13\cdots 89}{22\cdots 25}a^{13}+\frac{72\cdots 67}{22\cdots 25}a^{12}-\frac{10\cdots 69}{11\cdots 25}a^{11}-\frac{10\cdots 13}{22\cdots 25}a^{10}+\frac{79\cdots 34}{22\cdots 25}a^{9}-\frac{11\cdots 04}{11\cdots 25}a^{8}-\frac{46\cdots 12}{11\cdots 25}a^{7}+\frac{37\cdots 74}{11\cdots 25}a^{6}+\frac{18\cdots 88}{11\cdots 25}a^{5}-\frac{46\cdots 89}{11\cdots 25}a^{4}+\frac{19\cdots 71}{11\cdots 25}a^{3}+\frac{95\cdots 88}{22\cdots 25}a^{2}-\frac{357946020788601}{45\cdots 05}a-\frac{268705462053374}{900275927177101}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{10}\times C_{10}$, which has order $400$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}\times C_{10}\times C_{10}$, which has order $400$ (assuming GRH) |
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| Relative class number: | $400$ (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( \frac{34279214135504}{112534490897137625} a^{19} - \frac{33746819736904}{112534490897137625} a^{18} + \frac{438708656579752}{112534490897137625} a^{17} - \frac{134987278947616}{112534490897137625} a^{16} + \frac{4657061123692752}{112534490897137625} a^{15} - \frac{16532430736329149}{112534490897137625} a^{14} + \frac{13802449272393736}{22506898179427525} a^{13} - \frac{33611832457956384}{22506898179427525} a^{12} + \frac{651347367741984104}{112534490897137625} a^{11} - \frac{49034129077721512}{4501379635885505} a^{10} + \frac{454859818816663842}{22506898179427525} a^{9} - \frac{3533764481930165456}{112534490897137625} a^{8} + \frac{5877616099937095872}{112534490897137625} a^{7} - \frac{1492486849684316304}{112534490897137625} a^{6} + \frac{1619273651435864632}{112534490897137625} a^{5} - \frac{2003540773400225681}{112534490897137625} a^{4} + \frac{358256238326972864}{112534490897137625} a^{3} + \frac{3003466956584456}{22506898179427525} a^{2} + \frac{3577162892111824}{4501379635885505} a - \frac{33746819736904}{900275927177101} \)
(order $10$)
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| Fundamental units: |
$\frac{9199005007004}{11\cdots 25}a^{19}-\frac{52556976583279}{11\cdots 25}a^{18}+\frac{158646321311502}{11\cdots 25}a^{17}-\frac{596889726796416}{11\cdots 25}a^{16}+\frac{13\cdots 52}{11\cdots 25}a^{15}-\frac{10\cdots 24}{11\cdots 25}a^{14}+\frac{78\cdots 61}{22\cdots 25}a^{13}-\frac{26\cdots 84}{22\cdots 25}a^{12}+\frac{38\cdots 04}{11\cdots 25}a^{11}-\frac{46\cdots 32}{45\cdots 05}a^{10}+\frac{42\cdots 92}{22\cdots 25}a^{9}-\frac{38\cdots 06}{11\cdots 25}a^{8}+\frac{58\cdots 97}{11\cdots 25}a^{7}-\frac{75\cdots 04}{11\cdots 25}a^{6}+\frac{16\cdots 32}{11\cdots 25}a^{5}-\frac{20\cdots 56}{11\cdots 25}a^{4}+\frac{35\cdots 39}{11\cdots 25}a^{3}+\frac{19\cdots 81}{22\cdots 25}a^{2}-\frac{317669455741684}{900275927177101}a-\frac{44196906873779}{900275927177101}$, $\frac{13695749901067}{11\cdots 25}a^{19}+\frac{10818431960093}{11\cdots 25}a^{18}+\frac{176484631808006}{11\cdots 25}a^{17}+\frac{240660561918182}{11\cdots 25}a^{16}+\frac{20\cdots 81}{11\cdots 25}a^{15}-\frac{32\cdots 77}{11\cdots 25}a^{14}+\frac{38\cdots 26}{22\cdots 25}a^{13}-\frac{58\cdots 61}{22\cdots 25}a^{12}+\frac{18\cdots 02}{11\cdots 25}a^{11}-\frac{27\cdots 44}{22\cdots 25}a^{10}+\frac{97\cdots 68}{22\cdots 25}a^{9}-\frac{56\cdots 68}{11\cdots 25}a^{8}+\frac{13\cdots 06}{11\cdots 25}a^{7}+\frac{14\cdots 88}{11\cdots 25}a^{6}+\frac{32\cdots 86}{11\cdots 25}a^{5}+\frac{40\cdots 17}{11\cdots 25}a^{4}+\frac{84\cdots 67}{11\cdots 25}a^{3}+\frac{15\cdots 68}{22\cdots 25}a^{2}+\frac{89682462888563}{900275927177101}a+\frac{484703246834640}{900275927177101}$, $\frac{144500771880809}{11\cdots 25}a^{19}+\frac{40297564990986}{11\cdots 25}a^{18}+\frac{17\cdots 87}{11\cdots 25}a^{17}+\frac{17\cdots 39}{11\cdots 25}a^{16}+\frac{20\cdots 12}{11\cdots 25}a^{15}-\frac{44\cdots 29}{11\cdots 25}a^{14}+\frac{43\cdots 57}{22\cdots 25}a^{13}-\frac{76\cdots 57}{22\cdots 25}a^{12}+\frac{20\cdots 54}{11\cdots 25}a^{11}-\frac{41\cdots 98}{22\cdots 25}a^{10}+\frac{97\cdots 21}{22\cdots 25}a^{9}-\frac{53\cdots 86}{11\cdots 25}a^{8}+\frac{11\cdots 12}{11\cdots 25}a^{7}+\frac{17\cdots 51}{11\cdots 25}a^{6}+\frac{12\cdots 22}{11\cdots 25}a^{5}+\frac{49\cdots 84}{11\cdots 25}a^{4}+\frac{71\cdots 09}{11\cdots 25}a^{3}+\frac{11\cdots 11}{22\cdots 25}a^{2}+\frac{10\cdots 51}{900275927177101}a+\frac{857308279509462}{900275927177101}$, $\frac{118223184162964}{11\cdots 25}a^{19}-\frac{103405440354084}{11\cdots 25}a^{18}+\frac{14\cdots 72}{11\cdots 25}a^{17}-\frac{244045083771211}{11\cdots 25}a^{16}+\frac{15\cdots 72}{11\cdots 25}a^{15}-\frac{54\cdots 14}{11\cdots 25}a^{14}+\frac{46\cdots 32}{22\cdots 25}a^{13}-\frac{10\cdots 56}{22\cdots 25}a^{12}+\frac{21\cdots 64}{11\cdots 25}a^{11}-\frac{76\cdots 97}{22\cdots 25}a^{10}+\frac{14\cdots 17}{22\cdots 25}a^{9}-\frac{10\cdots 36}{11\cdots 25}a^{8}+\frac{16\cdots 12}{11\cdots 25}a^{7}+\frac{60\cdots 31}{11\cdots 25}a^{6}-\frac{23\cdots 28}{11\cdots 25}a^{5}+\frac{34\cdots 49}{11\cdots 25}a^{4}+\frac{10\cdots 44}{11\cdots 25}a^{3}+\frac{85\cdots 76}{22\cdots 25}a^{2}+\frac{268796472763286}{900275927177101}a+\frac{14\cdots 29}{900275927177101}$, $\frac{61714558446633}{11\cdots 25}a^{19}-\frac{158105136515648}{11\cdots 25}a^{18}+\frac{873846937859124}{11\cdots 25}a^{17}-\frac{14\cdots 12}{11\cdots 25}a^{16}+\frac{85\cdots 74}{11\cdots 25}a^{15}-\frac{42\cdots 03}{11\cdots 25}a^{14}+\frac{33\cdots 62}{22\cdots 25}a^{13}-\frac{98\cdots 02}{22\cdots 25}a^{12}+\frac{16\cdots 38}{11\cdots 25}a^{11}-\frac{79\cdots 34}{22\cdots 25}a^{10}+\frac{58\cdots 30}{900275927177101}a^{9}-\frac{12\cdots 92}{11\cdots 25}a^{8}+\frac{19\cdots 54}{11\cdots 25}a^{7}-\frac{16\cdots 98}{11\cdots 25}a^{6}+\frac{30\cdots 99}{11\cdots 25}a^{5}-\frac{45\cdots 27}{11\cdots 25}a^{4}+\frac{11\cdots 08}{11\cdots 25}a^{3}+\frac{72\cdots 82}{22\cdots 25}a^{2}-\frac{632432762858203}{900275927177101}a-\frac{35\cdots 15}{900275927177101}$, $\frac{248023891132}{22\cdots 25}a^{19}-\frac{3224310584716}{22\cdots 25}a^{18}+\frac{992095564528}{22\cdots 25}a^{17}-\frac{34227296976216}{22\cdots 25}a^{16}-\frac{20533178640532}{22\cdots 25}a^{15}-\frac{101441771472988}{45\cdots 05}a^{14}+\frac{247031795567472}{45\cdots 05}a^{13}-\frac{47\cdots 32}{22\cdots 25}a^{12}+\frac{360378713814796}{900275927177101}a^{11}-\frac{40\cdots 62}{22\cdots 25}a^{10}+\frac{25\cdots 48}{22\cdots 25}a^{9}-\frac{43\cdots 76}{22\cdots 25}a^{8}+\frac{10\cdots 32}{22\cdots 25}a^{7}-\frac{11\cdots 56}{22\cdots 25}a^{6}-\frac{70\cdots 03}{22\cdots 25}a^{5}-\frac{26\cdots 12}{22\cdots 25}a^{4}-\frac{22074126310748}{45\cdots 05}a^{3}-\frac{26290532459992}{900275927177101}a^{2}+\frac{1240119455660}{900275927177101}a-\frac{12\cdots 31}{900275927177101}$, $\frac{611082793982}{22\cdots 25}a^{19}-\frac{7944076321766}{22\cdots 25}a^{18}+\frac{2444331175928}{22\cdots 25}a^{17}-\frac{84329425569516}{22\cdots 25}a^{16}-\frac{50732607264107}{22\cdots 25}a^{15}-\frac{249932862738638}{45\cdots 05}a^{14}+\frac{608638462806072}{45\cdots 05}a^{13}-\frac{11\cdots 82}{22\cdots 25}a^{12}+\frac{887903299655846}{900275927177101}a^{11}-\frac{10\cdots 37}{22\cdots 25}a^{10}+\frac{63\cdots 48}{22\cdots 25}a^{9}-\frac{10\cdots 76}{22\cdots 25}a^{8}+\frac{27\cdots 32}{22\cdots 25}a^{7}-\frac{29\cdots 06}{22\cdots 25}a^{6}-\frac{17\cdots 78}{22\cdots 25}a^{5}-\frac{64\cdots 12}{22\cdots 25}a^{4}-\frac{54386368664398}{45\cdots 05}a^{3}-\frac{64774776162092}{900275927177101}a^{2}+\frac{3055413969910}{900275927177101}a-\frac{37\cdots 86}{900275927177101}$, $\frac{6702222049}{45\cdots 05}a^{19}-\frac{87128886637}{45\cdots 05}a^{18}+\frac{26808888196}{45\cdots 05}a^{17}-\frac{924906642762}{45\cdots 05}a^{16}-\frac{581386755294}{45\cdots 05}a^{15}-\frac{2741208818041}{900275927177101}a^{14}+\frac{6675413160804}{900275927177101}a^{13}-\frac{129359587767749}{45\cdots 05}a^{12}+\frac{48691643185985}{900275927177101}a^{11}-\frac{10\cdots 29}{45\cdots 05}a^{10}+\frac{701816479638986}{45\cdots 05}a^{9}-\frac{11\cdots 32}{45\cdots 05}a^{8}+\frac{296412472339074}{45\cdots 05}a^{7}-\frac{321592720577167}{45\cdots 05}a^{6}-\frac{17\cdots 11}{45\cdots 05}a^{5}-\frac{71150789272184}{45\cdots 05}a^{4}-\frac{596497762361}{900275927177101}a^{3}-\frac{3552177685970}{900275927177101}a^{2}+\frac{167555551225}{900275927177101}a+\frac{434165717085533}{900275927177101}$, $\frac{53922439434}{22\cdots 25}a^{19}-\frac{700991712642}{22\cdots 25}a^{18}+\frac{215689757736}{22\cdots 25}a^{17}-\frac{7441296641892}{22\cdots 25}a^{16}-\frac{4445978774109}{22\cdots 25}a^{15}-\frac{22054277728506}{45\cdots 05}a^{14}+\frac{53706749676264}{45\cdots 05}a^{13}-\frac{10\cdots 34}{22\cdots 25}a^{12}+\frac{78349304497602}{900275927177101}a^{11}-\frac{88\cdots 44}{22\cdots 25}a^{10}+\frac{56\cdots 76}{22\cdots 25}a^{9}-\frac{93\cdots 12}{22\cdots 25}a^{8}+\frac{23\cdots 84}{22\cdots 25}a^{7}-\frac{25\cdots 22}{22\cdots 25}a^{6}-\frac{15\cdots 36}{22\cdots 25}a^{5}-\frac{572440617031344}{22\cdots 25}a^{4}-\frac{4799097109626}{45\cdots 05}a^{3}-\frac{5715778580004}{900275927177101}a^{2}+\frac{269612197170}{900275927177101}a-\frac{502917343734554}{900275927177101}$
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| Regulator: | \( 24173706.8324 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 24173706.8324 \cdot 400}{10\cdot\sqrt{22199191947851112787734405517578125}}\cr\approx \mathstrut & 0.622348063586 \end{aligned}\] (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
| \(\Q(\sqrt{5}) \), \(\Q(\zeta_{5})\), 5.5.923521.1, 10.10.2665284492003125.1 |
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 | $20$ | $20$ | R | $20$ | ${\href{/padicField/11.5.0.1}{5} }^{4}$ | $20$ | $20$ | ${\href{/padicField/19.10.0.1}{10} }^{2}$ | $20$ | ${\href{/padicField/29.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.5.0.1}{5} }^{4}$ | $20$ | $20$ | $20$ | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 5.1.4.3a1.1 | $x^{4} + 5$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
|
\(31\)
| 31.1.5.4a1.1 | $x^{5} + 31$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ |
| 31.1.5.4a1.1 | $x^{5} + 31$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ | |
| 31.1.5.4a1.1 | $x^{5} + 31$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ | |
| 31.1.5.4a1.1 | $x^{5} + 31$ | $5$ | $1$ | $4$ | $C_5$ | $$[\ ]_{5}$$ |