Normalized defining polynomial
\( x^{20} - 4 x^{19} + 96 x^{18} + 122 x^{17} + 4620 x^{16} + 18596 x^{15} + 184045 x^{14} + \cdots + 1082598671174 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(78148390643938933109518475030237312085224587264\)
\(\medspace = 2^{20}\cdot 17^{15}\cdot 53^{13}\)
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| Root discriminant: | \(221.13\) |
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| Galois root discriminant: | $2\cdot 17^{3/4}53^{3/4}\approx 328.90735776316194$ | ||
| Ramified primes: |
\(2\), \(17\), \(53\)
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| Discriminant root field: | \(\Q(\sqrt{901}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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| This field is not Galois over $\Q$. | |||
| This is a CM field. | |||
| Reflex fields: | unavailable$^{512}$ | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $\frac{1}{3}a^{15}+\frac{1}{3}a^{13}-\frac{1}{3}a^{12}+\frac{1}{3}a^{11}-\frac{1}{3}a^{10}+\frac{1}{3}a^{7}-\frac{1}{3}a^{6}-\frac{1}{3}a^{5}+\frac{1}{3}a^{4}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{51}a^{16}-\frac{2}{51}a^{15}-\frac{20}{51}a^{14}+\frac{3}{17}a^{13}-\frac{8}{17}a^{12}+\frac{6}{17}a^{11}+\frac{1}{3}a^{10}-\frac{3}{17}a^{9}-\frac{8}{51}a^{8}+\frac{2}{17}a^{7}-\frac{1}{3}a^{6}+\frac{7}{17}a^{5}+\frac{19}{51}a^{4}-\frac{6}{17}a^{3}-\frac{1}{51}a^{2}+\frac{8}{17}a+\frac{1}{51}$, $\frac{1}{51}a^{17}-\frac{7}{51}a^{15}+\frac{20}{51}a^{14}+\frac{11}{51}a^{13}+\frac{4}{51}a^{12}+\frac{19}{51}a^{11}+\frac{8}{51}a^{10}+\frac{25}{51}a^{9}-\frac{10}{51}a^{8}+\frac{4}{17}a^{7}+\frac{7}{17}a^{6}-\frac{7}{51}a^{5}-\frac{14}{51}a^{4}+\frac{14}{51}a^{3}+\frac{22}{51}a^{2}-\frac{19}{51}a+\frac{19}{51}$, $\frac{1}{153}a^{18}-\frac{1}{153}a^{16}+\frac{25}{153}a^{15}+\frac{44}{153}a^{14}-\frac{3}{17}a^{13}+\frac{11}{153}a^{12}-\frac{71}{153}a^{11}-\frac{43}{153}a^{10}+\frac{38}{153}a^{9}-\frac{4}{17}a^{8}+\frac{74}{153}a^{7}-\frac{25}{51}a^{6}+\frac{44}{153}a^{5}-\frac{8}{153}a^{4}-\frac{35}{153}a^{3}-\frac{76}{153}a^{2}-\frac{58}{153}a+\frac{74}{153}$, $\frac{1}{25\cdots 12}a^{19}+\frac{47\cdots 35}{25\cdots 12}a^{18}-\frac{24\cdots 75}{25\cdots 12}a^{17}-\frac{92\cdots 37}{83\cdots 04}a^{16}-\frac{19\cdots 65}{14\cdots 48}a^{15}-\frac{60\cdots 59}{25\cdots 12}a^{14}+\frac{14\cdots 76}{62\cdots 53}a^{13}-\frac{33\cdots 12}{69\cdots 17}a^{12}+\frac{47\cdots 15}{83\cdots 04}a^{11}+\frac{38\cdots 19}{25\cdots 12}a^{10}+\frac{26\cdots 74}{62\cdots 53}a^{9}-\frac{98\cdots 75}{62\cdots 53}a^{8}-\frac{30\cdots 72}{62\cdots 53}a^{7}+\frac{18\cdots 20}{62\cdots 53}a^{6}-\frac{89\cdots 59}{46\cdots 78}a^{5}+\frac{25\cdots 58}{62\cdots 53}a^{4}+\frac{29\cdots 11}{83\cdots 04}a^{3}-\frac{72\cdots 13}{25\cdots 12}a^{2}+\frac{17\cdots 56}{62\cdots 53}a-\frac{32\cdots 81}{12\cdots 06}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{5562}$, which has order $1601856$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}\times C_{2}\times C_{6}\times C_{6}\times C_{5562}$, which has order $1601856$ (assuming GRH) |
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| Relative class number: | $1601856$ (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{51\cdots 65}{99\cdots 11}a^{19}+\frac{30\cdots 24}{99\cdots 11}a^{18}+\frac{17\cdots 50}{33\cdots 37}a^{17}+\frac{42\cdots 17}{99\cdots 11}a^{16}+\frac{50\cdots 87}{99\cdots 11}a^{15}+\frac{34\cdots 94}{99\cdots 11}a^{14}+\frac{88\cdots 73}{33\cdots 37}a^{13}+\frac{15\cdots 51}{99\cdots 11}a^{12}+\frac{94\cdots 33}{99\cdots 11}a^{11}+\frac{44\cdots 17}{99\cdots 11}a^{10}+\frac{21\cdots 38}{99\cdots 11}a^{9}+\frac{26\cdots 49}{33\cdots 37}a^{8}+\frac{11\cdots 08}{33\cdots 37}a^{7}+\frac{85\cdots 39}{99\cdots 11}a^{6}+\frac{28\cdots 40}{99\cdots 11}a^{5}+\frac{82\cdots 76}{99\cdots 11}a^{4}-\frac{46\cdots 34}{33\cdots 37}a^{3}-\frac{73\cdots 79}{99\cdots 11}a^{2}-\frac{38\cdots 10}{33\cdots 37}a-\frac{57\cdots 55}{33\cdots 37}$, $\frac{38\cdots 49}{29\cdots 33}a^{19}-\frac{19\cdots 56}{29\cdots 33}a^{18}+\frac{42\cdots 98}{29\cdots 33}a^{17}-\frac{14\cdots 76}{29\cdots 33}a^{16}+\frac{20\cdots 35}{29\cdots 33}a^{15}+\frac{57\cdots 87}{29\cdots 33}a^{14}+\frac{74\cdots 25}{29\cdots 33}a^{13}+\frac{26\cdots 10}{29\cdots 33}a^{12}+\frac{23\cdots 92}{29\cdots 33}a^{11}+\frac{14\cdots 61}{99\cdots 11}a^{10}+\frac{61\cdots 32}{29\cdots 33}a^{9}+\frac{81\cdots 78}{29\cdots 33}a^{8}+\frac{10\cdots 00}{29\cdots 33}a^{7}-\frac{81\cdots 85}{29\cdots 33}a^{6}+\frac{91\cdots 60}{29\cdots 33}a^{5}-\frac{54\cdots 01}{99\cdots 11}a^{4}+\frac{20\cdots 75}{29\cdots 33}a^{3}-\frac{35\cdots 38}{99\cdots 11}a^{2}-\frac{35\cdots 38}{99\cdots 11}a+\frac{84\cdots 81}{29\cdots 33}$, $\frac{16\cdots 40}{32\cdots 83}a^{19}-\frac{94\cdots 72}{32\cdots 83}a^{18}+\frac{14\cdots 44}{32\cdots 83}a^{17}+\frac{18\cdots 24}{10\cdots 61}a^{16}+\frac{27\cdots 02}{16\cdots 63}a^{15}+\frac{10\cdots 12}{19\cdots 99}a^{14}+\frac{18\cdots 70}{32\cdots 83}a^{13}+\frac{15\cdots 02}{10\cdots 61}a^{12}+\frac{20\cdots 63}{12\cdots 29}a^{11}-\frac{81\cdots 56}{32\cdots 83}a^{10}+\frac{11\cdots 72}{32\cdots 83}a^{9}-\frac{16\cdots 45}{32\cdots 83}a^{8}+\frac{87\cdots 68}{19\cdots 99}a^{7}-\frac{44\cdots 06}{32\cdots 83}a^{6}+\frac{37\cdots 16}{10\cdots 61}a^{5}-\frac{60\cdots 57}{32\cdots 83}a^{4}+\frac{20\cdots 75}{10\cdots 61}a^{3}+\frac{35\cdots 20}{32\cdots 83}a^{2}+\frac{23\cdots 94}{32\cdots 83}a-\frac{11\cdots 89}{32\cdots 83}$, $\frac{23\cdots 42}{32\cdots 83}a^{19}-\frac{28\cdots 33}{32\cdots 83}a^{18}+\frac{34\cdots 08}{32\cdots 83}a^{17}-\frac{33\cdots 35}{64\cdots 33}a^{16}+\frac{21\cdots 26}{56\cdots 21}a^{15}-\frac{35\cdots 19}{32\cdots 83}a^{14}+\frac{21\cdots 40}{32\cdots 83}a^{13}-\frac{78\cdots 96}{36\cdots 87}a^{12}+\frac{18\cdots 27}{10\cdots 61}a^{11}-\frac{46\cdots 66}{32\cdots 83}a^{10}+\frac{28\cdots 86}{32\cdots 83}a^{9}-\frac{17\cdots 18}{32\cdots 83}a^{8}+\frac{73\cdots 42}{32\cdots 83}a^{7}-\frac{28\cdots 56}{32\cdots 83}a^{6}+\frac{86\cdots 22}{36\cdots 87}a^{5}-\frac{19\cdots 59}{32\cdots 83}a^{4}+\frac{10\cdots 13}{10\cdots 61}a^{3}-\frac{22\cdots 82}{32\cdots 83}a^{2}+\frac{36\cdots 34}{32\cdots 83}a+\frac{31\cdots 27}{32\cdots 83}$, $\frac{74\cdots 27}{89\cdots 99}a^{19}-\frac{27\cdots 23}{89\cdots 99}a^{18}+\frac{72\cdots 14}{89\cdots 99}a^{17}+\frac{35\cdots 85}{29\cdots 33}a^{16}+\frac{40\cdots 31}{99\cdots 11}a^{15}+\frac{15\cdots 56}{89\cdots 99}a^{14}+\frac{14\cdots 03}{89\cdots 99}a^{13}+\frac{23\cdots 10}{33\cdots 37}a^{12}+\frac{16\cdots 52}{29\cdots 33}a^{11}+\frac{13\cdots 25}{89\cdots 99}a^{10}+\frac{11\cdots 58}{89\cdots 99}a^{9}+\frac{13\cdots 05}{89\cdots 99}a^{8}+\frac{18\cdots 30}{89\cdots 99}a^{7}+\frac{26\cdots 53}{89\cdots 99}a^{6}+\frac{16\cdots 86}{99\cdots 11}a^{5}-\frac{19\cdots 45}{89\cdots 99}a^{4}+\frac{57\cdots 79}{29\cdots 33}a^{3}-\frac{22\cdots 84}{89\cdots 99}a^{2}-\frac{26\cdots 30}{89\cdots 99}a-\frac{38\cdots 19}{89\cdots 99}$, $\frac{22\cdots 80}{78\cdots 39}a^{19}-\frac{13\cdots 26}{78\cdots 39}a^{18}+\frac{22\cdots 56}{78\cdots 39}a^{17}-\frac{75\cdots 10}{26\cdots 13}a^{16}+\frac{14\cdots 60}{13\cdots 93}a^{15}+\frac{24\cdots 84}{78\cdots 39}a^{14}+\frac{28\cdots 56}{78\cdots 39}a^{13}+\frac{86\cdots 62}{87\cdots 71}a^{12}+\frac{29\cdots 12}{26\cdots 13}a^{11}+\frac{17\cdots 66}{78\cdots 39}a^{10}+\frac{20\cdots 76}{78\cdots 39}a^{9}-\frac{25\cdots 64}{78\cdots 39}a^{8}+\frac{28\cdots 28}{78\cdots 39}a^{7}-\frac{70\cdots 68}{78\cdots 39}a^{6}+\frac{84\cdots 08}{29\cdots 57}a^{5}-\frac{99\cdots 70}{78\cdots 39}a^{4}+\frac{32\cdots 28}{26\cdots 13}a^{3}-\frac{16\cdots 84}{78\cdots 39}a^{2}+\frac{28\cdots 68}{78\cdots 39}a-\frac{25\cdots 15}{78\cdots 39}$, $\frac{23\cdots 67}{32\cdots 83}a^{19}-\frac{12\cdots 39}{32\cdots 83}a^{18}+\frac{20\cdots 27}{32\cdots 83}a^{17}+\frac{70\cdots 23}{10\cdots 61}a^{16}+\frac{40\cdots 13}{16\cdots 63}a^{15}+\frac{29\cdots 19}{32\cdots 83}a^{14}+\frac{29\cdots 72}{32\cdots 83}a^{13}+\frac{30\cdots 96}{10\cdots 61}a^{12}+\frac{33\cdots 39}{12\cdots 29}a^{11}+\frac{77\cdots 34}{32\cdots 83}a^{10}+\frac{11\cdots 77}{19\cdots 99}a^{9}-\frac{43\cdots 81}{32\cdots 83}a^{8}+\frac{24\cdots 03}{32\cdots 83}a^{7}-\frac{46\cdots 22}{32\cdots 83}a^{6}+\frac{62\cdots 07}{10\cdots 61}a^{5}-\frac{61\cdots 65}{32\cdots 83}a^{4}+\frac{25\cdots 49}{10\cdots 61}a^{3}-\frac{13\cdots 47}{32\cdots 83}a^{2}-\frac{24\cdots 48}{32\cdots 83}a+\frac{34\cdots 81}{32\cdots 83}$, $\frac{78\cdots 71}{32\cdots 83}a^{19}+\frac{11\cdots 75}{32\cdots 83}a^{18}+\frac{75\cdots 24}{32\cdots 83}a^{17}+\frac{15\cdots 97}{10\cdots 61}a^{16}+\frac{93\cdots 88}{56\cdots 21}a^{15}+\frac{37\cdots 90}{32\cdots 83}a^{14}+\frac{28\cdots 83}{32\cdots 83}a^{13}+\frac{57\cdots 00}{12\cdots 29}a^{12}+\frac{31\cdots 79}{10\cdots 61}a^{11}+\frac{37\cdots 57}{32\cdots 83}a^{10}+\frac{18\cdots 94}{32\cdots 83}a^{9}+\frac{49\cdots 24}{32\cdots 83}a^{8}+\frac{19\cdots 68}{32\cdots 83}a^{7}+\frac{23\cdots 35}{32\cdots 83}a^{6}+\frac{30\cdots 39}{12\cdots 29}a^{5}-\frac{16\cdots 42}{32\cdots 83}a^{4}-\frac{72\cdots 48}{10\cdots 61}a^{3}-\frac{19\cdots 10}{32\cdots 83}a^{2}-\frac{14\cdots 16}{19\cdots 99}a-\frac{31\cdots 65}{32\cdots 83}$, $\frac{14\cdots 87}{32\cdots 83}a^{19}-\frac{35\cdots 50}{32\cdots 83}a^{18}+\frac{34\cdots 11}{32\cdots 83}a^{17}-\frac{10\cdots 49}{10\cdots 61}a^{16}+\frac{18\cdots 35}{56\cdots 21}a^{15}-\frac{94\cdots 44}{32\cdots 83}a^{14}-\frac{14\cdots 54}{19\cdots 99}a^{13}-\frac{27\cdots 18}{36\cdots 87}a^{12}-\frac{77\cdots 68}{10\cdots 61}a^{11}-\frac{98\cdots 96}{32\cdots 83}a^{10}+\frac{22\cdots 81}{32\cdots 83}a^{9}-\frac{29\cdots 12}{32\cdots 83}a^{8}+\frac{87\cdots 93}{32\cdots 83}a^{7}-\frac{44\cdots 94}{32\cdots 83}a^{6}+\frac{11\cdots 99}{36\cdots 87}a^{5}-\frac{24\cdots 31}{32\cdots 83}a^{4}+\frac{15\cdots 32}{10\cdots 61}a^{3}-\frac{20\cdots 24}{32\cdots 83}a^{2}+\frac{59\cdots 84}{32\cdots 83}a-\frac{15\cdots 35}{32\cdots 83}$
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| Regulator: | \( 4097659676.87 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 4097659676.87 \cdot 1601856}{2\cdot\sqrt{78148390643938933109518475030237312085224587264}}\cr\approx \mathstrut & 1.12581758228 \end{aligned}\] (assuming GRH)
Galois group
$C_{20}:C_4$ (as 20T18):
| A solvable group of order 80 |
| The 14 conjugacy class representatives for $C_{20}:C_4$ |
| Character table for $C_{20}:C_4$ |
Intermediate fields
| \(\Q(\sqrt{17}) \), \(\Q(\sqrt{-51 +10 \sqrt{17}})\), 5.5.2382032.1, 10.10.8056377164681869568.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 sibling: | data not computed |
| Degree 40 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 | ${\href{/padicField/3.4.0.1}{4} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | $20$ | $20$ | ${\href{/padicField/13.10.0.1}{10} }^{2}$ | R | ${\href{/padicField/19.4.0.1}{4} }^{4}{,}\,{\href{/padicField/19.2.0.1}{2} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{5}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.5.0.1}{5} }^{4}$ | ${\href{/padicField/47.2.0.1}{2} }^{10}$ | R | ${\href{/padicField/59.10.0.1}{10} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ | |
| 2.4.2.8a1.1 | $x^{8} + 2 x^{5} + 4 x^{4} + x^{2} + 4 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $$[2]^{4}$$ | |
| 2.4.2.8a1.1 | $x^{8} + 2 x^{5} + 4 x^{4} + x^{2} + 4 x + 5$ | $2$ | $4$ | $8$ | $C_4\times C_2$ | $$[2]^{4}$$ | |
|
\(17\)
| 17.1.4.3a1.3 | $x^{4} + 153$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 17.2.4.6a1.2 | $x^{8} + 64 x^{7} + 1548 x^{6} + 16960 x^{5} + 74806 x^{4} + 50880 x^{3} + 13932 x^{2} + 1728 x + 98$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ | |
| 17.2.4.6a1.2 | $x^{8} + 64 x^{7} + 1548 x^{6} + 16960 x^{5} + 74806 x^{4} + 50880 x^{3} + 13932 x^{2} + 1728 x + 98$ | $4$ | $2$ | $6$ | $C_4\times C_2$ | $$[\ ]_{4}^{2}$$ | |
|
\(53\)
| $\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{53}$ | $x + 51$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 53.1.2.1a1.1 | $x^{2} + 53$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 53.1.4.3a1.1 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 53.1.4.3a1.1 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 53.1.4.3a1.1 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 53.1.4.3a1.1 | $x^{4} + 53$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |