Normalized defining polynomial
\( x^{20} - 4 x^{19} + 40 x^{17} - 96 x^{16} - 44 x^{15} + 576 x^{14} - 616 x^{13} - 524 x^{12} + \cdots - 41824 \)
Invariants
| Degree: | $20$ |
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| Signature: | $[4, 8]$ |
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| Discriminant: |
\(124759655006667637122698051584\)
\(\medspace = 2^{30}\cdot 47^{12}\)
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| Root discriminant: | \(28.50\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(2\), \(47\)
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| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_2^2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}$, $\frac{1}{2}a^{5}$, $\frac{1}{2}a^{6}$, $\frac{1}{2}a^{7}$, $\frac{1}{4}a^{8}$, $\frac{1}{4}a^{9}$, $\frac{1}{4}a^{10}$, $\frac{1}{4}a^{11}$, $\frac{1}{8}a^{12}$, $\frac{1}{8}a^{13}$, $\frac{1}{8}a^{14}$, $\frac{1}{8}a^{15}$, $\frac{1}{16}a^{16}$, $\frac{1}{16}a^{17}$, $\frac{1}{16}a^{18}$, $\frac{1}{13\cdots 00}a^{19}-\frac{13\cdots 13}{13\cdots 00}a^{18}+\frac{24\cdots 21}{68\cdots 00}a^{17}+\frac{35\cdots 37}{13\cdots 00}a^{16}+\frac{30\cdots 23}{68\cdots 00}a^{15}-\frac{36\cdots 29}{68\cdots 00}a^{14}+\frac{15\cdots 37}{34\cdots 00}a^{13}-\frac{18\cdots 87}{34\cdots 00}a^{12}+\frac{24\cdots 77}{34\cdots 00}a^{11}+\frac{27\cdots 61}{34\cdots 50}a^{10}-\frac{19\cdots 77}{68\cdots 90}a^{9}-\frac{33\cdots 89}{34\cdots 00}a^{8}+\frac{39\cdots 43}{68\cdots 90}a^{7}+\frac{52\cdots 83}{34\cdots 50}a^{6}+\frac{49\cdots 93}{85\cdots 75}a^{5}+\frac{58\cdots 71}{17\cdots 50}a^{4}-\frac{82\cdots 81}{17\cdots 75}a^{3}-\frac{40\cdots 88}{85\cdots 75}a^{2}+\frac{21\cdots 26}{85\cdots 75}a+\frac{19\cdots 29}{85\cdots 75}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ (assuming GRH) |
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Unit group
| Rank: | $11$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: |
$\frac{44\cdots 57}{68\cdots 00}a^{19}+\frac{23\cdots 41}{68\cdots 00}a^{18}-\frac{32\cdots 69}{68\cdots 00}a^{17}-\frac{27\cdots 93}{13\cdots 00}a^{16}+\frac{63\cdots 03}{68\cdots 00}a^{15}-\frac{69\cdots 69}{68\cdots 00}a^{14}-\frac{43\cdots 09}{17\cdots 50}a^{13}+\frac{70\cdots 92}{85\cdots 75}a^{12}-\frac{70\cdots 57}{85\cdots 75}a^{11}-\frac{48\cdots 29}{34\cdots 50}a^{10}+\frac{11\cdots 14}{34\cdots 95}a^{9}-\frac{27\cdots 51}{85\cdots 75}a^{8}-\frac{54\cdots 56}{34\cdots 95}a^{7}+\frac{11\cdots 19}{17\cdots 75}a^{6}+\frac{55\cdots 73}{85\cdots 75}a^{5}+\frac{21\cdots 81}{17\cdots 50}a^{4}+\frac{28\cdots 09}{17\cdots 75}a^{3}+\frac{45\cdots 82}{85\cdots 75}a^{2}+\frac{53\cdots 36}{85\cdots 75}a+\frac{78\cdots 44}{85\cdots 75}$, $\frac{29\cdots 11}{37\cdots 52}a^{19}+\frac{31\cdots 49}{74\cdots 04}a^{18}-\frac{10\cdots 73}{18\cdots 26}a^{17}-\frac{37\cdots 67}{14\cdots 08}a^{16}+\frac{10\cdots 40}{93\cdots 63}a^{15}-\frac{40\cdots 29}{37\cdots 52}a^{14}-\frac{62\cdots 83}{18\cdots 26}a^{13}+\frac{17\cdots 13}{18\cdots 26}a^{12}-\frac{15\cdots 71}{18\cdots 26}a^{11}-\frac{17\cdots 01}{93\cdots 63}a^{10}+\frac{36\cdots 71}{93\cdots 63}a^{9}-\frac{10\cdots 13}{37\cdots 52}a^{8}-\frac{28\cdots 08}{93\cdots 63}a^{7}+\frac{69\cdots 80}{93\cdots 63}a^{6}+\frac{99\cdots 38}{93\cdots 63}a^{5}+\frac{11\cdots 59}{93\cdots 63}a^{4}+\frac{16\cdots 00}{93\cdots 63}a^{3}+\frac{59\cdots 96}{93\cdots 63}a^{2}+\frac{67\cdots 68}{93\cdots 63}a+\frac{13\cdots 32}{93\cdots 63}$, $\frac{97\cdots 07}{68\cdots 00}a^{19}+\frac{26\cdots 33}{34\cdots 00}a^{18}-\frac{69\cdots 69}{68\cdots 00}a^{17}-\frac{76\cdots 71}{17\cdots 50}a^{16}+\frac{13\cdots 03}{68\cdots 00}a^{15}-\frac{14\cdots 19}{68\cdots 00}a^{14}-\frac{50\cdots 92}{85\cdots 75}a^{13}+\frac{30\cdots 09}{17\cdots 50}a^{12}-\frac{27\cdots 89}{17\cdots 50}a^{11}-\frac{11\cdots 79}{34\cdots 50}a^{10}+\frac{25\cdots 29}{34\cdots 95}a^{9}-\frac{20\cdots 29}{34\cdots 00}a^{8}-\frac{15\cdots 76}{34\cdots 95}a^{7}+\frac{24\cdots 19}{17\cdots 75}a^{6}+\frac{14\cdots 23}{85\cdots 75}a^{5}+\frac{41\cdots 31}{17\cdots 50}a^{4}+\frac{59\cdots 09}{17\cdots 75}a^{3}+\frac{99\cdots 82}{85\cdots 75}a^{2}+\frac{11\cdots 36}{85\cdots 75}a+\frac{20\cdots 44}{85\cdots 75}$, $\frac{91\cdots 53}{13\cdots 00}a^{19}+\frac{26\cdots 57}{68\cdots 00}a^{18}-\frac{45\cdots 63}{68\cdots 00}a^{17}-\frac{12\cdots 93}{68\cdots 00}a^{16}+\frac{71\cdots 81}{68\cdots 00}a^{15}-\frac{10\cdots 13}{68\cdots 00}a^{14}-\frac{18\cdots 59}{85\cdots 75}a^{13}+\frac{32\cdots 11}{34\cdots 00}a^{12}-\frac{40\cdots 81}{34\cdots 00}a^{11}-\frac{79\cdots 41}{68\cdots 00}a^{10}+\frac{28\cdots 31}{68\cdots 90}a^{9}-\frac{14\cdots 83}{34\cdots 00}a^{8}-\frac{42\cdots 27}{34\cdots 95}a^{7}+\frac{26\cdots 01}{34\cdots 50}a^{6}+\frac{50\cdots 46}{85\cdots 75}a^{5}+\frac{68\cdots 87}{17\cdots 50}a^{4}+\frac{29\cdots 68}{17\cdots 75}a^{3}+\frac{40\cdots 14}{85\cdots 75}a^{2}+\frac{29\cdots 72}{85\cdots 75}a-\frac{57\cdots 12}{85\cdots 75}$, $\frac{44\cdots 41}{85\cdots 75}a^{19}+\frac{40\cdots 03}{13\cdots 00}a^{18}-\frac{30\cdots 51}{68\cdots 00}a^{17}-\frac{10\cdots 11}{68\cdots 00}a^{16}+\frac{13\cdots 53}{17\cdots 50}a^{15}-\frac{79\cdots 97}{85\cdots 75}a^{14}-\frac{34\cdots 61}{17\cdots 50}a^{13}+\frac{59\cdots 18}{85\cdots 75}a^{12}-\frac{23\cdots 87}{34\cdots 00}a^{11}-\frac{76\cdots 57}{68\cdots 00}a^{10}+\frac{10\cdots 26}{34\cdots 95}a^{9}-\frac{41\cdots 83}{17\cdots 50}a^{8}-\frac{55\cdots 74}{34\cdots 95}a^{7}+\frac{94\cdots 01}{17\cdots 75}a^{6}+\frac{51\cdots 42}{85\cdots 75}a^{5}+\frac{45\cdots 87}{85\cdots 75}a^{4}+\frac{14\cdots 61}{17\cdots 75}a^{3}+\frac{30\cdots 28}{85\cdots 75}a^{2}+\frac{32\cdots 94}{85\cdots 75}a-\frac{57\cdots 49}{85\cdots 75}$, $\frac{30\cdots 61}{68\cdots 00}a^{19}+\frac{32\cdots 11}{13\cdots 00}a^{18}-\frac{33\cdots 99}{13\cdots 00}a^{17}-\frac{29\cdots 33}{17\cdots 50}a^{16}+\frac{43\cdots 19}{68\cdots 00}a^{15}-\frac{22\cdots 37}{68\cdots 00}a^{14}-\frac{22\cdots 91}{85\cdots 75}a^{13}+\frac{17\cdots 89}{34\cdots 00}a^{12}-\frac{11\cdots 19}{34\cdots 00}a^{11}-\frac{51\cdots 17}{34\cdots 50}a^{10}+\frac{24\cdots 73}{13\cdots 80}a^{9}+\frac{11\cdots 33}{34\cdots 00}a^{8}-\frac{99\cdots 58}{34\cdots 95}a^{7}+\frac{45\cdots 87}{17\cdots 75}a^{6}+\frac{69\cdots 54}{85\cdots 75}a^{5}+\frac{83\cdots 13}{17\cdots 50}a^{4}+\frac{14\cdots 07}{17\cdots 75}a^{3}+\frac{15\cdots 11}{85\cdots 75}a^{2}+\frac{40\cdots 03}{85\cdots 75}a+\frac{11\cdots 62}{85\cdots 75}$, $\frac{39\cdots 21}{68\cdots 00}a^{19}+\frac{40\cdots 71}{13\cdots 00}a^{18}-\frac{48\cdots 89}{13\cdots 00}a^{17}-\frac{12\cdots 77}{68\cdots 00}a^{16}+\frac{53\cdots 59}{68\cdots 00}a^{15}-\frac{23\cdots 41}{34\cdots 00}a^{14}-\frac{42\cdots 77}{17\cdots 50}a^{13}+\frac{57\cdots 76}{85\cdots 75}a^{12}-\frac{17\cdots 59}{34\cdots 00}a^{11}-\frac{97\cdots 99}{68\cdots 00}a^{10}+\frac{36\cdots 43}{13\cdots 80}a^{9}-\frac{60\cdots 37}{34\cdots 00}a^{8}-\frac{14\cdots 11}{68\cdots 90}a^{7}+\frac{81\cdots 82}{17\cdots 75}a^{6}+\frac{70\cdots 94}{85\cdots 75}a^{5}+\frac{19\cdots 93}{17\cdots 50}a^{4}+\frac{25\cdots 77}{17\cdots 75}a^{3}+\frac{42\cdots 96}{85\cdots 75}a^{2}+\frac{52\cdots 33}{85\cdots 75}a+\frac{18\cdots 32}{85\cdots 75}$, $\frac{46\cdots 21}{68\cdots 00}a^{19}-\frac{42\cdots 71}{13\cdots 00}a^{18}+\frac{10\cdots 07}{68\cdots 00}a^{17}+\frac{22\cdots 19}{85\cdots 75}a^{16}-\frac{53\cdots 59}{68\cdots 00}a^{15}+\frac{17\cdots 33}{17\cdots 50}a^{14}+\frac{26\cdots 33}{68\cdots 00}a^{13}-\frac{21\cdots 79}{34\cdots 00}a^{12}-\frac{13\cdots 91}{34\cdots 00}a^{11}+\frac{16\cdots 99}{68\cdots 00}a^{10}-\frac{14\cdots 49}{68\cdots 90}a^{9}-\frac{23\cdots 63}{34\cdots 00}a^{8}+\frac{35\cdots 01}{68\cdots 90}a^{7}-\frac{16\cdots 39}{34\cdots 50}a^{6}-\frac{12\cdots 69}{85\cdots 75}a^{5}-\frac{30\cdots 93}{17\cdots 50}a^{4}-\frac{43\cdots 02}{17\cdots 75}a^{3}-\frac{58\cdots 46}{85\cdots 75}a^{2}-\frac{96\cdots 58}{85\cdots 75}a-\frac{52\cdots 82}{85\cdots 75}$, $\frac{19\cdots 27}{13\cdots 00}a^{19}-\frac{15\cdots 01}{13\cdots 00}a^{18}+\frac{46\cdots 09}{13\cdots 00}a^{17}-\frac{20\cdots 97}{17\cdots 50}a^{16}-\frac{18\cdots 79}{68\cdots 00}a^{15}+\frac{60\cdots 17}{68\cdots 00}a^{14}-\frac{48\cdots 77}{68\cdots 00}a^{13}-\frac{42\cdots 37}{17\cdots 50}a^{12}+\frac{14\cdots 77}{17\cdots 50}a^{11}-\frac{10\cdots 14}{17\cdots 75}a^{10}-\frac{44\cdots 22}{34\cdots 95}a^{9}+\frac{53\cdots 61}{17\cdots 50}a^{8}-\frac{13\cdots 19}{68\cdots 90}a^{7}-\frac{47\cdots 42}{17\cdots 75}a^{6}+\frac{24\cdots 47}{17\cdots 50}a^{5}+\frac{62\cdots 71}{85\cdots 75}a^{4}-\frac{23\cdots 12}{17\cdots 75}a^{3}-\frac{79\cdots 76}{85\cdots 75}a^{2}+\frac{32\cdots 27}{85\cdots 75}a-\frac{59\cdots 67}{85\cdots 75}$, $\frac{33\cdots 37}{13\cdots 00}a^{19}-\frac{12\cdots 19}{13\cdots 00}a^{18}-\frac{45\cdots 29}{13\cdots 00}a^{17}+\frac{10\cdots 89}{34\cdots 00}a^{16}-\frac{48\cdots 01}{68\cdots 00}a^{15}-\frac{10\cdots 77}{68\cdots 00}a^{14}+\frac{60\cdots 37}{68\cdots 00}a^{13}-\frac{49\cdots 87}{68\cdots 00}a^{12}-\frac{63\cdots 87}{17\cdots 50}a^{11}+\frac{95\cdots 84}{17\cdots 75}a^{10}-\frac{49\cdots 07}{13\cdots 80}a^{9}-\frac{57\cdots 57}{34\cdots 00}a^{8}+\frac{12\cdots 39}{68\cdots 90}a^{7}+\frac{35\cdots 79}{34\cdots 50}a^{6}-\frac{94\cdots 41}{85\cdots 75}a^{5}+\frac{12\cdots 73}{17\cdots 50}a^{4}+\frac{35\cdots 22}{17\cdots 75}a^{3}+\frac{20\cdots 56}{85\cdots 75}a^{2}-\frac{82\cdots 87}{85\cdots 75}a-\frac{13\cdots 98}{85\cdots 75}$, $\frac{52\cdots 79}{13\cdots 00}a^{19}-\frac{16\cdots 47}{85\cdots 75}a^{18}+\frac{35\cdots 43}{13\cdots 00}a^{17}+\frac{74\cdots 49}{68\cdots 00}a^{16}-\frac{16\cdots 79}{34\cdots 00}a^{15}+\frac{46\cdots 98}{85\cdots 75}a^{14}+\frac{10\cdots 37}{85\cdots 75}a^{13}-\frac{27\cdots 71}{68\cdots 00}a^{12}+\frac{15\cdots 83}{34\cdots 00}a^{11}+\frac{43\cdots 63}{68\cdots 00}a^{10}-\frac{54\cdots 19}{34\cdots 95}a^{9}+\frac{51\cdots 19}{34\cdots 00}a^{8}+\frac{83\cdots 41}{34\cdots 95}a^{7}-\frac{49\cdots 59}{17\cdots 75}a^{6}-\frac{86\cdots 31}{17\cdots 50}a^{5}-\frac{53\cdots 08}{85\cdots 75}a^{4}-\frac{16\cdots 74}{17\cdots 75}a^{3}-\frac{22\cdots 52}{85\cdots 75}a^{2}-\frac{22\cdots 71}{85\cdots 75}a-\frac{27\cdots 34}{85\cdots 75}$
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| Regulator: | \( 3180413.876043657 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\cdot 3180413.876043657 \cdot 1}{2\cdot\sqrt{124759655006667637122698051584}}\cr\approx \mathstrut & 0.174974794000039 \end{aligned}\] (assuming GRH)
Galois group
$C_2^8:D_5$ (as 20T240):
| A solvable group of order 2560 |
| The 52 conjugacy class representatives for $C_2^8:D_5$ |
| Character table for $C_2^8:D_5$ |
Intermediate fields
| 5.1.2209.1, 10.2.4996793344.2, 10.2.11037916496896.10, 10.2.11037916496896.11 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 siblings: | data not computed |
| Degree 32 siblings: | 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 | ${\href{/padicField/3.5.0.1}{5} }^{4}$ | ${\href{/padicField/5.4.0.1}{4} }^{4}{,}\,{\href{/padicField/5.2.0.1}{2} }^{2}$ | ${\href{/padicField/7.5.0.1}{5} }^{4}$ | ${\href{/padicField/11.4.0.1}{4} }^{4}{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }^{4}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }^{6}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }^{6}$ | ${\href{/padicField/29.4.0.1}{4} }^{4}{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{4}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.5.0.1}{5} }^{4}$ | ${\href{/padicField/41.4.0.1}{4} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/53.5.0.1}{5} }^{4}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.4.30a25.1 | $x^{20} + 4 x^{17} + 4 x^{15} + 6 x^{14} + 12 x^{12} + 6 x^{11} + 6 x^{10} + 12 x^{9} + 5 x^{8} + 12 x^{7} + 8 x^{6} + 6 x^{5} + 6 x^{4} + 4 x^{3} + 4 x^{2} + 2 x + 3$ | $4$ | $5$ | $30$ | 20T17 | not computed |
|
\(47\)
| 47.1.2.1a1.2 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 47.1.2.1a1.2 | $x^{2} + 235$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 47.2.2.2a1.2 | $x^{4} + 90 x^{3} + 2035 x^{2} + 450 x + 72$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 47.2.2.2a1.2 | $x^{4} + 90 x^{3} + 2035 x^{2} + 450 x + 72$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
| 47.2.4.6a1.2 | $x^{8} + 180 x^{7} + 12170 x^{6} + 367200 x^{5} + 4222275 x^{4} + 1836000 x^{3} + 304250 x^{2} + 22500 x + 672$ | $4$ | $2$ | $6$ | $D_4$ | $$[\ ]_{4}^{2}$$ |