Normalized defining polynomial
\( x^{20} + 35 x^{18} - 35 x^{17} + 560 x^{16} - 644 x^{15} + 3500 x^{14} - 7980 x^{13} + 4900 x^{12} + \cdots + 9992605 \)
Invariants
| Degree: | $20$ |
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| Signature: | $(0, 10)$ |
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| Discriminant: |
\(389961449903647873889687061309814453125\)
\(\medspace = 3^{15}\cdot 5^{22}\cdot 7^{19}\)
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| Root discriminant: | \(85.03\) |
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| Galois root discriminant: | not computed | ||
| Ramified primes: |
\(3\), \(5\), \(7\)
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| Discriminant root field: | \(\Q(\sqrt{21}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $\frac{1}{2}a^{11}-\frac{1}{2}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{12}-\frac{1}{2}a^{9}-\frac{1}{2}a^{8}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}-\frac{1}{2}a$, $\frac{1}{2}a^{13}-\frac{1}{2}a^{10}-\frac{1}{2}a^{9}-\frac{1}{2}a^{7}-\frac{1}{2}a^{5}-\frac{1}{2}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}a^{2}$, $\frac{1}{2}a^{14}-\frac{1}{2}a^{10}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{1}{2}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{2}a^{15}-\frac{1}{2}$, $\frac{1}{2}a^{16}-\frac{1}{2}a$, $\frac{1}{2}a^{17}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{18}-\frac{1}{4}a^{17}-\frac{1}{4}a^{16}-\frac{1}{4}a^{15}-\frac{1}{4}a^{13}-\frac{1}{4}a^{12}+\frac{1}{4}a^{10}-\frac{1}{2}a^{9}-\frac{1}{4}a^{8}-\frac{1}{4}a^{7}+\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{2}a^{4}+\frac{1}{4}a^{3}-\frac{1}{4}a^{2}+\frac{1}{4}$, $\frac{1}{93\cdots 36}a^{19}-\frac{29\cdots 55}{93\cdots 36}a^{18}+\frac{42\cdots 75}{93\cdots 36}a^{17}-\frac{11\cdots 21}{93\cdots 36}a^{16}-\frac{31\cdots 93}{23\cdots 59}a^{15}-\frac{52\cdots 71}{93\cdots 36}a^{14}+\frac{12\cdots 67}{93\cdots 36}a^{13}+\frac{17\cdots 82}{23\cdots 59}a^{12}-\frac{18\cdots 21}{93\cdots 36}a^{11}+\frac{48\cdots 22}{23\cdots 59}a^{10}+\frac{56\cdots 27}{93\cdots 36}a^{9}+\frac{45\cdots 33}{93\cdots 36}a^{8}-\frac{37\cdots 47}{93\cdots 36}a^{7}+\frac{23\cdots 93}{93\cdots 36}a^{6}+\frac{35\cdots 38}{23\cdots 59}a^{5}+\frac{13\cdots 03}{93\cdots 36}a^{4}-\frac{23\cdots 97}{93\cdots 36}a^{3}+\frac{10\cdots 01}{23\cdots 59}a^{2}+\frac{28\cdots 89}{93\cdots 36}a+\frac{34\cdots 36}{23\cdots 59}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | $C_{2}\times C_{2}\times C_{540}$, which has order $2160$ (assuming GRH) |
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| Narrow class group: | $C_{2}\times C_{2}\times C_{540}$, which has order $2160$ (assuming GRH) |
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| Relative class number: | data not computed (assuming GRH) |
Unit group
| Rank: | $9$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| Fundamental units: |
$\frac{27\cdots 61}{21\cdots 92}a^{19}-\frac{21\cdots 92}{54\cdots 23}a^{18}-\frac{22\cdots 02}{54\cdots 23}a^{17}-\frac{47\cdots 31}{54\cdots 23}a^{16}-\frac{10\cdots 55}{21\cdots 92}a^{15}-\frac{28\cdots 43}{21\cdots 92}a^{14}-\frac{56\cdots 11}{10\cdots 46}a^{13}-\frac{84\cdots 67}{21\cdots 92}a^{12}+\frac{78\cdots 51}{21\cdots 92}a^{11}+\frac{10\cdots 53}{21\cdots 92}a^{10}+\frac{97\cdots 87}{21\cdots 92}a^{9}+\frac{31\cdots 75}{10\cdots 46}a^{8}+\frac{66\cdots 73}{54\cdots 23}a^{7}-\frac{10\cdots 25}{54\cdots 23}a^{6}-\frac{11\cdots 29}{21\cdots 92}a^{5}-\frac{38\cdots 23}{21\cdots 92}a^{4}-\frac{17\cdots 67}{54\cdots 23}a^{3}-\frac{10\cdots 93}{21\cdots 92}a^{2}-\frac{10\cdots 95}{21\cdots 92}a-\frac{14\cdots 77}{21\cdots 92}$, $\frac{28\cdots 93}{21\cdots 92}a^{19}-\frac{13\cdots 02}{54\cdots 23}a^{18}-\frac{39\cdots 03}{10\cdots 46}a^{17}-\frac{24\cdots 32}{54\cdots 23}a^{16}-\frac{77\cdots 93}{21\cdots 92}a^{15}-\frac{19\cdots 13}{21\cdots 92}a^{14}+\frac{59\cdots 61}{54\cdots 23}a^{13}-\frac{60\cdots 93}{21\cdots 92}a^{12}+\frac{62\cdots 27}{21\cdots 92}a^{11}+\frac{11\cdots 73}{21\cdots 92}a^{10}+\frac{53\cdots 35}{21\cdots 92}a^{9}+\frac{77\cdots 48}{54\cdots 23}a^{8}+\frac{17\cdots 13}{54\cdots 23}a^{7}-\frac{87\cdots 24}{54\cdots 23}a^{6}-\frac{89\cdots 15}{21\cdots 92}a^{5}-\frac{21\cdots 29}{21\cdots 92}a^{4}-\frac{81\cdots 99}{54\cdots 23}a^{3}-\frac{47\cdots 17}{21\cdots 92}a^{2}-\frac{42\cdots 55}{21\cdots 92}a-\frac{21\cdots 33}{21\cdots 92}$, $\frac{14\cdots 79}{21\cdots 92}a^{19}-\frac{72\cdots 98}{54\cdots 23}a^{18}+\frac{12\cdots 06}{54\cdots 23}a^{17}-\frac{35\cdots 87}{54\cdots 23}a^{16}+\frac{92\cdots 87}{21\cdots 92}a^{15}-\frac{22\cdots 31}{21\cdots 92}a^{14}+\frac{33\cdots 21}{10\cdots 46}a^{13}-\frac{16\cdots 15}{21\cdots 92}a^{12}+\frac{20\cdots 29}{21\cdots 92}a^{11}-\frac{10\cdots 91}{21\cdots 92}a^{10}+\frac{43\cdots 91}{21\cdots 92}a^{9}-\frac{72\cdots 20}{54\cdots 23}a^{8}+\frac{18\cdots 81}{10\cdots 46}a^{7}+\frac{58\cdots 58}{54\cdots 23}a^{6}+\frac{22\cdots 37}{21\cdots 92}a^{5}+\frac{40\cdots 49}{21\cdots 92}a^{4}+\frac{30\cdots 23}{10\cdots 46}a^{3}+\frac{85\cdots 61}{21\cdots 92}a^{2}+\frac{86\cdots 63}{21\cdots 92}a+\frac{31\cdots 67}{21\cdots 92}$, $\frac{30\cdots 97}{10\cdots 46}a^{19}+\frac{12\cdots 35}{21\cdots 92}a^{18}+\frac{22\cdots 17}{21\cdots 92}a^{17}-\frac{19\cdots 83}{21\cdots 92}a^{16}+\frac{36\cdots 97}{21\cdots 92}a^{15}-\frac{10\cdots 46}{54\cdots 23}a^{14}+\frac{23\cdots 69}{21\cdots 92}a^{13}-\frac{53\cdots 79}{21\cdots 92}a^{12}+\frac{55\cdots 22}{54\cdots 23}a^{11}-\frac{50\cdots 81}{21\cdots 92}a^{10}-\frac{26\cdots 07}{10\cdots 46}a^{9}-\frac{22\cdots 59}{21\cdots 92}a^{8}+\frac{57\cdots 45}{21\cdots 92}a^{7}+\frac{36\cdots 03}{21\cdots 92}a^{6}+\frac{19\cdots 21}{21\cdots 92}a^{5}+\frac{12\cdots 36}{54\cdots 23}a^{4}+\frac{80\cdots 59}{21\cdots 92}a^{3}+\frac{12\cdots 73}{21\cdots 92}a^{2}+\frac{28\cdots 16}{54\cdots 23}a+\frac{12\cdots 87}{21\cdots 92}$, $\frac{50\cdots 71}{21\cdots 92}a^{19}+\frac{29\cdots 47}{21\cdots 92}a^{18}-\frac{17\cdots 33}{21\cdots 92}a^{17}+\frac{27\cdots 27}{21\cdots 92}a^{16}-\frac{68\cdots 64}{54\cdots 23}a^{15}+\frac{43\cdots 99}{21\cdots 92}a^{14}-\frac{16\cdots 43}{21\cdots 92}a^{13}+\frac{20\cdots 75}{10\cdots 46}a^{12}-\frac{20\cdots 89}{21\cdots 92}a^{11}+\frac{84\cdots 28}{54\cdots 23}a^{10}+\frac{19\cdots 67}{21\cdots 92}a^{9}+\frac{11\cdots 03}{21\cdots 92}a^{8}-\frac{72\cdots 35}{21\cdots 92}a^{7}-\frac{26\cdots 79}{21\cdots 92}a^{6}-\frac{59\cdots 35}{10\cdots 46}a^{5}-\frac{23\cdots 99}{21\cdots 92}a^{4}-\frac{35\cdots 75}{21\cdots 92}a^{3}-\frac{24\cdots 35}{10\cdots 46}a^{2}-\frac{46\cdots 83}{21\cdots 92}a-\frac{22\cdots 21}{10\cdots 46}$, $\frac{20\cdots 15}{54\cdots 23}a^{19}+\frac{43\cdots 69}{21\cdots 92}a^{18}-\frac{28\cdots 15}{21\cdots 92}a^{17}+\frac{41\cdots 15}{21\cdots 92}a^{16}-\frac{45\cdots 55}{21\cdots 92}a^{15}+\frac{17\cdots 93}{54\cdots 23}a^{14}-\frac{27\cdots 47}{21\cdots 92}a^{13}+\frac{66\cdots 13}{21\cdots 92}a^{12}-\frac{17\cdots 67}{10\cdots 46}a^{11}+\frac{56\cdots 07}{21\cdots 92}a^{10}+\frac{19\cdots 63}{10\cdots 46}a^{9}+\frac{20\cdots 51}{21\cdots 92}a^{8}-\frac{59\cdots 49}{21\cdots 92}a^{7}-\frac{38\cdots 09}{21\cdots 92}a^{6}-\frac{19\cdots 25}{21\cdots 92}a^{5}-\frac{22\cdots 69}{10\cdots 46}a^{4}-\frac{72\cdots 05}{21\cdots 92}a^{3}-\frac{10\cdots 41}{21\cdots 92}a^{2}-\frac{25\cdots 19}{54\cdots 23}a-\frac{97\cdots 27}{21\cdots 92}$, $\frac{87\cdots 37}{21\cdots 92}a^{19}-\frac{50\cdots 29}{21\cdots 92}a^{18}+\frac{29\cdots 71}{21\cdots 92}a^{17}-\frac{45\cdots 89}{21\cdots 92}a^{16}+\frac{11\cdots 62}{54\cdots 23}a^{15}-\frac{70\cdots 01}{21\cdots 92}a^{14}+\frac{24\cdots 85}{21\cdots 92}a^{13}-\frac{32\cdots 25}{10\cdots 46}a^{12}+\frac{21\cdots 11}{21\cdots 92}a^{11}-\frac{13\cdots 35}{54\cdots 23}a^{10}-\frac{35\cdots 85}{21\cdots 92}a^{9}-\frac{16\cdots 33}{21\cdots 92}a^{8}+\frac{12\cdots 93}{21\cdots 92}a^{7}+\frac{49\cdots 05}{21\cdots 92}a^{6}+\frac{99\cdots 03}{10\cdots 46}a^{5}+\frac{38\cdots 97}{21\cdots 92}a^{4}+\frac{56\cdots 53}{21\cdots 92}a^{3}+\frac{39\cdots 07}{10\cdots 46}a^{2}+\frac{73\cdots 97}{21\cdots 92}a+\frac{91\cdots 11}{10\cdots 46}$, $\frac{18\cdots 43}{21\cdots 92}a^{19}+\frac{32\cdots 61}{21\cdots 92}a^{18}-\frac{67\cdots 67}{21\cdots 92}a^{17}+\frac{17\cdots 27}{21\cdots 92}a^{16}-\frac{60\cdots 75}{10\cdots 46}a^{15}+\frac{28\cdots 67}{21\cdots 92}a^{14}-\frac{89\cdots 09}{21\cdots 92}a^{13}+\frac{11\cdots 27}{10\cdots 46}a^{12}-\frac{26\cdots 73}{21\cdots 92}a^{11}+\frac{69\cdots 45}{10\cdots 46}a^{10}-\frac{36\cdots 23}{21\cdots 92}a^{9}+\frac{44\cdots 55}{21\cdots 92}a^{8}-\frac{48\cdots 85}{21\cdots 92}a^{7}-\frac{40\cdots 03}{21\cdots 92}a^{6}-\frac{84\cdots 33}{54\cdots 23}a^{5}-\frac{60\cdots 61}{21\cdots 92}a^{4}-\frac{89\cdots 15}{21\cdots 92}a^{3}-\frac{60\cdots 63}{10\cdots 46}a^{2}-\frac{12\cdots 43}{21\cdots 92}a-\frac{12\cdots 87}{54\cdots 23}$, $\frac{41\cdots 03}{21\cdots 92}a^{19}+\frac{37\cdots 77}{10\cdots 46}a^{18}+\frac{70\cdots 53}{10\cdots 46}a^{17}-\frac{34\cdots 22}{54\cdots 23}a^{16}+\frac{21\cdots 97}{21\cdots 92}a^{15}-\frac{25\cdots 15}{21\cdots 92}a^{14}+\frac{58\cdots 49}{10\cdots 46}a^{13}-\frac{31\cdots 95}{21\cdots 92}a^{12}-\frac{80\cdots 25}{21\cdots 92}a^{11}-\frac{28\cdots 27}{21\cdots 92}a^{10}-\frac{31\cdots 73}{21\cdots 92}a^{9}-\frac{48\cdots 77}{10\cdots 46}a^{8}+\frac{16\cdots 38}{54\cdots 23}a^{7}+\frac{91\cdots 35}{54\cdots 23}a^{6}+\frac{12\cdots 95}{21\cdots 92}a^{5}+\frac{22\cdots 01}{21\cdots 92}a^{4}+\frac{15\cdots 49}{10\cdots 46}a^{3}+\frac{39\cdots 13}{21\cdots 92}a^{2}+\frac{35\cdots 57}{21\cdots 92}a+\frac{15\cdots 23}{21\cdots 92}$
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| Regulator: | \( 632150489.6787821 \) (assuming GRH) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot 632150489.6787821 \cdot 2160}{2\cdot\sqrt{389961449903647873889687061309814453125}}\cr\approx \mathstrut & 3.31537058171741 \end{aligned}\] (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
| \(\Q(\sqrt{105}) \), \(\Q(\sqrt{-21 +2 \sqrt{105}})\), 5.5.67528125.1, deg 10 |
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: | 20.0.77992289980729574777937412261962890625.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{4}{,}\,{\href{/padicField/2.2.0.1}{2} }{,}\,{\href{/padicField/2.1.0.1}{1} }^{2}$ | R | R | R | ${\href{/padicField/11.4.0.1}{4} }^{5}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{4}{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{5}$ | ${\href{/padicField/23.4.0.1}{4} }^{4}{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | $20$ | $20$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{8}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{4}{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.5.0.1}{5} }^{4}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.4.3.1 | $x^{4} + 3$ | $4$ | $1$ | $3$ | $D_{4}$ | $$[\ ]_{4}^{2}$$ |
| 3.16.12.1 | $x^{16} + 8 x^{15} + 24 x^{14} + 32 x^{13} + 24 x^{12} + 48 x^{11} + 96 x^{10} + 64 x^{9} + 24 x^{8} + 96 x^{7} + 96 x^{6} + 32 x^{4} + 64 x^{3} + 19$ | $4$ | $4$ | $12$ | $C_4:C_4$ | $$[\ ]_{4}^{4}$$ | |
|
\(5\)
| 5.10.11.1 | $x^{10} + 20 x^{2} + 5$ | $10$ | $1$ | $11$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ |
| 5.10.11.1 | $x^{10} + 20 x^{2} + 5$ | $10$ | $1$ | $11$ | $F_5$ | $$[\frac{5}{4}]_{4}$$ | |
|
\(7\)
| 7.20.19.2 | $x^{20} + 21$ | $20$ | $1$ | $19$ | 20T18 | $$[\ ]_{20}^{4}$$ |