Normalized defining polynomial
\( x^{20} + 884 x^{18} - 120 x^{17} + 320632 x^{16} - 47040 x^{15} + 62498912 x^{14} + \cdots + 23\!\cdots\!10 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $[0, 10]$ |
| |
| Discriminant: |
\(5657954428456562353084024324409254309146809860096000000000000000\)
\(\medspace = 2^{55}\cdot 5^{15}\cdot 197^{16}\)
|
| |
| Root discriminant: | \(1540.40\) |
| |
| Galois root discriminant: | $2^{11/4}5^{3/4}197^{4/5}\approx 1540.3980928551418$ | ||
| Ramified primes: |
\(2\), \(5\), \(197\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{10}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | 4.0.256000.4 | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10840}a^{12}-\frac{1}{2}a^{11}-\frac{1659}{5420}a^{10}+\frac{16}{271}a^{9}+\frac{4509}{10840}a^{8}+\frac{118}{271}a^{7}+\frac{1609}{5420}a^{6}-\frac{41}{542}a^{5}+\frac{4681}{10840}a^{4}+\frac{93}{271}a^{3}+\frac{185}{542}a^{2}-\frac{48}{271}a+\frac{225}{1084}$, $\frac{1}{10840}a^{13}-\frac{1659}{5420}a^{11}+\frac{16}{271}a^{10}+\frac{4509}{10840}a^{9}-\frac{35}{542}a^{8}+\frac{1609}{5420}a^{7}-\frac{41}{542}a^{6}+\frac{4681}{10840}a^{5}-\frac{85}{542}a^{4}+\frac{185}{542}a^{3}-\frac{48}{271}a^{2}+\frac{225}{1084}a$, $\frac{1}{10840}a^{14}+\frac{16}{271}a^{11}-\frac{403}{2168}a^{10}-\frac{91}{542}a^{9}+\frac{122}{271}a^{8}-\frac{183}{542}a^{7}+\frac{921}{2168}a^{6}-\frac{81}{542}a^{5}+\frac{769}{5420}a^{4}+\frac{128}{271}a^{3}-\frac{287}{1084}a^{2}+\frac{84}{271}a-\frac{163}{542}$, $\frac{1}{10840}a^{15}-\frac{403}{2168}a^{11}-\frac{147}{542}a^{10}-\frac{91}{271}a^{9}+\frac{243}{542}a^{8}-\frac{535}{2168}a^{7}-\frac{77}{542}a^{6}-\frac{2411}{5420}a^{5}+\frac{28}{271}a^{4}+\frac{113}{1084}a^{3}-\frac{38}{271}a^{2}+\frac{31}{542}a+\frac{43}{271}$, $\frac{1}{21680}a^{16}-\frac{1}{21680}a^{14}+\frac{23}{271}a^{11}-\frac{1991}{4336}a^{10}+\frac{79}{271}a^{9}-\frac{73}{271}a^{8}-\frac{115}{542}a^{7}+\frac{3363}{21680}a^{6}+\frac{112}{271}a^{5}-\frac{9833}{21680}a^{4}+\frac{239}{542}a^{3}-\frac{977}{2168}a^{2}+\frac{257}{542}a-\frac{495}{2168}$, $\frac{1}{21680}a^{17}-\frac{1}{21680}a^{15}-\frac{1991}{4336}a^{11}-\frac{29}{271}a^{10}+\frac{112}{271}a^{9}+\frac{57}{542}a^{8}-\frac{9437}{21680}a^{7}+\frac{81}{271}a^{6}+\frac{3047}{21680}a^{5}+\frac{87}{542}a^{4}-\frac{369}{2168}a^{3}+\frac{245}{542}a^{2}-\frac{599}{2168}a+\frac{11}{271}$, $\frac{1}{12\cdots 20}a^{18}+\frac{20\cdots 13}{12\cdots 20}a^{17}-\frac{32\cdots 97}{12\cdots 20}a^{16}-\frac{34\cdots 77}{12\cdots 20}a^{15}+\frac{21\cdots 33}{60\cdots 60}a^{14}-\frac{25\cdots 41}{12\cdots 52}a^{13}-\frac{27\cdots 81}{12\cdots 20}a^{12}+\frac{11\cdots 29}{24\cdots 04}a^{11}+\frac{42\cdots 59}{60\cdots 60}a^{10}+\frac{96\cdots 39}{12\cdots 52}a^{9}-\frac{11\cdots 71}{12\cdots 20}a^{8}+\frac{25\cdots 59}{12\cdots 20}a^{7}+\frac{25\cdots 21}{12\cdots 20}a^{6}-\frac{34\cdots 11}{12\cdots 20}a^{5}-\frac{69\cdots 01}{15\cdots 90}a^{4}+\frac{49\cdots 47}{12\cdots 52}a^{3}-\frac{29\cdots 37}{12\cdots 52}a^{2}-\frac{27\cdots 89}{12\cdots 52}a+\frac{49\cdots 09}{60\cdots 76}$, $\frac{1}{67\cdots 80}a^{19}-\frac{11\cdots 71}{33\cdots 40}a^{18}+\frac{11\cdots 99}{67\cdots 80}a^{17}-\frac{11\cdots 69}{13\cdots 96}a^{16}+\frac{16\cdots 37}{83\cdots 10}a^{15}-\frac{12\cdots 39}{67\cdots 80}a^{14}-\frac{58\cdots 17}{67\cdots 80}a^{13}+\frac{31\cdots 87}{33\cdots 40}a^{12}-\frac{58\cdots 81}{16\cdots 20}a^{11}-\frac{16\cdots 57}{67\cdots 80}a^{10}-\frac{10\cdots 99}{13\cdots 96}a^{9}-\frac{44\cdots 23}{67\cdots 48}a^{8}-\frac{13\cdots 69}{67\cdots 80}a^{7}+\frac{38\cdots 47}{67\cdots 80}a^{6}-\frac{14\cdots 74}{41\cdots 05}a^{5}-\frac{82\cdots 73}{67\cdots 80}a^{4}+\frac{17\cdots 61}{67\cdots 48}a^{3}+\frac{16\cdots 07}{67\cdots 48}a^{2}-\frac{58\cdots 77}{33\cdots 24}a-\frac{33\cdots 55}{67\cdots 48}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | not computed |
| |
| Narrow class group: | not computed |
|
Unit group
| Rank: | $9$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
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| Fundamental units: | not computed |
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| Regulator: | not computed |
|
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 = \mathstrut &\frac{2^{0}\cdot(2\pi)^{10}\cdot R \cdot h}{2\cdot\sqrt{5657954428456562353084024324409254309146809860096000000000000000}}\cr\mathstrut & \text{
Galois group
$C_4\times F_5$ (as 20T20):
| A solvable group of order 80 |
| The 20 conjugacy class representatives for $C_4\times F_5$ |
| Character table for $C_4\times F_5$ |
Intermediate fields
| \(\Q(\sqrt{10}) \), 4.0.256000.4, 5.1.12049107848000.1, 10.2.5807239997309407644160000000.1 |
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 40 siblings: | data not computed |
| Minimal sibling: | 20.4.45263635427652498824672194595274034473174478880768000000000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}{,}\,{\href{/padicField/3.2.0.1}{2} }^{2}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.4.0.1}{4} }^{5}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | $20$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | $20$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.11a1.15 | $x^{4} + 4 x^{2} + 10$ | $4$ | $1$ | $11$ | $C_4$ | $$[3, 4]$$ |
| 2.4.4.44a1.1155 | $x^{16} + 4 x^{13} + 12 x^{12} + 6 x^{10} + 36 x^{9} + 34 x^{8} + 4 x^{7} + 36 x^{6} + 68 x^{5} + 37 x^{4} + 12 x^{3} + 34 x^{2} + 36 x + 15$ | $4$ | $4$ | $44$ | $C_4^2$ | $$[3, 4]^{4}$$ | |
|
\(5\)
| 5.1.4.3a1.2 | $x^{4} + 10$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |
| 5.4.4.12a1.4 | $x^{16} + 16 x^{14} + 16 x^{13} + 104 x^{12} + 192 x^{11} + 448 x^{10} + 864 x^{9} + 1432 x^{8} + 2048 x^{7} + 2624 x^{6} + 2752 x^{5} + 2208 x^{4} + 1280 x^{3} + 512 x^{2} + 128 x + 21$ | $4$ | $4$ | $12$ | $C_4^2$ | $$[\ ]_{4}^{4}$$ | |
|
\(197\)
| 197.2.5.8a1.1 | $x^{10} + 960 x^{9} + 368650 x^{8} + 70786560 x^{7} + 6796984360 x^{6} + 261202401792 x^{5} + 13593968720 x^{4} + 283146240 x^{3} + 2949200 x^{2} + 15360 x + 229$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ |
| 197.2.5.8a1.1 | $x^{10} + 960 x^{9} + 368650 x^{8} + 70786560 x^{7} + 6796984360 x^{6} + 261202401792 x^{5} + 13593968720 x^{4} + 283146240 x^{3} + 2949200 x^{2} + 15360 x + 229$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ |