Normalized defining polynomial
\( x^{20} + 684 x^{18} - 120 x^{17} + 210872 x^{16} - 27840 x^{15} + 37744672 x^{14} + \cdots + 61\!\cdots\!10 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $[4, 8]$ |
| |
| 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. | |||
| This field has no CM subfields. | |||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{10840}a^{12}-\frac{1}{2}a^{11}-\frac{1719}{5420}a^{10}+\frac{16}{271}a^{9}+\frac{1229}{10840}a^{8}-\frac{78}{271}a^{7}+\frac{609}{5420}a^{6}+\frac{243}{542}a^{5}+\frac{521}{10840}a^{4}-\frac{111}{271}a^{3}-\frac{207}{542}a^{2}+\frac{65}{271}a-\frac{419}{1084}$, $\frac{1}{10840}a^{13}-\frac{1719}{5420}a^{11}+\frac{16}{271}a^{10}+\frac{1229}{10840}a^{9}+\frac{115}{542}a^{8}+\frac{609}{5420}a^{7}+\frac{243}{542}a^{6}+\frac{521}{10840}a^{5}+\frac{49}{542}a^{4}-\frac{207}{542}a^{3}+\frac{65}{271}a^{2}-\frac{419}{1084}a$, $\frac{1}{10840}a^{14}+\frac{16}{271}a^{11}-\frac{603}{2168}a^{10}+\frac{105}{542}a^{9}-\frac{27}{271}a^{8}-\frac{47}{542}a^{7}+\frac{753}{2168}a^{6}+\frac{261}{542}a^{5}-\frac{771}{5420}a^{4}+\frac{15}{271}a^{3}-\frac{459}{1084}a^{2}-\frac{105}{271}a+\frac{57}{542}$, $\frac{1}{32520}a^{15}+\frac{1}{32520}a^{13}+\frac{4387}{32520}a^{11}+\frac{223}{542}a^{10}+\frac{823}{10840}a^{9}+\frac{51}{271}a^{8}+\frac{7223}{32520}a^{7}+\frac{92}{271}a^{6}+\frac{10499}{32520}a^{5}-\frac{111}{542}a^{4}+\frac{1447}{3252}a^{3}-\frac{65}{271}a^{2}+\frac{1315}{3252}a-\frac{56}{271}$, $\frac{1}{65040}a^{16}+\frac{1}{65040}a^{14}-\frac{1}{32520}a^{12}+\frac{247}{542}a^{11}-\frac{9983}{21680}a^{10}-\frac{51}{542}a^{9}+\frac{5731}{32520}a^{8}-\frac{78}{271}a^{7}+\frac{30497}{65040}a^{6}-\frac{17}{271}a^{5}+\frac{4201}{65040}a^{4}-\frac{1}{542}a^{3}-\frac{2759}{6504}a^{2}+\frac{241}{542}a+\frac{537}{2168}$, $\frac{1}{65040}a^{17}-\frac{1}{65040}a^{15}+\frac{1}{32520}a^{13}+\frac{5689}{65040}a^{11}+\frac{86}{271}a^{10}-\frac{14531}{32520}a^{9}-\frac{93}{271}a^{8}+\frac{49}{240}a^{7}-\frac{11}{542}a^{6}-\frac{157}{21680}a^{5}-\frac{37}{271}a^{4}+\frac{935}{6504}a^{3}-\frac{215}{542}a^{2}-\frac{2693}{6504}a-\frac{89}{271}$, $\frac{1}{77\cdots 40}a^{18}+\frac{23\cdots 05}{77\cdots 64}a^{17}-\frac{87\cdots 61}{38\cdots 20}a^{16}+\frac{80\cdots 53}{96\cdots 30}a^{15}-\frac{35\cdots 29}{77\cdots 40}a^{14}+\frac{32\cdots 21}{19\cdots 60}a^{13}+\frac{77\cdots 89}{77\cdots 40}a^{12}+\frac{35\cdots 09}{64\cdots 20}a^{11}+\frac{18\cdots 13}{77\cdots 40}a^{10}+\frac{23\cdots 09}{19\cdots 60}a^{9}+\frac{36\cdots 79}{77\cdots 40}a^{8}-\frac{17\cdots 08}{48\cdots 65}a^{7}+\frac{53\cdots 51}{12\cdots 40}a^{6}-\frac{11\cdots 17}{38\cdots 20}a^{5}-\frac{32\cdots 33}{77\cdots 40}a^{4}+\frac{20\cdots 32}{96\cdots 33}a^{3}-\frac{80\cdots 75}{19\cdots 66}a^{2}+\frac{53\cdots 93}{12\cdots 44}a-\frac{41\cdots 51}{25\cdots 88}$, $\frac{1}{10\cdots 20}a^{19}+\frac{73\cdots 99}{17\cdots 20}a^{18}-\frac{45\cdots 19}{35\cdots 40}a^{17}+\frac{42\cdots 71}{23\cdots 96}a^{16}-\frac{52\cdots 87}{52\cdots 60}a^{15}+\frac{16\cdots 11}{70\cdots 88}a^{14}-\frac{41\cdots 89}{10\cdots 20}a^{13}-\frac{41\cdots 99}{29\cdots 62}a^{12}+\frac{90\cdots 51}{52\cdots 60}a^{11}-\frac{29\cdots 03}{70\cdots 88}a^{10}+\frac{90\cdots 93}{11\cdots 80}a^{9}+\frac{11\cdots 47}{29\cdots 20}a^{8}-\frac{92\cdots 01}{35\cdots 40}a^{7}-\frac{33\cdots 95}{70\cdots 88}a^{6}-\frac{56\cdots 59}{26\cdots 80}a^{5}+\frac{11\cdots 17}{23\cdots 96}a^{4}+\frac{40\cdots 55}{10\cdots 32}a^{3}+\frac{78\cdots 53}{35\cdots 44}a^{2}+\frac{51\cdots 71}{19\cdots 96}a+\frac{16\cdots 01}{35\cdots 44}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $3$ |
Class group and class number
| Ideal class group: | not computed |
| |
| Narrow class group: | not computed |
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Unit group
| Rank: | $11$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: | not computed |
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| Regulator: | not computed |
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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 = \mathstrut &\frac{2^{4}\cdot(2\pi)^{8}\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
| \(\Q(\sqrt{10}) \), 4.4.256000.1, 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: | 20.0.45263635427652498824672194595274034473174478880768000000000000.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.4.0.1}{4} }^{4}{,}\,{\href{/padicField/3.1.0.1}{1} }^{4}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | ${\href{/padicField/11.4.0.1}{4} }^{5}$ | ${\href{/padicField/13.4.0.1}{4} }^{4}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | $20$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | $20$ | ${\href{/padicField/31.10.0.1}{10} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{4}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.10.0.1}{10} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }^{4}{,}\,{\href{/padicField/53.1.0.1}{1} }^{4}$ | $20$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.4.11a1.17 | $x^{4} + 8 x^{3} + 4 x^{2} + 10$ | $4$ | $1$ | $11$ | $C_4$ | $$[3, 4]$$ |
| 2.4.4.44a1.1147 | $x^{16} + 4 x^{13} + 4 x^{12} + 6 x^{10} + 12 x^{9} + 10 x^{8} + 4 x^{7} + 12 x^{6} + 20 x^{5} + 13 x^{4} + 4 x^{3} + 10 x^{2} + 12 x + 7$ | $4$ | $4$ | $44$ | $C_4^2$ | $$[3, 4]^{4}$$ | |
|
\(5\)
| 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 | $x^{10} + 960 x^{9} + 368650 x^{8} + 70786560 x^{7} + 6796984360 x^{6} + 261202401792 x^{5} + 13593968720 x^{4} + 283146240 x^{3} + 2949200 x^{2} + 15360 x + 229$ | $5$ | $2$ | $8$ | $F_5$ | $$[\ ]_{5}^{4}$$ |