Normalized defining polynomial
\( x^{20} - 5 x^{19} - 60 x^{18} + 340 x^{17} + 1420 x^{16} + 33985 x^{15} - 200850 x^{14} + \cdots + 13\!\cdots\!25 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $[4, 8]$ |
| |
| Discriminant: |
\(7700850521128818686981461752498528179712593555450439453125\)
\(\medspace = 5^{31}\cdot 11^{10}\cdot 41^{16}\)
|
| |
| Root discriminant: | \(784.02\) |
| |
| Galois root discriminant: | $5^{31/20}11^{1/2}41^{4/5}\approx 784.0196266547648$ | ||
| Ramified primes: |
\(5\), \(11\), \(41\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $\frac{1}{205}a^{10}+\frac{9}{41}a^{9}-\frac{13}{41}a^{8}-\frac{19}{41}a^{7}-\frac{17}{41}a^{6}+\frac{17}{41}a^{5}+\frac{19}{41}a^{4}+\frac{6}{41}a^{3}+\frac{5}{41}a^{2}+\frac{18}{41}a+\frac{13}{41}$, $\frac{1}{205}a^{11}-\frac{8}{41}a^{9}-\frac{8}{41}a^{8}+\frac{18}{41}a^{7}+\frac{3}{41}a^{6}-\frac{8}{41}a^{5}+\frac{12}{41}a^{4}-\frac{19}{41}a^{3}-\frac{2}{41}a^{2}-\frac{18}{41}a-\frac{11}{41}$, $\frac{1}{2050}a^{12}-\frac{181}{410}a^{9}+\frac{36}{205}a^{8}+\frac{93}{205}a^{7}-\frac{31}{82}a^{6}+\frac{20}{41}a^{5}-\frac{12}{41}a^{4}+\frac{23}{82}a^{3}-\frac{32}{205}a^{2}-\frac{7}{41}a-\frac{19}{82}$, $\frac{1}{2050}a^{13}-\frac{1}{410}a^{10}-\frac{14}{205}a^{9}-\frac{17}{205}a^{8}-\frac{7}{82}a^{7}+\frac{7}{41}a^{6}+\frac{1}{41}a^{5}-\frac{1}{82}a^{4}+\frac{3}{205}a^{3}-\frac{8}{41}a^{2}+\frac{23}{82}a-\frac{19}{41}$, $\frac{1}{6150}a^{14}+\frac{1}{1230}a^{11}+\frac{1}{615}a^{10}+\frac{1}{205}a^{9}-\frac{1}{82}a^{8}-\frac{55}{123}a^{7}-\frac{5}{123}a^{6}+\frac{1}{246}a^{5}+\frac{53}{615}a^{4}-\frac{19}{123}a^{3}-\frac{77}{246}a^{2}+\frac{28}{123}a+\frac{61}{123}$, $\frac{1}{6150}a^{15}-\frac{1}{6150}a^{12}+\frac{1}{615}a^{11}-\frac{143}{410}a^{9}-\frac{296}{615}a^{8}-\frac{298}{615}a^{7}+\frac{43}{246}a^{6}-\frac{187}{615}a^{5}-\frac{4}{123}a^{4}-\frac{5}{246}a^{3}+\frac{257}{615}a^{2}+\frac{49}{123}a+\frac{6}{41}$, $\frac{1}{6150}a^{16}-\frac{1}{6150}a^{13}+\frac{1}{6150}a^{12}-\frac{1}{410}a^{10}+\frac{527}{1230}a^{9}+\frac{293}{615}a^{8}-\frac{109}{1230}a^{7}+\frac{481}{1230}a^{6}-\frac{7}{123}a^{5}-\frac{59}{246}a^{4}-\frac{1}{30}a^{3}-\frac{292}{615}a^{2}-\frac{7}{41}a+\frac{17}{82}$, $\frac{1}{6150}a^{17}+\frac{1}{6150}a^{13}-\frac{1}{615}a^{11}+\frac{1}{1230}a^{10}+\frac{101}{615}a^{9}-\frac{122}{615}a^{8}-\frac{113}{410}a^{7}+\frac{16}{41}a^{6}+\frac{34}{123}a^{5}+\frac{67}{246}a^{4}+\frac{101}{205}a^{3}-\frac{53}{246}a^{2}-\frac{49}{246}a-\frac{50}{123}$, $\frac{1}{252150}a^{18}+\frac{1}{126075}a^{17}-\frac{3}{42025}a^{16}-\frac{17}{252150}a^{15}+\frac{3}{42025}a^{14}+\frac{28}{126075}a^{13}-\frac{19}{126075}a^{12}+\frac{47}{25215}a^{11}-\frac{7}{25215}a^{10}+\frac{11069}{25215}a^{9}+\frac{2401}{8405}a^{8}-\frac{518}{25215}a^{7}+\frac{10981}{25215}a^{6}+\frac{333}{8405}a^{5}-\frac{8876}{25215}a^{4}-\frac{3259}{50430}a^{3}+\frac{1316}{25215}a^{2}-\frac{690}{1681}a+\frac{2843}{10086}$, $\frac{1}{20\cdots 50}a^{19}-\frac{45\cdots 83}{10\cdots 75}a^{18}-\frac{28\cdots 09}{20\cdots 50}a^{17}+\frac{57\cdots 26}{10\cdots 75}a^{16}+\frac{11\cdots 13}{34\cdots 25}a^{15}-\frac{14\cdots 51}{20\cdots 50}a^{14}+\frac{16\cdots 87}{11\cdots 50}a^{13}+\frac{15\cdots 13}{69\cdots 50}a^{12}-\frac{10\cdots 89}{41\cdots 30}a^{11}+\frac{13\cdots 79}{27\cdots 02}a^{10}+\frac{45\cdots 09}{13\cdots 10}a^{9}+\frac{93\cdots 23}{83\cdots 06}a^{8}-\frac{31\cdots 31}{83\cdots 06}a^{7}-\frac{20\cdots 21}{13\cdots 10}a^{6}-\frac{38\cdots 69}{13\cdots 10}a^{5}-\frac{94\cdots 48}{20\cdots 15}a^{4}-\frac{11\cdots 41}{73\cdots 90}a^{3}+\frac{39\cdots 37}{20\cdots 15}a^{2}+\frac{40\cdots 59}{83\cdots 06}a-\frac{33\cdots 15}{27\cdots 02}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | not computed |
| |
| Narrow class group: | not computed |
|
Unit group
| Rank: | $11$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| 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^{4}\cdot(2\pi)^{8}\cdot R \cdot h}{2\cdot\sqrt{7700850521128818686981461752498528179712593555450439453125}}\cr\mathstrut & \text{
Galois group
$C_2\times F_5$ (as 20T9):
| A solvable group of order 40 |
| The 10 conjugacy class representatives for $C_2\times F_5$ |
| Character table for $C_2\times F_5$ |
Intermediate fields
| \(\Q(\sqrt{5}) \), 4.4.15125.1, 5.1.220762578125.3, 10.2.243680579501983642578125.4 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Galois closure: | deg 40 |
| Degree 10 siblings: | deg 10, 10.0.7849000201874793524169921875.1 |
| Degree 20 sibling: | deg 20 |
| Minimal sibling: | 10.0.7849000201874793524169921875.1 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | ${\href{/padicField/2.4.0.1}{4} }^{5}$ | ${\href{/padicField/3.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/7.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/13.4.0.1}{4} }^{5}$ | ${\href{/padicField/17.4.0.1}{4} }^{5}$ | ${\href{/padicField/19.2.0.1}{2} }^{8}{,}\,{\href{/padicField/19.1.0.1}{1} }^{4}$ | ${\href{/padicField/23.4.0.1}{4} }^{5}$ | ${\href{/padicField/29.2.0.1}{2} }^{8}{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.5.0.1}{5} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }^{5}$ | R | ${\href{/padicField/43.4.0.1}{4} }^{5}$ | ${\href{/padicField/47.4.0.1}{4} }^{5}$ | ${\href{/padicField/53.4.0.1}{4} }^{5}$ | ${\href{/padicField/59.2.0.1}{2} }^{10}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(5\)
| 5.1.20.31a1.1 | $x^{20} + 10 x^{12} + 5$ | $20$ | $1$ | $31$ | not computed | not computed |
|
\(11\)
| 11.5.2.5a1.2 | $x^{10} + 20 x^{7} + 18 x^{5} + 100 x^{4} + 180 x^{2} + 92$ | $2$ | $5$ | $5$ | $C_{10}$ | $$[\ ]_{2}^{5}$$ |
| 11.5.2.5a1.2 | $x^{10} + 20 x^{7} + 18 x^{5} + 100 x^{4} + 180 x^{2} + 92$ | $2$ | $5$ | $5$ | $C_{10}$ | $$[\ ]_{2}^{5}$$ | |
|
\(41\)
| 41.2.5.8a1.1 | $x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$ | $5$ | $2$ | $8$ | $C_{10}$ | $$[\ ]_{5}^{2}$$ |
| 41.2.5.8a1.1 | $x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$ | $5$ | $2$ | $8$ | $C_{10}$ | $$[\ ]_{5}^{2}$$ |