Normalized defining polynomial
\( x^{20} - 5279406x^{15} - 172313849975839x^{10} - 2105583058003554756x^{5} + 2161732704559354333451 \)
Invariants
| Degree: | $20$ |
| |
| Signature: | $[4, 8]$ |
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| Discriminant: |
\(1650745700457440430593235409009883454750158456154167652130126953125\)
\(\medspace = 5^{31}\cdot 11^{18}\cdot 41^{16}\)
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| Root discriminant: | \(2045.90\) |
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| Galois root discriminant: | $5^{31/20}11^{9/10}41^{4/5}\approx 2045.8981457874604$ | ||
| Ramified primes: |
\(5\), \(11\), \(41\)
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| Discriminant root field: | \(\Q(\sqrt{5}) \) | ||
| $\Aut(K/\Q)$: | $C_4$ |
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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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{451}a^{5}$, $\frac{1}{451}a^{6}$, $\frac{1}{451}a^{7}$, $\frac{1}{451}a^{8}$, $\frac{1}{4961}a^{9}-\frac{1}{11}a^{4}$, $\frac{1}{10170050}a^{10}-\frac{3}{22550}a^{5}+\frac{1}{50}$, $\frac{1}{10170050}a^{11}-\frac{3}{22550}a^{6}+\frac{1}{50}a$, $\frac{1}{416972050}a^{12}-\frac{103}{924550}a^{7}+\frac{901}{2050}a^{2}$, $\frac{1}{188054394550}a^{13}-\frac{410103}{416972050}a^{8}-\frac{34209}{84050}a^{3}$, $\frac{1}{7710230176550}a^{14}-\frac{44923}{1554168550}a^{9}-\frac{10462299}{37906550}a^{4}$, $\frac{1}{19\cdots 50}a^{15}-\frac{57\cdots 43}{44\cdots 50}a^{10}+\frac{83\cdots 21}{97\cdots 50}a^{5}+\frac{131539847067251}{314631098905895}$, $\frac{1}{19\cdots 50}a^{16}-\frac{57\cdots 43}{44\cdots 50}a^{11}+\frac{83\cdots 21}{97\cdots 50}a^{6}+\frac{131539847067251}{314631098905895}a$, $\frac{1}{89\cdots 50}a^{17}+\frac{32\cdots 63}{79\cdots 18}a^{12}-\frac{32\cdots 33}{44\cdots 50}a^{7}+\frac{13\cdots 19}{64\cdots 75}a^{2}$, $\frac{1}{18\cdots 50}a^{18}-\frac{1}{22\cdots 75}a^{17}-\frac{1}{49\cdots 25}a^{16}+\frac{1}{99\cdots 50}a^{15}-\frac{24\cdots 51}{20\cdots 25}a^{13}+\frac{31\cdots 99}{99\cdots 50}a^{12}+\frac{49\cdots 02}{11\cdots 75}a^{11}-\frac{24\cdots 51}{11\cdots 75}a^{10}+\frac{40\cdots 49}{45\cdots 75}a^{8}-\frac{38\cdots 31}{22\cdots 50}a^{7}+\frac{22\cdots 27}{24\cdots 25}a^{6}+\frac{15\cdots 74}{24\cdots 25}a^{5}+\frac{11\cdots 21}{26\cdots 50}a^{3}-\frac{20\cdots 97}{64\cdots 50}a^{2}+\frac{34\cdots 94}{78\cdots 75}a+\frac{43\cdots 81}{15\cdots 50}$, $\frac{1}{75\cdots 50}a^{19}-\frac{1}{44\cdots 50}a^{17}+\frac{1}{49\cdots 25}a^{16}+\frac{1}{49\cdots 25}a^{15}-\frac{24\cdots 51}{83\cdots 25}a^{14}-\frac{10\cdots 73}{99\cdots 50}a^{12}-\frac{49\cdots 02}{11\cdots 75}a^{11}-\frac{49\cdots 02}{11\cdots 75}a^{10}+\frac{78\cdots 19}{15\cdots 75}a^{9}+\frac{23\cdots 27}{22\cdots 50}a^{7}-\frac{22\cdots 27}{24\cdots 25}a^{6}-\frac{22\cdots 27}{24\cdots 25}a^{5}-\frac{29\cdots 69}{11\cdots 50}a^{4}-\frac{12\cdots 73}{32\cdots 75}a^{2}-\frac{34\cdots 94}{78\cdots 75}a-\frac{34\cdots 94}{78\cdots 75}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | not computed |
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| Narrow class group: | not computed |
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Unit group
| Rank: | $11$ |
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| Torsion generator: |
\( -1 \)
(order $2$)
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| 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{1650745700457440430593235409009883454750158456154167652130126953125}}\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, 5.1.3232184906328125.16, 10.2.52235096343476750903350830078125.1 |
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: | 10.0.114917211955648851987371826171875.1, deg 10 |
| Degree 20 sibling: | deg 20 |
| Minimal sibling: | 10.0.114917211955648851987371826171875.1 |
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 |
|
\(11\)
| 11.1.10.9a1.3 | $x^{10} + 33$ | $10$ | $1$ | $9$ | $C_{10}$ | $$[\ ]_{10}$$ |
| 11.1.10.9a1.3 | $x^{10} + 33$ | $10$ | $1$ | $9$ | $C_{10}$ | $$[\ ]_{10}$$ | |
|
\(41\)
| 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 | $x^{10} + 190 x^{9} + 14470 x^{8} + 553280 x^{7} + 10685960 x^{6} + 85860848 x^{5} + 64115760 x^{4} + 19918080 x^{3} + 3125520 x^{2} + 246281 x + 7776$ | $5$ | $2$ | $8$ | $C_{10}$ | $$[\ ]_{5}^{2}$$ |