Normalized defining polynomial
\( x^{10} - 5x^{9} + 15x^{8} - 10x^{7} - 85x^{6} + 331x^{5} - 645x^{4} + 650x^{3} - 115x^{2} - 315x + 179 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $[2, 4]$ |
| |
| Discriminant: |
\(256000000000000\)
\(\medspace = 2^{20}\cdot 5^{12}\)
|
| |
| Root discriminant: | \(27.59\) |
| |
| Galois root discriminant: | $2^{3}5^{271/200}\approx 70.82626546363946$ | ||
| Ramified primes: |
\(2\), \(5\)
|
| |
| Discriminant root field: | \(\Q\) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}-\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{47774}a^{9}-\frac{3053}{47774}a^{8}+\frac{6758}{23887}a^{7}-\frac{7795}{23887}a^{6}-\frac{16895}{47774}a^{5}-\frac{4081}{47774}a^{4}-\frac{3442}{23887}a^{3}+\frac{5148}{23887}a^{2}+\frac{5195}{47774}a-\frac{21481}{47774}$
| Monogenic: | No | |
| Index: | Not computed | |
| Inessential primes: | $2$ |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{21832}{23887}a^{9}-\frac{80027}{23887}a^{8}+\frac{220184}{23887}a^{7}+\frac{76644}{23887}a^{6}-\frac{1756224}{23887}a^{5}+\frac{4875066}{23887}a^{4}-\frac{7542776}{23887}a^{3}+\frac{4066392}{23887}a^{2}+\frac{2963752}{23887}a-\frac{2890048}{23887}$, $a-1$, $\frac{29133}{23887}a^{9}-\frac{107296}{23887}a^{8}+\frac{294964}{23887}a^{7}+\frac{99496}{23887}a^{6}-\frac{2351326}{23887}a^{5}+\frac{6538864}{23887}a^{4}-\frac{10124408}{23887}a^{3}+\frac{5474432}{23887}a^{2}+\frac{4010919}{23887}a-\frac{3884041}{23887}$, $\frac{10677}{23887}a^{9}-\frac{38900}{23887}a^{8}+\frac{104513}{23887}a^{7}+\frac{37960}{23887}a^{6}-\frac{853223}{23887}a^{5}+\frac{2314090}{23887}a^{4}-\frac{3607106}{23887}a^{3}+\frac{2176135}{23887}a^{2}+\frac{1315186}{23887}a-\frac{1542318}{23887}$, $\frac{5710595}{47774}a^{9}-\frac{10853677}{23887}a^{8}+\frac{29816481}{23887}a^{7}+\frac{7195298}{23887}a^{6}-\frac{468187019}{47774}a^{5}+\frac{664468317}{23887}a^{4}-\frac{1045004663}{23887}a^{3}+\frac{602890074}{23887}a^{2}+\frac{789833369}{47774}a-\frac{426151721}{23887}$
|
| |
| Regulator: | \( 2534.34927274 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 2534.34927274 \cdot 1}{2\cdot\sqrt{256000000000000}}\cr\approx \mathstrut & 0.493737318041 \end{aligned}\]
Galois group
$A_5^2:C_4$ (as 10T42):
| A non-solvable group of order 14400 |
| The 22 conjugacy class representatives for $A_5^2 : C_4$ |
| Character table for $A_5^2 : C_4$ |
Intermediate fields
| \(\Q(\sqrt{5}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 25 sibling: | data not computed |
| Degree 30 sibling: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 40 siblings: | data not computed |
| Minimal sibling: | This field is its own minimal sibling |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.6.0.1}{6} }{,}\,{\href{/padicField/3.4.0.1}{4} }$ | R | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.4.0.1}{4} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.4.0.1}{4} }$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.2.1.0a1.1 | $x^{2} + x + 1$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 2.2.4.20a1.8 | $x^{8} + 4 x^{7} + 10 x^{6} + 28 x^{5} + 43 x^{4} + 52 x^{3} + 34 x^{2} + 16 x + 3$ | $4$ | $2$ | $20$ | $(((C_4 \times C_2): C_2):C_2):C_2$ | $$[2, 3, 3, \frac{7}{2}]^{4}$$ | |
|
\(5\)
| 5.1.10.12a1.1 | $x^{10} + 5 x^{3} + 5$ | $10$ | $1$ | $12$ | $(C_5^2 : C_8):C_2$ | $$[\frac{11}{8}, \frac{11}{8}]_{8}^{2}$$ |