Normalized defining polynomial
\( x^{10} - 4x^{9} + 2x^{8} + 12x^{7} - 8x^{6} - 84x^{5} + 292x^{4} - 516x^{3} + 569x^{2} - 376x + 118 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $[0, 5]$ |
| |
| Discriminant: |
\(-127401984000000\)
\(\medspace = -\,2^{25}\cdot 3^{5}\cdot 5^{6}\)
|
| |
| Root discriminant: | \(25.73\) |
| |
| Galois root discriminant: | $2^{11/4}3^{1/2}5^{13/10}\approx 94.41785309490363$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-6}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-6}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{17}a^{8}-\frac{7}{17}a^{7}-\frac{6}{17}a^{6}-\frac{5}{17}a^{5}-\frac{6}{17}a^{4}-\frac{6}{17}a^{3}+\frac{8}{17}a^{2}+\frac{8}{17}a+\frac{7}{17}$, $\frac{1}{17}a^{9}-\frac{4}{17}a^{7}+\frac{4}{17}a^{6}-\frac{7}{17}a^{5}+\frac{3}{17}a^{4}-\frac{4}{17}a^{2}-\frac{5}{17}a-\frac{2}{17}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$2a^{9}-\frac{78}{17}a^{8}-\frac{83}{17}a^{7}+\frac{298}{17}a^{6}+\frac{305}{17}a^{5}-\frac{2456}{17}a^{4}+\frac{5483}{17}a^{3}-\frac{7033}{17}a^{2}+\frac{5190}{17}a-\frac{1685}{17}$, $\frac{6}{17}a^{9}-\frac{45}{17}a^{8}+\frac{36}{17}a^{7}+\frac{175}{17}a^{6}-\frac{140}{17}a^{5}-\frac{902}{17}a^{4}+\frac{2905}{17}a^{3}-\frac{4753}{17}a^{2}+\frac{4574}{17}a-\frac{2061}{17}$, $\frac{7}{17}a^{9}-\frac{19}{17}a^{8}-\frac{14}{17}a^{7}+\frac{74}{17}a^{6}+\frac{46}{17}a^{5}-\frac{545}{17}a^{4}+\frac{1287}{17}a^{3}-\frac{1710}{17}a^{2}+78a-\frac{453}{17}$, $\frac{93}{17}a^{9}-\frac{242}{17}a^{8}-\frac{208}{17}a^{7}+\frac{940}{17}a^{6}+\frac{780}{17}a^{5}-\frac{7177}{17}a^{4}+\frac{16327}{17}a^{3}-\frac{21297}{17}a^{2}+\frac{15806}{17}a-\frac{4991}{17}$
|
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| Regulator: | \( 3020.56452393 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 3020.56452393 \cdot 2}{2\cdot\sqrt{127401984000000}}\cr\approx \mathstrut & 2.62059171430 \end{aligned}\]
Galois group
$D_5^2.C_2^2$ (as 10T27):
| A solvable group of order 400 |
| The 16 conjugacy class representatives for $(D_5 \wr C_2):C_2$ |
| Character table for $(D_5 \wr C_2):C_2$ |
Intermediate fields
| \(\Q(\sqrt{-6}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 siblings: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 25 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 | R | R | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.2.0.1}{2} }^{4}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.4 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.1.8.22d1.13 | $x^{8} + 4 x^{7} + 4 x^{6} + 2 x^{4} + 8 x^{3} + 4 x^{2} + 10$ | $8$ | $1$ | $22$ | $Q_8:C_2$ | $$[2, 3, \frac{7}{2}]^{2}$$ | |
|
\(3\)
| 3.1.2.1a1.2 | $x^{2} + 6$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 3.4.2.4a1.2 | $x^{8} + 4 x^{7} + 4 x^{6} + 4 x^{4} + 8 x^{3} + 7$ | $2$ | $4$ | $4$ | $C_4\times C_2$ | $$[\ ]_{2}^{4}$$ | |
|
\(5\)
| $\Q_{5}$ | $x + 3$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 5.1.5.6a1.1 | $x^{5} + 10 x^{2} + 5$ | $5$ | $1$ | $6$ | $D_{5}$ | $$[\frac{3}{2}]_{2}$$ |