Normalized defining polynomial
\( x^{10} - 100x^{6} - 140x^{5} - 625x^{2} - 500x - 100 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $[4, 3]$ |
| |
| Discriminant: |
\(-500000000000000000\)
\(\medspace = -\,2^{17}\cdot 5^{18}\)
|
| |
| Root discriminant: | \(58.87\) |
| |
| Galois root discriminant: | $2^{65/24}5^{203/100}\approx 171.474158552352$ | ||
| Ramified primes: |
\(2\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{50}a^{5}-\frac{1}{2}a-\frac{2}{5}$, $\frac{1}{50}a^{6}-\frac{1}{2}a^{2}-\frac{2}{5}a$, $\frac{1}{200}a^{7}+\frac{1}{200}a^{5}+\frac{1}{4}a^{4}-\frac{3}{8}a^{3}+\frac{2}{5}a^{2}+\frac{1}{8}a-\frac{7}{20}$, $\frac{1}{200}a^{8}+\frac{1}{200}a^{6}-\frac{1}{100}a^{5}-\frac{3}{8}a^{4}+\frac{2}{5}a^{3}+\frac{1}{8}a^{2}+\frac{3}{20}a+\frac{1}{5}$, $\frac{1}{200}a^{9}-\frac{1}{100}a^{6}+\frac{3}{20}a^{4}-\frac{1}{2}a^{3}-\frac{1}{4}a^{2}-\frac{17}{40}a-\frac{1}{4}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ (assuming GRH) |
| |
| Narrow class group: | $C_{2}$, which has order $2$ (assuming GRH) |
|
Unit group
| Rank: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{200}a^{9}-\frac{1}{200}a^{8}+\frac{1}{100}a^{7}+\frac{1}{200}a^{6}-\frac{21}{50}a^{5}+\frac{1}{40}a^{4}+\frac{7}{20}a^{3}-\frac{3}{40}a^{2}-\frac{109}{40}a+\frac{33}{20}$, $\frac{33}{200}a^{9}-\frac{3}{40}a^{8}+\frac{3}{40}a^{7}-\frac{21}{200}a^{6}-\frac{131}{8}a^{5}-\frac{627}{40}a^{4}+\frac{23}{8}a^{3}+\frac{3}{8}a^{2}-\frac{507}{5}a-\frac{91}{2}$, $\frac{9}{40}a^{9}-\frac{9}{200}a^{8}-\frac{3}{100}a^{7}+\frac{1}{40}a^{6}-\frac{1119}{50}a^{5}-\frac{219}{8}a^{4}+\frac{203}{20}a^{3}+\frac{9}{40}a^{2}-\frac{1229}{8}a-\frac{1363}{20}$, $\frac{43}{50}a^{9}-\frac{83}{200}a^{8}+\frac{41}{200}a^{7}-\frac{19}{200}a^{6}-\frac{17197}{200}a^{5}-\frac{3153}{40}a^{4}+\frac{1497}{40}a^{3}-\frac{759}{40}a^{2}-\frac{21029}{40}a-\frac{3611}{20}$, $\frac{89}{50}a^{9}-\frac{4}{5}a^{8}+\frac{71}{200}a^{7}-\frac{3}{25}a^{6}-\frac{35613}{200}a^{5}-\frac{3377}{20}a^{4}+\frac{603}{8}a^{3}-\frac{173}{5}a^{2}-\frac{43909}{40}a-\frac{7849}{20}$, $\frac{29}{40}a^{9}-\frac{47}{200}a^{8}+\frac{1}{5}a^{7}-\frac{1}{200}a^{6}-\frac{7233}{100}a^{5}-\frac{629}{8}a^{4}+\frac{137}{10}a^{3}-\frac{249}{8}a^{2}-\frac{18411}{40}a-\frac{3993}{20}$
|
| |
| Regulator: | \( 753999.034354 \) (assuming GRH) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 753999.034354 \cdot 1}{2\cdot\sqrt{500000000000000000}}\cr\approx \mathstrut & 2.11599861727 \end{aligned}\] (assuming GRH)
Galois group
| A non-solvable group of order 3628800 |
| The 42 conjugacy class representatives for $S_{10}$ |
| Character table for $S_{10}$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling fields
| Degree 20 sibling: | data not computed |
| Degree 45 sibling: | 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 | ${\href{/padicField/3.9.0.1}{9} }{,}\,{\href{/padicField/3.1.0.1}{1} }$ | R | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.9.0.1}{9} }{,}\,{\href{/padicField/11.1.0.1}{1} }$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.3.0.1}{3} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.9.0.1}{9} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.7.0.1}{7} }{,}\,{\href{/padicField/19.3.0.1}{3} }$ | ${\href{/padicField/23.5.0.1}{5} }{,}\,{\href{/padicField/23.2.0.1}{2} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{3}$ | ${\href{/padicField/29.7.0.1}{7} }{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.9.0.1}{9} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.9.0.1}{9} }{,}\,{\href{/padicField/59.1.0.1}{1} }$ |
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 | $x^{2} + 4 x + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.1.4.8a1.2 | $x^{4} + 4 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $S_4$ | $$[\frac{8}{3}, \frac{8}{3}]_{3}^{2}$$ | |
| 2.2.2.6a1.3 | $x^{4} + 6 x^{3} + 7 x^{2} + 6 x + 3$ | $2$ | $2$ | $6$ | $D_{4}$ | $$[2, 3]^{2}$$ | |
|
\(5\)
| 5.2.5.18a1.15 | $x^{10} + 20 x^{9} + 170 x^{8} + 800 x^{7} + 2280 x^{6} + 4064 x^{5} + 4560 x^{4} + 3250 x^{3} + 1660 x^{2} + 820 x + 237$ | $5$ | $2$ | $18$ | $(C_5^2 : C_4) : C_2$ | $$[\frac{5}{4}, \frac{9}{4}]_{4}^{2}$$ |