Normalized defining polynomial
\( x^{10} - 10x^{8} - 150x^{6} + 1860x^{4} - 3150x^{2} - 5202 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(6, 2)$ |
| |
| Discriminant: |
\(2332800000000000000\)
\(\medspace = 2^{19}\cdot 3^{6}\cdot 5^{14}\)
|
| |
| Root discriminant: | \(68.67\) |
| |
| Galois root discriminant: | $2^{227/80}3^{3/4}5^{31/20}\approx 197.43177973614576$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{2}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $\frac{1}{3}a^{6}-\frac{1}{3}a^{4}$, $\frac{1}{3}a^{7}-\frac{1}{3}a^{5}$, $\frac{1}{394707}a^{8}+\frac{61351}{394707}a^{6}+\frac{106333}{394707}a^{4}-\frac{22925}{131569}a^{2}+\frac{33773}{131569}$, $\frac{1}{6710019}a^{9}-\frac{286544}{2236673}a^{7}-\frac{156805}{6710019}a^{5}-\frac{22925}{2236673}a^{3}-\frac{492503}{2236673}a$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | Trivial group, which has order $1$ |
| |
| Narrow class group: | $C_{2}$, which has order $2$ |
|
Unit group
| Rank: | $7$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{877}{394707}a^{8}-\frac{2298}{131569}a^{6}-\frac{159949}{394707}a^{4}+\frac{419539}{131569}a^{2}+\frac{410603}{131569}$, $\frac{1484}{394707}a^{8}-\frac{288}{131569}a^{6}-\frac{216197}{394707}a^{4}+\frac{187240}{131569}a^{2}+\frac{254481}{131569}$, $\frac{73}{394707}a^{8}+\frac{1759}{131569}a^{6}-\frac{262}{394707}a^{4}-\frac{226266}{131569}a^{2}-\frac{165951}{131569}$, $\frac{25876}{6710019}a^{9}+\frac{2972}{394707}a^{8}-\frac{41549}{2236673}a^{7}-\frac{19462}{394707}a^{6}-\frac{4634704}{6710019}a^{5}-\frac{533338}{394707}a^{4}+\frac{8457737}{2236673}a^{3}+\frac{1203763}{131569}a^{2}+\frac{22921856}{2236673}a+\frac{1828175}{131569}$, $\frac{1013}{131569}a^{8}+\frac{12416}{394707}a^{6}-\frac{382184}{394707}a^{4}-\frac{69074}{131569}a^{2}+\frac{12327}{131569}$, $\frac{544436}{6710019}a^{9}+\frac{31358}{131569}a^{8}+\frac{271006}{6710019}a^{7}+\frac{42740}{131569}a^{6}-\frac{25666266}{2236673}a^{5}-\frac{4304730}{131569}a^{4}+\frac{71013576}{2236673}a^{3}+\frac{8979290}{131569}a^{2}+\frac{98882890}{2236673}a+\frac{13321459}{131569}$, $\frac{252082}{6710019}a^{9}-\frac{602}{394707}a^{8}+\frac{73108}{6710019}a^{7}+\frac{12529}{131569}a^{6}-\frac{37019522}{6710019}a^{5}+\frac{456344}{394707}a^{4}+\frac{29659931}{2236673}a^{3}-\frac{671740}{131569}a^{2}+\frac{37890406}{2236673}a-\frac{990703}{131569}$
|
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| Regulator: | \( 1584413.70604 \) |
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Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{2}\cdot 1584413.70604 \cdot 1}{2\cdot\sqrt{2332800000000000000}}\cr\approx \mathstrut & 1.31050762554 \end{aligned}\]
Galois group
$C_2\wr F_5$ (as 10T29):
| A solvable group of order 640 |
| The 22 conjugacy class representatives for $((C_2^4 : C_5):C_4)\times C_2$ |
| Character table for $((C_2^4 : C_5):C_4)\times C_2$ |
Intermediate fields
| 5.5.11250000.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 10 sibling: | data not computed |
| Degree 20 siblings: | data not computed |
| Degree 32 siblings: | 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 | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }^{3}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.10.19a1.61 | $x^{10} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 4 x + 2$ | $10$ | $1$ | $19$ | $((C_2^4 : C_5):C_4)\times C_2$ | $$[\frac{14}{5}, \frac{14}{5}, \frac{14}{5}, \frac{14}{5}, 3]_{5}^{4}$$ |
|
\(3\)
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{3}$ | $x + 1$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 3.2.4.6a1.1 | $x^{8} + 8 x^{7} + 32 x^{6} + 80 x^{5} + 136 x^{4} + 160 x^{3} + 128 x^{2} + 67 x + 16$ | $4$ | $2$ | $6$ | $C_8:C_2$ | $$[\ ]_{4}^{4}$$ | |
|
\(5\)
| 5.2.5.14a1.1 | $x^{10} + 20 x^{9} + 170 x^{8} + 800 x^{7} + 2295 x^{6} + 4244 x^{5} + 5370 x^{4} + 4880 x^{3} + 2980 x^{2} + 1040 x + 157$ | $5$ | $2$ | $14$ | $F_{5}\times C_2$ | $$[\frac{7}{4}]_{4}^{2}$$ |