Normalized defining polynomial
\( x^{10} - 10x^{8} - 150x^{6} + 300x^{4} + 2250x^{2} - 450 \)
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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{5}a^{5}$, $\frac{1}{15}a^{6}+\frac{1}{3}a^{4}$, $\frac{1}{15}a^{7}-\frac{1}{15}a^{5}$, $\frac{1}{155985}a^{8}+\frac{658}{155985}a^{6}-\frac{7792}{31197}a^{4}-\frac{1832}{10399}a^{2}+\frac{3456}{10399}$, $\frac{1}{155985}a^{9}+\frac{658}{155985}a^{7}-\frac{7763}{155985}a^{5}-\frac{1832}{10399}a^{3}+\frac{3456}{10399}a$
| 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: |
$\frac{21}{51995}a^{8}-\frac{142}{155985}a^{6}-\frac{2143}{31197}a^{4}-\frac{1027}{10399}a^{2}-\frac{651}{10399}$, $\frac{100}{31197}a^{8}-\frac{1256}{51995}a^{6}-\frac{17173}{31197}a^{4}-\frac{888}{10399}a^{2}+\frac{53761}{10399}$, $\frac{233}{155985}a^{8}-\frac{2671}{155985}a^{6}-\frac{6110}{31197}a^{4}+\frac{9902}{10399}a^{2}+\frac{46121}{10399}$, $\frac{3916}{155985}a^{8}-\frac{8764}{31197}a^{6}-\frac{96397}{31197}a^{4}+\frac{84390}{10399}a^{2}-\frac{16201}{10399}$, $\frac{1117}{10399}a^{9}-\frac{490}{10399}a^{8}-\frac{164534}{155985}a^{7}+\frac{72028}{155985}a^{6}-\frac{2546128}{155985}a^{5}+\frac{222286}{31197}a^{4}+\frac{304259}{10399}a^{3}-\frac{136692}{10399}a^{2}+\frac{2582600}{10399}a-\frac{1140733}{10399}$, $\frac{5862}{51995}a^{9}-\frac{196}{51995}a^{8}-\frac{52828}{51995}a^{7}-\frac{22939}{155985}a^{6}-\frac{915794}{51995}a^{5}+\frac{47732}{31197}a^{4}+\frac{154535}{10399}a^{3}+\frac{328488}{10399}a^{2}+\frac{2230846}{10399}a+\frac{931587}{10399}$, $\frac{917}{155985}a^{9}-\frac{4307}{155985}a^{8}+\frac{10643}{155985}a^{7}-\frac{2445}{10399}a^{6}-\frac{1151}{31197}a^{5}+\frac{12970}{31197}a^{4}-\frac{16104}{10399}a^{3}+\frac{39179}{10399}a^{2}+\frac{7856}{10399}a-\frac{4023}{10399}$
|
| |
| Regulator: | \( 1319522.14875 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{6}\cdot(2\pi)^{2}\cdot 1319522.14875 \cdot 1}{2\cdot\sqrt{2332800000000000000}}\cr\approx \mathstrut & 1.09140929002 \end{aligned}\]
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 | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.2.0.1}{2} }^{4}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }^{2}{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 | $x^{10} + 20 x^{9} + 170 x^{8} + 800 x^{7} + 2295 x^{6} + 4244 x^{5} + 5370 x^{4} + 4880 x^{3} + 2980 x^{2} + 1040 x + 157$ | $5$ | $2$ | $14$ | $F_{5}\times C_2$ | $$[\frac{7}{4}]_{4}^{2}$$ |