Normalized defining polynomial
\( x^{10} - 3x^{8} + 2x^{6} + 10x^{4} - 3x^{2} + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(0, 5)$ |
| |
| Discriminant: |
\(-486645760000\)
\(\medspace = -\,2^{16}\cdot 5^{4}\cdot 109^{2}\)
|
| |
| Root discriminant: | \(14.75\) |
| |
| Galois root discriminant: | $2^{187/80}5^{2/3}109^{1/2}\approx 154.29464967757525$ | ||
| Ramified primes: |
\(2\), \(5\), \(109\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-1}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}$, $\frac{1}{2}a^{5}-\frac{1}{2}a$, $\frac{1}{2}a^{6}-\frac{1}{2}a^{2}$, $\frac{1}{4}a^{7}-\frac{1}{4}a^{6}-\frac{1}{4}a^{5}-\frac{1}{4}a^{4}-\frac{1}{4}a^{3}+\frac{1}{4}a^{2}+\frac{1}{4}a+\frac{1}{4}$, $\frac{1}{4}a^{8}-\frac{1}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a$
| 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: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{9}-3a^{7}+2a^{5}+10a^{3}-3a$, $\frac{1}{4}a^{8}-\frac{1}{2}a^{6}+\frac{5}{2}a^{2}+\frac{3}{4}$, $\frac{1}{4}a^{9}-\frac{1}{4}a^{8}-\frac{1}{2}a^{7}+\frac{1}{2}a^{6}+\frac{5}{2}a^{3}-\frac{5}{2}a^{2}+\frac{7}{4}a-\frac{3}{4}$, $a^{9}-\frac{1}{2}a^{8}-3a^{7}+\frac{3}{2}a^{6}+2a^{5}-a^{4}+10a^{3}-\frac{9}{2}a^{2}-2a+\frac{3}{2}$
|
| |
| Regulator: | \( 80.8094689763 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 80.8094689763 \cdot 1}{2\cdot\sqrt{486645760000}}\cr\approx \mathstrut & 0.567185509720 \end{aligned}\]
Galois group
$C_2\wr S_5$ (as 10T39):
| A non-solvable group of order 3840 |
| The 36 conjugacy class representatives for $C_2 \wr S_5$ |
| Character table for $C_2 \wr S_5$ |
Intermediate fields
| 5.1.43600.1 |
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 30 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 | ${\href{/padicField/3.10.0.1}{10} }$ | R | ${\href{/padicField/7.3.0.1}{3} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }{,}\,{\href{/padicField/7.1.0.1}{1} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }{,}\,{\href{/padicField/37.4.0.1}{4} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.8.0.1}{8} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.10.16a1.16 | $x^{10} + 2 x^{9} + 2 x^{7} + 4 x^{4} + 4 x^{3} + 4 x + 2$ | $10$ | $1$ | $16$ | $((C_2^4 : C_5):C_4)\times C_2$ | $$[2, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$$ |
|
\(5\)
| 5.4.1.0a1.1 | $x^{4} + 4 x^{2} + 4 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
| 5.2.3.4a1.2 | $x^{6} + 12 x^{5} + 54 x^{4} + 112 x^{3} + 108 x^{2} + 48 x + 13$ | $3$ | $2$ | $4$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
|
\(109\)
| $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{109}$ | $x + 103$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 109.2.1.0a1.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 109.1.2.1a1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 109.1.2.1a1.2 | $x^{2} + 654$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 109.2.1.0a1.1 | $x^{2} + 108 x + 6$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |