Normalized defining polynomial
\( x^{10} - 3x^{8} + 2x^{6} - 6x^{4} + 13x^{2} + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(0, 5)$ |
| |
| Discriminant: |
\(-447464669184\)
\(\medspace = -\,2^{16}\cdot 3^{2}\cdot 13^{2}\cdot 67^{2}\)
|
| |
| Root discriminant: | \(14.62\) |
| |
| Galois root discriminant: | $2^{187/80}3^{1/2}13^{1/2}67^{1/2}\approx 258.36121588504267$ | ||
| Ramified primes: |
\(2\), \(3\), \(13\), \(67\)
|
| |
| 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: |
$a$, $\frac{1}{2}a^{9}-\frac{3}{2}a^{7}+a^{5}-\frac{5}{2}a^{3}+\frac{11}{2}a$, $\frac{1}{2}a^{9}-\frac{5}{4}a^{7}+\frac{1}{4}a^{6}+\frac{3}{4}a^{5}-\frac{1}{4}a^{4}-\frac{11}{4}a^{3}-\frac{1}{4}a^{2}+\frac{19}{4}a-\frac{3}{4}$, $\frac{1}{4}a^{9}+\frac{1}{4}a^{8}-\frac{1}{2}a^{7}-\frac{1}{2}a^{6}-\frac{3}{2}a^{3}-\frac{3}{2}a^{2}+\frac{7}{4}a+\frac{3}{4}$
|
| |
| Regulator: | \( 75.1346630598 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 75.1346630598 \cdot 1}{2\cdot\sqrt{447464669184}}\cr\approx \mathstrut & 0.549959000813 \end{aligned}\]
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
| 5.1.41808.1 |
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 | R | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.10.0.1}{10} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.4.0.1}{4} }$ | ${\href{/padicField/19.4.0.1}{4} }^{2}{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.3.0.1}{3} }^{2}$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.3.0.1}{3} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.10.16a1.16 | $x^{10} + 2 x^{9} + 2 x^{7} + 4 x^{4} + 4 x^{3} + 4 x + 2$ | $10$ | $1$ | $16$ | $((C_2^4 : C_5):C_4)\times C_2$ | $$[2, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$$ |
|
\(3\)
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
| 3.6.1.0a1.1 | $x^{6} + 2 x^{4} + x^{2} + 2 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 13.2.2.2a1.2 | $x^{4} + 24 x^{3} + 148 x^{2} + 48 x + 17$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(67\)
| 67.1.2.1a1.2 | $x^{2} + 134$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 67.1.2.1a1.2 | $x^{2} + 134$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 67.6.1.0a1.1 | $x^{6} + 63 x^{3} + 49 x^{2} + 55 x + 2$ | $1$ | $6$ | $0$ | $C_6$ | $$[\ ]^{6}$$ |