Normalized defining polynomial
\( x^{10} + 3x^{8} + 135x^{2} - 135 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(2, 4)$ |
| |
| Discriminant: |
\(100776960000000\)
\(\medspace = 2^{16}\cdot 3^{9}\cdot 5^{7}\)
|
| |
| Root discriminant: | \(25.14\) |
| |
| Galois root discriminant: | $2^{187/80}3^{9/10}5^{7/8}\approx 55.547625380156134$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{15}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{3}a^{4}$, $\frac{1}{6}a^{5}-\frac{1}{6}a^{4}-\frac{1}{2}a-\frac{1}{2}$, $\frac{1}{6}a^{6}-\frac{1}{6}a^{4}-\frac{1}{2}a^{2}-\frac{1}{2}$, $\frac{1}{18}a^{7}-\frac{1}{6}a^{4}-\frac{1}{2}a^{3}-\frac{1}{2}$, $\frac{1}{522}a^{8}-\frac{2}{29}a^{6}+\frac{2}{87}a^{4}+\frac{3}{29}a^{2}+\frac{13}{58}$, $\frac{1}{522}a^{9}-\frac{7}{522}a^{7}+\frac{2}{87}a^{5}-\frac{1}{6}a^{4}-\frac{23}{58}a^{3}+\frac{13}{58}a-\frac{1}{2}$
| 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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{5}{174}a^{8}+\frac{23}{174}a^{6}+\frac{31}{174}a^{4}+\frac{3}{58}a^{2}+\frac{83}{29}$, $\frac{1}{174}a^{9}+\frac{2}{261}a^{8}+\frac{4}{261}a^{7}+\frac{5}{87}a^{6}+\frac{2}{29}a^{5}+\frac{8}{87}a^{4}+\frac{9}{29}a^{3}+\frac{12}{29}a^{2}+\frac{39}{58}a+\frac{26}{29}$, $\frac{5}{522}a^{9}+\frac{1}{87}a^{8}+\frac{23}{522}a^{7}+\frac{5}{58}a^{6}+\frac{10}{87}a^{5}+\frac{4}{29}a^{4}+\frac{1}{58}a^{3}+\frac{7}{58}a^{2}+\frac{7}{58}a-\frac{19}{29}$, $\frac{1}{261}a^{9}-\frac{5}{261}a^{8}-\frac{7}{261}a^{7}+\frac{2}{87}a^{6}-\frac{25}{87}a^{5}+\frac{38}{87}a^{4}-\frac{23}{29}a^{3}+\frac{28}{29}a^{2}-\frac{16}{29}a-\frac{94}{29}$, $\frac{5}{522}a^{9}-\frac{1}{87}a^{8}+\frac{23}{522}a^{7}-\frac{5}{58}a^{6}+\frac{10}{87}a^{5}-\frac{4}{29}a^{4}+\frac{1}{58}a^{3}-\frac{7}{58}a^{2}+\frac{7}{58}a+\frac{19}{29}$
|
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| Regulator: | \( 2345.57971388 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 2345.57971388 \cdot 1}{2\cdot\sqrt{100776960000000}}\cr\approx \mathstrut & 0.728314638992 \end{aligned}\]
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
| 5.1.162000.1 |
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 | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.5.0.1}{5} }^{2}$ | ${\href{/padicField/13.8.0.1}{8} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.2.0.1}{2} }^{5}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.4.0.1}{4} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.10.0.1}{10} }$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.10.0.1}{10} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 | $x^{10} + 2 x^{9} + 2 x^{7} + 4 x^{4} + 4 x^{3} + 4 x + 2$ | $10$ | $1$ | $16$ | $((C_2^4 : C_5):C_4)\times C_2$ | $$[2, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}]_{5}^{4}$$ |
|
\(3\)
| 3.1.10.9a1.1 | $x^{10} + 3$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $$[\ ]_{10}^{4}$$ |
|
\(5\)
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 5.1.8.7a1.1 | $x^{8} + 5$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $$[\ ]_{8}^{2}$$ |