Normalized defining polynomial
\( x^{10} + 2x^{8} + 10x^{6} - 40x^{4} + 40x^{2} - 10 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $[2, 4]$ |
| |
| Discriminant: |
\(21767823360000000\)
\(\medspace = 2^{19}\cdot 3^{12}\cdot 5^{7}\)
|
| |
| Root discriminant: | \(43.03\) |
| |
| Galois root discriminant: | $2^{53/20}3^{11/6}5^{7/8}\approx 192.33155488303092$ | ||
| Ramified primes: |
\(2\), \(3\), \(5\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{10}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{51}a^{8}-\frac{20}{51}a^{6}-\frac{3}{17}a^{4}+\frac{5}{51}a^{2}-\frac{19}{51}$, $\frac{1}{51}a^{9}-\frac{20}{51}a^{7}-\frac{3}{17}a^{5}+\frac{5}{51}a^{3}-\frac{19}{51}a$
| 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: | $5$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{2}{51}a^{8}+\frac{11}{51}a^{6}+\frac{11}{17}a^{4}-\frac{41}{51}a^{2}+\frac{13}{51}$, $\frac{116}{51}a^{9}-\frac{107}{51}a^{8}+\frac{332}{51}a^{7}-\frac{308}{51}a^{6}+\frac{485}{17}a^{5}-\frac{444}{17}a^{4}-\frac{3398}{51}a^{3}+\frac{3086}{51}a^{2}+\frac{1672}{51}a-\frac{1333}{51}$, $\frac{22}{51}a^{9}+\frac{6}{17}a^{8}+\frac{70}{51}a^{7}+\frac{16}{17}a^{6}+\frac{70}{17}a^{5}+\frac{14}{17}a^{4}-\frac{808}{51}a^{3}-\frac{106}{17}a^{2}+\frac{602}{51}a+\frac{107}{17}$, $\frac{412}{17}a^{9}+\frac{261}{17}a^{8}+\frac{974}{17}a^{7}+\frac{628}{17}a^{6}+\frac{4469}{17}a^{5}+\frac{2870}{17}a^{4}-\frac{14872}{17}a^{3}-\frac{9252}{17}a^{2}+\frac{10974}{17}a+\frac{6771}{17}$, $\frac{50}{51}a^{9}+\frac{29}{51}a^{8}+\frac{122}{51}a^{7}+\frac{83}{51}a^{6}+\frac{190}{17}a^{5}+\frac{117}{17}a^{4}-\frac{1841}{51}a^{3}-\frac{1181}{51}a^{2}+\frac{1345}{51}a+\frac{877}{51}$
|
| |
| Regulator: | \( 57620.4973023 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{4}\cdot 57620.4973023 \cdot 1}{2\cdot\sqrt{21767823360000000}}\cr\approx \mathstrut & 1.21735983571 \end{aligned}\]
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
| 5.1.1458000.1 |
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 | R | ${\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.10.0.1}{10} }$ | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.3.0.1}{3} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.5.0.1}{5} }^{2}$ | ${\href{/padicField/37.8.0.1}{8} }{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.5.0.1}{5} }^{2}$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.10.19a1.32 | $x^{10} + 4 x^{9} + 4 x^{7} + 4 x^{5} + 4 x^{3} + 10$ | $10$ | $1$ | $19$ | $((C_2^4 : C_5):C_4)\times C_2$ | $$[\frac{12}{5}, \frac{12}{5}, \frac{12}{5}, \frac{12}{5}, 3]_{5}^{4}$$ |
|
\(3\)
| 3.1.3.5a1.2 | $x^{3} + 9 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $$[\frac{5}{2}]_{2}$$ |
| 3.1.3.5a1.2 | $x^{3} + 9 x + 3$ | $3$ | $1$ | $5$ | $S_3$ | $$[\frac{5}{2}]_{2}$$ | |
| 3.2.2.2a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 8 x + 7$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(5\)
| 5.2.1.0a1.1 | $x^{2} + 4 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 5.1.8.7a1.4 | $x^{8} + 20$ | $8$ | $1$ | $7$ | $C_8:C_2$ | $$[\ ]_{8}^{2}$$ |