Normalized defining polynomial
\( x^{10} - 4x^{9} + 8x^{8} - 14x^{7} + 17x^{6} - 12x^{5} + 17x^{4} - 24x^{3} + 17x^{2} - 6x + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $[0, 5]$ |
| |
| Discriminant: |
\(-81491689472\)
\(\medspace = -\,2^{15}\cdot 19^{2}\cdot 83^{2}\)
|
| |
| Root discriminant: | \(12.33\) |
| |
| Galois root discriminant: | $2^{3/2}19^{1/2}83^{2/3}\approx 234.58875418713964$ | ||
| Ramified primes: |
\(2\), \(19\), \(83\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-2}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{11}a^{8}+\frac{1}{11}a^{7}-\frac{3}{11}a^{6}-\frac{1}{11}a^{5}+\frac{5}{11}a^{4}-\frac{4}{11}a^{3}+\frac{5}{11}a^{2}-\frac{1}{11}a-\frac{2}{11}$, $\frac{1}{11}a^{9}-\frac{4}{11}a^{7}+\frac{2}{11}a^{6}-\frac{5}{11}a^{5}+\frac{2}{11}a^{4}-\frac{2}{11}a^{3}+\frac{5}{11}a^{2}-\frac{1}{11}a+\frac{2}{11}$
| 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: |
$a$, $a^{9}-3a^{8}+5a^{7}-9a^{6}+8a^{5}-4a^{4}+13a^{3}-11a^{2}+6a-1$, $\frac{5}{11}a^{9}-\frac{15}{11}a^{8}+\frac{20}{11}a^{7}-3a^{6}+\frac{23}{11}a^{5}+\frac{12}{11}a^{4}+\frac{50}{11}a^{3}-\frac{50}{11}a^{2}-\frac{23}{11}a+\frac{18}{11}$, $\frac{16}{11}a^{9}-\frac{59}{11}a^{8}+\frac{108}{11}a^{7}-17a^{6}+\frac{210}{11}a^{5}-\frac{120}{11}a^{4}+\frac{237}{11}a^{3}-\frac{314}{11}a^{2}+\frac{164}{11}a-\frac{37}{11}$
|
| |
| Regulator: | \( 17.4767132602 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 17.4767132602 \cdot 1}{2\cdot\sqrt{81491689472}}\cr\approx \mathstrut & 0.299759148316 \end{aligned}\]
Galois group
$\SOPlus(4,4)$ (as 10T40):
| A non-solvable group of order 7200 |
| The 20 conjugacy class representatives for $A_5 \wr C_2$ |
| Character table for $A_5 \wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-2}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 12 sibling: | data not computed |
| Degree 20 sibling: | data not computed |
| Degree 24 sibling: | data not computed |
| Degree 25 sibling: | data not computed |
| Degree 30 sibling: | data not computed |
| Degree 36 sibling: | data not computed |
| Degree 40 sibling: | 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 | ${\href{/padicField/3.5.0.1}{5} }^{2}$ | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.2.0.1}{2} }^{2}$ | ${\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.2.0.1}{2} }^{2}{,}\,{\href{/padicField/11.1.0.1}{1} }^{3}$ | ${\href{/padicField/13.6.0.1}{6} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}$ | ${\href{/padicField/17.5.0.1}{5} }{,}\,{\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | R | ${\href{/padicField/23.6.0.1}{6} }{,}\,{\href{/padicField/23.2.0.1}{2} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{2}{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.10.0.1}{10} }$ | ${\href{/padicField/41.3.0.1}{3} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.3.0.1}{3} }{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{3}$ | ${\href{/padicField/47.6.0.1}{6} }{,}\,{\href{/padicField/47.2.0.1}{2} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}{,}\,{\href{/padicField/59.1.0.1}{1} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.2.15a1.1 | $x^{10} + 2 x^{7} + 2 x^{5} + x^{4} + 2 x^{2} + 3$ | $2$ | $5$ | $15$ | $C_{10}$ | $$[3]^{5}$$ |
|
\(19\)
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{19}$ | $x + 17$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 19.3.1.0a1.1 | $x^{3} + 4 x + 17$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
| 19.2.2.2a1.2 | $x^{4} + 36 x^{3} + 328 x^{2} + 72 x + 23$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ | |
|
\(83\)
| 83.2.1.0a1.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 83.1.3.2a1.1 | $x^{3} + 83$ | $3$ | $1$ | $2$ | $S_3$ | $$[\ ]_{3}^{2}$$ | |
| 83.5.1.0a1.1 | $x^{5} + 9 x + 81$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ |