Normalized defining polynomial
\( x^{10} - 4x^{9} + 12x^{8} - 21x^{7} + 27x^{6} - 24x^{5} + 18x^{4} - 12x^{3} + 9x^{2} - 4x + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(0, 5)$ |
| |
| Discriminant: |
\(-5216172147\)
\(\medspace = -\,3^{9}\cdot 43\cdot 6163\)
|
| |
| Root discriminant: | \(9.37\) |
| |
| Galois root discriminant: | $3^{9/10}43^{1/2}6163^{1/2}\approx 1383.6920346900363$ | ||
| Ramified primes: |
\(3\), \(43\), \(6163\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-795027}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-3}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{29}a^{9}+\frac{12}{29}a^{7}-\frac{2}{29}a^{6}-\frac{10}{29}a^{5}-\frac{6}{29}a^{4}-\frac{6}{29}a^{3}-\frac{7}{29}a^{2}+\frac{10}{29}a+\frac{7}{29}$
| 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: |
\( -\frac{33}{29} a^{9} + 4 a^{8} - \frac{338}{29} a^{7} + \frac{530}{29} a^{6} - \frac{627}{29} a^{5} + \frac{488}{29} a^{4} - \frac{324}{29} a^{3} + \frac{202}{29} a^{2} - \frac{156}{29} a + \frac{59}{29} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{30}{29}a^{9}-3a^{8}+\frac{244}{29}a^{7}-\frac{292}{29}a^{6}+\frac{280}{29}a^{5}-\frac{93}{29}a^{4}+\frac{52}{29}a^{3}-\frac{36}{29}a^{2}+\frac{68}{29}a+\frac{36}{29}$, $\frac{28}{29}a^{9}-3a^{8}+\frac{249}{29}a^{7}-\frac{346}{29}a^{6}+\frac{387}{29}a^{5}-\frac{284}{29}a^{4}+\frac{209}{29}a^{3}-\frac{167}{29}a^{2}+\frac{106}{29}a-\frac{7}{29}$, $\frac{42}{29}a^{9}-5a^{8}+\frac{417}{29}a^{7}-\frac{635}{29}a^{6}+\frac{740}{29}a^{5}-\frac{571}{29}a^{4}+\frac{444}{29}a^{3}-\frac{323}{29}a^{2}+\frac{246}{29}a-\frac{54}{29}$, $\frac{9}{29}a^{9}-a^{8}+\frac{79}{29}a^{7}-\frac{105}{29}a^{6}+\frac{113}{29}a^{5}-\frac{83}{29}a^{4}+\frac{120}{29}a^{3}-\frac{121}{29}a^{2}+\frac{90}{29}a-\frac{24}{29}$
|
| |
| Regulator: | \( 15.3090297009 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 15.3090297009 \cdot 1}{6\cdot\sqrt{5216172147}}\cr\approx \mathstrut & 0.345955149284 \end{aligned}\]
Galois group
$S_5\wr C_2$ (as 10T43):
| A non-solvable group of order 28800 |
| The 35 conjugacy class representatives for $S_5^2 \wr C_2$ |
| Character table for $S_5^2 \wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-3}) \) |
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 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 25 sibling: | data not computed |
| Degree 30 sibling: | data not computed |
| Degree 36 sibling: | 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 | ${\href{/padicField/2.10.0.1}{10} }$ | R | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.5.0.1}{5} }{,}\,{\href{/padicField/7.4.0.1}{4} }{,}\,{\href{/padicField/7.1.0.1}{1} }$ | ${\href{/padicField/11.6.0.1}{6} }{,}\,{\href{/padicField/11.4.0.1}{4} }$ | ${\href{/padicField/13.5.0.1}{5} }^{2}$ | ${\href{/padicField/17.10.0.1}{10} }$ | ${\href{/padicField/19.3.0.1}{3} }{,}\,{\href{/padicField/19.2.0.1}{2} }^{2}{,}\,{\href{/padicField/19.1.0.1}{1} }^{3}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.6.0.1}{6} }{,}\,{\href{/padicField/29.2.0.1}{2} }^{2}$ | ${\href{/padicField/31.2.0.1}{2} }^{3}{,}\,{\href{/padicField/31.1.0.1}{1} }^{4}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.3.0.1}{3} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{3}$ | ${\href{/padicField/41.8.0.1}{8} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | R | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.6.0.1}{6} }{,}\,{\href{/padicField/53.2.0.1}{2} }^{2}$ | ${\href{/padicField/59.6.0.1}{6} }{,}\,{\href{/padicField/59.4.0.1}{4} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.1.10.9a1.1 | $x^{10} + 3$ | $10$ | $1$ | $9$ | $F_{5}\times C_2$ | $$[\ ]_{10}^{4}$$ |
|
\(43\)
| $\Q_{43}$ | $x + 40$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 43.2.1.0a1.1 | $x^{2} + 42 x + 3$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 43.1.2.1a1.1 | $x^{2} + 43$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 43.5.1.0a1.1 | $x^{5} + 8 x + 40$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ | |
|
\(6163\)
| $\Q_{6163}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{6163}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{6163}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| $\Q_{6163}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |