Normalized defining polynomial
\( x^{10} - 2x^{9} - 3x^{8} + 12x^{7} - 8x^{6} - 8x^{5} + 5x^{4} + 6x^{3} - x^{2} - 2x + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $(4, 3)$ |
| |
| Discriminant: |
\(-55288230912\)
\(\medspace = -\,2^{10}\cdot 3^{5}\cdot 83\cdot 2677\)
|
| |
| Root discriminant: | \(11.86\) |
| |
| Galois root discriminant: | $2\cdot 3^{1/2}83^{1/2}2677^{1/2}\approx 1632.878440056087$ | ||
| Ramified primes: |
\(2\), \(3\), \(83\), \(2677\)
|
| |
| Discriminant root field: | $\Q(\sqrt{-666573}$) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{221}a^{9}-\frac{5}{221}a^{8}+\frac{12}{221}a^{7}-\frac{24}{221}a^{6}+\frac{64}{221}a^{5}+\frac{21}{221}a^{4}-\frac{58}{221}a^{3}-\frac{41}{221}a^{2}-\frac{99}{221}a+\frac{74}{221}$
| 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: | $6$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a$, $\frac{14}{221}a^{9}-\frac{70}{221}a^{8}-\frac{53}{221}a^{7}+\frac{327}{221}a^{6}-\frac{209}{221}a^{5}-\frac{369}{221}a^{4}+\frac{72}{221}a^{3}+\frac{752}{221}a^{2}-\frac{60}{221}a-\frac{69}{221}$, $\frac{20}{221}a^{9}-\frac{100}{221}a^{8}+\frac{19}{221}a^{7}+\frac{404}{221}a^{6}-\frac{709}{221}a^{5}+\frac{199}{221}a^{4}+\frac{387}{221}a^{3}+\frac{64}{221}a^{2}-\frac{212}{221}a+\frac{154}{221}$, $\frac{111}{221}a^{9}-\frac{113}{221}a^{8}-\frac{436}{221}a^{7}+\frac{872}{221}a^{6}+\frac{32}{221}a^{5}-\frac{763}{221}a^{4}-\frac{692}{221}a^{3}+\frac{532}{221}a^{2}+\frac{503}{221}a-\frac{184}{221}$, $\frac{155}{221}a^{9}-\frac{333}{221}a^{8}-\frac{571}{221}a^{7}+\frac{2026}{221}a^{6}-\frac{909}{221}a^{5}-\frac{2049}{221}a^{4}+\frac{734}{221}a^{3}+\frac{1822}{221}a^{2}+\frac{346}{221}a-\frac{464}{221}$, $\frac{6}{221}a^{9}-\frac{30}{221}a^{8}+\frac{72}{221}a^{7}+\frac{77}{221}a^{6}-\frac{500}{221}a^{5}+\frac{568}{221}a^{4}+\frac{315}{221}a^{3}-\frac{688}{221}a^{2}-\frac{152}{221}a+\frac{223}{221}$
|
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| Regulator: | \( 28.263854194 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{3}\cdot 28.263854194 \cdot 1}{2\cdot\sqrt{55288230912}}\cr\approx \mathstrut & 0.23853089032 \end{aligned}\]
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 | R | R | ${\href{/padicField/5.10.0.1}{10} }$ | ${\href{/padicField/7.6.0.1}{6} }{,}\,{\href{/padicField/7.4.0.1}{4} }$ | ${\href{/padicField/11.5.0.1}{5} }{,}\,{\href{/padicField/11.3.0.1}{3} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.6.0.1}{6} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.3.0.1}{3} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{4}$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.2.0.1}{2} }^{2}{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.6.0.1}{6} }{,}\,{\href{/padicField/41.4.0.1}{4} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.5.0.1}{5} }{,}\,{\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}{,}\,{\href{/padicField/59.2.0.1}{2} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.5.2.10a1.2 | $x^{10} + 2 x^{7} + 4 x^{5} + x^{4} + 4 x^{2} + 9$ | $2$ | $5$ | $10$ | $C_{10}$ | $$[2]^{5}$$ |
|
\(3\)
| 3.5.2.5a1.1 | $x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 7 x + 1$ | $2$ | $5$ | $5$ | $C_{10}$ | $$[\ ]_{2}^{5}$$ |
|
\(83\)
| $\Q_{83}$ | $x + 81$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 83.2.1.0a1.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 83.1.2.1a1.1 | $x^{2} + 83$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 83.2.1.0a1.1 | $x^{2} + 82 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 83.3.1.0a1.1 | $x^{3} + 3 x + 81$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ | |
|
\(2677\)
| $\Q_{2677}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| Deg $2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | ||
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | ||
| Deg $5$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ |