Normalized defining polynomial
\( x^{10} - 2x^{9} + 8x^{8} - 8x^{7} + 20x^{6} - 8x^{5} + 24x^{4} + 18x^{2} + 4x + 4 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $[0, 5]$ |
| |
| Discriminant: |
\(-102294880256\)
\(\medspace = -\,2^{22}\cdot 29^{3}\)
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| |
| Root discriminant: | \(12.62\) |
| |
| Galois root discriminant: | $2^{23/8}29^{3/4}\approx 91.67686126362786$ | ||
| Ramified primes: |
\(2\), \(29\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-29}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| This field is not Galois over $\Q$. | |||
| This is not a CM field. | |||
| Maximal CM subfield: | \(\Q(\sqrt{-1}) \) | ||
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{2}a^{8}$, $\frac{1}{14}a^{9}+\frac{1}{14}a^{8}+\frac{2}{7}a^{7}+\frac{2}{7}a^{6}+\frac{2}{7}a^{5}+\frac{2}{7}a^{4}-\frac{3}{7}a^{3}-\frac{2}{7}a^{2}+\frac{3}{7}a-\frac{3}{7}$
| 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 $1$ |
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Unit group
| Rank: | $4$ |
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| Torsion generator: |
\( -\frac{1}{14} a^{9} + \frac{3}{7} a^{8} - \frac{9}{7} a^{7} + \frac{19}{7} a^{6} - \frac{30}{7} a^{5} + \frac{33}{7} a^{4} - \frac{32}{7} a^{3} + \frac{16}{7} a^{2} - \frac{17}{7} a + \frac{3}{7} \)
(order $4$)
|
| |
| Fundamental units: |
$\frac{5}{14}a^{9}-\frac{8}{7}a^{8}+\frac{24}{7}a^{7}-\frac{39}{7}a^{6}+\frac{59}{7}a^{5}-\frac{60}{7}a^{4}+\frac{55}{7}a^{3}-\frac{45}{7}a^{2}+\frac{22}{7}a-\frac{15}{7}$, $\frac{1}{14}a^{9}-\frac{3}{7}a^{8}+\frac{9}{7}a^{7}-\frac{19}{7}a^{6}+\frac{30}{7}a^{5}-\frac{33}{7}a^{4}+\frac{32}{7}a^{3}-\frac{9}{7}a^{2}+\frac{10}{7}a+\frac{11}{7}$, $\frac{1}{14}a^{9}-\frac{3}{7}a^{8}+\frac{9}{7}a^{7}-\frac{19}{7}a^{6}+\frac{30}{7}a^{5}-\frac{40}{7}a^{4}+\frac{39}{7}a^{3}-\frac{37}{7}a^{2}+\frac{17}{7}a-\frac{17}{7}$, $\frac{2}{7}a^{9}-\frac{5}{7}a^{8}+\frac{15}{7}a^{7}-\frac{20}{7}a^{6}+\frac{36}{7}a^{5}-\frac{34}{7}a^{4}+\frac{37}{7}a^{3}-\frac{15}{7}a^{2}+\frac{5}{7}a-\frac{5}{7}$
|
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| Regulator: | \( 124.32431863 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 124.32431863 \cdot 1}{4\cdot\sqrt{102294880256}}\cr\approx \mathstrut & 0.95163079843 \end{aligned}\]
Galois group
$F_5\wr C_2$ (as 10T33):
| A solvable group of order 800 |
| The 20 conjugacy class representatives for $F_5 \wr C_2$ |
| Character table for $F_5 \wr C_2$ |
Intermediate fields
| \(\Q(\sqrt{-1}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
| Degree 20 siblings: | data not computed |
| Degree 25 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 | ${\href{/padicField/3.8.0.1}{8} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.5.0.1}{5} }^{2}$ | ${\href{/padicField/7.4.0.1}{4} }^{2}{,}\,{\href{/padicField/7.2.0.1}{2} }$ | ${\href{/padicField/11.8.0.1}{8} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.4.0.1}{4} }^{2}{,}\,{\href{/padicField/13.1.0.1}{1} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }{,}\,{\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | R | ${\href{/padicField/31.8.0.1}{8} }{,}\,{\href{/padicField/31.2.0.1}{2} }$ | ${\href{/padicField/37.5.0.1}{5} }{,}\,{\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }$ | ${\href{/padicField/41.5.0.1}{5} }{,}\,{\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.8.0.1}{8} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.8.0.1}{8} }{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.4.0.1}{4} }^{2}{,}\,{\href{/padicField/53.1.0.1}{1} }^{2}$ | ${\href{/padicField/59.10.0.1}{10} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.1 | $x^{2} + 2 x + 2$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.8.20b1.12 | $x^{8} + 4 x^{7} + 2 x^{6} + 4 x^{5} + 8 x^{4} + 8 x^{3} + 8 x + 2$ | $8$ | $1$ | $20$ | $C_4\wr C_2$ | $$[2, 2, 3, \frac{7}{2}]^{2}$$ | |
|
\(29\)
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{29}$ | $x + 27$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 29.4.1.0a1.1 | $x^{4} + 2 x^{2} + 15 x + 2$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 29.1.4.3a1.4 | $x^{4} + 232$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |