Normalized defining polynomial
\( x^{10} - 4x^{9} + 11x^{8} - 20x^{7} + 16x^{6} - 6x^{5} + 16x^{4} - 20x^{3} + 11x^{2} - 4x + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $[0, 5]$ |
| |
| Discriminant: |
\(-271845882880\)
\(\medspace = -\,2^{10}\cdot 5\cdot 11\cdot 13^{6}\)
|
| |
| Root discriminant: | \(13.91\) |
| |
| Galois root discriminant: | $2^{47/40}5^{1/2}11^{1/2}13^{3/4}\approx 114.64341223591194$ | ||
| Ramified primes: |
\(2\), \(5\), \(11\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-55}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $\frac{1}{43}a^{8}+\frac{2}{43}a^{7}-\frac{21}{43}a^{6}-\frac{19}{43}a^{5}+\frac{9}{43}a^{4}-\frac{19}{43}a^{3}-\frac{21}{43}a^{2}+\frac{2}{43}a+\frac{1}{43}$, $\frac{1}{43}a^{9}+\frac{18}{43}a^{7}-\frac{20}{43}a^{6}+\frac{4}{43}a^{5}+\frac{6}{43}a^{4}+\frac{17}{43}a^{3}+\frac{1}{43}a^{2}-\frac{3}{43}a-\frac{2}{43}$
| 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: |
$\frac{51}{43}a^{9}-\frac{163}{43}a^{8}+\frac{420}{43}a^{7}-\frac{650}{43}a^{6}+\frac{205}{43}a^{5}+\frac{739}{43}a^{3}-\frac{396}{43}a^{2}+\frac{123}{43}a-\frac{50}{43}$, $a^{9}-4a^{8}+11a^{7}-20a^{6}+16a^{5}-6a^{4}+16a^{3}-20a^{2}+11a-4$, $\frac{24}{43}a^{9}-\frac{99}{43}a^{8}+\frac{277}{43}a^{7}-\frac{508}{43}a^{6}+\frac{429}{43}a^{5}-\frac{145}{43}a^{4}+\frac{311}{43}a^{3}-\frac{520}{43}a^{2}+\frac{289}{43}a-\frac{61}{43}$, $\frac{27}{43}a^{9}-\frac{61}{43}a^{8}+\frac{149}{43}a^{7}-\frac{162}{43}a^{6}-\frac{152}{43}a^{5}+\frac{414}{43}a^{3}+\frac{190}{43}a^{2}+\frac{12}{43}a+\frac{14}{43}$
|
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| Regulator: | \( 73.2475793442 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 73.2475793442 \cdot 1}{2\cdot\sqrt{271845882880}}\cr\approx \mathstrut & 0.687861881072 \end{aligned}\]
Galois group
$C_2\wr F_5$ (as 10T29):
| A solvable group of order 640 |
| The 22 conjugacy class representatives for $((C_2^4 : C_5):C_4)\times C_2$ |
| Character table for $((C_2^4 : C_5):C_4)\times C_2$ |
Intermediate fields
| 5.1.35152.1 |
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 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 | ${\href{/padicField/3.10.0.1}{10} }$ | R | ${\href{/padicField/7.8.0.1}{8} }{,}\,{\href{/padicField/7.2.0.1}{2} }$ | R | R | ${\href{/padicField/17.4.0.1}{4} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.8.0.1}{8} }{,}\,{\href{/padicField/19.1.0.1}{1} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }^{2}{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.10.0.1}{10} }$ | ${\href{/padicField/31.4.0.1}{4} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }^{2}{,}\,{\href{/padicField/37.2.0.1}{2} }$ | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.2.0.1}{2} }^{4}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.4.0.1}{4} }^{2}{,}\,{\href{/padicField/47.2.0.1}{2} }$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.10.10a1.1 | $x^{10} + 2 x + 2$ | $10$ | $1$ | $10$ | $(C_2^4 : C_5):C_4$ | $$[\frac{6}{5}, \frac{6}{5}, \frac{6}{5}, \frac{6}{5}]_{5}^{4}$$ |
|
\(5\)
| 5.1.2.1a1.2 | $x^{2} + 10$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 5.8.1.0a1.1 | $x^{8} + x^{4} + 3 x^{2} + 4 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $$[\ ]^{8}$$ | |
|
\(11\)
| 11.1.2.1a1.2 | $x^{2} + 22$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 11.8.1.0a1.1 | $x^{8} + 7 x^{4} + 7 x^{3} + x^{2} + 7 x + 2$ | $1$ | $8$ | $0$ | $C_8$ | $$[\ ]^{8}$$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 13.1.4.3a1.1 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 13.1.4.3a1.1 | $x^{4} + 13$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ |