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Normalized defining polynomial
\( x^{10} - x^{9} + 7x^{8} - 3x^{7} + 17x^{6} + 3x^{5} + 22x^{4} + 6x^{3} + 10x^{2} + x + 1 \)
Invariants
| Degree: | $10$ |
| |
| Signature: | $[0, 5]$ |
| |
| Discriminant: |
\(-101928170799\)
\(\medspace = -\,3^{5}\cdot 7^{2}\cdot 13^{2}\cdot 37^{3}\)
|
| |
| Root discriminant: | \(12.61\) |
| |
| Galois root discriminant: | $3^{1/2}7^{1/2}13^{1/2}37^{3/4}\approx 247.87493685030424$ | ||
| Ramified primes: |
\(3\), \(7\), \(13\), \(37\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-111}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $\frac{1}{8}a^{9}-\frac{1}{8}a^{7}-\frac{1}{2}a^{6}-\frac{3}{8}a^{5}-\frac{1}{4}a^{3}-\frac{1}{2}a^{2}-\frac{1}{4}a-\frac{1}{8}$
| 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: |
\( -\frac{17}{8} a^{9} + 3 a^{8} - \frac{119}{8} a^{7} + \frac{21}{2} a^{6} - \frac{253}{8} a^{5} - a^{4} - \frac{111}{4} a^{3} - \frac{7}{2} a^{2} - \frac{23}{4} a + \frac{1}{8} \)
(order $6$)
|
| |
| Fundamental units: |
$\frac{7}{8}a^{9}+\frac{33}{8}a^{7}+\frac{9}{2}a^{6}+\frac{43}{8}a^{5}+19a^{4}+\frac{37}{4}a^{3}+\frac{31}{2}a^{2}+\frac{9}{4}a+\frac{17}{8}$, $\frac{21}{8}a^{9}-2a^{8}+\frac{123}{8}a^{7}-\frac{1}{2}a^{6}+\frac{217}{8}a^{5}+27a^{4}+\frac{119}{4}a^{3}+\frac{41}{2}a^{2}+\frac{23}{4}a+\frac{19}{8}$, $\frac{1}{8}a^{9}+a^{8}-\frac{9}{8}a^{7}+\frac{15}{2}a^{6}-\frac{43}{8}a^{5}+15a^{4}+\frac{3}{4}a^{3}+\frac{15}{2}a^{2}-\frac{5}{4}a-\frac{9}{8}$, $\frac{5}{2}a^{9}-a^{8}+\frac{27}{2}a^{7}+6a^{6}+\frac{43}{2}a^{5}+41a^{4}+27a^{3}+36a^{2}+6a+\frac{9}{2}$
|
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| Regulator: | \( 75.155539666 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{5}\cdot 75.155539666 \cdot 1}{6\cdot\sqrt{101928170799}}\cr\approx \mathstrut & 0.38420408122 \end{aligned}\]
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
| \(\Q(\sqrt{-3}) \) |
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 | ${\href{/padicField/2.8.0.1}{8} }{,}\,{\href{/padicField/2.2.0.1}{2} }$ | R | ${\href{/padicField/5.8.0.1}{8} }{,}\,{\href{/padicField/5.2.0.1}{2} }$ | R | ${\href{/padicField/11.2.0.1}{2} }^{5}$ | R | ${\href{/padicField/17.8.0.1}{8} }{,}\,{\href{/padicField/17.2.0.1}{2} }$ | ${\href{/padicField/19.5.0.1}{5} }{,}\,{\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.1.0.1}{1} }$ | ${\href{/padicField/23.8.0.1}{8} }{,}\,{\href{/padicField/23.2.0.1}{2} }$ | ${\href{/padicField/29.8.0.1}{8} }{,}\,{\href{/padicField/29.2.0.1}{2} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | R | ${\href{/padicField/41.4.0.1}{4} }^{2}{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.10.0.1}{10} }$ | ${\href{/padicField/53.10.0.1}{10} }$ | ${\href{/padicField/59.8.0.1}{8} }{,}\,{\href{/padicField/59.2.0.1}{2} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.5.2.5a1.2 | $x^{10} + 4 x^{6} + 2 x^{5} + 4 x^{2} + 4 x + 4$ | $2$ | $5$ | $5$ | $C_{10}$ | $$[\ ]_{2}^{5}$$ |
|
\(7\)
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{7}$ | $x + 4$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 7.4.1.0a1.1 | $x^{4} + 5 x^{2} + 4 x + 3$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
| 7.2.2.2a1.1 | $x^{4} + 12 x^{3} + 42 x^{2} + 43 x + 9$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 13.2.2.2a1.1 | $x^{4} + 24 x^{3} + 148 x^{2} + 61 x + 4$ | $2$ | $2$ | $2$ | $C_4$ | $$[\ ]_{2}^{2}$$ | |
|
\(37\)
| $\Q_{37}$ | $x + 35$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 37.1.4.3a1.3 | $x^{4} + 148$ | $4$ | $1$ | $3$ | $C_4$ | $$[\ ]_{4}$$ | |
| 37.5.1.0a1.1 | $x^{5} + 10 x + 35$ | $1$ | $5$ | $0$ | $C_5$ | $$[\ ]^{5}$$ |