Normalized defining polynomial
\( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x - 13 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(2, 2)$ |
| |
| Discriminant: |
\(1193297\)
\(\medspace = 7^{5}\cdot 71\)
|
| |
| Root discriminant: | \(10.30\) |
| |
| Galois root discriminant: | $7^{5/6}71^{1/2}\approx 42.64592522013582$ | ||
| Ramified primes: |
\(7\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{497}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{127}a^{5}-\frac{55}{127}a^{4}+\frac{50}{127}a^{3}-\frac{34}{127}a^{2}+\frac{59}{127}a-\frac{12}{127}$
| 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: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{2}{127}a^{5}+\frac{17}{127}a^{4}-\frac{27}{127}a^{3}+\frac{59}{127}a^{2}-\frac{9}{127}a-\frac{24}{127}$, $\frac{5}{127}a^{5}-\frac{21}{127}a^{4}-\frac{4}{127}a^{3}-\frac{43}{127}a^{2}+\frac{41}{127}a-\frac{60}{127}$, $\frac{32}{127}a^{5}+\frac{18}{127}a^{4}+\frac{76}{127}a^{3}+\frac{55}{127}a^{2}+\frac{110}{127}a+\frac{251}{127}$
|
| |
| Regulator: | \( 5.19019197896 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 5.19019197896 \cdot 1}{2\cdot\sqrt{1193297}}\cr\approx \mathstrut & 0.375144756141 \end{aligned}\]
Galois group
$C_2\times A_4$ (as 6T6):
| A solvable group of order 24 |
| The 8 conjugacy class representatives for $A_4\times C_2$ |
| Character table for $A_4\times C_2$ |
Intermediate fields
| \(\Q(\zeta_{7})^+\) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Galois closure: | deg 24 |
| Twin sextic algebra: | \(\Q(\sqrt{497}) \) $\times$ 4.0.247009.1 |
| Degree 8 sibling: | 8.0.2989658857969.3 |
| Degree 12 siblings: | deg 12, deg 12 |
| 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 | ${\href{/padicField/2.3.0.1}{3} }^{2}$ | ${\href{/padicField/3.6.0.1}{6} }$ | ${\href{/padicField/5.6.0.1}{6} }$ | R | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.1.0.1}{1} }^{6}$ | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.6.0.1}{6} }$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }^{2}{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(7\)
| 7.1.6.5a1.6 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |
|
\(71\)
| 71.2.1.0a1.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |
| 71.2.1.0a1.1 | $x^{2} + 69 x + 7$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ | |
| 71.1.2.1a1.1 | $x^{2} + 71$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |