Normalized defining polynomial
\( x^{6} - 3x^{5} + 9x^{4} - 13x^{3} - 3x^{2} + 9x - 41 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(2, 2)$ |
| |
| Discriminant: |
\(5899257\)
\(\medspace = 3^{3}\cdot 7^{5}\cdot 13\)
|
| |
| Root discriminant: | \(13.44\) |
| |
| Galois root discriminant: | $3^{1/2}7^{5/6}13^{1/2}\approx 31.606810323667133$ | ||
| Ramified primes: |
\(3\), \(7\), \(13\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{273}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{3}a^{3}+\frac{1}{3}$, $\frac{1}{9}a^{4}+\frac{1}{9}a^{3}+\frac{1}{9}a+\frac{1}{9}$, $\frac{1}{9}a^{5}-\frac{1}{9}a^{3}+\frac{1}{9}a^{2}-\frac{1}{9}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{2}$, which has order $2$ |
| |
| Narrow class group: | $C_{2}\times C_{2}$, which has order $4$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{1}{3}a^{2}-\frac{1}{3}a+\frac{1}{3}$, $\frac{1}{9}a^{4}-\frac{2}{9}a^{3}+\frac{1}{3}a^{2}-\frac{2}{9}a-\frac{8}{9}$, $\frac{1}{9}a^{5}-\frac{1}{9}a^{4}+\frac{7}{9}a^{3}+\frac{1}{9}a^{2}-\frac{1}{9}a+\frac{16}{9}$
|
| |
| Regulator: | \( 5.55815547952 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 5.55815547952 \cdot 2}{2\cdot\sqrt{5899257}}\cr\approx \mathstrut & 0.361369721487 \end{aligned}\]
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: | \(\Q(\sqrt{273}) \) $\times$ 4.0.8281.1 |
| Degree 8 sibling: | 8.0.272174020209.5 |
| 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 | ${\href{/padicField/2.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/5.6.0.1}{6} }$ | R | ${\href{/padicField/11.3.0.1}{3} }^{2}$ | R | ${\href{/padicField/17.3.0.1}{3} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.2.0.1}{2} }{,}\,{\href{/padicField/29.1.0.1}{1} }^{4}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.6.0.1}{6} }$ | ${\href{/padicField/53.6.0.1}{6} }$ | ${\href{/padicField/59.6.0.1}{6} }$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(3\)
| 3.3.2.3a1.1 | $x^{6} + 4 x^{4} + 2 x^{3} + 4 x^{2} + 7 x + 1$ | $2$ | $3$ | $3$ | $C_6$ | $$[\ ]_{2}^{3}$$ |
|
\(7\)
| 7.1.6.5a1.6 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |
|
\(13\)
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{13}$ | $x + 11$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 13.1.2.1a1.1 | $x^{2} + 13$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 13.2.1.0a1.1 | $x^{2} + 12 x + 2$ | $1$ | $2$ | $0$ | $C_2$ | $$[\ ]^{2}$$ |