Normalized defining polynomial
\( x^{6} - 2x^{5} - 13x^{4} + 80x^{3} - 256x^{2} + 472x - 316 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(2, 2)$ |
| |
| Discriminant: |
\(1396498432\)
\(\medspace = 2^{13}\cdot 7^{4}\cdot 71\)
|
| |
| Root discriminant: | \(33.43\) |
| |
| Galois root discriminant: | $2^{11/4}7^{2/3}71^{1/2}\approx 207.42464514045537$ | ||
| Ramified primes: |
\(2\), \(7\), \(71\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{142}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{2}a^{4}-\frac{1}{2}a^{2}$, $\frac{1}{2484}a^{5}+\frac{35}{207}a^{4}-\frac{379}{2484}a^{3}-\frac{49}{138}a^{2}-\frac{551}{1242}a-\frac{16}{621}$
| Monogenic: | Not computed | |
| Index: | $1$ | |
| Inessential primes: | None |
Class group and class number
| Ideal class group: | $C_{9}$, which has order $9$ |
| |
| Narrow class group: | $C_{9}$, which has order $9$ |
|
Unit group
| Rank: | $3$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$\frac{17}{621}a^{5}-\frac{1}{414}a^{4}-\frac{233}{621}a^{3}+\frac{187}{138}a^{2}-\frac{2588}{621}a+\frac{3880}{621}$, $\frac{59}{1242}a^{5}-\frac{10}{207}a^{4}-\frac{1247}{1242}a^{3}+\frac{214}{69}a^{2}-\frac{1459}{621}a-\frac{25}{621}$, $\frac{77}{828}a^{5}-\frac{4}{69}a^{4}+\frac{1031}{828}a^{3}-\frac{183}{46}a^{2}+\frac{4339}{414}a-\frac{1873}{207}$
|
| |
| Regulator: | \( 61.111820441060296 \) |
| |
| Unit signature rank: | \( 2 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 61.111820441060296 \cdot 9}{2\cdot\sqrt{1396498432}}\cr\approx \mathstrut & 1.16208323710442 \end{aligned}\]
Galois group
$\SOPlus(4,2)$ (as 6T13):
| A solvable group of order 72 |
| The 9 conjugacy class representatives for $C_3^2:D_4$ |
| Character table for $C_3^2:D_4$ |
Intermediate fields
| \(\Q(\sqrt{2}) \) |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | data not computed |
| Degree 6 sibling: | data not computed |
| Degree 9 sibling: | data not computed |
| Degree 12 siblings: | data not computed |
| Degree 18 siblings: | data not computed |
| Degree 24 siblings: | data not computed |
| Degree 36 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 | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.6.0.1}{6} }$ | R | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.4.0.1}{4} }{,}\,{\href{/padicField/13.2.0.1}{2} }$ | ${\href{/padicField/17.3.0.1}{3} }{,}\,{\href{/padicField/17.2.0.1}{2} }{,}\,{\href{/padicField/17.1.0.1}{1} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.2.0.1}{2} }^{2}{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.2.0.1}{2} }^{2}{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.2.0.1}{2} }^{3}$ | ${\href{/padicField/41.3.0.1}{3} }{,}\,{\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.3.0.1}{3} }{,}\,{\href{/padicField/47.1.0.1}{1} }^{3}$ | ${\href{/padicField/53.4.0.1}{4} }{,}\,{\href{/padicField/53.2.0.1}{2} }$ | ${\href{/padicField/59.2.0.1}{2} }^{3}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.3 | $x^{2} + 4 x + 2$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.1.4.10a1.2 | $x^{4} + 4 x^{3} + 8 x^{2} + 2$ | $4$ | $1$ | $10$ | $D_{4}$ | $$[2, 3, \frac{7}{2}]$$ | |
|
\(7\)
| 7.1.3.2a1.1 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ |
| 7.1.3.2a1.1 | $x^{3} + 7$ | $3$ | $1$ | $2$ | $C_3$ | $$[\ ]_{3}$$ | |
|
\(71\)
| $\Q_{71}$ | $x + 64$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| 71.1.2.1a1.2 | $x^{2} + 497$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| 71.3.1.0a1.1 | $x^{3} + 4 x + 64$ | $1$ | $3$ | $0$ | $C_3$ | $$[\ ]^{3}$$ |