Normalized defining polynomial
\( x^{6} + 25x^{4} + 97x^{2} - 79 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[2, 2]$ |
| |
| Discriminant: |
\(504871936\)
\(\medspace = 2^{10}\cdot 79^{3}\)
|
| |
| Root discriminant: | \(28.22\) |
| |
| Galois root discriminant: | $2^{2}79^{1/2}\approx 35.552777669262355$ | ||
| Ramified primes: |
\(2\), \(79\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{79}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $\frac{1}{106}a^{4}-\frac{10}{53}a^{2}+\frac{43}{106}$, $\frac{1}{106}a^{5}-\frac{10}{53}a^{3}+\frac{43}{106}a$
| 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: |
$\frac{1}{53}a^{4}+\frac{33}{53}a^{2}+\frac{149}{53}$, $\frac{1}{53}a^{4}+\frac{33}{53}a^{2}+\frac{255}{53}$, $\frac{602}{53}a^{5}-\frac{500}{53}a^{4}+\frac{15467}{53}a^{3}-\frac{12843}{53}a^{2}+\frac{69081}{53}a-\frac{57328}{53}$
|
| |
| Regulator: | \( 122.514386922 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 122.514386922 \cdot 2}{2\cdot\sqrt{504871936}}\cr\approx \mathstrut & 0.861025877178 \end{aligned}\]
Galois group
| A solvable group of order 24 |
| The 5 conjugacy class representatives for $S_4$ |
| Character table for $S_4$ |
Intermediate fields
| 3.3.316.1 |
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: | \(\Q\) $\times$ \(\Q\) $\times$ 4.0.5056.2 |
| Degree 4 sibling: | 4.0.5056.2 |
| Degree 6 sibling: | 6.2.1597696.3 |
| Degree 8 sibling: | 8.0.2552632508416.3 |
| Degree 12 siblings: | deg 12, deg 12 |
| Minimal sibling: | 4.0.5056.2 |
Frobenius cycle types
| $p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Cycle type | R | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.3.0.1}{3} }^{2}$ | ${\href{/padicField/7.3.0.1}{3} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.3.0.1}{3} }^{2}$ | ${\href{/padicField/17.2.0.1}{2} }^{3}$ | ${\href{/padicField/19.2.0.1}{2} }^{3}$ | ${\href{/padicField/23.2.0.1}{2} }^{3}$ | ${\href{/padicField/29.2.0.1}{2} }^{3}$ | ${\href{/padicField/31.2.0.1}{2} }^{3}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{2}$ | ${\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.2.0.1}{2} }^{3}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.2a1.2 | $x^{2} + 2 x + 6$ | $2$ | $1$ | $2$ | $C_2$ | $$[2]$$ |
| 2.1.4.8b1.3 | $x^{4} + 4 x^{3} + 2 x^{2} + 4 x + 2$ | $4$ | $1$ | $8$ | $D_{4}$ | $$[2, 3]^{2}$$ | |
|
\(79\)
| 79.1.2.1a1.2 | $x^{2} + 237$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ |
| 79.2.2.2a1.2 | $x^{4} + 156 x^{3} + 6090 x^{2} + 468 x + 88$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |