Normalized defining polynomial
\( x^{6} - 7x^{2} - 7 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $(2, 2)$ |
| |
| Discriminant: |
\(1075648\)
\(\medspace = 2^{6}\cdot 7^{5}\)
|
| |
| Root discriminant: | \(10.12\) |
| |
| Galois root discriminant: | $2^{7/4}7^{5/6}\approx 17.023578553967216$ | ||
| Ramified primes: |
\(2\), \(7\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{7}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
| Monogenic: | Yes | |
| 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: |
$a^{4}-a^{2}-5$, $a^{4}-2a^{2}-5$, $a^{5}-a^{3}+a^{2}-4a+1$
|
| |
| Regulator: | \( 4.36334588378 \) |
| |
| Unit signature rank: | \( 1 \) |
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{2}\cdot(2\pi)^{2}\cdot 4.36334588378 \cdot 1}{2\cdot\sqrt{1075648}}\cr\approx \mathstrut & 0.332180616183 \end{aligned}\]
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: | 4.0.3136.1 $\times$ \(\Q(\sqrt{7}) \) |
| Degree 8 sibling: | 8.0.1927561216.9 |
| Degree 12 siblings: | 12.4.74049191673856.3, 12.0.74049191673856.4 |
| 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 | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }$ | R | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.2.0.1}{2} }{,}\,{\href{/padicField/13.1.0.1}{1} }^{4}$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.2.0.1}{2} }^{2}{,}\,{\href{/padicField/29.1.0.1}{1} }^{2}$ | ${\href{/padicField/31.3.0.1}{3} }^{2}$ | ${\href{/padicField/37.3.0.1}{3} }^{2}$ | ${\href{/padicField/41.2.0.1}{2} }{,}\,{\href{/padicField/41.1.0.1}{1} }^{4}$ | ${\href{/padicField/43.2.0.1}{2} }{,}\,{\href{/padicField/43.1.0.1}{1} }^{4}$ | ${\href{/padicField/47.3.0.1}{3} }^{2}$ | ${\href{/padicField/53.3.0.1}{3} }^{2}$ | ${\href{/padicField/59.3.0.1}{3} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.6a2.2 | $x^{6} + 4 x^{4} + 2 x^{3} + 3 x^{2} + 4 x + 7$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $$[2, 2, 2]^{3}$$ |
|
\(7\)
| 7.1.6.5a1.6 | $x^{6} + 42$ | $6$ | $1$ | $5$ | $C_6$ | $$[\ ]_{6}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *24 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.28.2t1.a.a | $1$ | $ 2^{2} \cdot 7 $ | \(\Q(\sqrt{7}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| *24 | 1.7.3t1.a.a | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| 1.28.6t1.b.a | $1$ | $ 2^{2} \cdot 7 $ | \(\Q(\zeta_{28})^+\) | $C_6$ (as 6T1) | $0$ | $1$ | |
| 1.28.6t1.b.b | $1$ | $ 2^{2} \cdot 7 $ | \(\Q(\zeta_{28})^+\) | $C_6$ (as 6T1) | $0$ | $1$ | |
| *24 | 1.7.3t1.a.b | $1$ | $ 7 $ | \(\Q(\zeta_{7})^+\) | $C_3$ (as 3T1) | $0$ | $1$ |
| 3.3136.4t4.a.a | $3$ | $ 2^{6} \cdot 7^{2}$ | 4.0.3136.1 | $A_4$ (as 4T4) | $1$ | $-1$ | |
| *24 | 3.21952.6t6.b.a | $3$ | $ 2^{6} \cdot 7^{3}$ | 6.2.1075648.1 | $A_4\times C_2$ (as 6T6) | $1$ | $-1$ |
Data is given for all irreducible
representations of the Galois group for the Galois closure
of this field. Those marked with * are summands in the
permutation representation coming from this field. Representations
which appear with multiplicity greater than one are indicated
by exponents on the *.