Normalized defining polynomial
\( x^{6} - x^{5} + 2x^{4} + 2x^{2} - x + 1 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[0, 3]$ |
| |
| Discriminant: |
\(-27848\)
\(\medspace = -\,2^{3}\cdot 59^{2}\)
|
| |
| Root discriminant: | \(5.51\) |
| |
| Galois root discriminant: | $2^{3/2}59^{1/2}\approx 21.72556098240043$ | ||
| Ramified primes: |
\(2\), \(59\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-2}) \) | ||
| $\Aut(K/\Q)$: | $C_2$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $\frac{1}{2}a^{5}-\frac{1}{2}$
| 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: | Trivial group, which has order $1$ |
|
Unit group
| Rank: | $2$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a$, $\frac{1}{2}a^{5}-a^{4}+a^{3}-a^{2}-\frac{3}{2}$
|
| |
| Regulator: | \( 0.599228830431 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{0}\cdot(2\pi)^{3}\cdot 0.599228830431 \cdot 1}{2\cdot\sqrt{27848}}\cr\approx \mathstrut & 0.4453539458014 \end{aligned}\]
Galois group
$C_2\times S_4$ (as 6T11):
| A solvable group of order 48 |
| The 10 conjugacy class representatives for $S_4\times C_2$ |
| Character table for $S_4\times C_2$ |
Intermediate fields
| 3.1.59.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling algebras
| Twin sextic algebra: | 4.2.3776.2 $\times$ \(\Q(\sqrt{118}) \) |
| Degree 6 sibling: | 6.2.1643032.1 |
| Degree 8 siblings: | 8.0.14258176.2, 8.4.49632710656.3 |
| Degree 12 siblings: | 12.2.2928329928704.1, 12.0.2699554153024.1, 12.0.3176493481984.1, 12.4.11057373810786304.1, 12.0.11057373810786304.6, 12.0.11057373810786304.5 |
| Degree 16 sibling: | deg 16 |
| Degree 24 siblings: | deg 24, deg 24, deg 24, deg 24 |
| 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 | ${\href{/padicField/3.3.0.1}{3} }^{2}$ | ${\href{/padicField/5.6.0.1}{6} }$ | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.4.0.1}{4} }{,}\,{\href{/padicField/11.2.0.1}{2} }$ | ${\href{/padicField/13.2.0.1}{2} }^{3}$ | ${\href{/padicField/17.2.0.1}{2} }^{2}{,}\,{\href{/padicField/17.1.0.1}{1} }^{2}$ | ${\href{/padicField/19.3.0.1}{3} }^{2}$ | ${\href{/padicField/23.4.0.1}{4} }{,}\,{\href{/padicField/23.1.0.1}{1} }^{2}$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.1.0.1}{1} }^{2}$ | ${\href{/padicField/37.4.0.1}{4} }{,}\,{\href{/padicField/37.1.0.1}{1} }^{2}$ | ${\href{/padicField/41.3.0.1}{3} }^{2}$ | ${\href{/padicField/43.4.0.1}{4} }{,}\,{\href{/padicField/43.2.0.1}{2} }$ | ${\href{/padicField/47.2.0.1}{2} }^{3}$ | ${\href{/padicField/53.6.0.1}{6} }$ | R |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.1.2.3a1.2 | $x^{2} + 10$ | $2$ | $1$ | $3$ | $C_2$ | $$[3]$$ |
| 2.4.1.0a1.1 | $x^{4} + x + 1$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ | |
|
\(59\)
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ |
| $\Q_{59}$ | $x + 57$ | $1$ | $1$ | $0$ | Trivial | $$[\ ]$$ | |
| 59.2.2.2a1.2 | $x^{4} + 116 x^{3} + 3368 x^{2} + 232 x + 63$ | $2$ | $2$ | $2$ | $C_2^2$ | $$[\ ]_{2}^{2}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| *48 | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.472.2t1.a.a | $1$ | $ 2^{3} \cdot 59 $ | \(\Q(\sqrt{118}) \) | $C_2$ (as 2T1) | $1$ | $1$ | |
| 1.59.2t1.a.a | $1$ | $ 59 $ | \(\Q(\sqrt{-59}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 1.8.2t1.b.a | $1$ | $ 2^{3}$ | \(\Q(\sqrt{-2}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| 2.3776.6t3.d.a | $2$ | $ 2^{6} \cdot 59 $ | 6.2.105154048.1 | $D_{6}$ (as 6T3) | $1$ | $0$ | |
| *48 | 2.59.3t2.a.a | $2$ | $ 59 $ | 3.1.59.1 | $S_3$ (as 3T2) | $1$ | $0$ |
| 3.3776.4t5.d.a | $3$ | $ 2^{6} \cdot 59 $ | 4.2.3776.2 | $S_4$ (as 4T5) | $1$ | $1$ | |
| *48 | 3.472.6t11.a.a | $3$ | $ 2^{3} \cdot 59 $ | 6.0.27848.1 | $S_4\times C_2$ (as 6T11) | $1$ | $-1$ |
| 3.27848.6t11.a.a | $3$ | $ 2^{3} \cdot 59^{2}$ | 6.0.27848.1 | $S_4\times C_2$ (as 6T11) | $1$ | $1$ | |
| 3.222784.6t8.g.a | $3$ | $ 2^{6} \cdot 59^{2}$ | 4.2.3776.2 | $S_4$ (as 4T5) | $1$ | $-1$ |
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 *.