Normalized defining polynomial
\( x^{6} - 2x^{5} - x^{4} + 4x^{3} - 4x^{2} + 1 \)
Invariants
| Degree: | $6$ |
| |
| Signature: | $[4, 1]$ |
| |
| Discriminant: |
\(-974528\)
\(\medspace = -\,2^{6}\cdot 15227\)
|
| |
| Root discriminant: | \(9.96\) |
| |
| Galois root discriminant: | $2^{3/2}15227^{1/2}\approx 349.02148930975585$ | ||
| Ramified primes: |
\(2\), \(15227\)
|
| |
| Discriminant root field: | \(\Q(\sqrt{-15227}) \) | ||
| $\Aut(K/\Q)$: | $C_1$ |
| |
| 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 \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$
| 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: | $4$ |
| |
| Torsion generator: |
\( -1 \)
(order $2$)
|
| |
| Fundamental units: |
$a^{5}-2a^{4}-a^{3}+4a^{2}-4a$, $a-1$, $a^{5}-a^{4}-2a^{3}+2a^{2}-a-2$, $a^{4}-a^{3}-a^{2}+3a-2$
|
| |
| Regulator: | \( 6.66627176152 \) |
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{4}\cdot(2\pi)^{1}\cdot 6.66627176152 \cdot 1}{2\cdot\sqrt{974528}}\cr\approx \mathstrut & 0.339434286742 \end{aligned}\]
Galois group
| A non-solvable group of order 720 |
| The 11 conjugacy class representatives for $S_6$ |
| Character table for $S_6$ |
Intermediate fields
| The extension is primitive: there are no intermediate fields between this field and $\Q$. |
Sibling algebras
| Twin sextic algebra: | 6.0.225955552133312.1 |
| Degree 6 sibling: | 6.0.225955552133312.1 |
| Degree 10 sibling: | 10.4.14461155336531968.1 |
| Degree 12 siblings: | deg 12, deg 12 |
| Degree 15 siblings: | deg 15, deg 15 |
| Degree 20 siblings: | deg 20, deg 20, deg 20 |
| Degree 30 siblings: | deg 30, deg 30, deg 30, deg 30, deg 30, deg 30 |
| Degree 36 sibling: | deg 36 |
| Degree 40 siblings: | deg 40, deg 40, deg 40 |
| Degree 45 sibling: | deg 45 |
| 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 | ${\href{/padicField/3.4.0.1}{4} }{,}\,{\href{/padicField/3.2.0.1}{2} }$ | ${\href{/padicField/5.4.0.1}{4} }{,}\,{\href{/padicField/5.1.0.1}{1} }^{2}$ | ${\href{/padicField/7.6.0.1}{6} }$ | ${\href{/padicField/11.6.0.1}{6} }$ | ${\href{/padicField/13.5.0.1}{5} }{,}\,{\href{/padicField/13.1.0.1}{1} }$ | ${\href{/padicField/17.6.0.1}{6} }$ | ${\href{/padicField/19.4.0.1}{4} }{,}\,{\href{/padicField/19.2.0.1}{2} }$ | ${\href{/padicField/23.6.0.1}{6} }$ | ${\href{/padicField/29.6.0.1}{6} }$ | ${\href{/padicField/31.5.0.1}{5} }{,}\,{\href{/padicField/31.1.0.1}{1} }$ | ${\href{/padicField/37.6.0.1}{6} }$ | ${\href{/padicField/41.4.0.1}{4} }{,}\,{\href{/padicField/41.2.0.1}{2} }$ | ${\href{/padicField/43.5.0.1}{5} }{,}\,{\href{/padicField/43.1.0.1}{1} }$ | ${\href{/padicField/47.2.0.1}{2} }^{2}{,}\,{\href{/padicField/47.1.0.1}{1} }^{2}$ | ${\href{/padicField/53.5.0.1}{5} }{,}\,{\href{/padicField/53.1.0.1}{1} }$ | ${\href{/padicField/59.4.0.1}{4} }{,}\,{\href{/padicField/59.1.0.1}{1} }^{2}$ |
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 |
|---|---|---|---|---|---|---|---|
|
\(2\)
| 2.3.2.6a3.2 | $x^{6} + 4 x^{4} + 4 x^{3} + 7 x^{2} + 6 x + 5$ | $2$ | $3$ | $6$ | $A_4\times C_2$ | $$[2, 2]^{6}$$ |
|
\(15227\)
| Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $$[\ ]_{2}$$ | |
| Deg $4$ | $1$ | $4$ | $0$ | $C_4$ | $$[\ ]^{4}$$ |
Artin representations
| Label | Dimension | Conductor | Artin stem field | $G$ | Ind | $\chi(c)$ | |
|---|---|---|---|---|---|---|---|
| * | 1.1.1t1.a.a | $1$ | $1$ | \(\Q\) | $C_1$ | $1$ | $1$ |
| 1.15227.2t1.a.a | $1$ | $ 15227 $ | \(\Q(\sqrt{-15227}) \) | $C_2$ (as 2T1) | $1$ | $-1$ | |
| * | 5.974528.6t16.a.a | $5$ | $ 2^{6} \cdot 15227 $ | 6.4.974528.1 | $S_6$ (as 6T16) | $1$ | $3$ |
| 5.14839137856.12t183.a.a | $5$ | $ 2^{6} \cdot 15227^{2}$ | 6.4.974528.1 | $S_6$ (as 6T16) | $1$ | $1$ | |
| 5.344...824.12t183.a.a | $5$ | $ 2^{6} \cdot 15227^{4}$ | 6.4.974528.1 | $S_6$ (as 6T16) | $1$ | $-3$ | |
| 5.225...312.6t16.a.a | $5$ | $ 2^{6} \cdot 15227^{3}$ | 6.4.974528.1 | $S_6$ (as 6T16) | $1$ | $-1$ | |
| 9.144...968.10t32.a.a | $9$ | $ 2^{12} \cdot 15227^{3}$ | 6.4.974528.1 | $S_6$ (as 6T16) | $1$ | $3$ | |
| 9.510...344.20t145.a.a | $9$ | $ 2^{12} \cdot 15227^{6}$ | 6.4.974528.1 | $S_6$ (as 6T16) | $1$ | $-3$ | |
| 10.326...016.30t164.a.a | $10$ | $ 2^{18} \cdot 15227^{6}$ | 6.4.974528.1 | $S_6$ (as 6T16) | $1$ | $-2$ | |
| 10.140...104.30t164.a.a | $10$ | $ 2^{18} \cdot 15227^{4}$ | 6.4.974528.1 | $S_6$ (as 6T16) | $1$ | $2$ | |
| 16.484...696.36t1252.a.a | $16$ | $ 2^{24} \cdot 15227^{8}$ | 6.4.974528.1 | $S_6$ (as 6T16) | $1$ | $0$ |
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 *.