Basic invariants
| Dimension: | $9$ |
| Group: | $S_6$ |
| Conductor: | \(75772885410496\)\(\medspace = 2^{6} \cdot 71^{3} \cdot 149^{3} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.42316.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | $S_{6}$ |
| Parity: | odd |
| Determinant: | 1.10579.2t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.42316.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - x^{3} + x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 73 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 73 }$:
\( x^{2} + 70x + 5 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 12 a + 66 + \left(16 a + 31\right)\cdot 73 + \left(44 a + 20\right)\cdot 73^{2} + \left(34 a + 60\right)\cdot 73^{3} + \left(72 a + 71\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 70 a + 66 + \left(6 a + 52\right)\cdot 73 + \left(55 a + 39\right)\cdot 73^{2} + \left(47 a + 9\right)\cdot 73^{3} + \left(50 a + 36\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 3 a + 57 + \left(66 a + 3\right)\cdot 73 + \left(17 a + 52\right)\cdot 73^{2} + \left(25 a + 24\right)\cdot 73^{3} + \left(22 a + 67\right)\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 16 + 31\cdot 73 + 18\cdot 73^{2} + 5\cdot 73^{3} + 20\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 58 + 30\cdot 73 + 24\cdot 73^{2} + 72\cdot 73^{3} + 60\cdot 73^{4} +O(73^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 61 a + 29 + \left(56 a + 68\right)\cdot 73 + \left(28 a + 63\right)\cdot 73^{2} + \left(38 a + 46\right)\cdot 73^{3} + 35\cdot 73^{4} +O(73^{5})\)
|
Generators of the action on the roots $r_1, \ldots, r_{ 6 }$
| Cycle notation |
Character values on conjugacy classes
| Size | Order | Action on $r_1, \ldots, r_{ 6 }$ | Character value |
| $1$ | $1$ | $()$ | $9$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $3$ |
| $15$ | $2$ | $(1,2)$ | $3$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $1$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $0$ |
| $40$ | $3$ | $(1,2,3)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)$ | $-1$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $-1$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $0$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.