Basic invariants
| Dimension: | $10$ |
| Group: | $S_6$ |
| Conductor: | \(646166153810673664\)\(\medspace = 2^{18} \cdot 7^{4} \cdot 179^{4} \) |
| Frobenius-Schur indicator: | $1$ |
| Root number: | $1$ |
| Artin stem field: | Galois closure of 6.0.80192.1 |
| Galois orbit size: | $1$ |
| Smallest permutation container: | 30T164 |
| Parity: | even |
| Determinant: | 1.1.1t1.a.a |
| Projective image: | $S_6$ |
| Projective stem field: | Galois closure of 6.0.80192.1 |
Defining polynomial
| $f(x)$ | $=$ |
\( x^{6} - 2x^{5} + x^{4} + 2x^{2} - 2x + 1 \)
|
The roots of $f$ are computed in an extension of $\Q_{ 31 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 31 }$:
\( x^{2} + 29x + 3 \)
Roots:
| $r_{ 1 }$ | $=$ |
\( 29 a + 28 + \left(20 a + 4\right)\cdot 31 + \left(29 a + 28\right)\cdot 31^{2} + \left(2 a + 26\right)\cdot 31^{3} + \left(18 a + 27\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 2 }$ | $=$ |
\( 2 a + 24 + \left(10 a + 17\right)\cdot 31 + \left(a + 4\right)\cdot 31^{2} + \left(28 a + 3\right)\cdot 31^{3} + \left(12 a + 30\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 3 }$ | $=$ |
\( 14 a + 26 + \left(17 a + 25\right)\cdot 31 + 12 a\cdot 31^{2} + \left(12 a + 5\right)\cdot 31^{3} + \left(15 a + 16\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 4 }$ | $=$ |
\( 17 a + 23 + \left(13 a + 15\right)\cdot 31 + \left(18 a + 8\right)\cdot 31^{2} + \left(18 a + 17\right)\cdot 31^{3} + \left(15 a + 3\right)\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 5 }$ | $=$ |
\( 18 + 8\cdot 31 + 17\cdot 31^{2} + 12\cdot 31^{3} + 28\cdot 31^{4} +O(31^{5})\)
|
| $r_{ 6 }$ | $=$ |
\( 7 + 20\cdot 31 + 2\cdot 31^{2} + 28\cdot 31^{3} + 17\cdot 31^{4} +O(31^{5})\)
|
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 |
| $1$ | $1$ | $()$ | $10$ |
| $15$ | $2$ | $(1,2)(3,4)(5,6)$ | $-2$ |
| $15$ | $2$ | $(1,2)$ | $2$ |
| $45$ | $2$ | $(1,2)(3,4)$ | $-2$ |
| $40$ | $3$ | $(1,2,3)(4,5,6)$ | $1$ |
| $40$ | $3$ | $(1,2,3)$ | $1$ |
| $90$ | $4$ | $(1,2,3,4)(5,6)$ | $0$ |
| $90$ | $4$ | $(1,2,3,4)$ | $0$ |
| $144$ | $5$ | $(1,2,3,4,5)$ | $0$ |
| $120$ | $6$ | $(1,2,3,4,5,6)$ | $1$ |
| $120$ | $6$ | $(1,2,3)(4,5)$ | $-1$ |
The blue line marks the conjugacy class containing complex conjugation.